%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Common Theoretical Issues}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\authors{U.~Nierste, Z.~Ligeti, A.~S.~Kronfeld}

\section{Introduction}
\label{ch1:sec:intro}

This chapter provides some of the theoretical background needed to
interpret measurements studied by Working Groups~1--3
(and reported in Chapters~6--8, respectively).
These three working groups deal with the decay of $b$-flavored hadrons,
and they are independent of the production mechanism.
The theory of $b$ decays requires
some elementary concepts on symmetries and mixing,
some knowledge of the standard electroweak theory, and
some information on how the $b$ quark is bound into hadrons.
On the other hand, the theory of production, fragmentation, and
spectroscopy---the subjects of Working Group~4 (and Chapter~9)---is
separate, dealing entirely with aspects of the strong interactions.
Hence, the theoretical background needed for Working Group~4 is
entirely in Chapter~9, and theoretical issues common to the
working groups studying decays are collected together in this chapter.%
\footnote{There is, unavoidably, some overlap with theoretical material
in Chapters~6--8.  We have attempted to use consistent conventions and
notation throughout.}
Although the experimental study of $CP$ violation in the $B$ system is just
beginning~\cite{CDFs2b,Bas2b,Bes2b}, there are several theoretical
reviews~\cite{babook,BLS,book,bsbook,cern} that the reader may want to consult
for details not covered here.


We start in Sec.~\ref{ch1:sec:cpvsm} by reviewing how flavor mixing
and \CP\ violation arise in the Standard Model.
Experiments of the past decade have verified the
$SU(3)\times SU(2)\times U(1)$ gauge structure of elementary particle
interactions, in a comprehensive and very precise way.
By comparison, tests of the flavor interactions are not yet nearly as
broad or detailed.
The Standard Model, in which quark masses and mixing arise from
Yukawa interactions with the Higgs field, still serves as the current
foundation for discussing flavor physics.
Sec.~\ref{ch1:sec:cpvsm} discusses the standard flavor sector,
leading to the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which contains
a \CP\ violating parameter for three generations.
By construction, the CKM matrix is unitary, which implies several
relations among its entries and, hence, between \CP\ conserving and \CP\
violating observables.
Furthermore, the same construction shows how, in the Standard Model,
neutral currents conserve flavor at the tree level, which is known as
the Glashow-Iliopoulos-Maiani (GIM) effect.

We emphasize that in extensions of the Standard Model the CKM mechanisms
can exist side-by-side with other sources of \CP\ and flavor violation.
Many measurements are therefore needed to test whether the standard
patterns prevail.
Quark interactions are obscured by confinement, however.
Therefore, Sec.~\ref{ch1:sec:cpvsm} concludes with a brief summary of
the ways of avoiding or reducing uncertainties from nonperturbative
QCD, and the much of the rest of the chapter revisits various aspects
in greater detail.

Sec.~\ref{ch1:sect:gen} covers aspects of $B$ mesons that can be
discussed without reference to their underlying dynamics.
The strong interactions conserve the quantum numbers \p, \T, and \C\
of parity, time reversal, and charge conjugation.
We therefore start by discussing the transformation properties of
currents and hadrons under these discrete symmetry transformations.
Once these concepts---and associated phase conventions---are fixed,
one can discuss the mixing between neutral mesons.
(In the Standard Model, neutral meson mixing is induced through
one-loop effects.)
Although a flavored neutral meson and its anti-particle form a two-state
quantum mechanical system, the particles are not stable, so the two mass
eigenstates can have different decay widths in general.
Consequently, the physical description of decay during mixing contains
formulae that are not always simple and phase conventions that are not
always transparent.
Sec.~\ref{ch1:sect:gen} provides such a general set of formulae,
derived with a self-consistent set of conventions.

The general discussion of Sec.~\ref{ch1:sect:gen}, leads to a useful
classification of \CP\ violation.
There can be \CP\ violation in mixing, in decay, and in the interference
of decays with and without mixing.
Sec.~\ref{ch1:sec:aspects} gives the concrete mathematical definition
of these three types of \CP\ violation, illustrated with examples.
\CP\ violation in many $B$ decays is principally of one type or
another, although in general two or more of these types may be present,
as is the case with some kaon decays.
We also work through two important examples of \CP\ in the interference
of amplitudes: $B\to J/\psi K_S$ and $B_s\to D_s^\pm K^\mp$, where it is
possible to use the \CP\ invariance of QCD to show that the \CP\
asymmetry of these decays is independent of the hadronic transition
amplitude.

To gain a comprehensive view of flavor physics, and decide whether the
standard model correctly describes flavor-changing interactions, one
has to consider QCD.
Sec.~\ref{1:tools} discusses several theoretical tools to separate
scales, so that the nonperturbative hadronic physics can
be treated separately from physics at higher scales, where perturbation theory
is accurate.
Indeed, in $B$ decays several length scales are involved:
the scale of QCD, $\lqcd$, 
the mass of the $b$ quark, $m_b$,
the higher masses of the $W$ and $Z$ bosons and the top quark, and,
possibly, higher scales of new physics.
The first step is to separate out weak (and higher) scales with an
operator product expansion, leading to an effective weak Hamiltonian
for flavor-changing processes.
This formalism applies for all flavor physics.
For $b$ quarks, one finds further simplifications, because
$m_b\gg\lqcd$.
Two tools are used to separate these scales, heavy quark effective
theory for hadronic matrix elements, and the heavy quark expansion for
inclusive decay rates.
Sec.~\ref{1:tools} also includes a brief overview of lattice QCD,
which is a promising numerical method to compute hadronic matrix
elements of the electroweak Hamiltonian, when there is at most one hadron
in the final state.
In particular, we discuss how heavy quark effective theory can be used
to control uncertainties in the numerical calculation.

A further predictive aspect of the CKM mechanism, with only one
parameter to describe \CP\ violation, is that observables in the $B$
and $B_s$ systems are connected to those in the kaon system.
Sec.~\ref{1:sect:kaon} gives an overview of \kkm\ using the same
formalism as in our treatment of \bbm\ in Sec.~\ref{ch1:sect:gen}.
This also gives us the opportunity to introduce the most important
constraints from kaon physics: not only those currently available but
also those that could be measured in the coming decade.
Finally, Sec.~\ref{ch1:SM-expect} gives a summary of expectations for
measurements of the unitarity triangle, based on global fits of kaon
mixing and \CP\ conserving observables in $B$~physics.


\boldmath
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$CP$ Violation in the Standard Model}
\label{ch1:sec:cpvsm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\unboldmath

As mentioned in the introduction, in the Standard Model quark masses,
flavor violation, and $CP$ violation all arise from Yukawa interactions
among the quark fields and the Higgs field.
In this section we review how these phenomena appear, leading to the
Cabibbo-Kobayashi-Maskawa (CKM) matrix.
  From a theoretical point of view, the CKM mechanism could, and probably
does, exist along with other sources of $CP$ violation.
We therefore also discuss some of the important features of the CKM
model, to provide a framework for testing it.

\subsection{Yukawa interactions and the CKM matrix}
\label{1:subsec:yukawa}

Let us begin by recalling some of the most elementary aspects of
particle physics.
Experiments have demonstrated that there are several species, or
flavors, of quarks and leptons.
They are the down-type quarks ($d$, $s$, $b$), up-type quarks
($u$, $c$, $t$), charged leptons ($e$, $\mu$, $\tau$),
and neutrinos ($\nu_e$, $\nu_\mu$, $\nu_\tau$).
They interact through the exchange of gauge bosons:
the weak bosons $W^\pm$ and $Z^0$, the photon, and the gluons.
These interactions are dictated by local gauge invariance, with gauge
group $SU(3)\times SU(2)\times U(1)_Y$.
With this gauge symmetry, and the observed quantum numbers of the
fermions, at least one scalar field is needed to accommodate quark
masses, and, in turns out, the couplings to this field can generate
flavor and $CP$ violation.

One of the most striking features of the charged-current weak 
interactions is that they do not couple solely to a vector current 
(as in QED and QCD) but to the linear combination of vector and axial 
vector currents $V-A$.
As a consequence, the electroweak theory is a \emph{chiral} gauge theory,
which means that left- and right-handed fermions transform differently
under the electroweak gauge group $SU(2)\times U(1)_Y$.
The right-handed fermions do not couple to $W^\pm$, and they are
singlets under $SU(2)$:
\begin{eqnarray}
    E_R &=& (e_R, \mu_R, \tau_R)\,,\qquad Y_E = -1\,; \nonumber\\
    U_R &=& (u_R,  c_R,  t_R)\,,   \qquad Y_U = \frac{2}{3}\,; \\
    D_R &=& (d_R,  s_R,  b_R)\,,   \qquad Y_D = -\frac{1}{3}\,;\nonumber
\end{eqnarray}
where the hypercharge~$Y$ is given.
For convenience below, the three generations are grouped together.
The gauge and kinetic interactions for $G$ generations of these
fields are
\begin{equation}
    {\cal L}_R = \sum_{i=1}^G
        \bar{E}^i_R(i\kern+0.1em  /\kern-0.55em \partial
           -  g_1Y_E \kern+0.1em  /\kern-0.65em B)E^i_R +
        \bar{D}^i_R(i\kern+0.1em  /\kern-0.65em D
           -  g_1Y_D \kern+0.1em  /\kern-0.65em B)D^i_R +
        \bar{U}^i_R(i\kern+0.15em /\kern-0.65em D
           -  g_1Y_U \kern+0.1em  /\kern-0.65em B)U^i_R \,,
\end{equation}
where $B$ is the gauge boson of $U(1)_Y$, with coupling~$g_1$, and
$D^\mu$ is the covariant derivative of QCD:
quarks are triplets under color $SU(3)$.
On the other hand, the left-handed fermions do couple to~$W^\pm$, so
they are doublets under $SU(2)$:
\begin{eqnarray}
L_L &=& \left(
    \pmatrix{\nu_e \cr e}_L,
    \pmatrix{\nu_\mu \cr \mu}_L,
    \pmatrix{\nu_\tau \cr \tau}_L\,
    \right), \qquad Y_L = -\frac{1}{2} \,; \nonumber\\
Q_L &=& \left(
    \pmatrix{u \cr d}_L,
    \pmatrix{c \cr s}_L,
    \pmatrix{t \cr b}_L\,
    \right), \qquad\quad\  Y_Q = \frac{1}{6}\,.
\end{eqnarray}
The $SU(2)$ quantum number is called weak isospin, and the third
component~$I_3$ distinguishes upper and lower entries.
The gauge and kinetic interactions for $G$ generations of these
fields are
\begin{equation}
    {\cal L}_L = \sum_{i=1}^G
        \bar{L}^i_L(i\kern+0.1em /\kern-0.55em \partial
           -  g_1Y_L\kern+0.1em /\kern-0.65em B
           -  g_2   \kern+0.1em /\kern-0.65em W)L^i_L +
        \bar{Q}^i_L(i\kern+0.1em /\kern-0.65em D
           -  g_1Y_Q\kern+0.1em /\kern-0.65em B
           -  g_2   \kern+0.1em /\kern-0.65em W)Q^i_L \,,
\end{equation}
where $W=W^a\sigma_a/2$ are the gauge bosons of $SU(2)$, with gauge
coupling~$g_2$.
Note that as far as gauge interactions are concerned, the generations 
are simply copies of each other, and ${\cal L}_R+{\cal L}_L$ possesses 
a large global flavor symmetry.
For $G$ generations, the symmetry group is $U(G)^5$, that is, a $U(G)$ 
symmetry for each of~$E_R$, $U_R$, $D_R$, $L_L$, and~$Q_L$.

The assignments of $SU(2)$ and $U(1)_Y$ quantum numbers follow from simple,
experimentally determined properties of weak decays.
For example, by the mid-1980s measurements of decays of
$b$-flavored hadrons had shown the weak isospin of~$b_L$
to be~$-\frac{1}{2}$~\cite{Langacker:1988ce}.
Consequently, its isopartner~$t_L$ had to exist, for symmetry reasons,
although several years passed before the top quark was observed at the
Tevatron.
In contrast, gauge symmetry does not motivate the inclusion of
right-handed neutrinos, which would be neutral under all three gauge
groups.
For this reason, they are usually omitted from the ``standard'' model.

With only gauge fields and fermions, the model is incomplete.
In particular, it does not accommodate the observed non-zero masses of
the quarks, charged leptons, and weak gauge bosons $W^\pm$ and~$Z^0$.
For example, masses for the charged fermions%
\footnote{Because it is completely neutral, a right-handed neutrino
may have a so-called Majorana mass term, coupling neutrino to neutrino,
instead of---or in addition to---a Dirac mass term, coupling neutrino
to anti-neutrino.  For this reason neutrino masses are even more
perplexing than quark and charged lepton masses.}
normally would come from interactions that couple the
left- and right-handed components of the field, such as
\begin{equation}
    {\cal L}_m = -m\left( \bar{\psi}_R \psi_L +
        \bar{\psi}_L \psi_R \right),
\end{equation}
where, in the case at hand, $\psi\in\{e,\mu,\tau,d,s,b,u,c,t\}$.
With the fields introduced above, one would have to combine a component
of a doublet with a singlet, which would violate $SU(2)$.
Any pairing of left- and right-handed fields with the listed
hypercharges would violate $U(1)_Y$ as well.

To construct gauge invariant interactions coupling left- and
right-handed fermions, at least one additional field is necessary.
For simplicity, let us begin with only the first generation leptons.
Consider
\begin{equation}
    {\cal L}_Y = - ye^{i\delta}\, \bar{l}_L\phi\, e_R -
        ye^{-i\delta}\, \bar{e}_R\,\phi^\dagger l_L,
    \label{1:eq:Yuk}
\end{equation}
where $\bar{l}_L=\left(\bar{\nu}_L,\ \bar{e}_L\right)$,
and $y$ is real.
If the quantum numbers of~$\phi$ are chosen suitably, then
the interaction~${\cal L}_I$ would be gauge (and Lorentz) invariant.
To preserve Lorentz invariance, $\phi$ must have spin~0.
To preserve invariance under $U(1)_Y$, the hypercharge
of~$\phi$ must be $Y_\phi=Y_L-Y_E=+\frac{1}{2}$.
To preserve invariance under $SU(2)$ $\phi$, must be a
doublet,
\begin{equation}
    \phi = \pmatrix{ \phi^+ \cr \phi^0 }.
\end{equation}
The superscripts denote the electric charge $Q=Y+I_3$.
An interaction similar to~${\cal L}_Y$ was first introduced by Yukawa
to describe the decay $\pi^+\to\mu^+\nu_\mu$, so it is called a Yukawa
interaction, and the coupling~$y$ is called a Yukawa coupling.

At first glance, the interaction in Eq.~(\ref{1:eq:Yuk}) appears to 
violate $CP$, with a strength proportional to $y\sin\delta$.
One may, however, remove $\delta$, by exploiting the invariance of 
${\cal L}_R+{\cal L}_L$ under independent changes in the phases 
of~$e_R$ and~$l_L$.
Thus, the one-generation Yukawa interaction has only one real 
parameter, $y$, and it conserves~$CP$.

Since it is charged under $SU(2) \times U(1)_Y$, the field~$\phi$
has gauge interactions, which are dictated by symmetry.
The scalar field may also have self-interactions, which are not dictated
by symmetry.
If one limits one's attention to renormalizable interactions
\begin{equation}
    V(\phi) = - \lambda v^2\phi^\dagger\phi +
        \lambda(\phi^\dagger\phi)^2,
\end{equation}
with two new parameters, $v$ and~$\lambda$.
The state with no propagating particles, called the vacuum, is realized 
when $\phi$ minimizes~$V(\phi)$.
The quartic coupling~$\lambda$ must be positive; otherwise the
potential energy would be unbounded from below, and the vacuum would
be unstable.
If $v^2<0$, then there is a single minimum of the potential, with 
vacuum expectation value $\langle\phi\rangle=0$; this possibility does 
not interest us here.
If $v^2>0$, then~$V(\phi)$ takes the shape of a sombrero with a 
three-dimensional family of minima:
\begin{equation}
    \langle\phi\rangle = e^{i\langle\xi^a\rangle\sigma_a/2v}
        \pmatrix{ 0 \cr v/\sqrt{2}},
    \label{1:eq:vev}
\end{equation}
parametrized by~$\langle\xi^a\rangle$.
Through an $x$-independent $SU(2)$ transformation, one can
set~$\langle\xi^a\rangle=0$.
Although the Lagrangian fully respects local $SU(2) \times U(1)_Y$
gauge symmetry, the vacuum solution of the equations of motion given
in Eq.~(\ref{1:eq:vev}) does not:
this is called spontaneous (as opposed to explicit) symmetry breaking.

Physical particles arise from fluctuations around the solution of
the equations of motion, so one writes
\begin{equation}
    \phi(x) = e^{i\xi^a(x)\sigma_a/2v}
        \pmatrix{ 0 \cr [v+h(x)]/\sqrt{2} }.
    \label{1:eq:phi}
\end{equation}
The vacuum expectation values of the fluctuation fields are
$\langle\xi^a\rangle=\langle h\rangle=0$.
Masses of the physical particles are found by inserting
Eq.~(\ref{1:eq:phi}) into the expressions for the interactions in
the Lagrangian and examining the quadratic terms.
By comparing the $\bar{e}_Re_L$ terms in ${\cal L}_Y$ and~${\cal L}_m$,
one sees that the electron mass in this model is $m_e=yv/\sqrt{2}$.
Similarly, from $V(\phi)$ the field $h$ is seen to have a (squared)
mass $m_h^2=2\lambda v^2$, and from the kinetic energy of the
scalar field non-zero masses for three of the gauge bosons arise:
$m_{W^\pm}^2 = \frac{1}{4}g_2^2v^2$ and
$m_{Z^0}^2   = \frac{1}{4}(g_1^2+g_2^2)v^2$,
where $Z^0$ is the massive linear combination of $W^3$ and $B$.
(The orthogonal combination is the massless photon~$\gamma$.)
% Finally, the degrees of freedom from the fields $\xi^\pm$ and $\xi^3$
% become the longitudinally polarized $W^\pm$ and $Z^0$.
The amplitude for muon decay is, to excellent approximation,
proportional to $g_2^2/m_W^2$.
Therefore, one can obtain the vacuum expectation value from the Fermi
decay constant, finding $v=246$~GeV.

Repeating this construction with $\left(\bar{u}_L,\ \bar{d}_L\right)$
and $d_R$ requires a doublet with hypercharge $Y_Q-Y_D=+\frac{1}{2}$.
The Standard Model uses the same doublet as for leptons.
Repeating it with $\left(\bar{u}_L,\ \bar{d}_L\right)$
and $u_R$ requires a doublet with hypercharge~$Y_Q-Y_U=-\frac{1}{2}$.
The Standard Model uses the charge-conjugate 
\begin{equation}
    \tilde{\phi} \equiv i\sigma_2\phi^* =
        \pmatrix{ \overline{\phi^0} \cr  -\phi^- }
\end{equation}
of the doublet used for leptons.
In the one-generation case, three real Yukawa couplings are 
introduced, leading to masses for the electron, down quark, and 
up~quark.

With $G$ generations the full set of Yukawa interactions is complicated.
It is instructive to leave $G$ arbitrary for now, and to compare the 
physics for $G=2$, 3, 4, later on.
The generations may interact with each other as in
\begin{equation}
    {\cal L}_Y = - \sum_{i,j=1}^G \left[
        \hat{y}^e_{ij} \bar{L}^i_L\phi\, E^j_R +
        \hat{y}^d_{ij} \bar{Q}^i_L\phi\, D^j_R +
        \hat{y}^u_{ij} \bar{Q}^i_L\tilde{\phi}\, U^j_R +
		{\rm h.c.}\right],
\end{equation}
because no symmetry would enforce a simpler structure.
For $G$ generations, the Yukawa matrices are complex $G\times G$
matrices.
At first glance, each matrix~$\hat{y}^a$ seems to introduce $2G^2$ 
parameters: $G^2$ that are real and $CP$-conserving, and another $G^2$ 
that are imaginary and $CP$ violating.
But, as in the one-generation case, one must think carefully about 
physically equivalent matrices before understanding how many physical
parameters there really are.

Let us consider the leptons first.
As mentioned above, the non-Yukawa part of the Lagrangian is invariant 
under the following transformation of generations
\begin{eqnarray}
    E_R \mapsto R E_R\,, & \qquad & \bar{E}_R \mapsto \bar{E}_R R^\dagger, 
    \nonumber\\
    L_L \mapsto S L_L\,, & \qquad & \bar{L}_L \mapsto \bar{L}_L S^\dagger,
    \label{1:eq:UGlepton}
\end{eqnarray}
where $R\in U(G)_{E_R}$ and $S\in U(G)_{L_L}$.
That means that the Yukawa matrix $\hat{y}^e$ is equivalent to 
${y^e=S\hat{y}^eR^\dagger}$.
By suitable choice, $y^e$ can be made diagonal, real, and
non-negative.
The leptons' Yukawa interactions now read
\begin{equation}
    {\cal L}_{Yl} = -\sum_{i=1}^G \left[
        y^e_{i} \bar{L}^i_L\phi\, E^i_R + {\rm h.c.} \right].
    \label{1:eq:Yl}
\end{equation}
Note that if $S$ and $R$ achieve this structure, so do $S'=DS$ and
$R'=DR$, where $D=\diag(e^{i\varphi_1},\ldots,e^{i\varphi_G})$.
Thus, part of the transformation from $\hat{y}^e$ to $y^e$ is redundant
and must not be counted twice.
(The freedom to choose these phases leads to global conservation of
lepton flavor.)
Hence, the transformation removes $2G^2-G$ parameters, leaving $G$
independent entries in~$y^e$.
Since all are real, there is no $CP$ violation.

For the quarks the reasoning is the same but the algebra is trickier.
There are now three distinct $U(G)$ symmetries, and
the non-Yukawa Lagrangian is invariant under 
\begin{eqnarray}
    D_R \mapsto R_d D_R\,, &\qquad& \bar{D}_R \mapsto \bar{D}_R R_d^\dagger\,, 
    \nonumber\\
    U_R \mapsto R_u U_R\,, &\qquad& \bar{U}_R \mapsto \bar{U}_R R_u^\dagger\,, 
     \\
    Q_L \mapsto S_u Q_L\,, &\qquad& \bar{Q}_L \mapsto \bar{Q}_L S_u^\dagger\,.
    \nonumber
    \label{1:eq:UGquark}
\end{eqnarray}
One may again exploit $S_u$ and $R_u$ to transform $\hat{y}^u$ into the
diagonal, real, non-negative form~$y^u$.
Then the transform of $\hat{y}^d$ is, in general, neither real nor diagonal.
Instead
\begin{equation}
    S_u \hat{y}^d R_d^\dagger = V y^d \,,
    \label{1:eq:VydR}
\end{equation}
where $y^d=S_d\hat{y}^dR_d^\dagger$ \emph{is} diagonal, real, and
non-negative, and
\begin{equation}
    V = S_uS_d^\dagger
    \label{1:eq:CKM}
\end{equation}
is the Cabibbo-Kobayashi-Maskawa (CKM)
matrix~\cite{Cabibbo:1963yz,Kobayashi:1973fv}.
By construction, $V$ is a $G\times G$ unitary matrix.
The quarks' Yukawa interactions now read
\begin{equation}
    {\cal L}_{Yq} = 
        - \sum_{i,j=1}^G \left[y^d_{j}\bar{Q}^i_L\phi\,  V_{ij}D^j_R +
        {\rm h.c.} \right]
        - \sum_{i=1}^G   \left[y^u_{i}\bar{Q}^i_L\tilde{\phi}\,U^i_R +
        {\rm h.c.} \right].
    \label{1:eq:Yq}
\end{equation}
If $S_u$, $R_u$, and $R_d$ achieve this structure, so do
$e^{i\varphi}S_u$, $e^{i\varphi}R_u$, and $e^{i\varphi}R_d$.
(The freedom to choose this phase leads to global conservation of
total baryon number.)
Thus, the manipulations remove $3G^2-1$ parameters from the $4G^2$ in 
two arbitrary $G\times G$ matrices, leaving $G^2+1$.
Of these, $2G$ are in $y^u$ and $y^d$, and the other 
$(G-1)^2$ are in the CKM matrix~$V$.
One can also count separately the real and imaginary parameters.
Since a $G\times G$ unitary matrix has $\frac{1}{2}G(G-1)$ real and
$\frac{1}{2}G(G+1)$ imaginary components, one finds that the CKM matrix
has $\frac{1}{2}G(G-1)$ real, $CP$-conserving parameters,
and $\frac{1}{2}(G-1)(G-2)$ imaginary, $CP$ violating parameters.
For example, the case $G=2$ has no $CP$ violation from this mechanism, 
$G=3$ has a single $CP$ violating parameter,%
\index{CP violation@\CP\ violation!Kobayashi-Maskawa mechanism}
and $G=4$ has three.

The CKM matrix $V$ arises from the misalignment of the matrices~$S_u$
and $S_d$.
Under circumstances that preserve some of the flavor symmetry, they can
be partially aligned, and then $V$ contains even fewer physical
parameters.
In an example with three generations, if two entries either in~$y^u$
or in~$y^d$ are equal, partial re-alignment removes one real angle
and one phase.
Therefore, the CKM mechanism leads to $CP$ violation only if
like-charged quarks all have distinct masses.

Substituting Eq.~(\ref{1:eq:phi}) into ${\cal L}_Y$ and keeping
quadratic terms shows that the masses are
\begin{equation}
    m^a_{k} = \frac{v}{\sqrt{2}}\, y^a_{k} \,,
\end{equation}
for $k=1, 2, 3$, and $a=e, d, u$.
For quarks this is easiest to see if one sets
\begin{equation}
    Q_L = \pmatrix{ U_L \cr VD_L },
    \label{1:eq:D}
\end{equation}
which diagonalizes the mass terms for the down-like quarks in 
Eq.~(\ref{1:eq:Yq}).
In this basis the CKM matrix migrates to the charged-current vertex:
\begin{equation}
    {\cal L}_{\bar{U}WD} = - \frac{g_2}{\sqrt{2}} \left[ 
        \bar{U}_L            \kern+0.1em /\kern-0.65em W^+ V D_L +
        \bar{D}_L V^\dagger \kern+0.1em /\kern-0.65em W^- U_L
        \right],
    \label{1:eq:UWD}
\end{equation}
where $W^\pm=(W^1\mp iW^2)/\sqrt{2}$.
The basis in \eq{1:eq:D}, with diagonal mass matrices and the CKM matrix
in the charged currents of quarks, is usually adopted in phenomenology.

Note that the neutral current interactions are unaffected by writing
$V$ in \eq{1:eq:D}.
Thus, there are no flavor-changing neutral currents (FCNC) at the tree
level in the Standard Model.
This is known as the Glashow-Iliopoulus-Maiani (GIM) effect~\cite{gim}.
Even at the loop level, where the FCNCs do arise, the GIM mechanism
can suppress processes by a factor $m_q^2/m_W^2$, which is very small,
except in the case of the top quark.
GIM suppression and Cabibbo suppression (i.e., factors of $\lambda$)
both imply near-null predictions for several processes.
Observation of any of these would constitute a clear signal of
non-standard physics.

Note that quark and lepton masses arise from the same microscopic
interactions as CKM flavor violation.
In Nature, the quark and lepton masses vary over orders of magnitude.
Thus, the large flavor symmetry that would arise in the absence of
Yukawa interactions is severely broken.
In the Standard Model, the Yukawa couplings are simply chosen to
contrive the observed masses.
This is unsatisfactory, but we lack the detailed experimental
information needed to develop a deeper theory of flavor.


\subsection{General models}
\label{1:subsec:general}

The foregoing discussion makes clear that the unitary CKM matrix
arises in an algebraic way.
Therefore, the mechanism can survive in models with a more complicated
Higgs sector.
The Standard Model is a model of economy: a single doublet generates
mass for the gauge bosons, charged leptons, down-like quarks, and
up-like quarks.
In models with two doublets (and three or more generations), the CKM
source of $CP$ violations remains, but there can be additional $CP$
violation in the Higgs sector~\cite{Gunion:1989we}.

To emphasize this point, let us consider an extreme example with
\begin{equation}
    {\cal L}_Y =
        \bar{L}^i_L\Phi^e_{ij} E^j_R +
        \bar{Q}^i_L\Phi^d_{ij} D^j_R +
        \bar{Q}^i_L\tilde{\Phi}^u_{ij} U^j_R + {\rm h.c.},
    \label{1:eq:ugly}
\end{equation}
where $i,j$ run over generations, and we take the basis in which
gauge interactions do not change generations.
The tilde on $\tilde{\Phi}_u$ is introduced so that, with
$\tilde{\Phi}^u=i\sigma_2{\Phi^u}^*$, all $\Phi^a$ are (matrix) 
fields with hypercharge~$+\frac{1}{2}$.
Here the Yukawa couplings are absorbed into the
fields---Eq.~(\ref{1:eq:ugly}) is hideous enough as it is.
Since all~$\Phi^a$ are doublets under $SU(2)$, they all would participate 
in electroweak symmetry breaking.

Suppose the Higgs potential, now a complicated function of all the 
scalar fields, leads to vacuum expectation values of the form
\begin{equation}
    \langle\Phi^a_{ij}\rangle = \pmatrix{ 0 \cr \hat{m}^a_{ij} }.
\end{equation}
Then the $\hat{m}^a_{ij}$ are mass matrices, and the algebra leading to
the real, physical masses~$m^a_k$ and the CKM matrix is just as above.
The CKM matrix, $V$, survives and should lead to $CP$ violation, because
there is no good reason for the phase in~$V$ to be small.
There would, however, almost certainly be new sources of $CP$ 
violation from the Higgs sector.

\boldmath
\subsection{$CP$ violation from a unitary CKM matrix}
\label{ch1:sect:ut}
\unboldmath

In the standard, one-doublet, model, we see that flavor and $CP$
violation arise solely through the CKM matrix.
Furthermore, in more general settings, the CKM matrix can still
arise, but there may be other sources of $CP$ violation as well.
If the CKM matrix is the only source of $CP$ violation, there are
many relations between $CP$-conserving and $CP$ violating observables
that arise from the fact that $V$ is a unitary matrix.
This section outlines a framework for testing whether these
constraints are, in fact, realized.

A useful way~\cite{Jarlskog:1985ht} of gauging the size of $CP$
violation starts with the commutator of the mass matrices,
$C=[\hat{m}^u\hat{m}^{u\dagger}, \hat{m}^d\hat{m}^{d\dagger}]$, which
can be re-written
\begin{equation}
    C = S_u^\dagger \left[(m^u)^2,\, V(m^d)^2\, V^\dagger\right] S_u\,,
\end{equation}
to show that $\det C$ depends on the physical masses and~$V$.
After some algebra one finds
\begin{equation}
    \det C = - 2i F_u F_d J\,,
    \label{1:eq:CJ}
\end{equation}
where
\begin{eqnarray}
    F_u & = & (m_u^2-m_c^2) (m_c^2-m_t^2) (m_t^2-m_u^2)\,, \\
    F_d & = & (m_d^2-m_s^2) (m_s^2-m_b^2) (m_b^2-m_d^2)\,, \\
     J  & = & \imag[V_{11}V^*_{21}\,V_{22}V^*_{12}]\,.
\end{eqnarray}
To arrive at Eq.~(\ref{1:eq:CJ}) one makes repeated use of the
property $VV^\dagger=\openone$, especially that
\begin{equation}
     J = \imag[V_{ij}V^*_{kj}\,V_{kl}V^*_{il}]\,
        \sum_m \varepsilon_{ikm} \sum_n \varepsilon_{jln}\,,
\end{equation}
for all combinations of $i$, $j$, $k$, and~$l$.
The determinant $\det C$ captures several essential features of $CP$
violation from the CKM mechanism.
It is imaginary, reminding us that $CP$ violation stems from a complex
coupling.
More significantly, there is no $CP$ violation unless $F_u$, $F_d$,
and~$J$ are all different from zero.
Non-vanishing $F_u$ and $F_d$ codify the requirements on the quark
masses given above.
Non-vanishing~$J$ codifies requirements on~$V$, which are
clearest after choosing a specific parameterization.
The key point, however, is that the value taken by $J$ is independent of
the parameterization, by construction of $\det C$.

To emphasize the physical transitions associated with the CKM matrix,
it is usually written
\begin{equation}
    V = \left( \begin{array}{ccc}
                V_{ud} & V_{us} & V_{ub} \\
                V_{cd} & V_{cs} & V_{cb} \\
                V_{td} & V_{ts} & V_{tb} 
        \end{array} \right),
\end{equation}
so that the entries are labeled by the quark flavors.
  From Eq.~(\ref{1:eq:UWD}),
the vertex at which a $b$ quark decays to a $W^-$ and $c$ quark is 
proportional to $V_{cb}$; similarly, the vertex at which a $c$ quark 
decays to a $W^+$ and $s$ quark is proportional to $V_{cs}^*$.
Because $V$ is unitary, $|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 =1$, and
similarly for all other rows and columns.
These constraints give information on unmeasured (or poorly measured)
elements of~$V$.
For example, because $|V_{cb}|$ and $|V_{ub}|$ are known to be small,
$|V_{tb}|$ should be very close to 1---if, indeed, there are only
three generations.
Furthermore, $|V_{ts}|$ and $|V_{td}|$ must also be small.

Even more interesting constraints come from the orthogonality of
columns (or rows) of a unitary matrix.
Taking the first and third columns of~$V$, one has
\begin{equation}
     V_{ud}V^*_{ub} + V_{cd}V^*_{cb} + V_{td}V^*_{tb} = 0\,.
     \label{1:eq:UT}
\end{equation}
Equation~(\ref{1:eq:UT}) says that the three terms in the sum
trace out a triangle on the complex plane.
Because it is a consequence of the unitarity property of~$V$, this
triangle is called the ``unitarity triangle,''
shown in Fig.~\ref{1:fig:UT}.
\begin{figure}[bt]
\centerline{
    \epsfysize 4cm
	\epsffile{ch1/Vtriangle.eps}
	\hfill
    \epsfysize 4cm
	\epsffile{ch1/triangle.eps}
	}
\vskip 6pt
\caption[The unitarity triangle]{The unitarity triangle.
The version on the left directly expresses Eq.~(\ref{1:eq:UT}).
The rescaled version shows the definition of $(\bar{\rho},\bar{\eta})$.}
\label{1:fig:UT}
\end{figure}
The lengths of the sides are simply $|V_{ud}V^*_{ub}|$, etc.,
and the angles are
\begin{equation}
    \alpha = \arg\left[ - \frac{V_{td}V_{tb}^*}{V_{ud}V_{ub}^*}\right],
    \qquad
    \beta  = \arg\left[ - \frac{V_{cd}V_{cb}^*}{V_{td}V_{tb}^*}\right],
    \qquad
    \gamma = \arg\left[ - \frac{V_{ud}V_{ub}^*}{V_{cd}V_{cb}^*}\right].
    \label{1:eq:beta}
\end{equation}
The notation $\beta \equiv \phi_1$, $\alpha \equiv \phi_2$,
$\gamma \equiv \phi_3$ is also used.
By construction $\alpha+\beta+\gamma=\pi$.
The area of the triangle is $|J|/2$ and the terms trace out the triangle
in a counter-clockwise (clockwise) sense if $J$ is positive (negative).
In fact, there are five more unitarity triangles, all with area $|J|/2$
and orientation linked to the sign of~$J$.

The unitarity triangle(s) are useful because they provide a simple,
vivid summary of the CKM mechanism.
Separate measurements of lengths, through decay and mixing rates,
and angles, through $CP$ asymmetries, should fit together.
Furthermore, when one combines measurements---from the $B$, $B_s$, $K$,
and $D$ systems, as well as from hadronic $W$ decays---all triangles
should have the same area and orientation.
If there are non-CKM contributions to flavor or $CP$ violation,
however, the interpretation of rates and asymmetries as measurements
of the sides and angles no longer holds.
The triangle built from experimentally defined sides and angles will
not fit with the CKM picture.

In the parameterization favored by the Particle Data Book~\cite{pdg}
\begin{equation}
    V = \left( \begin{array}{ccc}
                c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta_{13}} \\
                -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{13}} &
                 c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{13}} &
                s_{23}c_{13} \\
                 s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{13}} &
                -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{13}} &
                c_{23}c_{13}
        \end{array} \right),
    \label{1:eq:VPDG}
\end{equation}
where $c_{ij}=\cos\theta_{ij}$ and $s_{ij}=\sin\theta_{ij}$.
The real angles~$\theta_{ij}$ may be chosen so that
$0\le\theta_{ij}\le\pi/2$, and the phase $\delta_{13}$ so that
$0\le\delta_{13}<2\pi$.
With Eq.~(\ref{1:eq:VPDG}) the Jarlskog invariant becomes
\begin{equation}
    J = c_{12} c_{23} c_{13}^2 s_{12} s_{23} s_{13} \sin\delta_{13}\,.
\end{equation}
The parameters must satisfy
\begin{equation}
    \delta_{13}\neq 0,\;\pi\,; \qquad \theta_{ij}\neq 0,\;\pi/2\,;
\end{equation}
otherwise $J$ vanishes.
Since $CP$ violation is proportional to $J$, 
the CKM matrix must not only have complex entries, but also 
non-trivial mixing;
otherwise the KM phase~$\delta_{13}$ can be removed.

A convenient parameterization of the CKM matrix is due to
Wolfenstein~\cite{wolf}.
It stems from the observation that the measured matrix obeys a
hierarchy, with diagonal elements close to~1, and progressively
smaller elements away from the diagonal.
This hierarchy can be formalized by defining
$\lambda$, $A$, $\rho$, and $\eta$ via
\begin{equation}
    \lambda \equiv s_{12}\,, \qquad % \approx0.22\,, \qquad
    A \equiv s_{23}/\lambda^2\,, \qquad
    \rho+i\eta \equiv s_{13}e^{i\delta_{13}}/A\lambda^3\,.
\end{equation}
  From experiment $\lambda\approx 0.22$, $A\approx 0.8$,
and $\sqrt{\rho^2+\eta^2}\approx 0.4$, so it is phenomenologically
useful to expand $V$ in powers of~$\lambda$:
\begin{equation}
    V = \left( \begin{array}{ccc}
                1-\frac{1}{2}\lambda^2 & \lambda & A\lambda^3(\rho-i\eta) \\
                -\lambda & 1-\frac{1}{2}\lambda^2 & A\lambda^2 \\
                 A\lambda^3(1-\rho-i\eta) & -A\lambda^2 & 1
        \end{array} \right) + {\cal O}(\lambda^4)\,.
\end{equation}
The most interesting correction at ${\cal O}(\lambda^4)$ for our purposes is
$\imag V_{ts}=-A\lambda^4\eta$.
The Jarlskog invariant can now be expressed $J = A^2\lambda^6\eta \approx
(7\times10^{-5})\eta$. One sees that CKM $CP$ violation is small not because
$\delta_{13}$ is small but because flavor violation must also occur, and flavor
violation is suppressed, empirically, by powers of~$\lambda$.

The unitarity triangle in Eq.~(\ref{1:eq:UT}) is special, because its
three sides are all of order $A\lambda^3$.
The triangle formed from the orthogonality of the first and third
rows also has this property, but it is not accessible, because
the top quark decays before the mesons needed to measure the angles are
bound.
The other triangles are all long and thin, with sides
$(\lambda,  \lambda,  A\lambda^5)$ (e.g., for the kaon) or
$(\lambda^2,\lambda^2,A\lambda^4)$ (e.g., for the $B_s$ meson).

It is customary to rescale Eq.~(\ref{1:eq:UT}) by the common factor
$A\lambda^3$, to focus on the less well-determined parameters
$(\rho,\eta)$.
In the context of the Wolfenstein parameterization, there are many ways
to do this.
Since we anticipate precision in experimental measurements, and also in
theoretical calculations of some important hadronic transition
amplitudes, it is useful to choose an exact rescaling.
We choose to divide all three terms in Eq.~(\ref{1:eq:UT}) by
$V_{cd}V_{cb}^*$ and define%
\footnote{This definition differs at ${\cal O}(\lambda^4)$ from the original
one in Ref.~\cite{blo}.}
\begin{equation}
	\bar{\rho} + i \bar{\eta} \equiv
		- \frac{ V_{ud} V_{ub}^* }{ V_{cd} V_{cb}^* }.
\end{equation}
Then the rescaled triangle, also shown in Fig.~\ref{1:fig:UT}, has its
apex in the complex plane at $(\bar{\rho},\bar{\eta})$.
The angles of the triangle are easily expressed
\begin{equation}
    \alpha = \tan^{-1}\left(
		\frac{\bar{\eta}}{\bar{\eta}^2+\bar{\rho}(\bar{\rho}-1)}
		\right)\,, \quad
    \beta  = \tan^{-1}\left(\frac{\bar{\eta}}{1-\bar{\rho}}\right)\,, \quad
    \gamma = \tan^{-1}\left(\frac{\bar{\eta}}{\bar{\rho}}\right)\,,
    \label{1:eq:betarhoeta}
\end{equation}
Since $\bar{\eta}$, $\bar{\rho}$, and $1-\bar{\rho}$ could easily be of
comparable size, the angles and, thus, the corresponding $CP$
asymmetries could be large.

At the Tevatron there is also copious production of $B_s$~mesons.
The corresponding unitarity triangle is
\begin{equation}
     V_{us}V^*_{ub} + V_{cs}V^*_{cb} + V_{ts}V^*_{tb} = 0 \,,
     \label{1:eq:UTBs}
\end{equation}
replacing the $d$~quark with~$s$.
In Eq.~(\ref{1:eq:UTBs}) the first side is much shorter than the other
two.
Therefore, the opposing angle
\begin{equation}
    \beta_s  = \arg\left[ - \frac{V_{ts}V_{tb}^*}{V_{cs}V_{cb}^*}\right]
        = \lambda^2 \eta + {\cal O}(\lambda^4)
    \label{1:eq:betasdef}
\end{equation}
is small, of order one degree.
Therefore, the asymmetries in $B_s\to \psi\eta^{(\prime)}$
and $B_s\to \psi\phi$ are much smaller than in the corresponding $B$
decays.
On the other hand, this asymmetry is sensitive to new physics
in $B_s^0 - \B0bar_s$ mixing.
In the standard model, as discussed below, mixing is induced by
loop processes.
When, as here, there is also Cabibbo suppression, it is easy for the
non-standard phenomena to compete.
Thus, in the short term a measurement of $\beta_s$ represents a search
for new physics, whereas in the long term it would be a verification of
the CKM picture.

The unitarity triangle for the $D$ system comes from the 
orthogonality of the top two rows of the CKM matrix.
It is even longer and thinner than the one for the $B_s$ system.
Consequently, a non-zero measurement of the $CP$ asymmetry associated with
the small angle is a clear sign of new physics.
It seems that experiments to measure the $D$-system unitarity triangle 
are not yet feasible.


\subsection{Hadronic uncertainties and clean measurements}
\label{1:subsec:clean}

At a superficial level, the way to test the CKM picture is to measure
rates and asymmetries that are sensitive to the sides and angles
(of all triangles), in as many ways as possible.
A~serious obstacle, however, is that the quarks are confined in
hadrons.
Consequently, most relations between experimental observables and
Lagrangian-level couplings, like the CKM matrix, involve hadronic matrix
elements.
In this subsection, we summarize briefly how to treat the matrix
elements, with an eye toward identifying processes that are relatively
free of hadronic uncertainty.

There are several approaches for treating the hadronic matrix elements,
none of which is universally useful.
In Sec.~\ref{1:tools} we introduce some of the essential tools in 
greater detail.
Here we illustrate the possibilities with examples.
\begin{itemize}
    \item Perfect (or essentially perfect) symmetries of QCD, such as $C$
        or $CP$: When a single CKM factor dominates a process, the QCD
        part of the amplitude cancels in ratios such as the $CP$
        asymmetry in interference between decays with and without
        mixing. The two best examples are in the asymmetries for%
\footnote{Here, and in the rest of this chapter, $\psi$ stands for any
charmonium state, $J/\psi$, $\psi'$, $\chi_c$, etc.}
	$B\to \psi K_S$ and $B_s\to D_s^\pm K^\mp$.
	Assuming that $CP$ violation comes only from the CKM matrix,
	the first class of modes cleanly yields~$\sin 2\beta$, and
	the other pair of modes cleanly yields~$\sin(\gamma-2\beta_s)$.
    \item Approximate symmetries, such as isospin, flavor $SU(3)$, chiral
	symmetry, or heavy quark symmetry: The best-known examples are when the
	symmetry restricts a form factor for semileptonic decays. Isospin and
	$n\to pe\bar{\nu}_e$ give $|V_{ud}|$; flavor $SU(3)$ and $K\to\pi
	e\bar{\nu}_e$ give $|V_{us}|$; and heavy quark symmetry and $B\to
	D^*\ell\bar\nu_\ell$ give $|V_{cb}|$. The hadronic uncertainty is now
	in the deviation from the symmetry limit. An even more intriguing use
	of isospin is to relate the form factor of $K^0\to\pi^+e\bar{\nu}_e$ to
	that of $K^{0,\pm}\to\pi^{0,\pm} \nu\bar\nu$. The rare $\nu\bar\nu$
	decays are, thus, essentially free of hadronic uncertainties.%
\footnote{In the charged mode there is an uncertainty stemming from the 
uncertainty in the charmed quark mass. It has been estimated to be around 5\%
in $|V_{ts}V_{td}|$~\cite{Buchalla:1996fp}.  Power suppressed corrections to
$K\to \pi\nu\bar\nu$ have also been estimated and found to be
small~\cite{mcsup}.}
    \item Lattice QCD:
        This computational method is sound, in principle, for hadronic
        matrix elements with at most one final-state hadron.
        Limitations in computer power have led to an approximation,
        called the quenched approximation, whose error is difficult
        to quantify.
        With increases in computer resources, lattice results should, in
        the future, play a more important role in determining the sides 
        of the unitarity triangles.
        For more details, see Sec.\ref{1:lattice}.
    \item Perturbative QCD for exclusive processes:
        It may be possible to calculate the strong phases of
		certain\index{strong phase shifts!from perturbative QCD}
        nonleptonic $B$ decays using perturbative QCD.
        This is in some ways analogous to computing cross sections
        in hadronic collisions, and the nonperturbative information
        is captured in light-cone distribution
        amplitudes~\cite{Lepage:1980fj}.
        There are, at present, two different
        approaches~\cite{Chang:1997dw,Beneke:1999br},
        whose practical relevance remains an open question.
    \item QCD sum rules:
        Like lattice and perturbative QCD, sum rules are based on QCD
        and field theory.
        Uncertainty estimates are usually semi-quantitative and it is
        difficult to reduce them in a controlled manner.
    \item Models of QCD, such as quark models, naive factorization, etc:
        These techniques can be applied for back-of-the-envelope
        estimates.
        There is no prospect for providing a quotable error and, thus,
        should not be used in quantitative work.
\end{itemize}
In summary, at the present time the cleanest observables are the
$CP$ asymmetries in $\B0bar_d$ decays to charmonium+kaon and 
in $\B0bar_s$ decays to $D_s^\pm K^\mp$.
The rare decay $K_L\to\pi^0\nu\bar\nu$ is free of theoretical 
uncertainties at a similar level, but presents a big experimental 
challenge.
Semileptonic decays restricted by symmetries as well as 
$K^\pm\to\pi^\pm\nu\bar\nu$ are a step down, but still good.
With enough computer power to overcome the quenched approximation,
lattice QCD could yield, during the course of Run~II, controlled
uncertainty estimates for neutral-meson mixing and leptonic and
semileptonic decays of a few percent.


\boldmath
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{General Formalism for Mixing and $CP$ Violation}
\label{ch1:sect:gen} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\unboldmath

This section is devoted to the general formalism for meson mixing and 
$CP$ violation. Much of the material can also be found in other
review articles and reports \cite{babook,BLS,cern}, but some
topics require a different viewpoint in the light of the $B$ physics
program at a hadron collider: unlike the $B$ factories the Tevatron
will be able to study $B_s^0$ mesons. The two mass eigenstates in the
$B_s^0$ system may involve a sizable width difference \dg, which must
be included in the formulae for the $B_s^0$ time evolution. We
consequently present these formulae including all effects from a
non-vanishing \dg.  

Many details of the formalism depend on conventions, particularly in 
the choice of the complex
phases that unavoidably appear in any \CP\ violating physical system.
We would like to discourage the reader from combining formulae from
different sources, so we try to give a comprehensive and self-contained
presentation of the subject. We start by introducing the discrete
transformation \C, \p\ and \T\ in Sec.~\ref{subs:disc}. 
Experimentally we know that \C, \p\ and \T\ are
symmetries of the electromagnetic and strong interactions, 
so the corresponding quantum numbers can be used to classify the
hadron states.
The description in Sec.~\ref{subs:mix} of the time evolution of the 
neutral $B$ meson system is
applicable to both the $B_d^0$ and the $B_s^0$ meson systems.
Sec.~\ref{subs:unt} deals with untagged $B^0$ decays and
Sec.~\ref{subs:cp} presents the formulae for \CP\ asymmetries.
Finally, in Sec.~\ref{subs:cp} we discuss phase conventions and
rephasing invariant quantities.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Discrete transformations}
\label{subs:disc}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\index{parity} \index{time reversal} \index{charge conjugation}
In this section we introduce the parity, \p, time reversal, \T, 
and charge conjugation, \C, transformations. \p\ and
\T\ are defined through their action on coordinate vectors
$x=(x^0, x^1, x^2, x^3)$: \p\ flips the sign of the spatial
coordinates $x^1$,$x^2$,$x^3$ and \T\ changes the time component
$t=x^0$ into $-t$. Adopting the convention $g_{\mu \nu} = \mbox{diag}
(1,-1,-1,-1) $ for the Lorentz metric, one can compactly express the
transformations in terms of $x_{\mu} = g_{\mu \nu} x^{\nu}$:
\bey
\p: && \quad x^{\mu} \to x_{\mu} \,, \nn 
\T: && \quad x^{\mu} \to -x_{\mu} \,. \label{defpt} 
\eey
The definition \eq{defpt} implies that the derivative operator
$\partial^\mu=\partial/\partial x_\mu$
and the momentum $p^\mu$ transform under $P$ and $T$ 
in the same way as $x^{\mu}$. 
Finally, \C\ interchanges particles and anti-particles.
Apart from the weak interactions, these transformations are symmetries
of the Standard Model.
It is therefore convenient to classify hadronic states by their \C, 
\p\ and \T\ quantum numbers, which are multiplicative and take the 
values~$\pm1$.
 

% In quantum field theories like the Standard
% Model elementary particles are represented by $x$-dependent scalar,
% spinor and vector fields, whose dynamics is defined by their
% interaction Lagrangian.  A Dirac spinor describes a fermion, two of
% its four components represent some spin-1/2 particle and the other two
% components describe the corresponding antiparticle. The \C\
% transformation exchanges the particle and antiparticle components of
% the spinor. If the action and the ground state of a theory are
% invariant under some group of transformations, one encounters a
% symmetry. In this case one can assign quantum numbers to the particle
% states, and the interactions conserve these quantum numbers.  In the
% case of \C, \p\ and \T\ the quantum numbers are
% multiplicative and assume only the two values $+1$ and $-1$.
% Experimentally we know that \C, \p\ and \T\ are
% symmetries of the electromagnetic and strong interactions, which
% enables us to use the corresponding quantum numbers to classify the
% hadron states.

The Lagrangian of the Standard Model and its possible extensions
contain bilinear currents of the quark fields, to which gauge bosons and
scalar fields couple. 
For example, as discussed in Sec.~\ref{ch1:sec:cpvsm}, 
the $W$ boson field $W^{\mu}$ couples
to the chiral vector current $\ov{b}_L \gamma_{\mu} c_L$.
%\!=\!\ov{b}\gamma_{\mu}(1-\gamma_5) c/2$.  
Quark bilinears also appear in composite operators which represent the 
Standard Model interactions in low energy effective Hamiltonians, 
cf., Sec.~\ref{1:Heff}.
To understand how these interactions work, it is helpful to list the
transformation of the quark bilinears under \C, \p, \T\ and the
combined transformations \CP\ and \CPT. For illustration we specify to
currents involving a $b$ and a $d$ field.  The generic transformation
under some discrete symmetry $\mathit{X}$ is
\beq
\mathit{X}: \qquad \ov{b}\, \Gamma d \;\; \to \;\;  
    \mathit{X} \, \ov{b}\, \Gamma d \, \mathit{X}^{-1}, 
\eeq
and \tab{cpt} lists the transformation for the chiral scalar, vector
and magnetic
currents.\index{CP transformation@\CP\ transformation!quark currents}
%
\begin{table}[tp]
\begin{center}
$ \begin{array}{r@{\quad}|@{\quad}l@{\quad}l@{\quad}l}
\hline\hline
\mbox{~~current} &\ds  \ov{b}_{R}\, d_{L} \,(x^{\rho}) &\ds  
    \phantom{-}  \ov{b}_{L}\, \gamma_{\mu} \, d_{L}\,  (x^{\rho}) &\ds   
    \phantom{-} \ov{b}_{R}\, \sigma_{\mu \nu} d_{L}\,  (x^{\rho}) \\ \hline 
\C\ &\ds   \ov{d}_{R}\, b_{L} \, (x^{\rho}) \, \eta_C &\ds  
      - \ov{d}_{R} \, \gamma_{\mu}\,  b_{R}  (x^{\rho}) \, \eta_C &\ds  
     - \ov{d}_{R}\, \sigma_{\mu \nu}\, b_{L}  (x^{\rho}) \\
\p\ &\ds    \ov{b}_{L}\, d_{R} \, (x_{\rho}) \, \eta_P &\ds  
   \phantom{-}\ov{b}_{R}\, \gamma^{\mu} \, d_{R}\,  (x_{\rho}) \, \eta_P &\ds  
   \phantom{-}\ov{b}_{L}\, \sigma^{\mu \nu}\, d_{R}\,  (x_{\rho})\, \eta_P \\
\CP\ &\ds   \ov{d}_{L}\, b_{R}\, (x_{\rho})\, \eta_C  \eta_P &\ds  
      - \ov{d}_{L}\, \gamma^{\mu}\,  b_{L}\,  (x_{\rho})\,  \eta_C \eta_P&\ds  
     - \ov{d}_{L}\, \sigma^{\mu \nu}\, b_{R}\,  (x_{\rho})\, \eta_C \eta_P \\
\T\ &\ds    \ov{b}_{R}\, d_{L}\, (-x_{\rho})\, \eta_T &\ds  
  \phantom{-}\ov{b}_{L}\, \gamma^{\mu}\, d_{L}\,  (-x_{\rho}) \,  \eta_T &\ds  
  \phantom{-}\ov{b}_{R}\, \sigma^{\mu \nu}\, d_{L}\,  (-x_{\rho})\, \eta_T \\
\CPT\ &\ds  
\ov{d}_{L}\, b_{R}\, (-x^{\rho}) \, \eta_C  \eta_P \eta_T &\ds   
 - \ov{d}_{L}\, \gamma_{\mu}\,b_{L}\,(-x^{\rho})\, \eta_C  \eta_P \eta_T 
&\ds 
\phantom{-}\ov{d}_{L}\, \sigma_{\mu \nu}\, b_{R}\,  (-x^{\rho}) 
    \, \eta_C  \eta_P \eta_T \\ \hline\hline
\end{array} $ \\[6pt]
\caption[\C, \p\ and \T\ transformation properties of quark bilinears]%
{\C, \p\ and \T\ transformation properties of the chiral scalar, vector
and magnetic currents.
The coordinate $x$ in parentheses is the argument of both quark fields.}
\label{cpt}
\end{center}
\end{table}
Here $\sigma_{\mu \nu}=(i/2)[ \gamma_{\mu}, \gamma_{\nu} ] $. 
%%\index{$\sigma_{\mu \nu}$, definition} 
The transformation laws for the currents with opposite chirality are
obtained by interchanging $L\leftrightarrow R$ in \tab{cpt}.
The phase factors
\beq
\eta_X = \eta_X^{bd} = e^{i \lt( \phi_X^d
  - \phi_X^b \rt)}, \qquad  X=C,P,T \,. \label{phs} 
\eeq
depend on the quark flavors, but for simplicity the flavor indices of 
the $\eta_X$s have been omitted in \tab{cpt}.
One can absorb these
arbitrary phase factors $\exp(i \phi_X^q)$ into the definitions 
of the discrete transformations for every quark field in the theory. 
This feature originates from the freedom to redefine any quark field 
by a phase transformation\index{phase transformation!quark field}
\beq
q \to q \, e^{i \phi^q } . \label{pht} 
\eeq 
In the absence of flavor-changing couplings the change in \eq{pht}
is a $U(1)$ symmetry transformation leaving the Lagrangian
invariant. The corresponding conserved quantum number is the flavor of
the quark $q$. After including the flavor-changing interactions,
the phase transformations in \eq{pht} change the phases of the
flavor-changing couplings. The flavor symmetry is broken and every
phase transformation \eq{pht} leads to a different, but physically
equivalent Lagrangian. In the case of the Standard Model these
transform the Yukawa couplings and, hence, the CKM matrix from one phase 
convention into another.\index{phase convention!CKM matrix}

The currents in \tab{cpt} create and destroy the meson states with the
appropriate quantum numbers. Since the QCD interaction, which binds
the quarks into mesons, conserves \C, \p\ and \T, the  
meson states transform like the corresponding currents in \tab{cpt}.%
\index{CP transformation@\CP\ transformation!B meson@$B$ meson}%
\index{B meson@$B$ meson!CP transformation@\CP\ transformation}%
\index{B meson@$B$ meson!decay constant $f_B$}
For example, the $B_d$ meson is pseudoscalar and transforms under \CP\ as
\beqa
\CP\, \ket{\ov{B}{}_d^0 \, (P^{\rho})} &=&
  - \eta_P^{bd}\, \eta_C^{bd}\, \ket{B_d^0 \, (P_{\rho})} \,, 
  \nonumber\\
\CP\, \ket{B_d^0 \, (P^{\rho})} &=&
  - \eta_P^{bd}{}^* \eta_C^{bd}{}^* \, \ket{\ov{B}{}_d^0 \, (P_{\rho}) } \,. 
\label{mcp}
\eeqa
The vacuum state $\ket{0}$ is invariant under \C, \p\ and \T.  
Hence one finds, for example,
\beq
\bra{0}\, \ov{b} \gamma_{\mu} \gamma_5 d (x)\, \ket{B_d^0 (P)} 
\stackrel{CP}{=}
\bra{0}\, \ov{d} \gamma_{\mu} \gamma_5 b (x)\, \ket{\ov{B}{}_d^0 (P)}
    = i f_{B_d} P_{\mu} e^{-i P\cdot x} \,, 
\label{deffb}
\eeq
which is the definition of the $B$ meson decay constant $f_{B_d}$.
The phases $\eta_P^{bd} \eta_C^{bd}$ from the \CP\ transformation
of the pseudovector current $ \ov{b} \gamma_{\mu} \gamma_5 d =
\ov{b}_R \gamma_{\mu} d_R-\ov{b}_L \gamma_{\mu} d_L$
and $\eta_P^{bd}{}^* \eta_C^{bd}{}^* $ from \eq{mcp} cancel in the
first relation in \eq{deffb}. We can further multiply $\ket{B_d^0
(P)}$ and $\ket{\ov{B}{}_d^0 (P)}$ by another common phase factor
(unrelated to \CP) to choose $f_{B_d}$ positive.
\index{parity}\index{charge conjugation}%

Although \C\ and \p\ are unitary transformations, \T\ is
anti-unitary \index{time reversal}\index{anti-unitary transformation}%
(i.e., $T^\dagger T=1$ and
$\langle T\phi | T\psi\rangle =\langle \psi | \phi \rangle$).  
Thus, for example,
\beq
\T\, \ket{B_d^0 (P^{\rho})} = %% \ket{B_d^0 (-P_{\rho})}^* = 
	\bra{B_d^0 (-P_{\rho})} \,. 
\eeq
The anti-unitary property of \T\ means also that $c$-numbers, such as 
the CKM matrix, are transformed into their complex conjugates.


\tab{cptg} lists the transformation properties of the vector 
bosons\index{CP transformation@\CP\ transformation!vector boson}
and the scalar Higgs field
$H$\index{CP transformation@\CP\ transformation!Higgs boson}
appearing in the Standard Model.
\begin{table}[tp]
\begin{center}
$ \begin{array}{r@{\quad}|@{\quad}c@{\quad}l@{\quad}l}
\hline\hline
& \mbox{photon, gluon, $Z$ boson} & \mbox{$W$ boson} & \mbox{Higgs} \\
\mbox{field}
&\ds  V^{\mu} \,(x^{\rho}) = A^{\mu} (x^{\rho}) \,, 
    A^{\mu,\, a} (x^{\rho}) \,, Z^{\mu} (x^{\rho}) 
  &\ds  W^{\pm, \mu}  \,(x^{\rho})  &\ds  H  \,(x^{\rho}) \\ \hline
\C\ &\ds   - V^{\mu} \,(x^{\rho}) 
    &\ds - W^{\mp, \mu} \,(x^{\rho})  
    &\ds  H \,(x^{\rho}) \\ 
\p\ &\ds  \phantom{-} V_{\mu} \,(x_{\rho}) 
    &\ds  \phantom{-} W^{\pm}_{\mu} \,(x_{\rho}) 
    &\ds  H \,(x_{\rho}) \\ 
\CP\ &\ds   - V_{\mu} \,(x_{\rho}) 
    &\ds   - W^{\mp}_{\mu}  \,(x_{\rho}) 
    &\ds  H \,(x_{\rho}) \\ 
\T\ &\ds  \phantom{-} V_{\mu} \,(-x_{\rho}) 
    &\ds  \phantom{-} W^{\pm}_{\mu}  \,( -x_{\rho}) 
    &\ds  H (- x_{\rho}) \\ 
\CPT\ &\ds -  V^{\mu} \,( - x^{\rho})
 &\ds  - W^{\mp, \mu}  \,( - x^{\rho})   
    &\ds   H  \,(- x^{\rho}) \\ \hline\hline
\end{array} $ \\[6pt]
\caption[\C, \p\ and \T\ transformation properties of bosons]%
{\C, \p and \T\ transformation properties of bosons in the 
Standard Model.}
\label{cptg}
\end{center}
\end{table}
The transformation properties of the photon and gluon field are
deduced from the experimental observation that QED and QCD conserve \C, 
\p\ and \T quantum numbers. 
For the weak gauge bosons the absence of \CP\ and \T\
violation in the gauge sector fixes the transformation properties of
$W^{\pm, \mu}$ and $Z^{\mu}$ under \CP\ and \T. The assignment of the
\C\ and \p\ transformations to the weak gauge bosons and the Higgs in
\tab{cptg} is chosen such that the Standard Model conserves \C\ and \p\
in the absence of fermion fields. These assignments do not impose
additional selection rules on the Standard Model interactions and
therefore have no observable consequences.

\index{CP violation@\CP\ violation!Standard Model}
\index{CP violation@\CP\ violation!complex couplings}
  From \tab{cpt} one can see why the weak interaction in the
Standard Model violates \C\ and \p. These transformations flip the
chirality of the quark fields, but left- and right-handed fields
belong to different representations of the $SU(2)$ gauge group. The
combined transformation \CP, however, maps the quark fields onto
fields with the same chirality. 
Still, the currents and their $CP$ conjugates (i.e., the first and 
fourth rows of \tab{cpt}) are not identical: instead they are 
Hermitian conjugates of each other.
Since the Lagrangian of any quantm field theory is Hermitian, it 
contains for each coupling of a quark current to a vector field its 
Hermitian conjugate coupling as well.
For example, the coupling of the $W$ to $b$ and $u$ quarks in the 
Standard Model is
\beq\label{1:Lagsample}
{\cal L} = - \frac{g_2}{\sqrt{2}} \lt[
    V_{ub} \, \ov{u}_L\, \gamma^{\mu} b_L \, W^{+}_{\mu} +
    V_{ub}^* \, \ov{b}_L\, \gamma^{\mu} u_L \, W^{-}_{\mu} \rt] . 
 \label{wex}  
\eeq
  From Tables~\ref{cpt} and \ref{cptg} one derives the \CP\ transformation
\beq
\CP \, {\cal L} \, (\CP)^{-1}
= - \frac{g_2}{\sqrt 2} \, 
    \lt[ V_{ub} \, \ov{b}_L\, \gamma^{\mu} u_L \, W^{-}_{\mu} 
     + V_{ub}^* \, \ov{u}_L\, \gamma^{\mu} b_L \, W^{+}_{\mu} \rt] ,
\eeq
which is the same only if $V_{ub}=V_{ub}^*$.
This illuminates why \CP\ violation is related to complex phases in
couplings. Yet complex couplings alone are not sufficient for a theory
to violate \CP.  A~phase rotation \eq{pht} of the quark fields in the
\CP\ transformed Lagrangian changes the phases of the couplings. If we
can in this way rotate the phases in $\CP \, {\cal L} \, (\CP)^{-1}$
back into those in ${\cal L}$, then \CP\ is conserved. In our example
\eq{wex} the choice $\phi^b-\phi^u=2 \arg V_{ub}$ would transform 
$\CP \, {\cal L} \, (\CP)^{-1}$ back into ${\cal L}$. 
As outlined in Sec.~\ref{ch1:sec:cpvsm}, Kobayashi and Maskawa%
\index{CP violation@\CP\ violation!Kobayashi-Maskawa mechanism}
realized that it is not possible to remove all the phases, once there 
are more than two quark generations~\cite{Kobayashi:1973fv}.

It is also illustrative to apply
the time reversal transformation to \eq{wex}.  It does not modify the 
currents, but, due to its anti-unitary\index{anti-unitary transformation}
character, it flips the phases
of the couplings and thereby leads to the same result as the \CP\
transformation. In our example we have disregarded the changes in the 
arguments $x^{\rho}$ of the fields shown in \eq{defpt}. Since physical 
observables depend on the action, $S=\int \!\d^4 x\, {\cal L} (x)$, rather 
than on ${\cal L}$, the sign of $x^{\rho}$ can be absorbed into a
change of the integration variables.

\index{CPT theorem@\CPT\ theorem|}%
From \tab{cpt} and \tab{cptg} one
can verify that the action of the
Standard Model is  invariant under the combined transformation~\CPT.
The \CPT\ transformation\index{CPT transformation@\CPT\ transformation}
simply turns the currents and the vector fields into their Hermitian
conjugates.
Due to $ {\cal L}= {\cal L}^\dagger$ one has
\beq 
S = \int\! \d^4 x \, {\cal L} (x) =  
    \int\! \d^4 x^\prime\,  {\cal L} (-x^\prime ) =
    \int\! \d^4 x^\prime\, \CPT \, {\cal L}( x^\prime)\, (\CPT)^{-1} 
     =  \CPT \, S \, (\CPT)^{-1}\, . 
\eeq
This \CPT\ {\it theorem} holds in any local Poincar\'e invariant
quantum field theory \cite{lp}.
It implies that particles and antiparticles have the same masses and
total decay widths.
In certain string
theories\index{CPT violation@\CPT\ violation!string theory} $CPT$
violation may be possible, and at low energies manifests itself in the
violation of Poincar\'e invariance or of quantum mechanics \cite{ks}.
In the standard framework of quantum field theory, however, the $CPT$
theorem is built in from the very beginning.
For example, the Feynman diagram for
any decay or scattering process and its $CPT$ conjugate diagram are
simply related by complex conjugation and give the same result for the
decay rate or cross section.  Unless stated otherwise it is always
assumed that $CPT$ invariance holds in all the formulae in this
report. In this context it is meaningless to distinguish $CP$ violation
and $T$ violation.%
\index{CP violation@\CP\ violation!T violation@\T\ violation}%
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Time evolution and mixing}
\label{subs:mix} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\index{B mixing@$B$ mixing|(}

In this section we list the necessary formulae to describe \bbmd\ and 
\bbms.
The formulae are general and apply to both $B_d^0$ and to $B_s^0$ 
mesons, although with different values of the parameters.
Eqs.~(\ref{mgmat})--(\ref{defphi}) are even correct for 
\kkm\index{K mixing@$K$ mixing} and \ddm.\index{D mixing@$D$ mixing}
In the following, the notation $B^0$ represents either of the two 
neutral $B$ meson species with the standard convention that $B^0$ 
($\ov{B}{}^0$) contains a $\ov{b}$~antiquark (a $b$~quark).

\bbm\ refers to transitions between the two flavor eigenstates
\ket{B^0} and \ket{\ov{B}{}^0}. In the Standard Model \bbm\ is caused
by the fourth order flavor-changing weak interaction described by the
box diagrams in \fig{ch1:fig:box}. 
\begin{figure}[bt]
\centerline{\epsfysize=3cm \epsffile{ch1/boxes.eps}}
\caption{Standard Model box diagrams inducing
  $B_d^0-\ov{B}{}_d^0$ mixing.}
\label{ch1:fig:box}
\end{figure}
Such transitions are called \dbt\
transitions, because they change the bottom quantum number by two
units.  In the Standard Model \dbt \index{$|\Delta B|=2$ transition} 
amplitudes are small, so measurements of \bbm\ could easily be 
sensitive to new physics.\index{B mixing@$B$ mixing!new physics}

\bbm\ induces oscillations between $B^0$ and $\ov{B}{}^0$.  
An initially produced $B^0$ or $\ov{B}{}^0$ evolves in time into a
superposition of $B^0$ and $\ov{B}{}^0$. Let \ket{B^0 (t)} denote the
state vector of a $B$ meson which is tagged as a $B^0$ at time $t=0$,
i.e., $\ket{B^0 (t=0)}=\ket{B^0}$.  Likewise \ket{\ov{B}{}^0 (t)}
represents a $B$ meson initially tagged as a $\ov{B}{}^0$.  The time
evolution of these states is governed by a Schr\"odinger equation:
\index{Schr\"odinger equation}
\beq
i\, \frac{\d}{\d\, t} \pmatrix{ \ds \ket{B(t)} \cr
  \ds \ket{\ov{B}(t)} \cr } 
= \left( M - i\, \frac{\Gamma}{2} \right) \pmatrix{ \ds \ket{B(t)} \cr
  \ds \ket{\ov{B}(t)} \cr } . 
\label{mgmat}
\eeq
The \textit{mass matrix}\ $M$ and the \textit{decay matrix}\ $\Gamma$
are $t$-independent, Hermitian $2\times 2$ matrices. 
\index{mass matrix $M$}\index{decay matrix $\Gamma$}
\CPT\ invariance implies that
\beq
M_{11} = M_{22} \,, \qquad \qquad  \Gamma_{11} = \Gamma_{22} \,. 
\eeq
\dbt\ transitions induce non-zero off-diagonal elements in \eq{mgmat}, 
so that the 
mass eigenstates of the neutral $B$ meson are different from the
flavor eigenstates \ket{B^0} and \ket{\ov{B}{}^0}. The mass
eigenstates are defined as the eigenvectors of $M - i\,\Gamma/2$. We
express them in terms of the flavor eigenstates as
\index{mass eigenstates}
\bey
\mbox{Lighter eigenstate:} \quad \ket{B_L} &=& 
    p \ket{B^0} + q \ket{\ov{B}{}^0} \,,  \nn
\mbox{Heavier eigenstate:} \quad \ket{B_H} &=& 
    p \ket{B^0} - q \ket{\ov{B}{}^0} \,,
\label{defpq}
\eey
with $\lt|p\rt|^2+\lt|q\rt|^2 = 1$.
Note that, in general, $\ket{B_L}$ and $\ket{B_H}$ are not orthogonal 
to each other.

The time evolution of the mass eigenstates is governed by
the two eigenvalues $M_H -i\,\Gamma_H/2$ and $M_L -i\,\Gamma_L/2$:
\beq
\ket{B_{H,L} (t) } = e^{-(i M_{H,L} + \Gamma_{H,L}/2)t} \, 
  \ket{B_{H,L} } \,, \label{thl}
\eeq
where $\ket{B_{H,L}}$ (without the time argument) denotes the mass 
eigenstates at time $t=0$: $\ket{B_{H,L}}=\ket{B_{H,L} (t=0)}$.  
We adopt the following definitions for the average mass and width and
the mass and width differences of the $B$ meson eigenstates: 
\index{B meson@$B$ meson!mass difference \dm}
\index{B meson@$B$ meson!width difference $\dg$}
\index{B meson@$B$ meson!width difference $\dg$!sign convention}
\begin{equation}
\begin{array}{rclrcl}
m & = & \displaystyle \frac{ M_H + M_L}{2} =  M_{11} \,, \qquad & 
  \Gamma & = & \displaystyle \frac{\Gamma_L + \Gamma_H}2 = \Gamma_{11} \,,
  \\[6pt] 
\dm & = & M_H - M_L \,, & \dg & = & \Gamma_L - \Gamma_H \,.
\end{array}\label{mg} 
\end{equation}
\dm\ is positive by definition. Note that the sign convention for 
\dg\ is opposite to the one used in Refs.~\cite{babook,BLS,book,cern}. 
In our convention the Standard Model prediction
for \dg\ is positive.

We can find the time evolution of \ket{B (t)} and \ket{\ov{B}(t)} as 
follows.
We first invert \eq{defpq} to express \ket{B^0} and \ket{\ov{B}{}^0} in
terms of the mass eigenstates and using their time evolution in
\eq{thl}:
\bey
\ket{B^0 (t)} &=& \frac{1}{2 p} 
    \lt[ \, e^{-i M_L t - \Gamma_L t/2} \, \ket{B_L} \, + \,  
     e^{-i M_H t - \Gamma_H t/2} \, \ket{B_H} \, \rt] , \nn
\ket{\ov{B}{}^0 (t)} &=& \frac{1}{2 q}
    \lt[ \, e^{-i M_L t - \Gamma_L t/2} \, \ket{B_L} \, - \,  
     e^{-i M_H t - \Gamma_H t/2} \, \ket{B_H} \, \rt] .
\label{tes}
\eey
These expressions will be very useful in the discussion of $B_s$ mixing.%
\footnote{The Schr\"odinger equation,\index{Schr\"odinger equation} 
\eq{mgmat}, is not exactly valid, but the result of the so-called 
\textit{Wigner-Weisskopf approximation}~\cite{wwa}
\index{Wigner-Weisskopf approximation} to the decay problem. 
In general, there are tiny corrections to the exponential decay laws in
\eq{tes} at very short and very large times \cite{kha}. 
These corrections are irrelevant for the mixing and
\CP\ studies at Run~II, but they must be taken into account in high
precision searches for \CPT\ violation \cite{azi}.  
\index{CPT violation@\CPT\ violation}}
With \eq{defpq} we next eliminate the mass eigenstates in \eq{tes} in
favor of the flavor eigenstates:
\bey 
\ket{B^0 (t)} &=& \phantom{\frac{p}{q}\,} 
  g_+ (t)\, \ket{B^0} + \frac{q}{p}\, g_- (t)\, \ket{\ov{B}{}^0} \,, \nn 
\ket{\ov{B}{}^0 (t)} &=& \frac{p}{q}\, g_- (t)\, \ket{B^0} 
  + \phantom{\frac{q}{p}\,} g_+(t)\, \ket{\ov{B}{}^0} \,,
  \label{tgg}    
\eey 
where
\bey
g_+ (t) &=& e^{-i m t} \, e^{-\Gamma t/2} 
   \lt[ \phantom{-} \cosh\frac{\dg \, t}{4}\, \cos\frac{\dm\, t}{2} -   
      i \sinh\frac{\dg \, t}{4}\, \sin \frac{\dm \, t}{2} \, \rt] , \nn
g_- (t) &=& e^{-i m t} \, e^{-\Gamma t/2} 
   \lt[- \sinh\frac{\dg \, t}{4}\, \cos \frac{\dm \, t}{2} +
      i \cosh\frac{\dg \, t}{4}\, \sin \frac{\dm \, t}{2} \, \rt] . 
\label{gpgm} 
\eey
Note that---owing to $\dg \neq 0$---the coefficient $g_+ (t)$ 
has no zeros, and $g_- (t)$ vanishes only at $t=0$. Hence an initially 
produced $B^0$ will never turn into a pure $\ov{B}{}^0$ or back into a
pure $B^0$. The coefficients in \eq{gpgm} will enter the formulae for
the decay asymmetries in the combinations
\bey
| g_\pm (t) |^2 & = & \frac{e^{- \Gamma t}}{2} 
  \lt[ \cosh \frac{\Delta \Gamma \, t}{2} \pm  
    \cos\lt( \Delta m\, t \rt) \rt] , \nn
g_+^* (t)\, g_- (t) & = & \frac{e^{- \Gamma t}}{2} 
  \lt[ - \sinh \frac{\Delta \Gamma \, t}{2} +
    i \sin \lt( \Delta m\, t \rt) \rt] .
\label{gpgms}
\eey 

In a given theory, such as the Standard Model, one can calculate the
off-diagonal elements $M_{12}$ and $\Gamma_{12}$ entering \eq{mgmat}
from \dbt\ diagrams. In order to exploit the formulae
(\ref{tes})--(\ref{gpgm}) for the time evolution we still need to express
$\dm$, $\dg$ and $q/p$ in terms of $M_{12}$ and $\Gamma_{12}$.  By
solving for the eigenvalues and eigenvectors of $M -i\,\Gamma/2$ one
finds%
\index{mass matrix $M$!eigenvalues}\index{mass matrix $M$!eigenvectors}%
\index{decay matrix $\Gamma$!eigenvalues}%
\index{decay matrix $\Gamma$!eigenvectors}%
\index{$q/p$}
\begin{mathletters}
\label{mgqp}
\bey
\lt( \dm \rt)^2 - \frac{1}{4} \lt( \dg \rt)^2 
  &=& 4  \lt| M_{12} \rt|^2 -  \lt| \Gamma_{12} \rt|^2 \,,
  \label{mgqp:a} \\[2mm]
\dm\, \dg &=& -4\, \real ( M_{12} \Gamma_{12}^* ) \,,
  \label{mgqp:b} \\[2mm]
\frac{q}{p} &=& - \frac{\dm + i \, \dg/2}{2 M_{12} -i\, \Gamma_{12} } 
  = - \frac{2 M_{12}^* -i\, \Gamma_{12}^*}{\dm + i \, \dg/2} \,.
\label{mgqp:c}
\eey
\end{mathletters}%
\index{phase!in mass matrix, $\phi_M$}\index{phase!in decay matrix}%
The relative phase between $M_{12}$ and $\Gamma_{12}$ appears in 
many observables related to $B$ mixing.
We introduce
\index{phase!of \CP\ violation in mixing, $\phi$}
\beq
%	\frac{M_{12}}{\Gamma_{12}} =
%		- \lt| \frac{M_{12}}{\Gamma_{12}}  \rt|\, 
%		e^{i \phi}
	\phi = \arg \left( -\frac{M_{12}}{\Gamma_{12}} \right).
	\label{defphi}
\eeq
Now one can solve \eq{mgqp} for \dm\ and \dg\ in terms of $|M_{12}|$, 
$|\Gamma_{12}|$ and $\phi$.

The general solution is not illuminating, but a simple,
approximate solution may be derived when
\beq
	\lt| \Gamma_{12} \rt| \ll \lt| M_{12} \rt | \,,
		\qquad \mbox{and} \qquad 
	\dg \ll \dm \,.
	\label{dgsmall} 
\eeq
These inequalities hold (empirically) for both $B^0$ systems.
We first note that $|\Gamma_{12}| \leq \Gamma$ always,
because $\Gamma_{12}$ stems from the decays into final states common
to $B^0$ and $\ov{B}{}^0$.  
For the $B_s^0$ meson the lower bound on $\dms$ establishes experimentally
that $\Gamma _{B_s}\ll \dms$.  
Hence $\Gamma_{12}^s \ll \dms$, and Eqs.~(\ref{mgqp:a}) and (\ref{mgqp:b}) 
imply $\dms \approx 2 |M_{12}^s|$ and $|\dgs| \leq 2|\Gamma_{12}^s|$, 
so that \eq{dgsmall} holds.  
For the $B_d^0$ meson the experiments give 
$\dmd \approx 0.75 \,\Gamma_{B_d}$.
The Standard Model predicts 
$|\Gamma_{12}^{d}|/\Gamma_{B_d} = {\cal O}(1\% )$, but $\Gamma_{12}^{d}$ 
stems solely from CKM-suppressed
decay channels (common to $B_d^0$ and $\ov{B}{}_d^0$) and could
therefore be affected by new physics. 
New decay channels would, however,
also increase $\Gamma_{B_d}$ and potentially  conflict with the precisely
measured semileptonic branching ratio.  A~conservative estimate is
$|\Gamma_{12}^{d}|/\Gamma_{B_d} < 10\% $. Hence for both the $B_s^0$ and
$B_d^0$ system an expansion in $\Gamma_{12}/M_{12}$ and
$\dg/\dm$ is a good approximation, and we easily find
\index{B meson@$B$ meson!mass difference \dm}
\index{B meson@$B$ meson!width difference $\dg$}
\begin{mathletters}
\label{mgsol}
\bey
\dm &=& 2\, |M_{12}| \lt[ 1 + 
  {\cal O} \lt( \lt| \frac{\Gamma_{12}}{M_{12}} \rt|^2 \rt) \rt] ,
  \label{mgsol:a} \\
\dg &=& 2\, |\Gamma_{12}| \cos \phi \lt[ 1+  
  {\cal O} \lt( \lt| \frac{\Gamma_{12}}{M_{12}} \rt|^2 \rt) \rt] .
  \label{mgsol:b}
\eey
\end{mathletters}%
We also need an approximate expression for $q/p$ in \eq{mgqp}.
It is convenient to define a small parameter \index{$A_{AA}$@$\ega$!definition}
\beq
\ega = \imag \frac{\Gamma_{12}}{M_{12}} 
  =  \lt| \frac{\Gamma_{12}}{M_{12}} \rt| \sin \phi \,, \label{defepsg} 
\eeq
because occasionally we need to keep terms of order $\ega$.
Then $q/p$ becomes
\beq
\frac{q}{p} = - e^{- i \phi_M} \lt[ 1 - \frac{\ega}{2} \rt]  
  + {\cal O} \lt( \lt| \frac{\Gamma_{12}}{M_{12}} \rt|^2 \rt) ,
\label{qpsol}  
\eeq
where $\phi_M$ is the phase of 
$M_{12}$,\index{phase!in mass matrix, $\phi_M$}
\beq
  M_{12} = |M_{12}|\, e^{i \phi_M} \,. \label{defphm}
\eeq 
Note that \eq{qpsol} and the normalization condition $|p|^2+|q|^2=1$
imply 
\beq
\lt| p \rt| = \frac{1}{\sqrt{2}} \lt( 1 + \frac{\ega}{4} \rt) 
  + {\cal O} \lt( \lt| \frac{\Gamma_{12}}{M_{12}} \rt|^2 \rt), \qquad
\lt| q \rt| = \frac{1}{\sqrt2} \lt( 1 - \frac{\ega}{4} \rt) 
  + {\cal O} \lt( \lt| \frac{\Gamma_{12}}{M_{12}} \rt|^2 \rt) .
\eeq

We are now prepared to exhibit the time-dependent decay rate 
\gtf\ of an initially tagged $B^0$ into some final state $f$. It is
defined as\index{time evolution!decay rate} 
\beq
\gtf = \frac{1}{N_B}\, \frac{\d N(B^0 (t) \to f)}{\d t} \,, 
\label{defgtf}
\eeq
where $\d N(B^0(t) \to f)$ denotes the number of decays of a
$B$ meson tagged as a $B^0$ at $t=0$ into the final state $f$ occurring
within the time interval between $t$ and $t+\d t$. $N_B$ is the total
number of $B^0$'s produced at time $t=0$. An analogous
definition holds for \gbtf .  
One has 
\beq
\gtf = {\cal N}_f \lt| \langle f \ket{B^0 (t)}  \rt|^2 , \qquad 
  \gbtf = {\cal N}_f \lt| \langle f \ket{\ov{B}{}^0 (t)}  \rt|^2 . 
\label{gtfaf}
\eeq
Here $ {\cal N}_f$ is a time-independent normalization factor. To calculate
\gtf\ we introduce the two decay amplitudes \index{decay amplitude}
\beq 
A_f = \langle f \ket{B^0} \,, \qquad \qquad
\ov{A}_f = \langle f \ket{\ov{B}{}^0} \,, \label{defaf}
\eeq
and the quantity \index{$\lambda_f$!definition}
\beq
\lambda_f = \frac{q}{p}\, \frac{\ov{A}_f}{A_f}
  \simeq - e^{- i \phi_M}\, \frac{\ov{A}_f}{A_f}\,
  \lt[ 1 - \frac{\ega}{2} \rt] . \label{deflaf}
\eeq
We will see in the following sections that $\lambda_f$ plays the
pivotal role in \CP\ asymmetries and other observables in $B$ mixing.
Finally with \eq{tgg}, \eq{gpgms} and $|p/q|^2 = (1+\ega)$
we find the desired formulae for the decay rates:
\index{time evolution!decay rate} 
\bey 
\gtf &=&  {\cal N}_f \, | A_f |^2 \, e^{-\Gamma t}\,
  \Bigg\{ \frac{1 + \lt| \lambda_f \rt|^2}2\, \cosh \frac{\dg \, t}{2} + 
  \frac{ 1 - \lt| \lambda_f \rt|^2}2\, \cos ( \dm \, t )  \no \\*
&& \qquad \qquad \qquad
  - \real \lambda_f \, \sinh \frac{\dg \, t}{2} 
  - \imag \lambda_f \, \sin \lt( \dm \, t \rt) \Bigg\} \,, 
\label{gtfres} \\
\gbtf &=& {\cal N}_f \, | A_f |^2 \, ( 1+ \ega ) \,
  e^{-\Gamma t}\, \Bigg\{ \frac{1 + \lt| \lambda_f \rt|^2}2\,
    \cosh \frac{\dg \, t}{2}  
  - \frac{1 - \lt| \lambda_f \rt|^2}2\, \cos ( \dm \, t ) \no \\*
&&  \qquad \qquad \qquad        
    - \real \lambda_f \, \sinh \frac{\dg \, t}{2} 
    + \imag \lambda_f \, \sin ( \dm \, t ) \Bigg\} \,.
   \label{gbtfres}
\eey
Next we consider the decay into $\ov{f}$, which denotes the 
\CP\ conjugate state to
$f$,\index{CP conjugate state@\CP\ conjugate state}
\beq
\ket{\ov{f}} = \CP\, \ket{f} \,. 
\eeq
For example, for $f=D_s^- \pi^+$ the \CP\ conjugate state is 
$\ov{f}=D_s^+\pi^-$. 
The decay rate into $\ov{f}$ can be obtained from \eq{gtfres} and
\eq{gbtfres} by simply replacing $f$ with $\ov{f}$. Yet 
$| A_{\ov{f}} |$ and $| A_f |$ are unrelated, unless $f$
is a \CP\ eigenstate, fulfilling $\ket{\ov{f}} = \pm \ket{f}$.  On the
other hand the \CP\ transformation relates $| \ov{A}_{\ov{f}} |$
to $| A_f |$, so it is more useful to factor out $| \ov{A}_{\ov{f}} |$,
\bey 
\gtfb &=&  {\cal N}_f \lt| \ov{A}_{\ov{f}} \rt|^2 e^{-\Gamma t}\,
  ( 1 - \ega) \, \Bigg\{ \frac{1 + 
  | \lambda_{\ov{f}} |^{-2}}{2}\, \cosh \frac{\dg \, t}{2} 
  - \frac{ 1 - | \lambda_{\ov{f}} |^{-2}}{2}\, \cos ( \dm \, t ) \no \\*
&& \qquad \qquad \qquad 
  - \real \frac{1}{\lambda_{\ov{f}}}\, \sinh \frac{\dg \, t}{2} \, 
  + \imag \frac{1}{\lambda_{\ov{f}}}\, \sin (\dm \, t) \Bigg\} \,, 
\label{gtfbres} \\[2pt]
\gbtfb &=& {\cal N}_f \lt| \ov{A}_{\ov{f}} \rt|^2 e^{-\Gamma t}\,
  \Bigg\{ \frac{1 + | \lambda_{\ov{f}} |^{-2}}2\,
    \cosh \frac{\dg \, t}{2}  
  + \frac{1 - | \lambda_{\ov{f}} |^{-2}}2\, \cos ( \dm \, t ) \no\\*    
&& \qquad \qquad \qquad 
  - \real \frac{1}{\lambda_{\ov{f}}}\, \sinh \frac{\dg \, t}{2} 
  - \imag \frac{1}{\lambda_{\ov{f}}}\, \sin ( \dm \, t ) \Bigg\} \,.
   \label{gbtfbres}
\eey
Here we set ${\cal N}_{\ov{f}}={\cal N}_f$, because these
normalization factors arise from kinematics.  In
Eqs.~\eq{gtfres}--\eq{gbtfbres} we consistently keep terms of order
$\ega$, which appear explicitly in the prefactor in
\eq{gbtfres}, \eq{gtfbres} and are implicit in $\lambda_f$ through 
\eq{deflaf}.%
\footnote{We have omitted terms of order $|\Gamma_{12}/M_{12}|^2$ in
Eqs.~(\ref{deflaf})--(\ref{gbtfbres}) and will do this throughout the
report. In most applications one can set $\ega$ to zero and
often also \dg\ can be neglected, so that the expressions in
Eqs.~(\ref{gtfres})--(\ref{gbtfbres}) simplify considerably.}

We now apply the derived formalism to the decay rate into a 
\textit{flavor-specific}\ final state $f$
\index{flavor-specific} meaning that a $B^0$ can
decay into $f$, while $\ov{B}{}^0$ cannot. 
Examples are $f=D_s^- \pi^+$ (from $B_s^0$) and $f=X \ell^+ \nu_\ell$.
\index{decay!$B_s \to D_s^- \pi^+$}
\index{decay!$B_s \to X \ell^+ \nu_\ell$}
\index{decay!$B_d \to X \ell^+ \nu_\ell$}
In such decays $\ov{A}_f=A_{\ov{f}}=0$ by definition and, hence,
$\lambda_f=1/\lambda_{\ov{f}}=0$.
Therefore,
\bey
\gtf &=&  {\cal N}_f \, \lt| A_f \rt|^2 \, e^{-\Gamma t} \,
  \frac{1}{2} \lt[ \cosh \frac{\dg \, t}{2} 
  + \cos ( \dm \, t ) \rt] \qquad\qquad \! \mbox{ for }  \ov{A}_f=0 \,,\\
\label{gtfs}
\gtfb &=& {\cal N}_f \, \lt| \ov{A}_{\ov{f}} \rt|^2 \,  
    ( 1- \ega ) \,
    e^{-\Gamma t} \,
    \frac{1}{2}  
    \lt[ \cosh \frac{\dg \, t}{2}  
    - \cos ( \dm \, t ) \rt] \quad  \mbox{for }  A_{\ov{f}}=0 \,.
    \label{gtfbs}
\eey
Flavor-specific decays can be used to measure \dm\ via the
asymmetry in decays from mixed and unmixed $B$s:
\index{mixing asymmetry}
\beq
{\cal A}_0 (t) = \frac{\gtf - \gtfb}{\gtf + \gtfb } \,. 
\label{defa0}
\eeq
The amplitudes $A_f$ and $\ov{A}_{\ov{f}}$ are related to each other by
\CP\ conjugation. If there is no \CP\ violation in the decay amplitude 
(i.e., no \textit{direct \CP\ violation}), $|A_f|$ and
$|\ov{A}_{\ov{f}}|$ are equal. This is the case for decays like 
$B_s \to D_s^- \pi^+$ and $B \to X \ell^+ \nu_\ell $
conventionally used to measure \dm. Then the 
mixing asymmetry in \eq{defa0} reads
\index{mixing asymmetry}
\index{decay!$B_s \to D_s^- \pi^+$}
\index{decay!$B_s \to X \ell^+ \nu_\ell$}
\index{decay!$B_d \to X \ell^+ \nu_\ell$}
\beq
{\cal A}_0 (t) = \frac{\cos ( \dm\, t ) }{\cosh (\dg \,t/2)} 
  + \frac{\ega }{2} 
  \lt[1- \frac{\cos^2 ( \dm\, t )}{\cosh^2 (\dg \, t/2)}  \rt] ,
\label{resa0} 
\eeq
where we have allowed for a non-zero width difference.%
\index{B mixing@$B$ mixing|)}

\boldmath
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Time evolution of untagged $B^0$ mesons}
\label{subs:unt} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\unboldmath
\index{time evolution!untagged $B$}


Since $B^0$'s and $\ov{B}{}^0$'s are produced in equal numbers at the
Tevatron, the untagged decay rate for the decay $\Bun \to f$ reads
\bey
\guntf &=& \gtf + \gbtf         \label{guntf}  \\*
&=& {\cal N}_f \lt| A_f \rt|^2  
    \lt( 1 + |\lambda_f|^2  \rt) e^{- \Gamma t}  \lt[
    \cosh \frac{\dg  \, t}{2} +  \, 
    \sinh \frac{\dg  t}{2} \, A_{\Delta \Gamma} \rt]
    + {\cal O} ( \ega ) \, \nonumber
\eey 
with\index{$A_{\Delta \Gamma}$!definition}
\beq
A_{\Delta \Gamma} = - \frac{2\, \real \lambda_f}{1+ \lt| \lambda_f \rt|^2} \,.
    \label{adeg}
\eeq
  From this equation one realizes that untagged samples are interesting
for the determination of \dg. The fit of an untagged decay distribution to 
\eq{guntf} involves the overall normalization factor 
${\cal N}_f\, |A_f|^2\, (1 + |\lambda_f|^2)$.  From
\eq{defgtf} one realizes that by integrating \guntf\ over all times
one obtains the branching ratio for the decay of an untagged $B^0$
into the final state $f$:
\index{untagged $B$!branching ratio}
\bey
{\cal B} \big( \!\Bun \to f \, \big) &=& \frac{1}{2}   
    \int_0^\infty \! \d t \, \guntf\
    = \frac{{\cal N}_f}{2} \, | A_f |^2 \, 
    \frac{\Gamma \lt( 1+ | \lambda_f |^2 \rt) - 
    \dg \, \real \lambda_f }{\Gamma^2-(\dg/2)^2}
    + {\cal O} ( \ega ) \no \\[2mm]
 &=&    \frac{{\cal N}_f}{2} \, | A_f |^2 \lt( 1 + |\lambda_f|^2 \rt)
    \frac{1}{\Gamma} \lt[ 1 + 
    \frac{\dg}{2 \Gamma} \, A_{\Delta \Gamma} \, + 
    {\cal O} \lt( \frac{(\dg)^2}{\Gamma^2} \rt) \rt] . 
    \label{untnorm} 
\eey
Relation \eq{untnorm} allows to eliminate $ {\cal N}_f \, | A_f|^2\,[
1 + |\lambda_f|^2]$ from \eq{guntf}, if the branching ratio is
known. If both ${\cal B}\, \big( \!\Bun \to f \big)$ and \dg\ are
known, a one-parameter fit to the measured untagged time evolution
\eq{guntf} allows to determine $ A_{\Delta \Gamma}$, which is of key
interest for \CP\ studies.

Finally we write down a more intuitive expression for \guntf. From 
\eq{gtfaf} and \eq{tes} one immediately finds
\beq
\guntf = {\cal N}_f
    \lt[ \,  e^{-\Gamma_L t} \lt| \langle f \ket{B_L} \rt|^2 + 
    e^{-\Gamma_H t} \lt| \langle f \ket{B_H} \rt|^2  \rt] +  
    {\cal O} ( \ega ) \,.
    \label{guntfeig}
\eeq
With \eq{defpq} one recovers \eq{guntf} from \eq{guntfeig}. Now
\eq{guntfeig} nicely shows that the decay of the untagged sample into
some final state $f$ is governed by two exponentials. If $B_s$ mixing
is correctly described by the Standard Model, the mass eigenstates 
$\ket{B_L}$ and $\ket{B_H}$ are to a high precision also
\CP\ eigenstates and \eq{guntfeig} proves useful for the description of 
decays into \CP\ eigenstates.


\boldmath
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Time-dependent and time-integrated $CP$ asymmetries}
\label{subs:cp}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\unboldmath
\index{CP asymmetry@\CP\ asymmetry!time-dependent}

The \CP\ asymmetry for the decay of a charged $B$ into the final state
$f$ reads
\index{CP asymmetry@\CP\ asymmetry!in charged $B$ decay}
\beq
a_f = \frac{\Gamma (B^- \to f )-\Gamma (B^+ \to \ov{f} ) }{
      \Gamma (B^- \to f )+\Gamma (B^+ \to
      \ov{f} ) }
    \qquad \quad \mbox{with~~} \ket{\ov{f}} = \CP\, \ket{f} 
    \label{defacpch}.
\eeq 
Defining 
\beq
A_f = \langle f \ket{B^+}  \qquad \mbox{and} \qquad
  \ov{A}_{\ov{f}} = \langle \ov{f} \ket{B^-}  \label{defafch}
\eeq
in analogy to \eq{defaf} one finds 
\beq
a_f = - \frac{1- \lt| \ov{A}_{\ov{f}}/A_f \rt|^2}{1+ \lt|
    \ov{A}_{\ov{f}}/A_f \rt|^2} \,. \label{acpdirch}
\eeq
Since charged $B$ mesons cannot mix, a non-zero $a_f$ can only occur
through \CP\ violation in the \dbo\ matrix elements $A_f$ and
$\ov{A}_{\ov{f}}$.  This is called \textit{direct}\ \CP\ violation and
stems from $|A_f| \neq |\ov{A}_{\ov{f}}|$. 
\index{CP violation@\CP\ violation!direct}
\index{CP violation@\CP\ violation!in decay}

Next we consider the decay of a  neutral $B$ meson into a \CP\
eigenstate $f=f_{CP}=\eta_f \ov{f}$. Here $\eta_f=\pm 1$ is the \CP\
quantum number of $f$. An example for this situation is the decay
$B_s^0 \to D_s^+ D_s^-$\index{decay!$B_s \to D_s^+ D_s^-$},
where $\eta_f=+1$.    
We define the time-dependent \CP\ asymmetry as  
\beq
a_f(t) = \frac{ \gbtf - \gtf }{ \gbtf + \gtf } \,. \label{defacp}
\eeq
Using \eq{gtfres} and \eq{gbtfres} one finds 
\beq
a_f(t) = - \frac{A_{CP}^{\rm dir} \cos ( \Delta m  \, t ) + 
       A_{CP}^{\rm mix} \sin ( \Delta m \, t )}{
       \cosh (\Delta \Gamma \, t/ 2) + 
       A_{\Delta \Gamma} \sinh  (\Delta \Gamma \, t / 2) }
    + {\cal O} ( \ega ) \,, 
    \label{acp} 
\eeq
where $A_{\Delta \Gamma}$ is defined in \eq{adeg}, and
the \textit{direct} and \textit{mixing-induced} (or
\textit{interference type}) \CP\ asymmetries are
\index{CP asymmetry@\CP\ asymmetry!mixing-induced}%
\index{CP asymmetry@\CP\ asymmetry!interference type}%
\index{CP violation@\CP\ violation!interference type}%
\index{$A_{CP}^{\rm dir}$ and $A_{CP}^{\rm mix}$ definitions}%
%%\index{$A_{CP}^{\rm mix}$!definition}
\beq
A_{CP}^{\rm dir} =
    \frac{1- \lt| \lambda_f \rt|^2}{1+ \lt| \lambda_f 
    \rt|^2} \,, \qquad
A_{CP}^{\rm mix} = - \frac{2\, \imag \lambda_f}{1+ \lt| \lambda_f \rt|^2} \,,
    \label{dirmix} 
\eeq
$A_{CP}^{\rm mix}$ stems from the interference of the decay amplitudes of
the unmixed and the mixed $B$, i.e., of $\ov{B}{}^0 \to f$ and
$B^0 \to f $. It is discussed in more detail in Sec.~\ref{ch1:sect:ty}.
Note that the quantities in \eq{dirmix} and \eq{adeg} are not
independent, 
$|A_{CP}^{\rm dir}|^2 + |A_{CP}^{\rm mix}|^2 + |A_{\Delta\Gamma}|^2 = 1$.

\index{CP asymmetry@\CP\ asymmetry!time-integrated}
The time integrated asymmetry reads%
\footnote{
Our sign conventions for the \CP\ asymmetries in \eq{defacpch} and 
\eq{defacp} are opposite to those in \cite{cern}. 
Our definitions of $A_{CP}^{\rm dir}$, $A_{CP}^{\rm mix}$ and 
$A_{\Delta \Gamma}$ are the same as in \cite{cern}, taking into 
account that the quantity $\xi_f^{(q)}$ of \cite{cern} equals 
$-\lambda_f$.}
\beq 
a_f^{\rm int} = \frac{ \int_0^\infty \d t \lt[ \gbtf - \gtf \rt] }{ 
             \int_0^\infty \d t \lt[ \gbtf + \gtf \rt] }
= - \frac{1 + y^2}{1 + x^2} \,
    \frac{ A_{CP}^{\rm dir} + A_{CP}^{\rm mix} \, x }{
    1 + A_{\Delta \Gamma} \, y } \,.
    \label{acpint}
\eeq
Here the quantities $x$ and $y$ are defined as 
\index{mixing parameter!$x$}
\index{mixing parameter!$y$}
\beq
x = \frac{\dm}{\Gamma} \,, \qquad 
  y = \frac{\dg}{2\, \Gamma} \,. \label{defxy}
\eeq
Thus, even without following the time evolution, a measurement of
$a_f^{\rm int}$ puts constraints on $\dm$ and $\dg$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Phase conventions}
\label{subs:phc}
\index{phase convention}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



In Sec.~\ref{subs:disc} we learned that there is no unique way to define the
\CP\ transformation, because it involves an arbitrary phase factor
$\eta_{CP}\equiv\eta_{C}\eta_{P}$ (see \tab{cpt} and Eq.~\eq{phs}). This
arbitrariness stems from the fact that phases of  quark fields are unobservable
and phase redefinitions as in \eq{pht} transform the Lagrangian into a
physically equivalent one. This feature implies that the phases of the
flavor-changing couplings in our Lagrangian are not fixed and the phase
rotation \eq{pht} transforms one phase convention for these couplings into
another one. 
% This is familiar from the CKM matrix, for which different phase
% conventions exist. If a theory conserves \CP, the phase convention for the
% flavor-changing couplings equally fixes the phase convention for the \CP\
% transformation, because \CP\ invariance means that \CP\ commutes with the
% Lagrangian $L$. For example in the Standard Model with two fermion generations,
% we can choose the Cabibbo matrix real. Requiring \CP\ to commute with $L$
% determines $\eta_{CP}=1$ for the quark bilinears. For any other chosen phase
% convention for the Cabibbo matrix one likewise finds a solution for the
% $\eta_{CP}$'s. Once some Lagrangian $L$ does not commute with \CP\ for any
% choice of $\eta_{CP}$, one concludes that \CP\ is explicitly broken in that
% theory. When this is the case, the phases in $L$ become unrelated to
% $\eta_{CP}$ and any choice is equally permitted.  
Of course, physical
observables are independent of these phase conventions. Hence it is
worth noting which of the quantities defined in the previous sections are
invariant, when $\eta_{CP}$ or the phases of the quark fields are changed. 
It is also important to identify the quantities that do depend on phase
conventions to avoid mistakes when combining convention dependent quantities
into an invariant observable.

The phases of 
\beq
M_{12} \,,\  \Gamma_{12} \,,\  \frac{q}{p} \,, 
  \mbox{ and } \frac{\ov{A}_f}{A_f} \,. 
\label{notinv}
\eeq
depend on the phase convention of the \CP\
transformation or the phase convention of the \CP\ violating couplings.
In particular, the phase $\phi_M$ of the mixing amplitude $M_{12}$ 
(defined in \eq{defphm}) is convention dependent.
When speaking informally, one often says that a given process, such as
$B_d^0\to\psi K_S$, measures the phase of the \dbt\ amplitude, 
i.e.,~$\phi_M$.
Such statements refer to a specific phase convention, in which the decay 
amplitude of the process has a vanishing (or negligible) phase.  
The following quantities are independent of phase conventions:
\beq
\lt|\frac{q}{p} \rt| \,,\ \lt| \frac{\ov{A}_f}{A_f} \rt| \,,\ 
  \ega \,,\  \phi \,,\  \dm \,,\  \dg \,,\ 
  \mbox{and } \lambda_f \,. 
\label{inv}
\eeq
The only complex quantity here is $\lambda_f$.
Its phase is a physical observable. 

We have shown that the arbitrary phases accompanying the 
\CP\ transformation stem from the freedom to rephase the quark fields,
see \eq{pht}. The corresponding phase factors $\eta_{CP}$ in the \CP\
transformed quark bilinears are sufficient to parameterize this
arbitrariness and likewise appear in the \CP\ transformations of the
mesons and the quantities in \eq{notinv}. In some discussions of this
issue authors allow for phases different from $\eta_{CP}$ accompanying
the \CP\ transformation \eq{mcp} of the meson states. This is simply
equivalent to using our transformation
\eq{mcp} followed by a multiplication of $\ket{B_d^0}$ and 
$\ket{\ov{B}{}_d^0}$ with extra phase factors (unrelated to \CP ),
which do not affect observables.  This would further introduce an
extra inconvenient phase into \eq{deffb}. The quantities in \eq{inv} are
still invariant under such an extra rephasing and no new information
is gained from this generalization.
Unless stated otherwise, we will use the phase convention
\index{phase convention}
$\eta_{CP}=1$, i.e.,
\beq
\CP\, \ket{\ov{B}{}^0 (P^{\rho})} =
    - \ket{B^0 (P_{\rho})} \,, \qquad 
\CP\, \ket{B^0 (P^{\rho})} =
    - \ket{\ov{B}{}^0 (P_{\rho}) } \,.  \label{mcpconv}
\eeq
For the phases of the CKM elements we use the convention of the 
Particle Data Group, \eq{1:eq:VPDG}. 


\boldmath
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Aspects of $CP$ Violation }
\label{ch1:sec:aspects}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\unboldmath

\OMIT{
Thoughts:
{\bf This section needs some work: it should have an intro tying it 
into the larger picture.  I also don't get the choice of subsections,
because $B_s\to D_sK$ is $CP$ violation in interference between decays 
with and without mixing.  Finally, is there overlap with WG1?  
%
There is also Uli's point about whether $\psi K$ gets inference 
through $K - \Kbar$ mixing or through $\pi\pi$ final states.}
}

\boldmath
\subsection{The three types of $CP$ violation}
\label{ch1:sect:ty}
\unboldmath
As discussed in Sec.~\ref{subs:phc}, there are three phase convention
independent physical $CP$ violating observables
\begin{equation}
\bigg| \frac qp \bigg|\,, \qquad 
  \lt| \frac{\ov{A}_{\ov{f}}}{A_f} \rt|\,, \qquad  
  \lambda_f = \frac qp\, \frac{\ov{A}_f}{A_f} \,.
\end{equation}
If any one of these quantities is not equal to 1 (or $-1$ for $\lambda_f$),
then $CP$ is violated in the particular decay.  In fact, there are decays 
where only one of these types of $CP$ violations occur (to a very good
approximation).  


%\boldmath
\subsubsection*{\boldmath $CP$ violation in mixing ($|q/p| \neq 1$) }
%\unboldmath
\index{CP violation@\CP\ violation!in mixing}


It follows from Eq.~(\ref{mgqp:c}) that 
\begin{equation}
\bigg| \frac qp \bigg|^2 = \bigg| { 2M_{12}^* - i\,\Gamma_{12}^* \over
  2M_{12} - i\,\Gamma_{12} } \bigg| \,.
\end{equation}
If $CP$ were conserved, then the relative phase between $M_{12}$ and
$\Gamma_{12}$ would vanish, and so $|q/p| = 1$.  If $|q/p| \neq 1$, then
$CP$ is violated.  This is called $CP$ violation in mixing, because it
results from the mass eigenstates being different from the $CP$
eigenstates.  It follows from Eq.~(\ref{defpq}) that
$\braket{B_H}{B_L} = |p|^2 - |q|^2$, and so the two physical states
are orthogonal if and only if $CP$ is conserved in $|\Delta B| = 2$
amplitudes.

The simplest example of this type of $CP$ violation is the neutral meson 
semileptonic decay asymmetry to ``wrong sign" leptons
\index{CP asymmetry@\CP\ asymmetry!in flavor-specific decay}
\index{CP asymmetry@\CP\ asymmetry!semileptonic}
\index{decay!$B_d \to X \ell^+ \nu$ }
\begin{eqnarray}
a_{\rm sl}(t) &=&
{\Gamma(\B0bar(t) \to \ell^+\nu X) - \Gamma(B^0(t) \to \ell^-\bar\nu X) \over
  \Gamma(\B0bar(t) \to \ell^+\nu X) + \Gamma(B^0(t) \to \ell^-\bar\nu X) }
  \nonumber\\*
&=& { |p/q|^2 - |q/p|^2 \over |p/q|^2 + |q/p|^2 }
  = { 1 - |q/p|^4 \over 1 + |q/p|^4 } 
  = \ega + {\cal O}(\ega^2) \,.
  \label{1:eq:asl(t)}
\end{eqnarray}
The second line follows from Eq.~(\ref{tgg}).  In $B$ meson decay such an
asymmetry is expected to be ${\cal O}(10^{-2})$.  The calculation of $|q/p|-1$
involves $\imag (\Gamma_{12} / M_{12})$, which suffers from hadronic
uncertainties.  Thus, it would be difficult to relate the observation of such
an asymmetry to CKM parameters.  This type of $CP$ violation can also be
observed in any decay for which $A_f \gg A_{\ov{f}}\,$, such as decays to 
flavor specific final states (for which $A_{\ov{f}} = 0$), e.g., $B_{(s)}\to
D_{(s)}^-\pi^+$.  In kaon decays this asymmetry was recently measured by
CPLEAR~\cite{cplear} in agreement with the expectation that it should be equal
to $4\,\real \e_K$.
\index{decay!$B_d \to D^- \pi^+$ }
\index{decay!$B_s \to D_s^- \pi^+$ }


\boldmath
\subsubsection*{$CP$ violation in decay ($|\ov{A}_{\ov{f}}/A_f| \neq 1$) }
\unboldmath
\index{CP violation@\CP\ violation!in decay}
\index{CP violation@\CP\ violation!direct}
\index{$\lambda_f$!$|\lambda_f|\neq 1$}

For any final state $f$, the quantity $|\ov{A}_{\ov{f}} / A_f|$ is a phase
convention independent physical observable.  There are two types of complex
phases which can appear in $\ov{A}_{\ov{f}}$ and $A_f$ defined in
Eq.~(\ref{defaf}).  Complex parameters in the Lagrangian which enter a decay
amplitude also enter the $CP$ conjugate amplitude but in complex conjugate
form.  In the Standard Model such parameters only occur in the CKM matrix. 
These so-called weak phases enter $\ov{A}_{\ov{f}}$ and $A_f$ with opposite
signs.  Another type of phase can arise even when the Lagrangian is real, from
absorptive parts of decay amplitudes.\index{absorptive part}  
These correspond to on-shell intermediate
states rescattering into the desired final state.  Such rescattering is usually
dominated by strong interactions, and give rise to $CP$ conserving
strong phases, which enter $\ov{A}_{\ov{f}}$ and $A_f$ with the same signs. 
Thus one can write $\ov{A}_{\ov{f}}$ and $A_f$ as
\begin{equation}
A_f = \sum_k A_k\, e^{i(\delta_k + \phi_k)} \,, \qquad
\ov{A}_{\ov{f}} = \sum_k A_k\, e^{i(\delta_k - \phi_k)} \,,
\end{equation}
where $k$ label the separate contributions to the amplitudes, $A_k$ are the
magnitudes of each term, $\delta_k$ are the strong phases, and $\phi_k$ are the
weak phases.  The individual phases $\delta_k$ and $\phi_k$ are convention
dependent, but the phase differences between different terms, $\delta_i -
\delta_j$ and $\phi_i - \phi_j$, are physical.

Clearly, if $|\ov{A}_{\ov{f}} / A_f| \neq 1$ then $CP$ is violated.  This is
called $CP$ violation in decay, or direct $CP$ violation.  It occurs due to
interference between various terms in the decay amplitude, and requires that at
least two terms differ both in their strong and in their weak phases.  The
simplest example of this is direct $CP$ violation in charged $B$ decays
\begin{equation}
{\Gamma(B^- \to f) - \Gamma(B^+ \to \ov{f}) \over
  \Gamma(B^- \to f) + \Gamma(B^+ \to \ov{f}) }
  = - { 1 - |\ov{A}_{\ov{f}}/A_f|^2 \over 1 + |\ov{A}_{\ov{f}}/A_f|^2 } \,.
\end{equation}
To extract the interesting weak phases from such $CP$ violating observables,
one needs to know the amplitudes $A_k$ and their strong phases $\delta_k$.  
The problem is that theorists do not know how to compute these from first
principles, and most estimates are unreliable.  The only experimental
observation of direct $CP$ violation so far is $\real \e_K'$ in kaon decays.

This type of $CP$ violation can also occur in neutral $B$ decays in conjunction
with the others.  In such cases direct $CP$ violation is rarely beneficial, and
is typically a source of hadronic uncertainties that are hard to control.

\boldmath
\subsubsection*{$CP$ violation in the interference between decay and mixing 
($\lambda_f \neq \pm1)$ }
\unboldmath
\index{CP violation@\CP\ violation!interference type}
\index{$\lambda_f$!$\imag \lambda_f\neq 0$}

Another type of $CP$ violation is possible in neutral $B$ decay into a $CP$
eigenstate final state, $f_{CP}$.  If $CP$ is conserved, then not only
$|q/p| = 1$ and $|\ov{A}_f/A_f| = 1$, but the relative phase between $q/p$
and $\ov{A}_f/A_f$ also vanishes.  In this case it is convenient to rewrite 
\beq
\lambda_{f_{CP}} = \frac qp\, \frac{\ov{A}_{f_{CP}}}{A_{f_{CP}}} 
  = \eta_{f_{CP}}\, \frac qp\, \frac{\ov{A}_{\ov{f}_{CP}}}{A_{f_{CP}}} \,,
\eeq
where $\eta_{f_{CP}} = \pm 1$ is the $CP$ eigenvalue of $f_{CP}$.  This form of
$\lambda_{f_{CP}}$ is useful for calculations, because $A_{f_{CP}}$ and
$\ov{A}_{\ov{f}_{CP}}$ are related by $CP$ as discussed in the previous
subsection.  If $\lambda_{f_{CP}} \neq \pm 1$ then $CP$ is violated.  This is
called $CP$ violation in the interference between decays with and without
mixing, because it results from the $CP$ violating interference between $B^0
\to f_{CP}$ and $B^0 \to \ov{B}{}^0 \to f_{CP}$.

As derived in Eq.~(\ref{acp}), the time dependent asymmetry is
\index{CP asymmetry@\CP\ asymmetry!time-dependent}
\begin{eqnarray}
a_f(t) &=&
{\Gamma(\B0bar(t) \to f) - \Gamma(B^0(t) \to f) \over
  \Gamma(\B0bar(t) \to f) + \Gamma(B^0(t) \to f) }
  \nonumber\\*
&=& - {(1-|\lambda_f|^2) \cos(\Delta m\, t) 
  - 2\, \imag \lambda_f \sin(\Delta m\, t) 
  \over (1+|\lambda_f|^2) \cosh(\Delta\Gamma\, t/2) 
  - 2\, \real \lambda_f \sinh(\Delta\Gamma\, t/2) }
  + {\cal O}(\ega) \,.
\label{1:cpmix}
\end{eqnarray}
This asymmetry is non-zero if any type of $CP$ violation occurs.  In
particular, it is possible that $\imag \lambda_f \neq 0$, but
$|\lambda_f| = 1$ to a good approximation, because $|q/p| \simeq 1$
and $|\ov{A}_f/A_f| \simeq 1$.  In both the $B_d$ and $B_s$ systems
$|q/p| - 1 \lesssim {\cal O}(10^{-2})$.
Furthermore, if only one amplitude contributes to a decay,
then $|\ov{A}_f/A_f| = 1$ automatically.
These modes are ``clean'', because in such cases $A_f$ drops out and
\index{CP violation@\CP\ violation!clean modes}
\begin{equation}
a_f(t) = { \imag \lambda_f \sin(\Delta m\, t) \over \cosh(\Delta\Gamma\, t/2) 
  - \real \lambda_f \sinh(\Delta\Gamma\, t/2) } \,,
\end{equation}
measures $\imag \lambda_f$, which is given by a weak phase.  In addition, if
$\Delta\Gamma$ can be neglected then $a_f(t)$ further simplifies to
$a_f(t) = \imag \lambda_f \sin(\Delta m\, t)$.

The best known example of this type of $CP$ violation (and also the one where
$|\lambda_f| = 1$ holds to a very good accuracy) is the asymmetry in $B \to
\psi K_S$,\index{decay!$B_d \to \psi K_S$} where $\psi$ denotes any
charmonium state.
The decay is dominated by the tree level $b\to c\bar cs$ transition
and its $CP$ conjugate.  In the phase convention (\ref{mcpconv}) one finds
\begin{equation}
{\ov{A}_{\psi K_S} \over A_{\psi K_S} } = 
  \bigg( {V_{cb} V_{cs}^* \over V_{cb}^* V_{cs}} \bigg)\,
  \bigg( {V_{cs} V_{cd}^* \over V_{cs}^* V_{cd}} \bigg) \,.
\end{equation}
The overall plus sign arises from \eq{mcpconv} and because $\psi K_S$
is $CP$ odd, $\eta_{\psi K_S} = -1$, and the last factor is $(q/p)^*$ in
$K^0 - \Kbar^0$ mixing.  This is crucial, because in the absence of
$K^0 - \Kbar^0$ mixing there could be no interference between 
$\B0bar\to \psi \Kbar^0$ and $B^0\to \psi K^0$.  There are also penguin
contributions to this decay, which have different weak and
strong phases.  These are discussed in detail in Chapter~6, where they
are shown to give rise to hadronic uncertainties suppressed by 
$\lambda^2$.  Then one finds
\begin{equation}
\lambda_{\psi K_S} = - \bigg( { V_{tb}^* V_{td} \over V_{tb} V_{td}^*} \bigg)\,
  \bigg( {V_{cb} V_{cs}^* \over V_{cb}^* V_{cs}} \bigg)\,
  \bigg( {V_{cs} V_{cd}^* \over V_{cs}^* V_{cd}} \bigg) 
= - e^{-2i\beta} \,,
\end{equation}
where the first factor is the Standard Model value of $q/p$ in $B_d$
mixing.  Thus, $a_{\psi K_S}(t)$ measures $\imag \lambda_{\psi K_S} =
\sin2\beta$ cleanly.
\index{CP asymmetry@\CP\ asymmetry!$\sin2\beta$}

Of significant interest are some final states which are not pure $CP$
eigenstates, but have $CP$ self conjugate particle content and can be
decomposed in $CP$ even and odd partial waves.  In some cases an angular
analysis can separate the various components, and may provide theoretically
clean information.  An example is $B_s\to \psi\,\phi$ discussed in Chapters 6
and 8.  There are many cases when $CP$ violation in decay occurs in addition to
$CP$ violation in the interference between mixing and decay.  Then the
asymmetry in Eq.~(\ref{1:cpmix}) depends on the ratio of different decay
amplitudes and their strong phases, which introduce hadronic uncertainties.  In
some cases it is possible to remove (or reduce) these by measuring several
rates related by isospin symmetry.  An example is $B_d\to \rho\,\pi$ (or
$\pi\,\pi$) discussed in Chapter~6.
\index{decay!$B_d \to \pi \pi $ }
\index{decay!$B_d \to \rho \pi $ }


\boldmath
\subsection{Decays to non-$CP$ eigenstates}
\unboldmath
\index{CP violation@\CP\ violation!clean modes}
\index{CP violation@\CP\ violation!in non-$CP$ eigenstates}


In certain decays to final states which are not $CP$ eigenstates, it is still
possible to extract weak phases model independently from the interference
between mixing and decay.  This occurs if both $B^0$ and $\B0bar$ can decay
into a particular final state and its $CP$ conjugate, but there is only one
contribution to each of these decay amplitudes.  In this case no assumptions
about hadronic physics are needed, even though $|\ov{A}_f/A_f| \neq 1$ and
$|\ov{A}_{\ov{f}}/A_{\ov{f}}| \neq 1$.

\index{decay!$B_s \to D_s^\pm K^\mp$ }
The most important example is $B_s\to D_s^\pm K^\mp$, which allows a model
independent determination of $\gamma$~\cite{ADK}.
Both $\B0bar_s$ and $B_s^0$
can decay to $D_s^+ K^-$ and $D_s^- K^+$, but the only decay processes are the
tree level $b\to c\bar u s$ and $b\to u \bar c s$ transitions, and their $CP$
conjugates.  One can easily see that
\begin{equation}
{\ov{A}_{D_s^+ K^-} \over A_{D_s^+ K^-} } = \frac{A_1}{A_2}
  \bigg( {V_{cb} V_{us}^* \over V_{ub}^* V_{cs}} \bigg) \,, \qquad
{\ov{A}_{D_s^- K^+} \over A_{D_s^- K^+} } = \frac{A_2}{A_1}
  \bigg( {V_{ub} V_{cs}^* \over V_{cb}^* V_{us}} \bigg) \,, 
\label{1:A12ratio}
\end{equation}
where the ratio of amplitudes, $A_1/A_2$, includes the strong phases, and is an
unknown complex number of order unity.  It is important for the utility of this
method that $|V_{cb} V_{us}|$ and $|V_{ub} V_{cs}|$ are comparable in
magnitude, since both are of order $\lambda^3$ in the Wolfenstein
parameterization.  Eqs.~(\ref{gtfres}) and (\ref{gbtfres}) show that measuring
the four time dependent decay rates determine both $\lambda_{D_s^+K^-}$ and
$\lambda_{D_s^-K^+}$.  The unknown $A_1/A_2$ ratio drops out from their product
\begin{equation}
\lambda_{D_s^+ K^-}\, \lambda_{D_s^- K^+} = 
  \bigg( {V_{tb}^* V_{ts} \over V_{tb} V_{ts}^*} \bigg)^{\!2}
  \bigg( {V_{cb} V_{us}^* \over V_{ub}^* V_{cs}} \bigg)
  \bigg( {V_{ub} V_{cs}^* \over V_{cb}^* V_{us}} \bigg)
= e^{-2i(\gamma-2\beta_s-\beta_K)}\,.
\end{equation}
The first factor is the Standard Model value of $q/p$ in $B_s$ mixing.  The
angles $\beta_s$ and $\beta_K$ occur in ``squashed" unitarity triangles;
$\beta_s$ defined in Eq.~(\ref{1:eq:betasdef}) is of order $\lambda^2$ and
$\beta_K = \arg(-V_{cs}V_{cd}^*/V_{us}V_{ud}^*)$ is of order $\lambda^4$.
Thus, this mode can provide a precise determination of $\gamma$ (or
$\gamma-2\beta_s)$; the determination of $\beta_s$ is discussed in Chapter 6,
e.g., from $B_s \to \psi\, \eta^{(\prime)}$.

\index{decay!$B_d \to D^\pm \pi^\mp$ }
\index{decay!$B_d \to D^{*\pm} \pi^\mp$ }
In exact analogy to the above, the $B_d\to D^{(*)}{}^\pm \pi^\mp$ decays can
determine $\gamma + 2\beta$, since $\lambda_{D^+\pi^-}\, \lambda_{D^-\pi^+} =
\exp\, [-2i(\gamma + 2\beta)]$.  In this case, however, the two decay
amplitudes differ in magnitude by order $\lambda^2$, and therefore the $CP$
asymmetries are expected to be much smaller, at the percent level.

\boldmath
\subsection{$\Delta F=2$ vs.~$\Delta F=1$ \CP\ violation}
\unboldmath
\index{CP violation@\CP\ violation!direct}
\index{CP violation@\CP\ violation!indirect}
\index{CP violation@\CP\ violation!in $\Delta F=1$ transitions}


At low energies flavor-changing transitions are described by effective
Hamiltonians, which are discussed in detail in Sec.~\ref{1:Heff}.
Decays are mediated by the $\Delta F=1$ Hamiltonian $H^{\dfo}$, whereas
mixing is induced by the $\Delta F=2$ Hamiltonian.
The changing flavor is $F=B$ for $B$ decays and $F=S$ for $K$ decays.
In kaon physics it is customary to distinguish $\Delta F=1$ \CP\
violation, which is often called \emph{direct} \CP\ violation, from
$\Delta F=2$ \CP\ violation, called \emph{indirect} \CP\ violation.
Here we compare this classification with the three types of \CP\
violation in $B$ decays discussed in Sec.~\ref{ch1:sect:ty}.

If we can find phase transformations of the quark fields in
\eq{pht} which leave the Hamiltonian invariant, $\CP\, H^{\dfo}
(\CP)^{-1} = H^{\dfo}$, then we conclude that the \dfo\ interaction
conserves \CP. Analogously we could define \CP\ violation and \CP\ 
conservation in $H^{\dft}$, but a $B$ physics experiment probes only
one matrix element of $H^{\dft}$, namely $M_{12}$. One can always find
a phase transformation which renders $M_{12}$ real and thereby shifts
the \CP\ violation from $H^{\dft}$ completely into $H^{\dfo}$. The
converse is not true, since one can explore the different couplings in
$H^{\dfo}$ by studying different decay modes. This leaves three
scenarios to be experimentally distinguished:%
\index{CP violation@\CP\ violation!superweak}
\begin{itemize}
\item[i)] With rephasing of the quark fields one can achieve
     $\CP\, H (\CP)^{-1} =  H$ for both $H^{\dft}$ and 
     $H^{\dfo}$: The theory conserves \CP.     
\item[ii)] One can rephase the quark fields such that 
     $\CP\, H^{\dfo} (\CP)^{-1} = H^{\dfo}$, but for this phase 
     transformation  $\CP\,H^{\dft} (\CP)^{-1} \neq H^{\dft}$.
     This scenario is called \emph{superweak} \cite{w62}.            
\item[iii)]  $\CP\,H^{\dfo} (\CP)^{-1} \neq H^{\dfo}$ for any phase 
    convention of the quark fields. This scenario is realized 
    in the CKM mechanism of the Standard Model.
\end{itemize}
Historically, after the discovery of \CP\ violation in 1964 \cite{ccft},
it was of prime interest to distinguish the second from the third 
scenario in kaon physics. The recent establishment of $\e_K^\prime
\neq 0$ has shown that possibility iii) is realized in kaon physics. 
\index{CP violation@\CP\ violation!kaon physics!$\e_K^\prime$}
\index{CP violation@\CP\ violation!kaon physics!$\e_K$}

It is difficult (but possible) to build a viable theory with
$\e_K^\prime \neq 0$ in which \CP\ violation in the $B$ system is of
the superweak type. Still we can play the rules of the kaon game and
ask, what must be measured to rule out the superweak scenario. 
Clearly, \CP\ violation in decay unambiguously proves $\dfo$ \CP\ violation. 
\CP\ violation in mixing purely
measures \CP\ violation in the \dft\ transition. It measures the
relative phase between $M_{12}$ and the decay matrix
$\Gamma_{12}$. $\Gamma_{12}$ arises at second order in the \dfo\
interaction, from $\sum_f A_f^* \ov{A}_f$, where $A_f$ and $\ov{A}_f$
are the \dfo\ decay amplitudes introduced in
\eq{defafch}. $M_{12}$ receives contributions at first order in
$H^{\dft}$ and at second order in $H^{\dfo}$. Interference type \CP\
violation measures the difference between the mixing phase
$\phi_M=\arg M_{12}$ and twice the weak phase $\phi_f$ of some decay
amplitude $\ov{A}_f$. Both types of \CP\ violations are therefore
sensitive to relative phases between $H^{\dft}$ and $H^{\dfo}$. Yet
the measurement of a single \CP\ violating observable of either type
is not sufficient to rule out the superweak scenario, because we can
always rephase $\Gamma_{12}$ or $A_f$ to be real. However, the measurement 
of interference type \CP\ violation in two different decay modes with
different results would prove that two weak
phases in $H^{\dfo}$ are different. Since $\phi_{f_1}-\phi_{f_2}$ is
a rephasing invariant observable, no field transformation in
\eq{pht} can render $H^{\dfo}$ real and \dfo\ \CP\ violation is
established. Hence for example the measurement of different \CP\
asymmetries in $B_d \to J/\psi K_S$ and $B_d \to \pi^+ \pi^-$ is
sufficient to rule out the superweak scenario. Interestingly,
$\e_K^\prime$ contains both of the discussed types of $\Delta S=1$
\CP\ violation: \CP\ violation in decay and the difference of two
interference type \CP\ violating phases.  Since in both $K$-
and $B$ physics the dominant decay modes have the same weak phases,
essentially no new information is gained by comparing \CP\ violation
in mixing with interference type
\CP\ violation in a dominant decay mode. We will see this in
Sec.~\ref{1:sect:kaon} when comparing $\e_K$ with the semileptonic
\CP\ asymmetry in $K_L$ decays.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theoretical Tools}
\label{1:tools}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


This section provides a brief review of the tools used to
derive theoretical predictions for $B$ mixing and decays.
The theory of $b$ production and fragmentation is discussed in
Chapter~9.

The principal aim of $B$ physics is to learn about the short distance
dynamics of nature.  Short distance physics couples to $b$ quarks,
while experiments detect $b$-flavored hadrons.  One therefore needs
to connect the properties of these hadrons in terms of the
underlying $b$ quark dynamics.  Except for a few special cases, this
requires an understanding of the long distance, nonperturbative
properties of QCD.  It is then useful to separate long distance
physics from short distance using an operator product expansion (OPE)
or an effective field theory.
The basic idea is that interactions at higher scales give rise to
local operators at lower scales. This allows us to think about the
short distance phenomena responsible for the flavor structure in
nature independent of the complications due to hadronic physics, which
can then be attacked separately.  This strategy can lead to very
practical results: the hadronic part of an interesting process may be
related by exact or approximate symmetries to the hadronic part of a
less interesting or more easily measured process.

In the description of $B$ decays several short distances arise.  
$CP$ and flavor violation stem from the weak scale and, probably, even shorter
distances.  These scales are separated from the scale $m_B$ with an OPE,
leading to an effective Hamiltonian for flavor changing
processes.\index{operator product expansion} This is reviewed in
Sec.~\ref{1:Heff}.  Furthermore, the $b$ and (to a lesser extent) the $c$
quark masses are much larger than $\Lambda_{\rm QCD}$.  In the limit
$\Lambda_{\rm QCD}/m_Q\to 0$, the bound state dynamics simplify.  Implications
for exclusive processes are discussed in Sec.~\ref{1:HQET}.  For inclusive
decays one can apply an OPE again, the so-called heavy quark expansion,
reviewed in Sec.~\ref{1:HQE}.
% Isospin and flavor $SU(3)$ in Sec.~\ref{1:isospin}.
Despite the simplifications, these expansions still require hadronic matrix
elements, so we briefly review lattice QCD in Sec.~\ref{1:lattice}.


\subsection{Effective Hamiltonians}
\label{1:Heff}

To predict the decay rate of a $B$ meson into some final
state $f$, one must calculate the transition amplitude ${\cal M}$
for $B\to f$. In general there are many contributions to
${\cal M}$, each of which is, at the quark level, pictorially
represented by Feynman diagrams such as those in
Fig.~\ref{ch1:fig:wpeng}.
%
\begin{figure}[bt]
\centerline{\epsfysize=4cm \epsffile{ch1/wexch.eps} \hspace{2cm}
  \epsfysize=3.5cm \epsffile{ch1/full-peng.eps} }\vspace{3ex}
\caption[Standard Model weak decay diagrams]%
{Standard Model $W$ exchange diagram and penguin diagram with 
internal top quark for the decay $b\to c \ov{c} d$.}
\label{ch1:fig:wpeng}
\end{figure}

Quark diagrams are a poor description for the decay amplitude of a
$B$ meson. The quarks feel the strong interaction, whose nature
changes drastically over the distances at which it is probed: At short
distances much smaller than $1/\Lambda_{\rm QCD}$ the strong
interaction can be described perturbatively by dressing the lowest
order diagrams in Fig.~\ref{ch1:fig:wpeng} with gluons.  When traveling
over a distance of order $1/\Lambda_{\rm QCD}$, however, quarks and
gluons hadronize and QCD becomes nonperturbative.  Therefore the
physics from different length scales, or, equivalently, from different
energy scales must be treated differently.  One theoretical tool for
this is the \emph{operator product expansion} (OPE)~\cite{wil}.
\index{operator product expansion}  
Schematically the decay amplitude ${\cal M}$ is expressed as
\beq
{\cal M} = - \frac{4\,G_F}{\sqrt{2}}\, V_{\rm CKM} 
    \sum_{j} C_j (\mu)\, \langle f | O_j (\mu) | B \rangle 
    \bigg[ 1+{\cal O} \bigg( \frac{m_b^2}{M_W^2} \bigg)\bigg] ,
\label{ope}
\eeq
where $\mu$ is a renormalization scale.
Physics from distances shorter than $\mu^{-1}$ is contained in the
Wilson coefficients\index{Wilson coefficient} $C_j$, and
physics from distances longer than $\mu^{-1}$ is accounted for by the
hadronic matrix elements $\bra{f}O_j\ket{B}$ of the
local operators~$O_j$.
In principle, there are infinitely many terms in the OPE, but higher
dimension operators yield contributions suppressed by powers of
$m_b^2/m_W^2$.
From a practical point of view, therefore, the sum in \eq{ope} ranges
over operators of dimension five and six.

All dependence on heavy masses $M\gg\mu$ such as $m_t$, $M_W$ or the 
masses of new undiscovered heavy
particles is contained in $C_j$. By convention one factors out
$4\,G_F/\sqrt{2}$ and the CKM factors, which are denoted by
$V_{\rm CKM}$ in \eq{ope}. 
On the other hand, the matrix element $\langle f | O_j | B\rangle$ 
of the $B\to f$ transition contains information from scales, such as 
$\Lambda_{\rm QCD}$, that are below~$\mu$. 
Therefore, they can only be evaluated using nonperturbative methods 
such as lattice calculations (cf., Sec.~\ref{1:lattice}), QCD sum 
rules, or by using related processes to obtain them from experiment.

An important feature of the OPE in (\ref{ope}) is the
universality of the coefficients $C_j$; they are independent of the
external states, i.e., their numerical value is the same for all final
states $f$ in (\ref{ope}). Therefore one can view the
$C_j$'s as effective coupling constants and the $O_j$'s as the
corresponding interaction vertices. Thus one can introduce
the \emph{effective Hamiltonian} 
\index{Hamiltonian!$|\Delta B|=1$}
\beq
\hbo = \frac{4\, G_F}{\sqrt{2}}\, V_{\rm CKM} 
  \sum_{j} C_j \, O_j + \mathrm{h.c.} \label{heff}
\eeq
An amplitude calculated from \hbo\, defined at a scale of order $m_b$,
reproduces the corresponding Standard Model result up to corrections 
of order $m_b^2/M_W^2$ as indicated in (\ref{ope}). 
Hard QCD effects can be included
perturbatively in the Wilson coefficients, i.e., by calculating
Feynman diagrams with quarks and gluons.

The set of
operators $O_j$ needed in (\ref{heff}) depends on the flavor
structure of the physical process under consideration. 
Pictorially the operators are
obtained by contracting the lines corresponding to heavy particles in
the Feynman diagrams to a point. The tree level diagram involving the
$W$ boson in Fig.~\ref{ch1:fig:wpeng} generates the operator $O_2^c$
shown in Fig.~\ref{ch1:fig:ops}.  
%
\begin{figure}[bt]
\centerline{\epsfxsize=0.98\textwidth \epsffile{ch1/ops.eps}}
\caption[Effective Hamiltonian weak decay diagrams]%
{Effective operators of \eq{basis}.
There are two types of fermion-gluon couplings associated
with the chromomagnetic operator $O_8$.}
\label{ch1:fig:ops}
\end{figure}
%
In the Standard Model only two operators occur for
$b\to c \ov{u} d$ transitions,
\index{operator!current-current}
\index{operator!$O_{1-2}$}
\beq \label{1:O12def}
O_1=\bar{b}_L^{\alpha} \gamma_{\mu} c_L^{\beta} \, 
  \bar{u}_L^{\beta} \gamma^{\mu} d_L^{\alpha} \,, \qquad 
O_2=\bar{b}_L^{\alpha} \gamma_{\mu} c_L^{\alpha} \, 
  \bar{u}_L^{\beta} \gamma^{\mu} d_L^{\beta} \,,
\eeq
where $\alpha$ and $\beta$ are color indices.  These arise from $W$
exchange shown in Fig.~\ref{ch1:fig:wpeng}, and QCD corrections to it.
Operators and Wilson coefficients at different scales $\mu_1$ and $\mu_2$
are related by a renormalization group transformation.
\index{renormalization group} $C_{1,2} (\mu_1)$ is not just a function 
of $C_{1,2} (\mu_2)$, but a linear combination of both $C_1 (\mu_2)$
and $C_2 (\mu_2)$. This feature is called operator mixing. It is
convenient to introduce the linear combinations $O_\pm = (O_2 \pm
O_1)/2$, which do not mix with each other.  Their coefficients can be
more easily calculated and are related to $C_1$ and $C_2$ by
$C_{\pm}=C_2\pm C_1$.
\index{operator!$O_\pm$}

The Hamiltonian for $\Delta B=1$ and $\Delta C=\Delta S=0$ transitions
consists of more operators, because it must also accommodate for
the so-called penguin diagram with an internal top quark, shown in
Fig.~\ref{ch1:fig:wpeng}.
The corresponding operator basis reads
\index{operator!$O_{3-6}$}
\index{operator!$O_{8}$}
\index{operator!QCD penguin}
\index{operator!chromomagnetic}
\begin{equation}
  \begin{array}{rclrcl}
O_1^{c} &=& 
  \bar{d}{}_L^{\alpha} \gamma_{\mu}  c_L^{\beta}  \, 
  \bar{c}{}_L^{\beta}  \gamma^{\mu}  b_L^{\alpha} \,, \qquad &
O_1^{u} &=& 
  \bar{d}{}_L^{\alpha} \gamma_{\mu}  u_L^{\beta}  \, 
  \bar{u}{}_L^{\beta}  \gamma^{\mu}  b_L^{\alpha} \,, \\[3mm]
O_2^{c} &=& 
  \bar{d}{}_L^{\alpha} \gamma_{\mu}  c_L^{\alpha}  \, 
  \bar{c}{}_L^{\beta}  \gamma^{\mu}  b_L^{\beta} \,, \qquad &
O_2^{u} &=& 
  \bar{d}{}_L^{\alpha} \gamma_{\mu}  u_L^{\alpha}  \, 
  \bar{u}{}_L^{\beta}  \gamma^{\mu}  b_L^{\beta} \,, \\[3mm]
O_3 &=& \displaystyle \sum_{q=u,d,s,c,b} \!\!
  \bar{d}{}_L^{\alpha} \gamma_{\mu}  b_L^{\alpha}  \, 
  \bar{q}{}_L^{\beta}  \gamma^{\mu}  q_L^{\beta} \,, \qquad & 
O_4 &=& \displaystyle \sum_{q=u,d,s,c,b} \!\!
  \bar{d}{}_L^{\alpha} \gamma_{\mu}  b_L^{\beta}  \, 
  \bar{q}{}_L^{\beta}  \gamma^{\mu}  q_L^{\alpha} \,, \\[5mm]
O_5 &=& \displaystyle \sum_{q=u,d,s,c,b} \!\!
  \bar{d}{}_L^{\alpha} \gamma_{\mu}  b_L^{\alpha}  \, 
  \bar{q}{}_R^{\beta}  \gamma^{\mu}  q_R^{\beta} \,, \qquad &
O_6 &=& \displaystyle \sum_{q=u,d,s,c,b} \!\!
  \bar{d}{}_L^{\alpha} \gamma_{\mu}  b_L^{\beta}  \, 
  \bar{q}{}_R^{\beta}  \gamma^{\mu}  q_R^{\alpha} \,, \\[5mm]
O_8 & = & \displaystyle - \frac{g}{16 \pi^2} \, m_b \, 
    \bar{d}_L \sigma^{\mu \nu} G^a_{\mu \nu} T^a b_R \,. 
  \end{array}
\label{basis}
\end{equation}
These operators are also depicted in Fig.~\ref{ch1:fig:ops}.
In $O_8$, $G_{\mu\nu}^{a}$ is the chromomagnetic field strength tensor.
The operators are grouped into classes, based on their origin:
$O_1$ and $O_2$ are called \emph{current-current operators},
$O_3$ through $O_6$ are called \emph{four-quark penguin operators},
and $O_8$ is called the \emph{chromomagnetic penguin operator}.%
\footnote{In the literature one also finds $O_7$ and $O_8$ with the
opposite signs. In QCD and QED the sign of the gauge coupling is
convention dependent, and \eq{basis} is consistent with the values
for $C_8$ in Table~\ref{tab}, if the Feynman rule for the quark-gluon
coupling is chosen as $+ig$.}

The operators in \eq{basis} arise from the lowest order in the
electroweak interaction, i.e., diagrams involving a single $W$ bosons
plus QCD corrections to it. In some cases, especially when isospin breaking
plays a role, one also needs to consider penguin diagrams which are of
higher order in the electroweak fine structure constant $\alpha_{ew}$.
They give rise to the \emph{electroweak penguin operators}:
\index{operator!electroweak penguin}
\index{operator!magnetic}
\index{operator!$O_{7}$}
\index{operator!$O_{7-10}^{\rm ew}$}
\begin{equation}
  \begin{array}{rclrcl}
O_7 &=& \displaystyle - {e \over 16\pi^2}\, m_b\, 
  \bar d{}_L^{\alpha} \, \sigma^{\mu\nu} F_{\mu\nu}\, b_R^{\alpha} \,, \\[3mm]
O_7^{\rm ew} &=& \displaystyle \frac{3}{2} 
\sum_{q=u,d,s,c,b} \!\! e_q \, 
  \bar{d}{}_L^{\alpha} \gamma_{\mu}  b_L^{\alpha}  \, 
  \bar{q}{}_R^{\beta}  \gamma^{\mu}  q_R^{\beta} \,,  \qquad &
O_8^{\rm ew} &=& \displaystyle \frac{3}{2} 
\sum_{q=u,d,s,c,b} \!\!  e_q \, 
  \bar{d}{}_L^{\alpha} \gamma_{\mu}  b_L^{\beta}  \, 
  \bar{q}{}_R^{\beta}  \gamma^{\mu}  q_R^{\alpha} \,, \\[5mm]
O_9^{\rm ew} &=& \displaystyle \frac{3}{2} 
   \sum_{q=u,d,s,c,b} \!\! e_q \, 
  \bar{d}{}_L^{\alpha} \gamma_{\mu}  b_L^{\alpha}  \, 
  \bar{q}{}_L^{\beta}  \gamma^{\mu}  q_L^{\beta} \,,  \qquad &
O_{10}^{\rm ew} &=& \displaystyle \frac{3}{2} 
   \sum_{q=u,d,s,c,b} \!\! e_q \, 
  \bar{d}{}_L^{\alpha} \gamma_{\mu}  b_L^{\beta}  \, 
  \bar{q}{}_L^{\beta}  \gamma^{\mu}  q_L^{\alpha} \,.
  \end{array}
\label{basis2}
\end{equation}
Here $F^{\mu\nu}$ is the electromagnetic field strength tensor, and
$e_q$ denotes the charge of quark $q$.  
The magnetic (penguin) operator $O_7$ is also of key importance
for the radiative decay $b \to d \gamma$.
Eqs.~(\ref{basis}) and (\ref{basis2}) reveal that
there is no consensus yet on how to number the operators
consecutively.

For semileptonic decays the following additional operators occur
\index{operator!semileptonic}
\index{operator!$O_{9-11}$}
\begin{equation}
  \begin{array}{rclrcl}
O_9 &=& \displaystyle {e^2 \over 16\pi^2}\, \bar d_L\, \gamma_\mu\, b_L \,
  \bar\ell\, \gamma^\mu\, \ell \,,  \qquad &
O_{10} &=& \displaystyle {e^2 \over 16\pi^2}\, \bar d_L\, \gamma_\mu\, b_L \,
  \bar\ell\, \gamma^\mu\gamma_5\, \ell \,, \\[3mm]
O_{11} &=& \displaystyle {e^2 \over 32\pi^2\sin^2\theta_W}\, 
  \bar d_L\, \gamma_\mu\, b_L \, \bar\nu_L \, \gamma^\mu \, \nu_L \,,
  \end{array}
\label{basis3}
\end{equation}
and the counterparts of these with $\bar d_L$ replaced by $\bar s_L$.

Hence the $\Delta B=1$ and $\Delta C=\Delta S=0$ Hamiltonian reads:
\beq
\hbo = \frac{4\, G_F}{\sqrt{2}}\, \bigg\{ 
    \sum_{j=1}^2 C_j \left( \xi_c O_j^c + \xi_u O_j^u \right) 
    - \xi_t \sum_{j=3}^{11} C_j \, O_j  
    - \xi_t \sum_{j =7}^{10} C_j^{\rm ew} \, O_j \bigg\} 
    + \mathrm{h.c.} \,, 
\label{hd}
\eeq
where
\beq
\xi_{q} = V_{q b}^* V_{q d} \,.
\eeq
Note that $  \xi_u + \xi_c + \xi_t = 0$ by unitarity of the CKM matrix.
The corresponding operator basis for $b\to s$
transitions is obtained by simply exchanging $d$ with $s$ in
\eq{basis}, \eq{basis2} and \eq{basis3} and changing $\xi_i$
accordingly.  

The operators introduced above are sufficient to describe nonleptonic
transitions in the Standard Model to order~$G_F$.
In extensions of the Standard Model, on the other hand, the
short distance structure can be very different.
Additional operators with new Dirac structures, whose standard Wilson 
coefficients vanish, could enter the effective Hamiltonian. 
A~list of these operators, including their RG evolution, can
be found in \cite{bmu}.

\index{renormalization group}
\index{Wilson coefficient}
In general the QCD corrections to the transition amplitude ${\cal M}
(B \to f)$ contain large logarithms such as $\ln (m_b/M_W)$
which need to be resummed to all orders in $\alpha_s$.  The OPE splits
these logarithms as $\ln (m_b/M_W) = \ln (\mu/M_W) - \ln (\mu/m_b)$. 
The former term resides in the Wilson coefficients, the latter
logarithm is contained in the matrix element.  Such large
logarithms can be summed to all orders by solving
\emph{renormalization group} (RG) equations for the
$C_j$'s. These RG-improved perturbation series are
well-behaved. The minimal way to include QCD corrections is the
\emph{leading logarithmic approximation}. The corresponding 
\emph{leading order} (LO) Wilson coefficients comprise $[\alpha_s \ln
(m_b/M_W)]^n$ to all orders $n=0,1,2,\ldots$ in perturbation
theory. This approximation has certain conceptual deficits and is too
crude for the precision of the experiments and the accuracy of present
day lattice calculations of the hadronic matrix elements. The
\emph{next-to-leading order}\ (NLO) results for the $C_j$'s comprises
in addition terms of order $\alpha_s \, [\alpha_s
\ln (m_b/M_W)]^n, n=0,1,2,\ldots$. The Wilson coefficients depend on
the unphysical scale $\mu$ at which the OPE is performed. Starting
from the NLO the $C_j$'s further depend on the
\emph{renormalization scheme}, which is related to the way one treats
divergent loops in Feynman diagrams.  In an exact calculation both the
scale and scheme dependence cancels between the coefficients and the
matrix elements, but in practice the calculation of matrix elements
with the correct scale and scheme dependence can be a non-trivial
task.  The appearance of the scale and scheme dependence in the
coefficients is inevitable.  The OPE enforces the short distance
physics involving heavy masses like $M_W$ and $m_t$ to belong to the
$C_j$'s, while the long distance physics is contained in the matrix
elements. But a constant number can be attributed to either of them.
Switching from one scheme to another or changing the scale $\mu$ just
shuffles constant terms between the Wilson coefficients and the matrix
elements.  There is no unique definition of ``scheme independent''
Wilson coefficients.

The numerical values for the renormalization group improved Wilson
coefficients can be found in Table~\ref{tab}.
%
\begin{table}[p]
\[
\begin{array}{clc|*{7}{@{\hspace{1em}}r}}
\hline\hline
\alpha_s (M_Z) &  \mathrm{scheme} &  
  \mu\,(\mathrm{GeV}) & 
	\multicolumn{1}{c}{C_1} & 
	\multicolumn{1}{c}{C_2~} & 
	\multicolumn{1}{c}{C_3~} & 
	\multicolumn{1}{c}{C_4} & 
	\multicolumn{1}{c}{C_5~} & 
	\multicolumn{1}{c}{C_6} & 
	\multicolumn{1}{c}{C_8}
\\ \hline
0.112 &
   {\rm LO} & 4.8 
&       -0.229    &        1.097    &         0.010   &       -0.024 
&        0.007    &       -0.029    &        -0.146 \\
&     & 2.4 
&        -0.325   &        1.149    &        0.015    &        -0.033
&        0.009    &        -0.043   &         -0.161 \\ 
&     & 9.6
&        -0.155   &        1.062    &        0.007    &        -0.016
&        0.005    &        -0.019   &        -0.133   \\ 
\cline{2-10}
& {\rm NDR} & 4.8 
&        -0.160   &        1.066    &        0.011    &        -0.031
&        0.008    &        -0.035   &        \\ 
&     & 2.4
&        -0.245   &        1.110    &        0.017    &        -0.043
&        0.009    &        -0.052   &        \\
&     & 9.6
&        -0.093   &        1.036    &        0.008    &        -0.021
&        0.006    &        -0.023   &        \\
\cline{2-10}
& {\rm HV}  & 4.8 
&        -0.177   &        0.993    &        0.009    &        -0.024
&        0.007    &        -0.026   &         \\
&     & 2.4 
&        -0.260    &        1.020     &        0.014    &        -0.033
&        0.010     &        -0.038   &        \\
&     & 9.6 
&        -0.111   &        0.975    &        0.006    &        -0.015
&        0.005    &        -0.017   &         \\
\hline
0.118 & 
LO & 4.8 &  -0.249 & 1.108 & 0.011 & -0.026 & 0.008 & -0.031 & -0.149 \\ 
&  & 2.4 & -0.361 &  1.169 & 0.017 & -0.036 & 0.010 & -0.048 & -0.166 \\
&  & 9.6 & -0.167 &  1.067 & 0.007 & -0.018 & 0.005 & -0.020 & -0.135 \\
\cline{2-10}
& NDR & 4.8  &  -0.174 & 1.073 & 0.013 & -0.034 & 0.009 & -0.038 &  \\
&     & 2.4  &  -0.272 & 1.124 & 0.020 & -0.047 & 0.010 & -0.060 &  \\
&     & 9.6  &  -0.100 & 1.039 & 0.008 & -0.024 & 0.006 & -0.025 &  \\
\cline{2-10}
& HV & 4.8 & -0.192 & 0.993 & 0.010 & -0.026 & 0.008 & -0.028 & \\ 
&    & 2.4 & -0.286 & 1.022 & 0.016 & -0.036 & 0.011 & -0.042 & \\
&    & 9.6 & -0.120 & 0.972 & 0.006 & -0.017 & 0.005 & -0.018 & \\
\hline
0.124 & 
  LO  & 4.8 
&        -0.272   &        1.120    &        0.012    &        -0.028
&        0.008    &        -0.035   &        -0.153   \\ 
&     & 2.4 
&        -0.403   &        1.194    &        0.019    &        -0.040
&        0.011    &        -0.055   &        -0.172    \\
&     & 9.6 
&        -0.180   &        1.073    &        0.008    &        -0.019
&        0.006    &        -0.022   &        -0.138    \\ 
\cline{2-10}
& NDR & 4.8 
&        -0.190   &        1.082    &        0.014    &        -0.037
&        0.009    &        -0.043   &        \\
&     & 2.4  
&        -0.303   &        1.142    &        0.022    &        -0.054
&        0.011    &        -0.069   &        \\
&     & 9.6
&        -0.108   &        1.042    &        0.009    &        -0.025
&        0.007    &        -0.028   &        \\
\cline{2-10}
& HV  & 4.8 
&        -0.208   &        0.993    &        0.011    &        -0.028
&        0.008    &        -0.031   &        \\
&     & 2.4 
&        -0.316   &        1.025    &        0.018    &        -0.040
&        0.012    &        -0.048   &        \\
&     & 9.6
&        -0.129   &        0.970    &        0.007    &        -0.018
&        0.006    &        -0.019   &        \\
\hline\hline
\end{array}
\] \vspace*{-2pt}
\caption[Wilson coefficients in the leading and next-to-leading
order]{QCD Wilson coefficients in the leading and next-to-leading
order. The NLO running of $\alpha_s$ has been used in both the LO
and NLO coefficients. $\alpha_s(M_Z) =0.112$, $0.118$, $0.124$ implies 
$\alpha_s(4.8\, \mathrm{GeV}) = 0.196$, $0.216$, $0.238$. The 
corresponding values of the five-flavor QCD scale parameter 
$\Lambda_{\overline{\mathrm{MS}}}$ are $159$, $226$ and $312$ MeV.
The dependence on $m_t (m_t)$, here taken as $168$ GeV, is negligible.   
The NLO coefficients are listed for the NDR and HV scheme. 
There are two different conventions for the HV scheme, here we use the
one adopted in \cite{nlo}. The HV coefficients tabulated in
\cite{bbl} are related to our $C_j^{HV}\!$'s by 
$C_j^{HV}(\!\!\mbox{\cite{bbl}}) = 
  [1+ 16/3 \cdot \alpha_s (\mu)/(4\pi) ] C_j^{HV} $. 
Small QED corrections have been omitted.\index{Wilson coefficient!table} }
\label{tab}
\end{table}
%
The NLO coefficients are
listed for two popular schemes, the \emph{naive dimensional
regularization}\ (NDR) scheme and the \emph{'t Hooft-Veltman}\ (HV)
scheme. These results have been independently obtained by the Rome and
Munich groups \cite{nlo}. The situation with $C_8$ is special: To
obtain the LO values for $C_{1-6}$ in Table~\ref{tab} one must
calculate one-loop diagrams. The calculation of $C_8$, however,
already involves two-loop diagrams in the leading order. This implies
that even the LO expression for $C_8$ is scheme dependent. The
tabulated value corresponds to the commonly used ``effective''
coefficient $C_8$ introduced in
\cite{cfmrs}, which is defined in a scheme independent way.
To know the NLO value for $C_8$ one must calculate three-loop
diagrams.  The operator basis in \eq{basis} is badly suited for this
calculation and hence a different one has been used \cite{cmm}. For
the basis in \eq{basis} the NLO value for $C_8$ is not known, we
therefore leave the corresponding rows open. In Table~\ref{tab} small
corrections proportional to $\alpha_{ew}$ have been omitted. For the 
Wilson coefficients of the electroweak penguin operators in
\eq{basis2} and the semileptonic operators in \eq{basis3}  we refer
the reader to \cite{bbl}.

\index{Hamiltonian!$|\Delta B|=2$}
We can derive an effective Hamiltonian for the \dbt\ transition, which
induces \bbmd, just in the same way as discussed above for \dbo. 
In the Standard Model only a single operator $Q$ arises:%
\footnote{Once again in \eq{ch1:uli:h2}, new short distance 
physics can generate Wilson coefficients for additional operators.}
\beq
H^{|\Delta B|=2} = \frac{G_F^2}{4 \pi^2}\, ( V_{tb} V_{td}^* )^2 \,
    C^{|\Delta B|=2}( m_t, M_W, \mu )\, Q (\mu) + h.c.
\label{ch1:uli:h2}
\eeq
with
\beq 
  Q = \ov{d}_L \gamma_{\nu} b_L \, \ov{d}_L \gamma^{\nu} b_L .
\label{ch1:uli:defq}
\eeq
The Wilson coefficient\index{Wilson coefficient!$|\Delta B|=2$} is 
\beq
C^{|\Delta B|=2} ( m_t, M_W, \mu ) = M_W^2\, 
    S \bigg( \frac{m_t^2}{M_W^2} \bigg)\, \eta_B\, b_B(\mu) \,.
\label{ch1:uli:c2} 
\eeq
It contains the \textit{Inami-Lim function}\ \cite{il}
\beq
S(x) = x \left[ \frac{1}{4} + \frac{9}{4}
    \frac{1}{1-x} - \frac{3}{2} \frac{1}{(1-x)^2} \right]
    - \frac{3}{2} \left[ \frac{x}{1-x} \right]^3 \ln x \,,
\label{ch1:uli:sxt}
\eeq
which is calculated from the box diagram in Fig.~\ref{ch1:fig:box}.
The coefficients $\eta_B$ and $b_B$ in (\ref{ch1:uli:c2}) account for
short distance QCD corrections. In the next-to-leading order of QCD
one finds $\eta_B=0.55$ \cite{bjw}. $b_B$ depends on the
renormalization scale $\mu={\cal O}(m_b)$, at which the matrix element
$\bra{B^0_d} Q \ket{\ov{B}{}^0_d}$ is calculated.  $b_B(\mu)$ equals
$[ \alpha_s (\mu) ]^{-6/23}$ in the LO. The $\mu$-dependence of
$b_B(\mu)$ cancels the $\mu$-dependence of the matrix element to the
calculated order. The same remark applies to the dependence of
$b_B(\mu)$ on the renormalization scheme in which the calculation is
carried out.  One parameterizes the hadronic matrix elements as
\bey
\bra{B^0} Q (\mu) \ket{\ov{B}{}^0} &=& 
    \frac{2}{3} f_B^2 m_B^2 \frac{\widehat{B}_B}{b_B (\mu)} 
 , \label{defbb}
\eey
so that $\widehat{B}_B$ \index{hadronic parameter!$\widehat{B}_B$}
is scale and scheme independent.  The
effective Hamiltonian for \bbms\ is obtained as usual by replacing $d$
with $s$ in \eq{ch1:uli:h2} and
\eq{ch1:uli:defq}.  The Wilson coefficient in \eq{ch1:uli:c2} does not
depend on the light quark flavor.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Heavy quark effective theory}
\label{1:HQET}
\index{heavy quark effective theory (HQET)|(}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In hadrons composed of a heavy quark and light degrees of freedom (light 
quarks, antiquarks, and gluons), the binding energy, which is of 
order $\lqcd$, is small compared to the heavy quark mass~$m_Q$.
In the limit $m_Q\gg\lqcd$, the heavy quark acts approximately as a 
static color-triplet source,%
\footnote{For the same reason, heavy quark symmetries also apply to 
hadrons composed of two heavy and a light quark, because the color 
quantum numbers of the two heavy quarks combine to an antitriplet.}
and its spin and flavor do not affect the light degrees of freedom.
This is analogous to atomic physics, where isotopes with different 
nuclei have nearly the same properties.
Thus, the properties of heavy-light hadrons are related by a 
symmetry, called heavy quark symmetry\index{heavy quark symmetry} 
(HQS)~\cite{HQS,Shur,Nuss,VS,PoWi,Eich,Grin,Geor}.
In practice, only the $b$ and $c$ quarks have masses large enough for 
HQS to be useful.%
\footnote{The top quark also satisfies $m_t\gg\lqcd$, but it 
decays before it hadronizes.}  
This results in an $SU(2N_h)$ spin-flavor symmetry, where 
$N_h=1$ or 2, depending on the problem at hand.

The heavy quark spin-flavor symmetries are helpful for understanding 
many aspects of the spectroscopy and decays of heavy hadrons from 
first principles.
For example, in the infinite mass limit, mass splittings between 
$b$-flavored hadrons can be related to those between charmed hadrons, and 
many semileptonic and radiative decay form factors can be related to one 
another.
There are corrections to the HQS limit from long distances and from 
short distances.
The former are suppressed by powers of~$\lqcd/m_Q$.
They must be calculated by nonperturbative methods, but HQS again
imposes relations among these terms.
The latter arise from the exchange of hard virtual gluons, so they can 
be calculated accurately in a perturbation series in~$\alpha_s(m_Q)$.
The heavy quark effective theory (HQET) provides a convenient 
framework for treating these effects~\cite{Eich,Grin,Geor,FGL,FGGW,Bjor,Luke}.
In leading order the effective theory reproduces the model independent 
predictions of HQS, and both series of symmetry breaking corrections are
developed in a systematic, consistent way.

To see how the heavy quark symmetries arise, it is instructive to 
look at the infinite-mass limit of the Feynman rules.
For momentum $p=m_Qv+k$, with $v^2=1$ and $k\ll m_Q$, the propagator 
of a heavy quark becomes 
\begin{equation}
{i\over p\!\!\!\slash - m_Q} = {i(p\!\!\!\slash + m_Q)\over p^2-m_Q^2}
= {i(m_Q v\!\!\!\slash + k\!\!\!\slash + m_Q)\over 2m_Q\,v\cdot k +k^2}
= {i\over v\cdot k}\,{1+ v\!\!\!\slash \over2} + \ldots \,.
\end{equation}
As $m_Q\to\infty$ it is independent of its mass, and in this way 
heavy quark \emph{flavor} symmetry emerges.
In a Feynman diagram, the quark-gluon vertex appears between two 
propagators and, hence, for $m_Q\to\infty$, sandwiched between the 
projection operator
\begin{equation}
	P_+(v) = \frac{1+ v\!\!\!\slash}{2}\,.
\end{equation}
Consequently the gamma matrix at the vertex becomes
\begin{equation}
	P_+ \gamma^\mu P_+ = v^\mu\,P_+\,.
\end{equation}
Thus, both the vertex and the propagator depend on gamma matrices
only through~$P_+$.
Since $P_+^2=P_+$, all these factors reduce to a single one, and in 
this way heavy quark \emph{spin} symmetry emerges.

The construction of HQET \cite{HQS} starts by removing the mass-dependent piece
of the momentum operator by a field redefinition. One introduces a field
$h_v(x)$, which annihilates a heavy quark with velocity $v$ \cite{Geor},
\begin{equation}
h_v(x) = e^{i m_Q v\cdot x}\, P_+(v)\, Q(x) \,,
\label{field}
\end{equation}
where $Q(x)$ denotes the quark field in full QCD.
Here the physical interpretation of the projection operator~$P_+$ is 
that $h_v$ represents just the heavy quark (rather than antiquark) 
components of~$Q$.  
If $p$ is the total momentum of the heavy quark, then the field $h_v$ 
carries the residual momentum $k=p-m_Qv$.
Inside a hadron, the residual momentum~$k\sim{\cal O}(\lqcd)$.
Since the phase factor in Eq.~(\ref{field}) effectively removes the
mass of the heavy quark from the states, it is the mass difference
\begin{equation}
	\bar\Lambda = m_H - m_Q \,,
	\label{Ldef}
\end{equation}
where $m_H$ is the hadron mass, that determines the $x$-dependence of 
hadronic matrix elements in HQET~\cite{Luke}.
It is also this parameter that sets the characteristic scale of the
$1/m_Q$ expansion.
Because of heavy quark flavor symmetry $\bar\Lambda=m_B-m_b=m_D-m_c$, 
and because of heavy quark spin symmetry $\bar\Lambda=m_{B^*}-m_b$,
in both cases up to ${\cal O}(\lqcd^2/m_Q)$ corrections.
Other heavy hadrons, for example heavy-flavored baryons, have a 
distinct ``$\bar{\Lambda}$'', but the flavor symmetry implies
$m_{\Lambda_b}-m_b=m_{\Lambda_c}-m_c$, 
up to ${\cal O}(\lqcd^2/m_Q)$.
 
The HQET Lagrangian is constructed from the field~$h_v$.
Including the leading $1/m_Q$ corrections, it is~\cite{Eich,Geor,FGL}
\begin{equation}
{\cal L}_{\rm HQET} = \bar h_v\,i v\!\cdot\!D\,h_v
+ {1\over 2 m_Q}\,\Big[ O_{\rm kin}
+ C_{\rm mag}(\mu)\,O_{\rm mag}(\mu) \Big] + {\cal O}(1/m_Q^2) \,,
\label{Lag}
\end{equation}
where $D^\mu = \partial^\mu - i g_s T_a A_a^\mu$ is the color $SU(3)$ covariant
derivative. 
The leading term respects both the spin and flavor symmetries, and 
reproduces the heavy quark propagator derived above. 
The symmetry breaking operators appearing at order $1/m_Q$ are
\begin{equation}
O_{\rm kin} = \bar h_v\,(i D)^2\, h_v \,, \qquad
O_{\rm mag} = {g_s\over 2}\,\bar h_v\,\sigma_{\mu\nu}\, G^{\mu\nu}\, h_v \,.
\label{Lag1}
\end{equation}
Here $G^{\mu\nu}$ is the gluon field strength tensor defined by
$[iD^\mu,iD^\nu] = i g_s G^{\mu\nu}$.  In the rest frame of the hadron, 
$O_{\rm kin}$ describes the kinetic energy resulting from the residual motion
of the heavy quark, whereas $O_{\rm mag}$ corresponds to the chromomagnetic
coupling of the heavy quark spin to the gluon field. 
While $O_{\rm kin}$ violates only the heavy quark flavor symmetry, 
$O_{\rm mag}$ violates the spin symmetry as well.

In the operators of the electroweak Hamiltonian, the QCD field~$Q$
must also be replaced with~$h_v$ and a series of higher-dimension
operators to describe $1/m_Q$ effects.
The short distance behavior can be matched using perturbation theory.
The matrix elements of the HQET operators still cannot be calculated in
perturbatively, but HQS restricts their form.
The best known example is in exclusive semileptonic $b\to c$
corrections.
In $B\to D^{(*)}\ell\nu$ and $\Lambda_b\to \Lambda_c\ell\nu$, let $v$
($v'$) be the velocity of the initial (final) heavy-light hadron.
HQS requires that the mesonic decays are described by a set of
heavy quark spin- and mass-independent functions of the kinematic
variable $w=v\cdot v'$.
The baryonic decay is described by another function of~$w$.
When $v=v'$ the symmetry becomes larger---from $SU(2)_v\times SU(2)_{v'}$
to~$SU(4)$---so there are further restrictions.
One is that symmetry limit of the form factor is completely determined
by symmetry (at $w=1$).
Furthermore, HQS also requires that the $1/m_Q$ corrections to
$B\to D^*\ell\nu$ and $\Lambda_b\to \Lambda_c\ell\nu$ vanish for $w=1$.

The utility of HQET is not limited to exclusive decays.
Matrix elements of the effective Lagrangian play an important
role in inclusive semileptonic and radiative decays.
One defines
\begin{eqnarray}
\lambda_1 &=& {1\over 2\,m_M}\, 
  \langle M(v)|\, O_{\rm kin}\, |M(v) \rangle \,, \nonumber\\
d_M\,\lambda_2 &=& {1\over 2\, m_M}\, 
  \langle M(v)|\, O_{\rm mag}\, |M(v)\rangle \,,
\label{l12def}
\end{eqnarray}
where $M$ denotes a $B$ or $B^*$ meson, and $d_M=3,\,-1$ for $B$ and $B^*$,
\index{heavy quark effective theory (HQET)!$\lambda_1$ and $\lambda_2$}
respectively.  
Strictly speaking, both $\lambda_1$ and $\lambda_2$ depend on the 
renormalization scale~$\mu$.
For $\lambda_1$, however, there is no $\mu$ dependence if 
$O_{\rm kin}$ is renormalized in the $\ov{\rm MS}$ scheme.
For $\lambda_2$, the $\mu$ dependence is canceled by the coefficient
$C_{\rm mag}(\mu)$ in \eq{Lag}.

HQET provides an expansion of the heavy meson masses
in terms of the  heavy quark masses,
\begin{equation}
m_B = m_b + \bar\Lambda - {\lambda_1 + 3\lambda_2 \over 2m_b} +\ldots \,, \qquad
m_{B^*} = m_b + \bar\Lambda - {\lambda_1-\lambda_2 \over 2m_b} +\ldots\,.
\label{quarkmass}
\end{equation}
Consequently, the value of $\lambda_2$ is related to the mass splitting 
between the vector and the pseudoscalar mesons, 
\begin{equation}
\lambda_2 = {m_{B^*}^2 - m_B^2 \over 4} 
  + {\cal O} \Big( \lqcd^3/ m_b \Big) \,,
\end{equation}
taking $\mu=m_b$ and $C_{\rm mag}(m_b)=1$.
From the measured $B$ and $B^*$ masses one finds
$\lambda_2(m_b) \simeq 0.12~{\rm GeV}^2$.
These formulae will play an important role in the description of both
inclusive and exclusive heavy meson decays in the following chapters.  

It was only recognized recently that HQS also yields important
simplifications in the description of heavy-to-light radiative
and semileptonic decays in the region of large recoil
(small~$q^2$)~\cite{Charles}.  In the infinite mass limit, the
three form factors which parameterize the vector and tensor current matrix
elements in $B\to K\ell^+\ell^-$ are related to a single function of $q^2$, and
the seven form factors which occur in $B\to K^*\ell^+\ell^-$ are related to
only two functions of $q^2$.  In contrast to the predictions of HQS in the
region of small recoil, in this case it is not known yet how to formulate the
subleading corrections suppressed by powers of $\lqcd/m_Q$.  Nevertheless,
these relations play a very important role in Chapter~7, where they will be
discussed in detail.%
\index{heavy quark effective theory (HQET)|)}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Heavy quark expansion}
\label{1:HQE}
\index{heavy quark expansion|(}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In inclusive $B$ decays, when many final states are summed over,
certain model independent formulae can be derived.
In this section we examine how the large $b$ quark mass, $m_b \gg \lqcd$, 
allows one to extract reliable information about such decays. In most of the
phase space the energy release, which can be as large as  ${\cal O}(m_b)$, is
much larger than the typical scale of hadronic interactions. The large energy
release implies a short distance, and we can use the same tools as before---an
operator product expansion~\cite{CGG,incl,MaWi}
(though not the same OPE as in Sec.~\ref{1:Heff}) and HQET---to
separate short and long distances.
In this way, inclusive decay rates can be described with a double
series in $\lqcd/m_b$ and $\alpha_s(m_b)$.

Inclusive decay widths are given by the sum over all final states.
Schematically, the width is given by
\begin{equation}
	\Gamma \sim \sum_X
		\langle B |O^\dagger | X\rangle\, 
		\langle X |O         | B\rangle\,.
\end{equation}
where $X$ is any final state.
One can also limit $X$ to $X_c$ or $X_u$, i.e., to final states with or
without a charmed quark, respectively.
  From Sec.~\ref{1:Heff}, we see that inclusive semileptonic 
$B$ decays are mediated by operators of the form
\begin{equation}
O_\ell \sim \bar{q}_L\gamma^\mu  b_L\,
  \bar{\ell}_{L1}\gamma_\mu  \ell_{L2}\,, 
\end{equation}
% radiative decays are mediated by penguin operators of the form
% \begin{equation}
% 	O_7 =  - {e \over 16\pi^2}\, m_b\, 
%   \bar{d}_L\, \sigma^{\mu\nu} F_{\mu\nu}\, b_R \,,
% \end{equation}
and nonleptonic decays are mediated by four-quark operators of the 
form
\begin{equation}
O_h \sim \bar{q}_L\gamma^\mu b_L\,
  \bar{q}_{L1}\gamma_\mu q_{L2}\,.
\end{equation}
Although these operators are superficially similar, we shall see that 
they have to be treated differently, because in $O_h$ hard gluons 
can be exchanged among all four quark fields.
We start by showing in detail how the OPE and HQET are used to 
describe inclusive semileptonic $B$ decays.
We then explain what restrictions arise for nonleptonic decay rates and 
lifetimes.
Finally, we treat the width difference in the $B_s$ system, which is 
of special interest to Tevatron experiments.

\boldmath
\subsubsection{Inclusive semileptonic $B$ decays} 
\unboldmath
\label{1:subsubsec:incsemi}
\index{B decay@$B$ decay!semileptonic}

In semileptonic decays, one may factorize the matrix 
element of the four-fermion operator
\begin{equation}
\langle X\ell\, \bar\nu_\ell|\, O_\ell\, |B\rangle = 
  \langle X|\, \bar q\, \gamma^\mu P_L\, b\, |B\rangle\, 
  \langle \ell\, \bar\nu_\ell|\, \bar\ell\, \gamma_\mu P_L\, \nu_\ell\, 
  |0\rangle\,,
\end{equation}
neglecting electroweak loop corrections.
Then the decay rate can be written in the form
\begin{equation}
\frac{\d^2\Gamma}{\d y\,\d q^2} \sim \int \d(q\cdot v)\, 
  L_{\mu\nu}(p_\ell,p_{\bar\nu})\, 
  W^{\mu\nu}(q\cdot v, q^2)\,,
  \label{1:eq:dGamma}
\end{equation}
where $L_{\mu\nu}$ is the lepton tensor and $W^{\mu\nu}$ is the hadron 
tensor.
The momentum of the decaying $b$ quark is written as $p_b^\mu=m_bv^\mu$,
$q^\mu = p_\ell^\mu + p_{\bar\nu}^\mu$, and we have introduced the
dimensionless variable $y=2E_\ell/m_b$.
Since the antineutrino is not detected, its energy or, equivalently,
$q\cdot v=E_\ell+E_{\bar\nu}$ is integrated over.
The lepton tensor
$L^{\mu\rho}=2(p^\mu_\ell p^\rho_{\bar\nu}+p^\rho_\ell p^\mu_{\bar\nu}%
- g^{\mu\rho} p_\ell \cdot p_{\bar\nu}%
-i\varepsilon^{\mu\rho\alpha\beta}p_{\ell\alpha}p_{\bar\nu\beta})$.
The hadron tensor $W^{\mu\nu}$ contains all strong interaction physics
relevant for the semileptonic decay, and it can be expressed as
\begin{equation}
W^{\mu\nu} = \sum_X\, (2\pi)^3\, \delta^4(p_B-q-p_X)\,
  { \langle \Bbar |\, {J^\mu}^\dagger\, | X \rangle\, 
  \langle X|\, J^\nu\, |\Bbar\rangle \over 2m_B } \,,
\end{equation}
where $J^\mu = \bar q\, \gamma^\mu P_L\, b$.  

The optical theorem can be used to relate $W^{\mu\nu}$ to the 
discontinuity across a cut of the forward scattering matrix element of 
a time ordered product
\begin{equation}
T^{\mu\nu} = - i \int \d^4 x\, e^{-iq\cdot x}\, { \langle \Bbar |\, 
  T \{ {J^\mu}^\dagger(x)\, J^\nu(0) \} |\Bbar\rangle \over 2 m_B }\,.
\label{1:top}
\end{equation}
To show that
\begin{equation}
	W^{\mu\nu} = - \frac1\pi\, \imag  T^{\mu\nu} \,,
	\label{1:Wmunu}
\end{equation}
one inserts a complete set of states between the currents
in the two possible time orderings in~$T^{\mu\nu}$. 
Using $\langle A| J(x) | B\rangle =
\langle A| J(0) | B\rangle\, e^{i(p_A-p_B)\cdot x}$ and the identity
$\theta(x^0) = i/(2\pi)\, \int_{-\infty}^{+\infty} \d\omega\, [ e^{-i\omega
x^0} \! / (\omega + i\varepsilon) ]$, the $\d^4x$ integration gives 
(in the $B$ rest frame, so $q\cdot v=q^0$)
\begin{eqnarray}
T^{\mu\nu} &=& \sum_{X_q}\, 
  { \langle \Bbar |\, J^\mu{}^\dagger\, | X_q \rangle\, 
  \langle X_q|\, J^\nu\, |\Bbar\rangle \over 
  2m_B\, (m_B - E_X - q^0 + i\varepsilon)}\,
  (2\pi)^3\, \delta^3(\vek{q} + \vek{p}_X) \nonumber\\*
&-& \sum_{X_{\bar qbb}}\, 
  { \langle \Bbar |\, J^\nu\, | X_{\bar qbb} \rangle\, 
  \langle X_{\bar qbb}|\, {J^\mu}^\dagger\, |\Bbar\rangle \over 
  2m_B\, (E_X - m_B - q^0 - i\varepsilon)}\, 
  (2\pi)^3\, \delta^3(\vek{q} - \vek{p}_X) \,.
\label{1:twoterms}
\end{eqnarray}
This form shows that, for fixed $q^2$,
$T^{\mu\nu}$ has cuts in the complex $q^0$ plane
corresponding to physical processes.
The first sum in Eq.~(\ref{1:twoterms}) corresponds to $B$ decay shown
in Fig.~\ref{1:ope1}, with intermediate states containing a $q$ quark
(and arbitrary number of gluons and light quark-antiquark pairs).
%
\begin{figure}[bt] 
\centerline{\epsfysize 2.5cm \epsffile{ch1/ope1.eps}}
\caption{OPE diagram for semileptonic and radiative $B$ decays.}
\label{1:ope1}
\end{figure} 
%
It leads to a cut for
$q^0=q\cdot v < (m_B^2 + q^2 - m_{X_q^{\rm min}}^2)/2m_B$,  
towards the left in Fig.~\ref{1:cuts}.
%
\begin{figure}[bt]
\centerline{\epsfysize 4cm \epsffile{ch1/6.3.eps}}
\caption[Analytic structure of $T^{\mu\nu}$ in the $q\cdot v$ plane.]{The
analytic structure of $T^{\mu\nu}$ in the $q\cdot v$ plane, with $q^2$
fixed.  The cuts corresponding to $B$ decay (left) and to an unphysical process 
(right) are both shown, together with the integration contour for computing 
the decay rate. }
\label{1:cuts}
\end{figure}
%
For charmed final states $m_{X_q^{\rm min}}^2=m_D^2$ and for charmless
final states $m_{X_q^{\rm min}}^2=m_\pi^2$.
The second sum in Eq.~(\ref{1:twoterms}) corresponds to an unphysical
process with a $\bar q$ and two $b$ quarks in the intermediate state.
It leads to another cut for
$q^0=q\cdot v > (m_{X_{\bar qbb}^{\rm min}}^2 - m_B^2 -q^2) / 2m_B$,
towards the right in Fig.~\ref{1:cuts}.
The imaginary part can be read off using 
$\imag(A+i\varepsilon)^{-1}=-\pi\delta(A)$, and \eq{1:Wmunu} follows 
immediately, because the kinematics of the decay process allow only the
first sum to contribute.

Because $W^{\mu\nu}$ is the discontinuity across the left cut in
Fig.~\ref{1:cuts}, the integral in \eq{1:eq:dGamma} can be replaced
with a contour integral of $L_{\mu\nu}T^{\mu\nu}$.
The two cuts are well separated (unless $m_q \to 0$ and $q^2 \to
m_b^2$), so one may deform the contour away from the cuts~\cite{PQW},
as shown in Fig.~\ref{1:cuts}.
The equivalence of the sum over hadronic states with a contour ranging
far from the physical region is called ``global duality".
This procedure is advantageous, because $T^{\mu\nu}$ can be
reliably described by an operator product expansion (OPE) far (compared
to $\lqcd$) from its singularities in the complex $q\cdot v$ plane
\cite{CGG,incl,MaWi}.
One simply replaces the time ordered product
\begin{equation}
- i \int \d^4 x\, e^{-iq\cdot x}\, T \{ J^\mu{}^\dagger(x)\, J^\nu(0) \} \,,
\label{1:topb}
\end{equation}
appearing in Eq.~(\ref{1:top}), with a series of local operators
multiplied with Wilson coefficients.
The Wilson coefficients of this OPE can again be evaluated in a
perturbation series in $\alpha_s(m_b)$.
Higher dimension operators in the OPE incorporate higher powers of
$\lqcd/m_b$.

Unfortunately, the contour $C$ must still approach the cut near the
low $q\cdot v$ endpoint of the integration.
Using the OPE directly in the physical region is an assumption called
``local duality".
It introduces an uncertainty to the calculation, which can be argued
to be small.
First, in semileptonic and radiative decays the fraction of the contour
which has to be within order $\lqcd$ from the cut scales as $\lqcd/m_b$.
Second, since the energy release to the hadronic final state is large
compared to $\lqcd$, the imaginary part of $T^{\mu\nu}$ is dominated
by multiparticle states, so it is expected to be a smooth function.
In the end, the violation of local duality is believed to be
exponentially suppressed in the $m_Q\to\infty$ limit, but it is not
well understood how well it works at the scale of the $b$ quark mass.
In semileptonic decay the agreement between the inclusive and exclusive
determinations of $|V_{cb}|$ suggests that duality violation is at
most a few percent.
But there is no known relation between the size of duality violation
in semileptonic and nonleptonic $B$ decays~\cite{FWD}, or between
these processes and others, such as $e^+e^-\to{\rm hadrons}$.

At lowest order in $\lqcd/m_b$ the OPE leads to operators of the form
$\bar b\Gamma b$ occur, where $\Gamma$ is any Dirac matrix.
For $\Gamma=\gamma^\mu$ or $\gamma^\mu\gamma_5$ their matrix elements
are known to all orders in $\lqcd/m_B$
\begin{eqnarray}
\langle \Bbar(p_B) |\, \bar b\, \gamma^\mu b \, |\Bbar(p_B)\rangle &=&
  2p_B^\mu = 2m_B\, v^\mu \,, \nonumber\\*
\langle \Bbar(p_B) |\, \bar b\, \gamma^\mu\gamma_5\, b \, |\Bbar(p_B)\rangle 
  &=& 0 \,.
  \label{1:eq:hqe3}
\end{eqnarray}
by conservation of the $b$ quark number current and parity invariance
of strong interactions, respectively.
The matrix elements for other gamma matrices can be related by heavy quark
symmetry to these plus order $\lqcd^2/m_b^2$ corrections.
Consequently, at the leading order in $\lqcd/m_b$ inclusive decay
rates are given by the rate for $b$ quark decay, multiplied with a
Wilson coefficient that does not depend on the decaying hadron.

To compute subleading corrections in $\lqcd/m_b$, it is convenient to
use HQET.
\index{heavy quark effective theory (HQET)!in inclusive decays}
There are no order $\lqcd/m_b$ corrections because the matrix element 
of any gauge invariant dimension-4 two-quark operator vanishes,
\begin{equation}
\langle \Bbar(v) |\, \bar h_v^{(b)}\, iD_\alpha \Gamma\, h_v^{(b)}\, 
  |\Bbar(v)\rangle = 0 \,,
  \label{1:eq:hqe4}
\end{equation}
because contracting the left-hand side by $v^\alpha$ gives zero due
to the equation of motion following from (\ref{Lag}).
Thus, the leading nonperturbative corrections to $b$ quark decay occur
at order $\lqcd^2/m_b^2$.
The operators that appear are again $O_{\rm kin}$ and $O_{\rm mag}$
so the same hadronic elements $\lambda_1$ and $\lambda_2$,%
\index{heavy quark effective theory (HQET)!$\lambda_1$ and $\lambda_2$}
defined in Eq.~(\ref{l12def}), appear again.

Combining the matrix elements from Eqs.~(\ref{1:eq:hqe3}),
(\ref{1:eq:hqe4}) and~(\ref{l12def}) with the Wilson coefficients leads
to expressions of the form
\begin{eqnarray}
\frac{\d^2\Gamma}{\d y\,\d q^2} =
	\pmatrix{ b{\rm ~quark} \cr {\rm decay}\cr } \times \bigg\{ 1 &+&
  \frac{\alpha_s}\pi\, A_1 + \frac{\alpha_s^2}{\pi^2}\, A_2 + \ldots 
  + \frac{f(\lambda_1,\lambda_2)}{m_B^2}\, 
  \Big[ 1 + {\cal O}(\alpha_s) + \ldots \Big] \nonumber\\*
  &+& {\cal O}(\lqcd^3/m_B^3) + \ldots \bigg\} \,.
\label{1:blahblah}
\end{eqnarray}
The differential rate may be integrated to obtain the full rate.
The description in \eq{1:blahblah} is model independent, although
$\lambda_1$ must be determined either from data~\cite{Gremm:1996yn}
or from lattice QCD~\cite{Crisafulli:1995pg}.
For most quantities of interest the functions $f$, $A_1$, and the
part of $A_2$ proportional to $\beta_0$, the first coefficient of
the $\beta$-function, are known.
Corrections to the $m_b\to\infty$ limit are expected to be under
control in parts of the $\Bbar\to X_q\, \ell\, \bar\nu$ phase space
where several hadronic final states are allowed (but not required)
to contribute with invariant mass and energy satisfying
$m_X^2 \gg m_q^2 + \lqcd E_X$.


\boldmath
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Inclusive nonleptonic $B$ decays}
\unboldmath
\index{B decay@$B$ decay!nonleptonic}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Inclusive nonleptonic decays can also be studied using the OPE,
and much of the discussion in Sec.~\ref{1:subsubsec:incsemi}
applies here also.
In this case, however, there are no ``external" variables, such as
$q^2$ and $q\cdot v$, since all particles in the final state interact
strongly.
For this reason, only the fully integrated inclusive width can be treated
with the OPE, term-by-term in the weak Hamiltonian.
For example, the $B$ decay width corresponding to the $b\to c\bar u d$
effective Hamiltonian in \eq{heff}--\eq{1:O12def} is given by
\begin{eqnarray}
\Gamma &=& \frac1{2m_B}\, \sum_X (2\pi)^4\, \delta^4(p_B-p_X)
  \left| \langle X(p_X)|\, H^{|\Delta B|=1}(0)\, |\Bbar(p_B)\rangle \right|^2 
  \nonumber\\*
&=& \frac1{2m_B}\, \imag  \langle \Bbar |\, i \int \d^4 x\,
  T \lt\{ H^{|\Delta B|=1}(x)\, H^{|\Delta B|=1}(0) \rt\} |\Bbar\rangle \,.
\label{1:nonlept}
\end{eqnarray}
Because one has to use the OPE directly in the physical region,
the results are more sensitive to violations of local duality than
in the case of semileptonic and radiative decays.
The leading term in the OPE corresponds to the left diagram in
Fig.~\ref{1:ope2},
%
\begin{figure}[bt]
\centerline{\epsfysize 2.5cm \epsffile{ch1/ope2.eps} \hspace*{1cm}
  \epsfysize 2.5cm \epsffile{ch1/ope2b.eps}}
\caption[OPE diagrams for nonleptonic $B$ decays.]{OPE diagrams for 
nonleptonic $B$ decays.  The left one is the leading contribution, while
the ``Pauli interference" diagram on the right corresponds to a dimension-6 
contribution of order $16\pi^2\, (\lqcd^3/m_B^3)$.}
\label{1:ope2}
\end{figure}
%
whose imaginary part gives the total nonleptonic width.

The result is again of the form shown in Eq.~(\ref{1:blahblah}).  
An important new ingredient at order $\lqcd^3/m_B^3$ are certain
contributions due to four-quark operators involving the spectator quark.
They are usually called ``weak annihilation", ``$W$~exchange", and
``Pauli interference" contributions.
(The last is sketched on the right in Fig.~\ref{1:ope2}).
They contain one less loop than the diagram on the left, so they
are enhanced by a relative factor of~$16\pi^2$.
They are expected to be
more important than the dimension-5 contributions proportional to
$\lambda_1$ and $\lambda_2$.  The matrix elements of the resulting
four-quark operators are poorly known.  Such contributions are expected
to explain the $D^\pm - D^0$ lifetime difference.

\boldmath
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{$B_s$ width difference, $\Delta\Gamma$}
\unboldmath
\index{width difference!$B_s$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Another important application, especially for the Tevatron, 
is for the $B_s$ width difference.  
The off-diagonal element of the width matrix (cf., Sec.~\ref{subs:mix}) 
is given by
\begin{eqnarray}
\Gamma_{12} &=& \frac1{2m_{B_s}}\, \sum_X (2\pi)^4\, \delta^4(p_{B_s}-p_X)\, 
  \langle B_s |\, H^{|\Delta B|=1}\, | X \rangle\, 
  \langle X|\, H^{|\Delta B|=1}\, | \Bbar_s\rangle  \nonumber\\*
&=& \frac1{2m_{B_s}}\, \imag  \langle B_s |\, i \int \d^4 x\, 
  T \left\{ H^{|\Delta B|=1}(x)\, H^{|\Delta B|=1}(0) \right\} | 
  \Bbar_s\rangle \,.
\label{1:delgam}
\end{eqnarray}
The first line defines $\Gamma_{12}$, and the second
line can be verified by inserting a complete set of intermediate states. 
The corresponding diagram is shown in Fig.~\ref{1:ope3}.
%
\begin{figure}[bt]
\centerline{\epsfysize 2.5cm \epsffile{ch1/ope3.eps}}
\caption{OPE diagram for the $B_s$ width difference.}
\label{1:ope3}
\end{figure}
%
$\Gamma_{12}$ arises from final states~$X$ which are common to both
$B_s$ and $\Bbar_s$ decay.  Therefore, the spectator quark is involved, and
Eq.~(\ref{1:delgam}) is dominated by the $b \to c \bar c s$ part of the weak
Hamiltonian, $O_1$ and $O_2$ in Eq.~(\ref{basis}), with the others,
$O_3$ through $O_6$, making very small contributions.

Thus, the naive estimate of the $B_s$ width difference is $\Delta
\Gamma_{B_s} / \Gamma_{B_s} = 2\, |\Gamma_{12}| \cos\phi / \Gamma_{B_s} \sim
16\pi^2 (\lqcd^3/m_B^3) \sim 0.1$.  In the $B_d$ system the common decay
modes of $B^0$ and $\B0bar$ are suppressed relative to the leading ones by
the Cabibbo angle, and therefore the naive estimate is $\Delta\Gamma_{B_d} /
\Gamma_{B_d} \lesssim 1\%$.
See the discussion following \eq{dgsmall} and Chapter~8 for more details.
\index{heavy quark expansion|)}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Isospin and $SU(3)$ symmetry}
%\label{1:isospin}
%\typeout{isospin was here}
%\index{isospin}
%\index{flavor $SU(3)$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Lattice QCD}
\label{1:lattice}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


If one considers the long term goal of ``measuring'' the Wilson
coefficients of the electroweak Hamiltonian, as outlined elsewhere,
then it is clear that it will be important to gain theoretical control
over hadronic matrix elements.
Since QCD is a completely well-defined quantum field theory, the
calculation of hadronic matrix elements should be, in principle,
possible.
The main difficulty is that hadronic wavefunctions are sensitive mostly
to the long distances where QCD becomes nonperturbative.

The difficulties of the bound-state problem in QCD led
Wilson~\cite{Wilson:1974sk} to formulate gauge field theory on a
discrete spacetime, or lattice.
The basic idea starts with the functional integral for correlation
functions in QCD
\begin{equation}
    \langle O_1\cdots O_n \rangle = \frac{1}{Z}
        \int \prod_{x,\mu} dA_\mu(x) \prod_x d\psi(x) d\bar{\psi}(x)
        \, O_1\cdots O_n \, e^{-S_{\rm QCD}}
    \label{1:eq:funcint}
\end{equation}
where $Z$ is defined so that $\langle 1\rangle=1$.
For QCD $A_\mu$ is the gluon field, $\psi$ and $\bar{\psi}$ are the
quark and antiquark fields, and $S_{\rm QCD}$ is the QCD action.
The $O_i$ are operators for creating and annihilating the hadrons of 
interest and also terms in the electroweak Hamiltonian.
The continuous spacetime is then replaced with a discrete grid of
points, or lattice.
Then the quark variables live on sites;
the gluons on links connecting the sites.
With quarks on sites and gluons on links, it is possible to devise
lattice actions that respect gauge symmetry.
As in discrete approximations to partial differential equations,
derivatives in the Lagrangian are replaced with difference operators.

The breakthrough of the lattice formulation is that it turns quantum
field theory into a mathematically well-defined problem in statistical
mechanics.
Condensed matter theorists and mathematical physicists have devised a
variety of methods for tackling such problems, only one of which is
weak-coupling perturbation theory.
In the years immediately following Wilson's work, many of these tools were
tried, for example analytical strong coupling expansions.
The strong coupling limit is especially appealing, because confinement
emerges immediately~\cite{Kogut:1983ds}.

Strong coupling is, however, not the whole story.
Owing to asymptotic freedom, the continuum limit of lattice QCD is
controlled by weak coupling.
Unfortunately, strong coupling expansions do not converge quickly
enough to reach into the weak-coupling regime, at least with the simple
discretizations that have been used till now.
Consequently, results from strong coupling expansions for hadron masses
and matrix elements are not close enough to continuum QCD to apply to
particle phenomenology.

Since such analytical methods have not borne out, the tool of choice
now is to compute (the discrete version of) Eq.~(\ref{1:eq:funcint})
numerically via Monte Carlo integration.
This numerical method has, over the years, developed several
specialized features, and corresponding jargon, that often make its
results impenetrable to non-experts.
Moreover, as with any numerical method, there are several 
sources of systematic uncertainty.
Most of the systematic effects can, however, be controlled with
effective field theories, i.e., with techniques like those explained
in the previous sections.
After reviewing the basic elements of the Monte Carlo method,
we cover the systematic effects.
First is the so-called quenched approximation, which is difficult to
control, but also not a fundamental limitation.
Other uncertainties, which can be controlled, are reviewed next,
emphasizing the role of effective field theories.
It is hoped that in this way non-experts can learn to make simple
estimates of size systematic uncertainties, without repeating all the
steps of the numerical analysis.
We end with a comment on the (unsatisfactory) status of computing
strong phase shifts for $B$ decays.


\subsubsection{Monte Carlo integration}

This part of the method is well understood and, these days, rarely
leads to controversy.
For completeness, however, we include a short explanation, focusing on
the points that limit the range of applicability of the method.
A~more thorough treatment aimed at experimenters can be found in
Ref.~\cite{DiPierro:2000nt}.

The first salient observation is that there are very many variables.
Continuum field theory has uncountably many degrees of freedom.
Field theory on an infinite lattice still has an infinite number of
degrees of freedom, but at least countably infinite.
(This makes the products over $x$ in Eq.~(\ref{1:eq:funcint})
well-defined.)
To keep the number finite (for a computer with finite memory),
one must also introduce a finite spacetime volume.
This may seem alarming, but what one has done is simply to introduce an
ultraviolet cutoff (the lattice) and an infrared cutoff (the finite
volume).
This is usual in quantum field theory, and field theoretic techniques
can be used to understand how to extract cutoff-free quantities from
numerically calculable cutoff quantities.

Even with a finite lattice, the number of integration variables is
large.
If one only demands a volume a few times the size of a hadron and also
several grid points within a hadron's diameter, one already requires
at least, say, 10 points along each direction.
In four-dimensional spacetime this leads to $\sim 32\times 10^4$ gluonic
variables.
With so many variables, the only feasible methods are based on Monte
Carlo integration.
The basic idea of Monte Carlo integration is simple: generate an
ensemble of random variables and approximate the integrals in
Eq.~(\ref{1:eq:funcint}) by ensemble averages.

Quarks pose special problems, principally because, to implement Fermi
statistics, fermi\-onic variables are Grassmann numbers.
In all cases of interest, the quark action can be written
\begin{equation}
    S_F = \sum_{\alpha\beta}
        \bar{\psi}_\alpha M_{\alpha\beta} \psi_\beta,
\end{equation}
where $\alpha$ and $\beta$ are multi-indices for (discrete) spacetime,
spin and internal quantum numbers.
The matrix $M_{\alpha\beta}$ is some discretization of the Dirac
operator $\kern+0.1em /\kern-0.65em D+m$.
Note that it depends on the gauge field, but one may integrate over the
gauge fields after integrating over the quark fields.
Then, because the quark action is a quadratic form, the integral can be
carried out exactly:
\begin{equation}
    \int \prod_{\alpha\beta} d\bar{\psi}_\alpha d\psi_\beta\,
        e^{- \bar{\psi} M \psi} = \det M .
    \label{1:eq:quarkdet}
\end{equation}
Similarly, products $\psi_\alpha\bar{\psi}_\beta$ in the integrand are
replaced with quark propagators~$[M^{-1}]_{\alpha\beta}$.
The computation of $M^{-1}$ is demanding, and the computation of $\det M$
(or, more precisely, changes in $\det M$ as the gauge field is changed)
is very demanding.

With the quarks integrated analytically, it is the gluons that are
subject to the Monte Carlo method.
The factor with the action is now $\det M e^{-S}$, where $S$ is now just
the gluons' action.
Both $\det M$ and $e^{-S}$ are the exponential of a number that scales
with the spacetime volume.
In Minkowski spacetime the exponent is an imaginary number, so
there are wild fluctuations for moderate changes in the gauge field.
On the other hand, in Euclidean spacetime, with an imaginary time
variable, $S$ is real.
In that case (assuming $\det M$ is positive definite) one can devise a
Monte Carlo with \emph{importance sampling}, which means that the
random number generator creates gauges field weighted according to
$\det M e^{-S}$.
Because importance sampling is essential, only in Euclidean spacetime is 
lattice QCD numerically tractable.

Importance sampling works well if $\det M$ is positive.
For pairs of equal-mass quarks, this is easy to achieve.
As a result, most calculations of $\det M$ are for 2 or 4 flavors.
Note that a physically desirable situation with three flavors, with the
strange quark's mass different from that of two lighter quarks, must
either cope with (occasional) non-positive weights, or find a (new)
discretization with $\det M$ positive flavor by flavor.

The choice of imaginary time has an important practical advantage.
Consider the two-point correlation function
\begin{equation}
    C_2(t) = \bra{0} \Phi_B(t) \Phi_B^\dagger(0) \ket{0},
\end{equation}
where $\Phi_B$ is an operator with the quantum numbers of the $B$ meson at
rest.
Inserting a complete set of states between $B$ and $B^\dagger$
\begin{equation}
    C_2(t) = \sum_n \frac{1}{2m_n} \bra{0} \Phi_B \ket{B_n}
        \bra{B_n} \Phi_B^\dagger \ket{0} e^{im_nt},
\end{equation}
where $m_n$ is the mass of $\ket{B_n}$, the $n$th radial excitation
of the $B$~meson.
For real $t$ it would be difficult to disentangle all these
contributions.
If, however, $t=ix_4$, with $x_4$ real and positive, then one has a sum
of damped exponentials.
For large $x_4$ the lowest-lying state dominates and
\begin{equation}
    C_2(x_4) = (2m_B)^{-1} |\bra{0} \Phi_B \ket{B}|^2 e^{-m_Bx_4} + \cdots,
    \label{1:eq:C2exp}
\end{equation}
where $\ket{B}$ is the lowest-lying state and $m_B$ its mass.
The omitted terms are exponentially suppressed.
It is straightforward to test when the first term dominates a
numerically computed correlation function, and then fit the exponential
form to obtain the mass.

This technique for isolating the lowest-lying state is essential also
for obtaining hadronic matrix elements.
For $B^0 - \B0bar$ mixing, for example, one must compute the matrix
element $\bra{B^0}Q\ket{\B0bar}$, given in Eq.~(\ref{ch1:uli:defq}).
One uses a three-point correlation function
\begin{equation}
    C_Q(x_4,y_4) = \bra{0} \Phi_B(x_4+y_4) Q(y_4) \Phi_B(0) \ket{0},
\end{equation}
where only the Euclidean times of the operators have been written out.
Inserting complete sets of states and taking $x_4$ and $y_4$ large
enough,
\begin{equation}
    C_Q(x_4,y_4) = (2m_B)^{-1} (2m_{\Bbar})^{-1}
    	\bra{0} \Phi_B \ket{B} \bra{B} Q \ket{\Bbar}
        \bra{\Bbar} \Phi_B\ket{0} e^{-m_Bx_4-m_{\Bbar}y_4}.
    \label{1:eq:CQexp}
\end{equation}
The amplitude ($\bra{0}\Phi_B\ket{B}=\bra{\Bbar}\Phi_B\ket{0}$)
and mass ($m_B=m_{\Bbar}$) are obtained from $C_2$, leaving
$\bra{B} Q \ket{\Bbar}$ to be determined from~$C_Q$.
Similarly, to obtain amplitudes for $B$ decays to a single hadron (plus
leptons or photons), simply replace one of the $\Phi_B$ operators with
one for the desired hadron and $Q$ with the desired operator.
To compute the purely leptonic decay, simply replace $\Phi_B$ in $C_2$
with the charged current.

These methods are conceptually clean and technically feasible for
the calculation of masses and hadronic matrix elements
with at most one hadron in the final state.
The procedure for computing correlation functions is as follows.
First generate an ensemble of lattice gluon fields with the appropriate
weight.
Next form the desired product $O_1\cdots O_n$, with quark variables
exactly integrated out to form propagators~$M^{-1}$.
Then take the average over the ensemble.
Finally, fit the Euclidean time dependence of 
Eqs.~(\ref{1:eq:C2exp}) and~(\ref{1:eq:CQexp}).
Note that since the same ensemble is used for many similar correlation
functions, the statistical fluctuations within the ensemble are
correlated.
This is not a concern, as long as the correlations are propagated
sensibly through the analysis.


\subsubsection{Quenched approximation}

Any perusal of the literature on lattice QCD quickly comes across
something called the\index{lattice QCD!quenched approximation}
``quenched approximation.''
As mentioned above, the factor $\det M$ in Eq.~(\ref{1:eq:quarkdet}) is
difficult to incorporate.
The determinant generates sea quarks inside a hadron.
The quenched approximation replaces $\det M$ with $1$ \emph{and}
compensates the corresponding omission of the sea quarks with shifts in
the bare couplings.
This is analogous to a dielectric approximation in electromagnetism,
and it fails under similar circumstances.
In particular, if one is interested in comparing two quantities that are
sensitive to somewhat different energy scales, one cannot expect the
same dielectric shift to suffice.
Another name for the quenched approximation is the ``valence''
approximation, which makes clearer that the valence quarks (and gluons)
in hadrons are treated fully, and the sea quarks merely modeled.

It is not easy to estimate quantitatively the effect of quenching.
For $\alpha_s$~\cite{El-Khadra:1992vn}
and the quark masses~\cite{Mackenzie:1994zw}
one can compute the short distance
contribution to the quenching shift, but that is only a start.
The quenched approximation can be cast as the first term in a
systematic expansion~\cite{Sexton:1997ud}, but it is about as difficult
to compute the next term as to restore the fermion determinant.
In the context of heavy quark physics one should note that the
CP-PACS~\cite{AliKhan:2000eg} and MILC~\cite{Bernard:2000nv} groups now
have unquenched calculations\index{lattice QCD!unquenched $f_B$}
of the heavy-light decay constants $f_B$,
$f_{B_s}$, $f_D$, and~$f_{D_s}$.
Both have results at several lattice spacings, so they can study the
$a$~dependence.
Their results are about 10--15\% higher than the most mature estimates
from the quenched approximation.

\subsubsection{Controllable systematic uncertainties}

By a controllable systematic uncertainty we mean an uncertainty that can
be incrementally improved in a well-defined way.
In lattice QCD they arise from the ultraviolet and infrared cutoffs,
and also from the fact that quark masses are freely adjustable and, for
technical reasons, not always adjusted to their physical values.
Because these effects are subject to theoretical control, the errors
they introduce can largely be reduced to a level that is essentially
statistical, given enough computing.

One of the least troublesome systematic effects comes from the finite
volume.
Finite-volume effects can be understood separately from lattice-spacing
effects with an effective massive quantum field
theory~\cite{Luscher:1986dn}.
In some cases adjusting the volume at will is, at least in principle,
a boon, yielding valuable information, such as scattering lengths and
resonance widths.

The computer algorithms for computing the quark propagator $M^{-1}$ 
converge more quickly at masses near that of the strange quark than 
for lighter masses.
Consequently, the Monte Carlo is run at a sequence of light quark masses
typically in the range $0.2m_s\lesssim m_q\lesssim m_s$.
(The up and down quark masses are far smaller still and not reached.)
The dependence on $m_q$ can be understood and controlled via the chiral
Lagrangian~\cite{Gasser:1984yg}, another effective field theory.
A~recent development is to show in detail how to extract physical
information from results at practical values of the light quark
masses~\cite{Sharpe:2000bc}.

A special difficulty with heavy quarks is the effect of non-zero lattice
spacing.
The bottom and charmed quark masses are large in lattice units.
For this reason it is frequently (but incorrectly) stated that heavy
quarks cannot be directly accommodated by a lattice.
  From the inception of HQET and NRQCD, these effective field theories
have been used to treat heavy quarks, and more recently it has been
shown how to use these tools to understand the discretization effects
of heavy quarks discretized with the original Wilson
formulation~\cite{Wilson:1977nj}.

\index{lattice QCD!lattice spacing effects!light quarks and gluons|(}
Let us first recall how lattice-spacing effects are controlled for
systems of light quarks.
Long ago, Symanzik introduced a local effective Lagrangian
(LE${\cal L}$) to describe cutoff effects~\cite{Symanzik:1979ph}.
One writes
\begin{equation}
    {\cal L}_{\rm lat} \doteq {\cal L}_{\rm cont} +
        \sum_{i} a^{s_{{\cal O}_i}} K_i(a;\mu) {\cal O}_i(\mu),
\end{equation}
where $s_{{\cal O}_i}={\rm dim\,}{\cal O}_i-4$.
The symbol $\doteq$ means
``has the same (on-shell) matrix elements as''.
For operators such as $Q$,
needed for mixing,\index{hadronic parameter!in lattice QCD}
\begin{equation}
    Q_{\rm lat} \doteq Z^{-1}_Q(a;\mu) Q_{\rm cont}(\mu) +
        \sum_{i} a^{s_{{\cal Q}_i}} C_i(a;\mu) {\cal Q}_i(\mu),
\end{equation}
where now $s_{{\cal Q}_i}={\rm dim\,}{\cal Q}_i-{\rm dim\,}Q$.
The continuum operators ${\cal O}_i$, $Q_{\rm cont}$, and
${\cal Q}_i$ are defined in a mass-independent scheme
at scale~$\mu$.
They do not depend on the lattice spacing~$a$.
The coefficients $K_i$, $Z_Q$, and $C_i$ account for short distance
effects, so they do depend on~$a$.

If $a$ is small enough the higher terms can be treated as
perturbations.
So, the $a$ dependence of $\bra{B} Q_{\rm lat} \ket{\Bbar}$ is
\begin{equation}
    \bra{B} Q_{\rm lat} \ket{\Bbar} =
         Z_Q ^{-1} \bra{B} Q_{\rm cont} \ket{\Bbar} +
        a K_{\sigma F} \bra{B}T\, Q_{\rm cont} \int\!d^4x\,
                \bar{\psi} \sigma\cdot F\psi \ket{\Bbar} +
        a C_1 \bra{B} {\cal Q}_1 \ket{\Bbar},
    \label{1:eq:Qlat}
\end{equation}
keeping only contributions of order~$a$.
To reduce the unwanted terms one might try to reduce $a$ greatly, but
CPU time goes as $a^{-\rm (5~or~6)}$.
It is more effective to use a sequence of lattice spacings and
extrapolate, with Eq.~(\ref{1:eq:Qlat}) as a guide.
It is even better to adjust things so $K_{\sigma F}$ and $K_1$ are
${\cal O}(\alpha_s^\ell)$~\cite{Sheikholeslami:1985ij} or
${\cal O}(a)$~\cite{Jansen:1996ck}, which is called Symanzik improvement.
For light hadrons, a combination of improvement and extrapolation is 
best.
Note that one still has to adjust $Q_{\rm lat}$ so that $Z_Q=1$.
In some cases the needed adjustment can be made nonpertubatively, even
though it is a short distance quantity.
When that is possible, lattice QCD can provide results with no
perturbative uncertainty, although perturbative uncertainty
may reenter through the electroweak Hamiltonian.
\index{lattice QCD!lattice spacing effects!light quarks and gluons|)}

\index{lattice QCD!lattice spacing effects!heavy quarks|(}
The Symanzik theory, as usually applied, assumes $m_qa\ll1$.
The bottom and charmed quarks' masses in lattice units are at present
large: $m_ba\sim1$--2 and $m_ca$ about a third of that.
It will not be possible to reduce $a$ enough to make $m_ba\ll1$
for many, many years.
So, other methods are needed to control the lattice spacing
effects of heavy quarks.
There are several alternatives:
\begin{enumerate}
    \item static approximation~\cite{Eichten:1990zv}
    \item lattice NRQCD~\cite{Lepage:1987gg}
    \item extrapolation from $m_Q\lesssim m_c$ up to $m_b$
    \item[$3'\!.$] combine 3 with 1
    \item normalize systematically to HQET~\cite{El-Khadra:1997mp}
    \item anisotropic lattices with temporal lattice spacing
        $a_t\ll a$~\cite{Kla99}
\end{enumerate}
All but the last use the heavy quark expansion in some way.
The first two discretize continuum HQET;
method~1 stops at the leading term, and method~2 carries the
heavy quark expansion out to the desired order.
Methods~3 and~$3'$ keep the heavy quark mass artificially small and
appeal to the $1/m_Q$ expansion to extrapolate back up to $m_b$.
Method~4 uses the same lattice action as method~3, but uses the
heavy quark expansion to normalize and improve it.
Methods~2 and~4 are able to calculate matrix elements directly at
the $b$-quark mass.
Method~5 has only recently been applied to heavy-light
mesons~\cite{Collins:2001pe}, and, like the other methods, it requires
that spatial momenta are much less than~$m_Q$.

The methods can be compared and contrasted by \emph{describing} the 
lattice theories with HQET~\cite{Kronfeld:2000ck}.%
\index{heavy quark effective theory (HQET)!for lattice QCD}
This is, in a sense, the opposite of \emph{discretizing} HQET.
One writes down a (continuum) effective Lagrangian
\begin{equation}
    {\cal L}_{\rm lat} \doteq
        \sum_{n} {\cal C}^{(n)}_{\rm lat}(m_Qa; \mu) 
                O^{(n)}_{\rm HQET}(\mu),
    \label{eq:hqet}
\end{equation}
with the operators $O^{(n)}_{\rm HQET}$ defined exactly as in
Sec.~\ref{1:HQET}, so they do not depend on $m_Q$ or~$a$.
As long as $m_Q\gg\Lambda_{\rm QCD}$ this description makes
sense.
There are two short distances, $1/m_Q$ and the lattice spacing~$a$,
so the short distance coefficients ${\cal C}^{(n)}_{\rm lat}$ depend 
on~$m_Qa$.
Since all dependence on $m_Qa$ is isolated into the coefficients,
this description shows that heavy quark lattice artifacts arise only
from the mismatch of the ${\cal C}^{(n)}_{\rm lat}$ and their continuum 
analogs~${\cal C}^{(n)}_{\rm cont}$.

For methods~1 and~2, Eq.~(\ref{eq:hqet}) is just a
Symanzik~LE${\cal L}$.
For lattice NRQCD we recover the result that some of the 
coefficients ${\cal C}^{(n)}_{\rm lat}$ have power-law divergences
as $a\to0$~\cite{Lepage:1987gg}.
So, to obtain continuum (NR)QCD, one must add more and more terms to
the action.
(This is just a generic feature of effective field theories, namely,
that accuracy is improved by adding more terms, rather than taking the 
cutoff too high.)
The truncation leaves a systematic error, which, in practice, is
usually accounted for conservatively.

Eq.~(\ref{eq:hqet}) is more illuminating for methods~3--5, which 
use the same actions, but with different normalization conditions.
The lattice quarks are Wilson fermions~\cite{Wilson:1977nj},
which have the same degrees of freedom and heavy quark symmetries as
continuum~quarks.
Thus, the HQET description is admissible for all~$m_Qa$.
Method~4 matches the coefficients of Eq.~(\ref{eq:hqet}) term by term
to Eq.~(\ref{Lag}), by adjusting the lattice action and operators.
In practice, this is possible only to finite order, so there are
errors $({\cal C}^{(n)}_{\rm lat}-{\cal C}^{(n)}_{\rm cont})%
\langle O^{(n)}_{\rm HQET}\rangle$.
The rough size of matrix element here is
$\Lambda_{\rm QCD}^{{\rm dim\,}O-4}$.
The coefficients balance the dimensions with $a$ and $1/m_Q$.
If ${\cal C}^{(n)}_{\rm lat}$ is matched to ${\cal C}^{(n)}_{\rm cont}$
in perturbation theory, the difference is of order~$\alpha_s^\ell$.
Method~3 artificially reduces $m_Qa$ until the mismatch is of 
order~$(m_Qa)^2$.
This would be fine if $m_Qa$ were small enough, but with currently
available lattices, $m_Qa$ is small only if $m_Q$ is reduced until the
heavy quark expansion falls apart.
In method~5 the temporal lattice spacing~$a_t$ is smaller than the
spatial lattice spacing.
The behavior of the mismatch
${\cal C}^{(n)}_{\rm lat}-{\cal C}^{(n)}_{\rm cont}$ for practical
values of~$m_Q$ and~$a_t$ is still an open
question~\cite{Harada:2001ei}.

The non-expert can get a feel for which methods are most appropriate by
asking himself what order in $\Lambda_{\rm QCD}/m_b$ is needed.
For zeroth order, method~1 will do.
Perhaps the only quantity where this is sufficiently accurate is the
mass of the $b$ quark, where the most advanced
calculation~\cite{Gimenez:2000cj} neglects the subleading
term~$\lambda_1/m_b$ in Eq.~(\ref{quarkmass}).
For matrix elements, the first non-trivial terms are those of
Eq.~(\ref{Lag1}), so the other methods must be used.
With method~3 one should check that $m_Q/\Lambda_{\rm QCD}$ is large
enough; so far, all work with this method is worrisome in this
respect.
\index{lattice QCD!lattice spacing effects!heavy quarks|)}

Most of the matrix elements that are of interest to $B$ physics will
soon be recalculated, like $f_B$~\cite{AliKhan:2000eg,Bernard:2000nv}
and $m_b$~\cite{Gimenez:2000cj}, with two flavors of sea quarks.
It seems, therefore, not useful tabulate quenched results.
One can consult recent reviews focusing on the status of matrix
elements instead~\cite{Hashimoto:2000bk}.


\subsubsection{Strong phases of nonleptonic decays}
\label{1:subsubsec:phases}

In considering $CP$ asymmetries one encounters strong
phase\index{strong phase shifts!from lattice QCD} shifts.
It is therefore interesting to consider computing them in lattice QCD.

A short summary is that this is still an unsolved problem, at least for
inelastic decays, such as $B$ decays.
This does not mean that it is an unsolvable problem, but at this time
numerical lattice calculations are not helpful for computing scattering
phases above the inelastic threshold.

Often an even bleaker picture is painted, based on a superficial
understanding a theorem of Maiani and Testa~\cite{Maiani:1990ca}.
The theorem assumes an infinite volume and is, thus, relevant only to
extremely large volumes.
In volumes of $(\mbox{2--6 fm})^3$ it is possible to disentangle phase
information, because the scattering phase shift enters into the
finite-volume boundary conditions of the final-state two-body wave
function~\cite{Luscher:1991ux}.
This works, however, only in the kinematic region with two-body final
states.
This has been worked out explicitly for kaon
decays~\cite{Lellouch:2000pv}, giving also references to earlier work.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Constraints from Kaon Physics}
\label{1:sect:kaon}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

There are two strong reasons for the discussion of the neutral kaon
system in a report on $B$ physics.
First, for more than 30 years the only observation of \CP\ 
violation was in the neutral kaon system. Over these years
the formalism used to describe \CP\ violation has changed, partly
because our theoretical understanding of the subject has improved.
One example of this development is the present classification of three,
rather than two, types of \CP\ violation, as explained in
Sec.~\ref{ch1:sect:ty}. 
Second, the Standard Model expresses all \CP\ violating quantities in
terms of the same CKM phase. The consistency of the experiments in $B$
physics with those in the kaon system therefore provides a stringent
test of the Standard Model. In practice both $B$ and $K$ data are used
to overconstrain the unitarity triangle: the indirect constraints on
$\sin 2\beta$, in particular its sign, rely largely on $\epsilon_K$. Any
future inconsistency in the overdetermined unitarity triangle
indicates new physics either in the $B$ or $K$ system or in both. 

Sec.\ref{subs:nks} describes the neutral kaon system with the modern
formalism and makes contact with the formalism traditionally used for
kaon physics.  To show how kaon measurements shape our expectations
for $B$ physics, Sec.~\ref{sect:epsk} discusses the already measured
\CP\ violating quantities $\e_K$ and $\e_K^\prime$.  In a similar
vein, Sec.~\ref{subs:kpnn} deals with the rare decays
$K^+\to\pi^+\nu\bar\nu$ and $K_L\to\pi^0\nu\bar\nu$, which are the
target of new high-precision kaon experiments.

\subsection{The neutral kaon system}
\label{subs:nks}
\index{K mixing@$K$ mixing}%
\index{CP violation@\CP\ violation!kaon physics} 

\CP\ violation in \kkm\ was discovered in 1964 \cite{ccft}. The quantity
$\epsilon_K$, which is discussed in Sec.~\ref{sect:epsk}, is of key
importance to test the CKM mechanism of \CP\ violation, because new
physics enters $K$ and $B$ physics in different ways. We introduce
the neutral kaon system using the same formalism as for the $B$-meson
system as derived in Sec.~\ref{ch1:sect:gen} and translate it to the
traditional notation.

The lighter mass eigenstate of the neutral kaon is $\ket{K_S}$
and the heavier one is $\ket{K_L}$, where the subscripts refer to their
short and long lifetimes.
They are\index{kaon!mass eigenstates}
\begin{eqnarray}
\ket{K_S} &=& p\, \ket{K^0} + q\, \ket{\Kbar^0} 
  = \frac{\lt( 1+ \ov{\e} \rt) \ket{K^0} -        
    \lt( 1- \ov{\e} \rt) \ket{\ov{K}{}^0}}{
    \sqrt{2 \lt( 1+|\ov{\e}|^2 \rt)}} \,,
    \nn 
\ket{K_L} &=& p\, \ket{K^0} - q\, \ket{\Kbar^0}
  = \frac{\lt( 1+ \ov{\e} \rt) \ket{K^0} +        
    \lt( 1- \ov{\e} \rt) \ket{\ov{K}{}^0}}{
    \sqrt{ 2\lt(  1+|\ov{\e}|^2 \rt)}} \,. 
\end{eqnarray}
The quantity 
\begin{equation}
  \ov{\e} = \frac{1+q/p}{1-q/p}
\end{equation}
depends on phase conventions.
(The parameter $\ov{\e}$ is not to be confused with the well-known
parameter~$\e_K$, defined in \eq{ek}.)

\CP\ conservation in \dst\ transitions
corresponds to $\ov{\e}=0$, in which case $\ket{K_S}$ and $\ket{K_L}$
become the \CP\ even and \CP\ odd eigenstates.  \CP\ violation in mixing
is well-established from the semileptonic \CP\ asymmetry
\index{decay!$K_L \to \ell^{\pm}\nu\,\pi^{\mp}$}
\bey
\delta (\ell) &=&   
  {\Gamma(K_L \to \ell^+\nu\,\pi^-) - \Gamma(K_L\to \ell^-\bar\nu\,\pi^+) \over
   \Gamma(K_L \to \ell^+\nu\,\pi^-) + \Gamma(K_L\to \ell^-\bar\nu\,\pi^+)} \nn
&=& \frac{1 - |q/p|^2}{1 + |q/p|^2} 
  = \frac{2\, \real \ov{\e} }{1 + |\ov{\e}|^2} \; = \; 
    \lt( 3.27 \pm 0.12 \rt) \times 10^{-3} \,.
   \label{ksl}
\eey 
\index{CP violation@\CP\ violation!kaon physics!semileptonic asymmetry}%
The quoted numerical value is the average for $\ell=e$ and
$\mu$ \cite{pdg}.  From \eq{ksl} it is clear that in the kaon
system $|q/p|$ is close to one.
In the $B$ systems $|q/p|$ is close to one because the width difference
is smaller than the mass difference.
Here, however, they are comparable~\cite{pdg}%
\index{kaon!mass difference $\dm_K$}%
\index{kaon!width difference $\dg_K$}%
\index{kaon!$q/p$}
\beq
  \Delta m_K = \lt( 0.5301 \pm 0.0014 \rt) \times 10^{10}\, s^{-1}, \qquad
  \Delta \Gamma_K = \lt( 1.1174 \pm 0.0010 \rt) \times 10^{10}\, s^{-1} \,.
\label{dmdgk}
\eeq
\index{phase!of \CP\ violation in mixing, $\phi$!kaon}%
Hence one concludes that $|q/p|-1$ is so small, because the relative
phase $\phi$ between $M_{12}$ and $-\Gamma_{12}$ (cf., \eq{defphi}) is
close to zero. Expanding in $\phi$ one easily finds from \eq{mgqp} that
\begin{mathletters}
\label{mgsolk}
\bey
\lt| M_{12} \rt| &=& \frac{\dm_K}{2} + 
                {\cal O} \lt( \phi^2 \rt), \qquad \qquad
\lt| \Gamma_{12} \rt| = \frac{\dg_K}{2} + 
                {\cal O} \lt( \phi^2 \rt), \label{mgsolk:a}\\
\frac{q}{p} &=& - e^{- i \phi_M} \lt[ 1 - \phi \, 
        \frac{\dg_K/2}{\dm_K+i \, \dg_K/2} + 
        {\cal O} \lt( \phi^2 \rt)  \rt] . 
    \label{mgsolk:b}
\eey
\end{mathletters}%
Hence \eq{mgsolk:b} and \eq{ksl} allow us to solve for the 
\CP\ violating phase $\phi$:
\bey
\phi &=& 
    \frac{ \lt( \dm_K \rt)^2+ \lt( \dg_K/2 \rt)^2}{\dm_K \dg_K/2} 
    \, \delta (\ell) \, + \, {\cal O} \lt( \phi^2 \rt) 
    = \lt( 6.6 \pm 0.2 \rt) \cdot 10^{-3} \,. \label{phknum1}
\eey 

\index{decay!$K \to \pi \pi$}
In the literature on $K\to\pi\pi$ decays the following 
amplitude ratios are introduced:
\begin{eqnarray}
\eta_{+-} &=& \frac{\bra{\pi^+\pi^-} K_L \rangle}{ 
        \bra{\pi^+\pi^-} K_S \rangle} \,,
 \qquad \qquad
\eta_{00} = \frac{\bra{\pi^0\pi^0} K_L \rangle}{ 
    \bra{\pi^0\pi^0} K_S \rangle}  \,.
\end{eqnarray}
If \CP\ were conserved, both would vanish. 
The moduli and phases of $\eta_{+-}$ and $\eta_{00}$ have been 
measured to be
\bey
\lt|\eta_{+-} \rt|
  = \lt( 2.285 \pm 0.019 \rt) \cdot 10^{-3} \,, && \qquad
\phi_{+-} 
  = 43.5^\circ \pm 0.6^\circ \,,  \nn 
\lt|\eta_{00} \rt|
  = \lt( 2.275 \pm 0.019 \rt) \cdot 10^{-3} \,, && \qquad\;\, 
\phi_{00} 
  = 43.4^\circ \pm 1.0^\circ \,,
    \label{epmnum}
\eey
according to the PDG fit \cite{pdg}. 
All three types of \CP\ violation lead to non-zero 
$\eta_{+-}$ and $\eta_{00}$.
To separate \dst\ from \dso\ \CP\ violation one introduces isospin 
states
\bey
\ket{\pi^0 \pi^0 } & = & 
      \sqrt{\frac{1}{3}} \, \ket{\lt(\pi \pi \rt)_{I=0}}  
    - \sqrt{\frac{2}{3}} \, \ket{\lt(\pi \pi \rt)_{I=2}} \,, \nn
\ket{\pi^+ \pi^- } & = & 
      \sqrt{\frac{2}{3}} \, \ket{\lt(\pi \pi \rt)_{I=0}}  
    + \sqrt{\frac{1}{3}} \, \ket{\lt(\pi \pi \rt)_{I=2}}  \,, \no
\eey
and isospin amplitudes
\beq
A_I = \bra{ \lt( \pi \pi  \rt)_I} K^0 \rangle \,, \qquad 
  \ov{A}_I = \bra{ \lt( \pi \pi  \rt)_I} \ov{K}{}^0 \rangle \,, 
  \qquad \qquad I=0,2 \,.
\eeq
\index{kaon!isospin}%
The strong final state interaction of the two-pion final states is
highly constrained by kinematics and conservation laws: the \CP\
invariance of the strong interaction forbids a two-pion state to
scatter into a three-pion state and the rescattering into a state with
four or more pions is kinematically forbidden. Furthermore, isospin is an
almost exact symmetry of QCD and forbids the rescattering between the
two isospin eigenstates. Hence the final state interaction of the
$I=0$ and $I=2$ states is only elastic and, thus, fully described by two
scattering phases.
This feature is known as \emph{Watson's theorem}~\cite{wat}.
Hence we can write
\bey
A_I & = & |A_I| e^{i \Phi_I} e^{i \delta_I} \,, \qquad \qquad
  \ov{A}_I = -|A_I| e^{-i \Phi_I} e^{i \delta_I} \,, \nn 
\lambda_I & = & \frac{q}{p} \frac{\ov{A}_I}{A_I} 
    = e^{-i \lt( 2   \Phi_I + \phi_M   \rt) } 
    \, \lt[ 1 - \phi \frac{\dg_K/2}{\dm_K + i\, \dg/2} \rt] + 
     {\cal O} (\phi^2) \,, \label{ai}
\eey
where the two scattering phases~$\delta_I$ are empirically determined
to be $\delta_0\approx 37^\circ$ and $\delta_2\approx -7^\circ$.
Several weak amplitudes (with different \CP\ violating
phases) contribute to $|A_I| e^{i \Phi_I}$, but the presence of a single
strong phase allows to write $A_I$ as in \eq{ai}, ensuring
$|\ov{A}_I/A_I|=1$.  Therefore, there is no direct \CP\ violation in $K
\to (\pi \pi)_I$. Note that our definition of $A_0$ and $A_2$ includes
both the weak and strong phases, in accordance with the formalism used
in $B$ physics. In the kaon literature the $A_I$'s are commonly defined
without the factors $e^{i\delta_I}$.   

A simplification arises from the experimental observation that 
$|A_0|\simeq 22 \, |A_2|$, which is called $\Delta I =1/2$ rule. This
enhancement of $|A_0|$ allows to expand in $|A_2/A_0|$. The \CP\
violating quantity $\e_K$ reads
\index{CP violation@\CP\ violation!kaon physics!$\e_K$}
\beq
\e_K = \frac{\eta_{00}+2\, \eta_{+-}}{3} =
    \frac{  \bra{(\pi\pi)_{I=0}} K_L \rangle }{ 
        \bra{(\pi\pi)_{I=0}} K_S \rangle  } 
    \lt[ 1+ {\cal O} \lt( \frac{A_2^2}{A_0^2}  \rt) \rt]
    = \frac{1-\lambda_0}{1+\lambda_0} \,. 
\label{ek} 
\eeq
Hence $\e_K$ is defined in a way that to zeroth and first order in
$A_2/A_0$ only a single strong amplitude contributes and therefore
\CP\ violation in decay is absent. The $I=0$ two-pion state 
dominates the $K_S$ width $\Gamma_S$. Thus 
$\bra{K^0}(\pi\pi)_{I=0} \rangle \bra{(\pi\pi)_{I=0}} \ov{K}{}^0
\rangle$ almost saturates $\Gamma_{12}$, so that the phase 
$\phi_M-\phi$ of $-\Gamma_{12}$ equals $-2 \Phi_0$ up to tiny
corrections of order $A_2^2/A_0^2$ and $\Gamma_L/\Gamma_S$. This
implies that $\e_K $ does not provide any additional information
compared to the semileptonic asymmetry in \eq{ksl}. We find from
\eq{ai}
\beq
\lambda_0 = 1 - i \, \phi \frac{\dm_K}{\dm_K + i \, \dg_K } + 
     {\cal O} \lt( \phi^2,\frac{A_2^2}{A_0^2},
               \frac{\Gamma_L}{\Gamma_S}  \rt) ,
\eeq
and \eq{ek} evaluates to
\index{CP violation@\CP\ violation!kaon physics!$\e_K$}
\beq
\e_K = \frac{\phi}{2} \frac{\dm_K}{\sqrt{(\dm_K)^2 + (\dg_K/2)^2  }} \, 
    e^{i \phi_{\e}} 
       +  {\cal O} \lt( \phi^2,\frac{A_2^2}{A_0^2},
               \frac{\Gamma_L}{\Gamma_S}  \rt)  
    \quad \mbox{ with~~}
    \phi_{\e} = \arctan \frac{\dm_K}{\dg_K/2} \,. 
   \label{ekres}
\eeq 
  From \eq{epmnum} one finds the experimental value:
\beq
\e_K = e^{i\, (0.97 \pm 0.02)\, \pi/4}\, (2.28 \pm 0.02) \times 10^{-3} \,.  
\eeq
Therefore \eq{ekres} yields
\index{phase!of \CP\ violation in mixing, $\phi$!kaon}
\beq
\phi = \lt( 6.63 \pm 0.06 \rt) \times 10^{-3} \,, \label{phknum}
\eeq
in perfect agreement with \eq{phknum1}.  A~numerical accident leads to
$\dm_K \approx \dg_K/2$, which explains why the phase $\phi_{\e}$ in
\eq{ekres} is so close to $\pi/4$.

To first order in $\phi$ one finds from \eq{ek} 
\beq
\e_K \simeq \frac{1}{2} \lt[ 1 -\lambda_0  \rt] \; \simeq \;
    \frac{1}{2} 
    \lt( 1 -\lt|\frac{q}{p}\rt| - i \, \imag \lambda_0 \rt) .
\eeq
Therefore $\real \e_K$ measures \CP\ violation in mixing and $\imag \e_K$ 
measures interference type \CP\ violation.
\index{CP violation@\CP\ violation!in mixing!kaon}
\index{CP violation@\CP\ violation!interference type!kaon}

\CP\ violation in \dso\ transitions is characterized by 
\index{CP violation@\CP\ violation!kaon physics!$\e_K^\prime$}
\bey
\e_K^{\prime} & = & 
 \frac{\eta_{+-} - \, \eta_{00}}{3} = 
    \frac{\e_K}{\sqrt{2}} \lt[ 
    \frac{  \bra{(\pi\pi)_{I=2}} K_L \rangle }{ 
        \bra{(\pi\pi)_{I=0}} K_L \rangle  } 
    - 
    \frac{  \bra{(\pi\pi)_{I=2}} K_S \rangle }{ 
        \bra{(\pi\pi)_{I=0}} K_S \rangle  } \rt] 
    \lt[ 1+ {\cal O} \lt( \frac{A_2 }{A_0 }  \rt) \rt] \nn 
&=& \frac{A_2}{A_0}\, \frac{1}{\sqrt{2}} \lt[ 
    \frac{1-\lambda_2}{1+ \lambda_0}  
    - \frac{(1-\lambda_0)(1+\lambda_2)}{(1+ \lambda_0)^2} \rt] 
    . \label{ekp} 
\eey
Next we use  
\beq
\lambda_2 = \lambda_0\, e^{2 \, i\, \lt(\Phi_0-\Phi_2 \rt)} \,,
\eeq
and expand to first order in the small phases:
\beq
\e_K^{\prime} = \frac{1}{2\, \sqrt{2}} \,  
     \frac{A_2}{A_0} \, 
    ( \lambda_0-\lambda_2 ) + 
    {\cal O} \lt( \frac{A_2^2 }{A_0^2 }\,, \phi^2\,, 
    (\Phi_0-\Phi_2)^2 \rt)  
   = \frac{1}{\sqrt{2}}\, \frac{A_2}{A_0}\, i\, 
    ( \Phi_2-\Phi_0 ) \,.   \label{ekpres}
\eeq
A non-vanishing value of $\e_K^{\prime}$ implies different \CP\
violating phases in the two isospin amplitudes and therefore 
\dso\ \CP\ violation.
\index{CP violation@\CP\ violation!in $\Delta F=1$ transitions}
Since experimentally $\real \e_K^\prime >0$,
one finds $ \Phi_2 > \Phi_0$. The phase of $\e_K^{\prime}$ is
$90^\circ +\delta_2-\delta_0 \simeq 46^\circ $ and
$\e_K^{\prime}/\e_K$ is almost real and positive.

Since \eq{ekpres} does not depend on $q/p$, there is no contribution
from \CP\ violation in mixing to $\e_K^{\prime}$.  The strong phases
drop out in the combination
\beq
\imag \frac{A_0}{A_2}\, \e_K^\prime \simeq   \frac{1}{2\, \sqrt{2}} \,  
    ( \imag \lambda_0 - \imag \lambda_2 ) \,. \label{ekpcpm}
\eeq
Since we work to first order in $\phi$, we can set $|\lambda_I|=1$,
and therefore \eq{ekpcpm} purely measures interference type \CP\
violation. From the definition in \eq{ekp} one further finds that
\index{CP violation@\CP\ violation!interference type!kaon}
\beq
\real  \e_K^\prime \simeq \frac{1}{6} 
    \lt( 1- \lt| \frac{A_{\pi^0\pi^0}\, \ov{A}_{\pi^+\pi^-} }{
                  \ov{A}_{\pi^0\pi^0}\, A_{\pi^+\pi^-} }
    \rt|\, \rt)
 \simeq \frac{1}{\sqrt{2}}\, \frac{|A_2|}{|A_0|}
    \, \sin \lt( \delta_0-\delta_2 \rt)
    \lt( \Phi_2 - \Phi_0 \rt)
\eeq
originates solely from $|\ov{A}_f/A_f|\neq 1$.  Hence $\real
\e_K^\prime $ measures \CP\ violation in decay.
\index{CP violation@\CP\ violation!in decay!kaon}

Experimentally the quantity 
$|\eta_{00}/\eta_{+-}|^2= 1- 6\, \real \e_K^\prime/\e_K $ has been
determined. Recent results are 
\bey
\real \frac{\e_K'}{\e_K}  &=& (20.7 \pm 2.8) \times 10^{-4} \qquad 
  {\rm (KTeV)}~\mbox{\cite{KTeV}} \,, \nonumber\\*
\real  \frac{\e_K'}{\e_K} &=& (15.3 \pm 2.6) \times 10^{-4} \qquad 
  {\rm (NA48)}~\mbox{\cite{NA48}} .  
\label{epnum}
\eey 
We therefore find from \eq{ekpres} that the difference of the \CP\
violating phases is tiny:
\beq
\Phi_2-\Phi_0 =  (1.5 \pm 0.2) \cdot 10^{-4} \quad  {\rm (KTeV)}\,, \qquad 
  \Phi_2-\Phi_0 = (1.1 \pm 0.2) \cdot 10^{-4} \quad  {\rm (NA48)}\,. 
\label{ph20num}
\eeq

\boldmath
\subsection{Phenomenology of $\e_K$ and $\e_K^\prime$}
\label{sect:epsk}
\unboldmath
\index{phase!in mass matrix, $\phi_M$!kaon}


In order to exploit the precise measurement of $\phi=-\arg
M_{12}/\Gamma_{12}$ from $\e_K$ in \eq{phknum} one must calculate
the phases of 
\beq
M_{12} = \frac{1}{2 m_K}\, \bra{K^0} H^{\dst} \ket{\ov{K}{}^0} 
    - \mbox{Disp}\, \frac{i}{4 m_K} \int \! \d^4 x \, 
    \bra{K^0} H^{\dso} (x)\, H^{\dso} (0) \ket{\ov{K}{}^0} \,.    
\label{m12k} 
\eeq
and\index{phase!in decay matrix!kaon}
\begin{eqnarray}
\Gamma_{12} & = & \mbox{Abs}\,  
    \frac{i}{2 m_K} \int \! \d^4 x \, 
    \bra{K^0} H^{\dso} (x)\, H^{\dso} (0) \ket{\ov{K}{}^0} 
\label{ga12k} \\
&=& \frac{1}{2 m_K} \sum_f (2\pi)^4 \delta^4 ( p_K-p_f )  
  \bra{K^0} H^{\dso}\ket{f}\, \bra{f} H^{\dso} \ket{\ov{K}{}^0}
  \simeq \frac{1}{2 m_K}\, A_0^*\, \ov{A}_0 \,. \no
\end{eqnarray}
Here Abs\index{absorptive part} denotes the absorptive part of the 
amplitude.  It is calculated by retaining only the imaginary part of 
the loop integration while keeping both real and imaginary parts of 
complex coupling constants.  Analogously, the dispersive part 
Disp\index{dispersive part} is obtained from the real part of the loop 
integral.

The second term in \eq{m12k} shows that, at second order, also the
\dso\ Hamiltonian contributes to $M_{12}$. In the $B$ system the
corresponding contribution is negligibly small.  The Standard Model
\dst\ Hamiltonian reads
\bey 
H^{\dst} = \frac{G_{F}^2}{4 \pi^2}\, M_W\, \Big[ &&
          \lambda_c^{*2} \, \eta_1  \, 
          S (x_c)   \:  + \:
          \lambda_t^{*2} \, \eta_2  \, 
          S (x_t)  \nn 
&& {}       + 2\, \lambda_{c}^* \, \lambda_t^* \, \eta_3 \, 
          S (x_c,x_t) \Big]  
    b_K (\mu)\, Q_K (\mu) +  \mathrm{h.c.} \label{hs2}
\eey
It involves the \dst\ operator 
\beq
Q_K (\mu) = \ov{d}_L \gamma_{\nu} s_L \, \ov{d}_L \gamma^{\nu} s_L \,.
\eeq
\index{Cabibbo-Kobayashi-Maskawa matrix!$V_{td}$}%
\index{Cabibbo-Kobayashi-Maskawa matrix!$V_{ts}$}%
In \eq{hs2} $\lambda_q=V_{qd}V_{qs}^*$, $x_q=m_q^2/M_W^2$ and $S(x)$ is
the Inami-Lim function introduced in \eq{ch1:uli:sxt}. The third
function $S(x_c,x_t)$ comes from the box diagram with one charmed and
one top quark.  One finds $S(x_c) \simeq x_c$, $S(x_c,x_t) \simeq x_c
(0.6-\ln x_c)$ and $S (x_t)\simeq 2.4$ for $m_t\simeq 167\,$GeV in
the ${\ov{\rm MS}}$ scheme. Short distance QCD corrections are
contained in the $\eta_i$'s. In the ${\ov{\rm MS}}$ scheme the
next-to-leading order results are $\eta_1=1.4 \pm 0.3$, $\eta_2=0.57
\pm 0.01$ and $\eta_3=0.47 \pm 0.04 $ \cite{hn1}. $\eta_1$ strongly
depends on $m_c$ and $\alpha_s$, the quoted range corresponds to
$m_c=1.3\,$GeV. A~common factor of the QCD coefficients is 
$b_K(\mu)$, the kaon analogue of $b_B(\mu)$ encountered in
\eq{ch1:uli:c2}. The matrix element of $Q_K$ is parameterized as
\beq
\bra{K^0} Q_K (\mu) \ket{\ov{K}{}^0} =
    \frac{2}{3}\, f_K^2\, m_K^2\, \frac{\widehat{B}_K}{b_K (\mu)} \,,
 \label{defbk} 
\eeq
where $f_K$ is the kaon decay constant.% 
\index{hadronic parameter!$\widehat{B}_K$}

\CP\ violation in the  kaon system  is related to the squashed 
unitarity triangle with sides $|\lambda_u|$, $|\lambda_c|$ and
$|\lambda_t|$. In the limit $\lambda_t=0$ all \CP\ violation vanishes,
thus \CP\ violation is governed by the small parameter $\imag
(\lambda_t/\lambda_u)$. This explains the smallness of the measured
phases in \eq{phknum} and \eq{ph20num}. This pattern is a feature of
the CKM mechanism of \CP\ violation and need not hold in extensions of
the Standard Model. Hence kaon physics provides a fertile testing
ground for non-standard \CP\ violation related to the first two
quark generations.  

The presence of the second term in \eq{m12k} impedes the clean
calculation of the mixing phase $\phi_M =\arg M_{12}$ in terms of the
CKM phases. It constitutes a long distance contribution, which is not
proportional to $\widehat{B}_K$. Since both terms in \eq{m12k} have
different weak phases, $\phi_M$ involves the ratio of the two hadronic
matrix elements. This is different from the case in \bbm\, where only
one hadronic matrix element contributes in the Standard Model, which
therefore cancels from $\phi_M$.  The long distance \dso\ piece is
hard to calculate and is usually eliminated with the help of the
experimental value of $\dm_K=2\, |M_{12}|$ in \eq{dmdgk}. Then,
however, our expression for the mixing phase $\phi_M$ still depends on
the hadronic parameter $\widehat{B}_K$. 

The phases of both $\phi_M$ and $\arg \Gamma_{12}$ are close to $\arg
\lambda_u$, which vanishes in the CKM phase convention. The dominant 
corrections to $\phi_M$ stems from the term proportional to
$\lambda_t^{*2}$ in $H^{\dst}$. For $\arg \Gamma_{12}$ we need the 
\dso\ Hamiltonian, which is obtained from the \dbo\ Hamiltonian in 
\eq{hd} by replacing $\xi_{u,c,t}$ with $\lambda_{u,c,t}$ and
replacing the $b$ quark field in the operators by an $s$ field.  The
leading contribution to $\arg (-\Gamma_{12})\approx - 2\, \Phi_0$, is
proportional to $\imag \lambda_t \, \bra{(\pi\pi)_{I=0}} Q_6
\ket{\ov{K}{}^0}/|A_0|$.  The $\Delta I =1/2$ enhancement of $|A_0|$
suppresses $\Phi_0$, which is calculated to $\Phi_0={\cal O} (2 \cdot
10^{-4})$ \cite{ep1}.  Hence in the CKM phase convention $\arg
(-\Gamma_{12})$ contributes roughly 6\% to the measured phase $\phi$
in \eq{phknum} and one can approximate $\phi\approx \phi_M$.  After
expressing the CKM elements in \eq{hs2} in terms of the improved
Wolfenstein parameters the constraint from the measured value in
\eq{phknum}  can be cast in the form \cite{n1,hn1}
\beq
5.3 \times 10^{-4} = \widehat{B}_K A^2 \, \ov{\eta} \,
    \lt\{ \lt[ 1 - \ov{\rho} + \Delta  \lt(\ov{\rho},\ov{\eta} \rt) \rt] 
     A^2 \lambda^4 \eta_2\, S(x_t)
       + \eta_3 S(x_c,x_t) - \eta_1 x_c \right\} . 
\label{cons2}  
\eeq
\index{CP violation@\CP\ violation!kaon physics!$\e_K$ constraint on $\ov{\rho},\ov{\eta}$}%
\index{unitarity triangle!kaon physics}%
In the absence of the small term $\Delta \lt( \ov{\rho},\ov{\eta}
\rt) = \lambda^2 \, \lt( \ov{\rho} -\ov{\rho}^2 -\ov{\eta}^2 \rt)$
this equation defines a hyperbola in the $(\ov{\rho},\ov{\eta})$ plane.
The largest uncertainties in \eq{cons2} stem from $\widehat{B}_K$ and 
$A=|V_{cb}|/\lambda^2$, which enters the largest term in \eq{cons2} raised
to the fourth power. Hence, reducing the error of $|V_{cb}|$ improves the 
$\e_K$ constraint.

It is more difficult to analyze $\e_K^\prime$, because the weak phases
$\Phi_0$ and $\Phi_2$ are much harder to compute than $\phi_M$.
$\Phi_2$ is essentially proportional to $\imag \lambda_t \,
\bra{(\pi\pi)_{I=2}}\, Q_8\, \ket{\ov{K}{}^0}/|A_2|$.  The two matrix
elements entering $\Phi_2-\Phi_0$ are difficult to calculate and
numerically tend to cancel each other. Especially there is a
controversy about $\bra{(\pi\pi)_{I=0}} Q_6 \ket{\ov{K}{}^0}$ and the
different theoretical estimates can accommodate for both the KTeV and
the NA48 result in \eq{ph20num} \cite{ep1,kns}. Even after the
experimental discrepancy in \eq{epnum} is resolved, $\e_K^\prime$ will
not immediately be useful to determine $\imag \lambda_t \simeq A^2
\lambda ^5 \ov{\eta}$.  Nevertheless, $\e_K^\prime$ can be useful to
constrain new physics contributions \cite{bs}. For recent overviews on
$\e_K^\prime$ we refer to \cite{eprev}.


\boldmath
\subsection{$K \to \pi \nu\bar\nu$}
\label{subs:kpnn}
\unboldmath
\index{kaon!rare decay}
\index{decay!$K^+ \to \pi^+ \nu\bar\nu$}
\index{Cabibbo-Kobayashi-Maskawa matrix!$V_{td}$}
\index{Cabibbo-Kobayashi-Maskawa matrix!$V_{ts}$}


Rare kaon decays triggered by loop-induced $s\to d$ transitions can
provide information on $\lambda_t$ and thereby on the shape of the
unitarity triangle. Final states with charged leptons are poorly
suited for a clean extraction of this information, because they
involve diagrams with photon-meson couplings. Such diagrams are
affected by long distance hadronic effects and are hard to evaluate.
The decays $K^+ \to \pi^+\nu\ov{\nu}$ and $K_L \to \pi^0\nu\ov{\nu}$,
however, are theoretically very clean, with negligible hadronic
uncertainties. The $K \to \pi$ form factors can be extracted from the
well-measured $K_{\ell3}$ decays. So far two 
$K^+ \to \pi^+\nu\ov{\nu}$ events have been observed \cite{bnl},
corresponding to a branching ratio
\beq
{\cal B} (K^+ \to \pi^+\nu\ov{\nu}) = \Big(1.57 \epm{1.75}{0.82}\Big) \times 10^{-10} 
    \,. \label{kppexp}
\eeq
Experimental proposals at BNL and Fermilab aim at a measurement  
of $K^+\to \pi^+\nu\ov{\nu}$ and $K_L \to \pi^0\nu\ov{\nu}$
at the 10\% level. The constraint on the improved
Wolfenstein parameters $(\ov{\rho},\ov{\eta})$ can be cast in the form
\cite{bbl}
\beq
\frac{{\cal B}(K^+ \to \pi^+ \nu \ov{\nu}) }{4.57 \cdot 10^{-11}} 
= A^4 X^2(x_t) \lt( 1- \lambda^2 \rt)
    \bigg[ \bigg( \frac{\ov{\eta}}{ 1 - \lambda^2 } \bigg)^2 
     + \lt( \rho_0 - \ov{\rho} \rt)^2 \bigg]  
    \label{kpp} .
\eeq
Here $X (x_t)\simeq 1.50$ comprises the dependence on $m_t$ and the
NLO short distance QCD corrections \cite{kpnncalc}.  $\rho_0 \approx
1+0.27/A^2$ contains the contribution from the internal charm loop
\cite{kpnncalc}.  The quoted numerical value corresponds to a $\ov{{\rm
MS}}$ mass of $m_c=1.3\,$GeV.  The largest theoretical uncertainty in
\eq{kpp}, of order 5\%, stems from the charm contribution. Further the
fourth power of $A$ introduces a sizable parametric uncertainty.  The
equation in \eq{kpp} describes an ellipse in the
$(\ov{\rho},\ov{\eta})$ plane centered at $(\rho_0,0)$. By
inserting typical values for the Wolfenstein parameters (e.g.,
$\lambda=0.22$, $A=0.8$, $\ov{\rho}=0.2$ and $\ov{\eta}=0.4$ ) into
\eq{kpp} one finds that \eq{kppexp} is compatible with the Standard
Model.

\index{CP violation@\CP\ violation!interference type!kaon}
\index{decay!$K_L \to \pi^0 \nu\bar\nu$}
In the Standard Model the decay $K_L \to \pi^0 \nu \ov{\nu}$ is \CP\
violating. It measures interference type \CP\ violation, the
associated phase $\arg \lambda_{\pi^0 \nu \ov{\nu}}$ is large, of
order $\ov{\eta}/(1-\ov{\rho})$. This is in sharp contrast to the
small phases we found in \eq{phknum} and \eq{ph20num}. A~measurement
of ${\cal B}(K_L \to \pi^0 \nu \ov{\nu})$ establishes $\arg \lambda_{\pi^0
\nu \ov{\nu}} \neq \phi $ and therefore implies \CP\ violation in the
\dso\ Hamiltonian. This is the same situation as with $\imag
(\e_K^\prime\, A_0/A_2)$ in \eq{ekpcpm}, which also proves \dso\ 
\CP\ violation from the difference of two interference type \CP\ 
violating phases. $K_L \to \pi^0 \nu \ov{\nu}$ is even cleaner than
$K^+\to \pi^+\nu\ov{\nu}$, because the charm contribution is
negligible.  ${\cal B}(K_L \to \pi^0 \nu \ov{\nu})$ is proportional to
$(\imag \lambda_t)^2 \propto \ov{\eta}^2$ and therefore determines
the height of the unitarity triangle
\beq
\frac{{\cal B}(K_L \to \pi^0 \nu \ov{\nu}) }{1.91 \cdot 10^{-10}} 
= A^4 X^2(x_t)\, \ov{\eta}^2 \lt( 1+ \lambda^2 \rt)
    \label{kpp0} .
\eeq
Hence the two discussed branching ratios allow for a precise
construction of the unitarity triangle from kaon physics alone.
Moreover the ratio of the two branching ratios is almost independent
of $A$ and $m_t$. It allows for a determination of $\sin 2\beta$ with a
similar precision as from $a_{CP} (B\to \psi K_S)$ \cite{kpnnphen}.
New physics may enter $s\to d$ transitions in a different way than $b \to
s$ and $b \to d$ transitions. Hence comparing of the unitarity
triangles from $K$ physics and from $B$ physics provides an excellent test
of the Standard Model.
\index{unitarity triangle!kaon physics}
\index{decay!$B_d \to \psi K_S$}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Standard Model Expectations }
\label{ch1:SM-expect}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\OMIT{
{\bf I believe that it has to be reconsidered now [after 1.5 years] whether
this is the right way to finish this section.  ---Z}
%
{\bf I think it's fine. -- A}
}

This section outlines what is known about the CKM matrix at the present
time, and what the pattern of expectations is for some of the most
interesting processes in the Standard Model.


%\subsection{Present constraints}


Since most of the existing data apart from $\sin2\beta$ come from $CP$
conserving measurements, it is convenient to present the constraints on the CKM
matrix using the Wolfenstein parameterization.  Magnitudes of CKM matrix
elements are simply related to $\lambda$, $A$, $\bar\rho$, and $\bar\eta$. The
best known of these is $\lambda$, the Cabibbo angle, which is known at the 1\%
level.  The parameter $A$ is determined by $|V_{cb}|$, which is known with a
5\% error.  The uncertainty in $\bar\rho$ and $\bar\eta$ is significantly
larger.  The most important constraints come
from\index{unitarity triangle!kaon physics}%
\index{CP violation@\CP\ violation!kaon physics!$\e_K$ constraint on $\ov{\rho},\ov{\eta}$}%
\index{unitarity triangle!and Vub@and $V_{ub}$}%
\index{unitarity triangle!and \bbmd}%
\index{unitarity triangle!and \bbms}%
\begin{itemize} \vspace*{-8pt} \itemsep -2pt

\item $CP$ violation in $K^0\!-\!\Kbar^0$ mixing described by the $\epsilon_K$
parameter;

\item $|V_{ub}/V_{cb}|$ measured from semileptonic $B$ decays;

\item $B^0\!-\!\B0bar$ mixing;

\item The lower limit on $B_s\!-\!\Bbar_s$ mixing.

\end{itemize} \vspace*{-8pt}

\begin{figure}[t]
\centerline{\epsfxsize 0.85\textwidth \epsffile{ch1/rhoeta.eps}}
\caption[The allowed region in the $\bar\rho-\bar\eta$ plane.]{The allowed
region in the $\bar\rho-\bar\eta$ plane. Also shown are the individual
constraints, and the world average $\sin2\beta$. (From Ref.~\cite{CKMfitter}.)
}
\label{1:refit}
\end{figure}

A problem in translating these to constraints on the CKM matrix is related to
theoretical uncertainties.  We follow the point of view adopted in the BaBar
book~\cite{babook} that no confidence level can be attached to model dependent
theory errors.  Fig.~\ref{1:refit} shows the result of such an analysis from
Ref.~\cite{CKMfitter}. Fig.~\ref{1:abfit} shows the same fit on the
$\sin2\alpha-\sin2\beta$ plane.  Note that any value for $\sin2\alpha$ would
still be allowed if $|V_{ub}|$ were slightly larger, or if $\Delta m_{B_s}$
were slightly smaller than their allowed ranges.  Fig.~\ref{1:bgfit} shows the
allowed range in the $\sin2\beta - \gamma$ plane, and that $\gamma$ is already
constrained.

\begin{figure}[t]
\centerline{\epsfxsize 0.85\textwidth \epsffile{ch1/s2as2b.eps}}
\caption[The allowed region in the $\sin2\alpha-\sin2\beta$ plane.]{The
allowed region in the $\sin2\alpha-\sin2\beta$ plane~\cite{CKMfitter}. }
\label{1:abfit}
\end{figure}

\begin{figure}[t]
\centerline{\epsfxsize 0.85\textwidth \epsffile{ch1/s2bgam.eps}}
\caption[The allowed region in the $\sin2\beta-\gamma$ plane.]{The
allowed region in the $\sin2\beta-\gamma$ plane~\cite{CKMfitter}. }
\label{1:bgfit}
\end{figure}

Some of the uncertainties entering these constraints will be significantly
reduced during Run~II.
The hadronic matrix elements $B_K$, $f_{B_d}^2 B_{B_d}$, and
$f_{B_s}^2 B_{B_s}$ need to be determined by unquenched lattice QCD
calculations.
The theoretical uncertainties in $|V_{ub}|$ and $|V_{cb}|$ will also be reduced
to the few percent level by unquenched lattice calculations of the exclusive
$\B0bar_d \to \pi\, \ell\, \bar\nu$%
\index{decay!$B_d \to \pi \ell \nu$}
and $\B0bar_d\to D^{(*)} \ell\, \bar\nu$%
\index{decay!$B_d \to  D^{(*)} \ell \nu$}
form factors in the region of phase space where the momentum of the final
hadron is small.
As discussed in Sec.~\ref{1:lattice}, these lattice calculations are
straightforward in principle, but a variety of uncertainties must be
brought under control.
The uncertainties in these two CKM matrix elements may be
reduced in the next few years, even without recourse to lattice QCD, 
using inclusive semileptonic decays.
The error in $|V_{cb}|$ may be reduced to 2--3\% with precise
determinations of a short distance $b$ quark mass and by gaining more
confidence about the smallness of quark-hadron duality violation.
On a similar timescale the error in $|V_{ub}|$ may be reduced to the 5--10\%
level \cite{Vub} by pursuing several model independent determinations.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Summary and Highlights}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

During the coming decade a wide range of measurements in the production and
decay of $b$ flavored hadrons will be made at the Tevatron. In this chapter, we
summarize the most important ones. Particularly in the decays, some observables
are key to pinning down the flavor structure of the Standard Model and,
possibly, uncovering new physics. Some are unique to the Tevatron, and some are
expected to be competitive with measurements at the $e^+e^-$ $B$ factories. The
Tevatron enjoys a few advantages that compensate for the clean environment in
$e^+e^-$ collisions.  First, the $b\bar{b}$ production cross section is much
higher at the Tevatron than at $e^+e^-$ machines. Thus, despite the higher
background rates at the Tevatron, it is possible to use selective triggers to
obtain high statistics samples with good signal-to-noise ratio.  Second, all
$b$ flavored hadrons are created ($B_s$, $\Lambda_b$, etc.), whereas an
$e^+e^-$ machine operating at the $\Upsilon(4S)$ resonance produces only
$B^\pm$, $\Bbar^0_d$ and $B^0_d$. Consequently, the $B$ decay program at hadron
colliders complements that at the $e^+e^-$ $B$ factories.

The CDF, D\O, and BTeV detectors and simulation tools used for this report are
described in Chapters 2--5. Since Run~I, both CDF and D\O\ have gone through
major upgrades, and both will be much more powerful for doing $B$ physics in
Run~II. Both detectors feature excellent charged particle tracking using
solenoidal magnetic fields, and both have silicon (Si) vertex detectors capable
of tracking in three dimensions. The magnetic field is 2\,T at D\O\ and 1.4\,T
at CDF. D\O\ has a smaller radius tracking volume ($50$\,cm), which allows for
Si and scintillating fiber tracking out to $\eta = 1.6$, and Si disk tracking
to $\eta=3$. CDF has a larger tracking radius ($140$\,cm), with full Si and
drift chamber tracking out to $\eta = 1$, and Si-only tracking (to $30$\,cm
radius) out to $\eta = 2$. D\O\ muon coverage extends to $\eta=2$; CDF to $\eta
= 1.5$. Both detectors have track based triggers at level-1 and Si vertex
impact triggers at level-2. Because of the larger level-1 bandwidth ($40$\,kHz)
and the use of deadtimeless SVX3 chip readout, CDF can deploy a Si vertex
hadronic decay trigger at level-2, which is directly sensitive to decays like
$B\to\pi^+\pi^-$ and $B_s\to D_s^-\pi^+$. With a smaller level-1 bandwidth
($10$\,kHz), D\O\ does not plan to implement such a strategy. CDF has two
methods for particle ID --- time-of-flight at low momentum ($p<2\,{\rm GeV}/c$)
and relativistic rise $dE/dx$ for high momentum ($p>2\,{\rm GeV}/c$); the
latter is crucial for channels like $B\to\pi^+\pi^-$, where a statistical
separation of $\pi\pi$ and $K\pi$ signals can be used. In summary, both CDF and
D\O\ have comparable $B$ physics reach; CDF has an advantage in particle ID and
in Si vertex triggering.

BTeV is slated to run in earnest after Run~II, following construction and some
running during Run~IIb. It will have competition from LHC-$b$ at CERN, and from
experiments at the $e^+e^-$ $B$ factories, which by then are expected deliver
an order of magnitude higher luminosity than at present. BTeV is not a central
detector (as are CDF and D\O), but a two-arm forward spectrometer. The key
features of its design are a silicon pixel detector, a flexible trigger based
on detached vertices, particle ID with excellent $K/\pi$ separation, and an
electromagnetic calorimeter capable of identifying $\pi^0$'s and photons.

In this workshop, simulations of the CDF, D\O\ and BTeV detectors have made
common assumptions about production rates, branching ratios, and flavor tagging
efficiencies. The common starting point is needed for comparing the reaches of
the three detectors, but it is nevertheless difficult to make meaningful
comparisons without Run~II data. At hadron colliders all signal channels, as
well as issues like flavor tagging, have backgrounds that will be detector
dependent and that, in many cases, cannot be reliably predicted from a heavy
flavor production Monte Carlo. These backgrounds affect both
signal-to-background statistics and also the strategies used to reject
background, which in turn affect signal yields. For example, for
$B\to\pi^+\pi^-$, to reject the combinatorics from all sources of backgrounds,
making harder cuts on the detachment of the secondary vertex and other
quantities will affect both the statistical accuracy and the ability to
separate the direct and mixing induced $CP$ asymmetries. While CDF has Run~I
data on many channels and flavor tags, D\O\ and BTeV will need real data to
understand fully the effects of backgrounds.

Most simulations carried out during this workshop considered an integrated
luminosity of 2\,fb$^{-1}$, corresponding to Run~IIa for D\O\ and CDF and to the
first year of running for BTeV. Other workshops in this series, which focused
on high $p_T$ physics, considered also the potential for 10\,fb$^{-1}$ and
30~fb$^{-1}$. In the case of $B$ physics, the role of real data in optimizing
event selection makes it difficult to make sensible estimates for such high
luminosity until more experience is at hand.

The most important measurements at the Tevatron can be divided roughly into two
categories. Some modes have a relatively simple theoretical interpretation, in
the context of the Standard Model, either because hadronic uncertainties are
under good control, or because loops, small CKM angles, or GIM effects suppress
the Standard Model rate. These modes test the CKM mechanism and probe for
non-CKM sources of \CP\ and flavor violation. Other measurements are more
sensitive to QCD in ways that are theoretically challenging. They do not (yet)
probe flavor dynamics, but, clearly, better understanding of QCD in $B$ physics
can be reinvested in understanding the flavor sector. We therefore present two
semi-prioritized lists labeled ``tests of flavor and \CP\ violation'' and
``test of QCD'', but both labels should be construed loosely. Moreover, the
composition of these lists is colored by our  understanding during the course
of the workshop. In the coming decade theoretical and experimental developments
are likely to spur changes in the lists. Many other interesting observables,
not discussed in this summary, are covered throughout Chapters 6--9.

\vspace*{8pt}
\noindent{\bf\em Tests of flavor and \boldmath\CP\ violation}
\begin{itemize} \vspace*{-8pt}

\item \bbms: Both CDF and D\O\ can measure $x_s \lesssim 30$ from events with a
semileptonic $B_s$ decay. This covers some of the expected range $x_s \lesssim
45$. With nonleptonic modes, CDF can measure $x_s \lesssim 59$--74, depending
on assumptions, and BTeV $x_s \lesssim 75$. As soon as a 5$\sigma$ observation
of mixing is made, the statistical error of $x_s$ is very small, about
$\pm0.14$. In the Standard Model, $x_s/x_d$ can be used to determine
$|V_{ts}/V_{td}|$, relying on input from lattice QCD for the size of $SU(3)$
breaking.

\item $CP$ asymmetries in $B_s \to\psi\phi$, $\psi\eta^{(\prime)}$: These
measure the relative phase between the amplitudes for \bbms\ and $b\to
c\bar{c}s$ decay, $\beta_s$.  In the Standard Model $\sin2\beta_s$ is a few
percent, so observation of a large $CP$ asymmetry would be a clear sign of new
physics.  The expected error at CDF is about $1.6$ times that of $\sin2\beta$,
further diluted by the $CP$-odd contribution to the $\psi\phi$ final state. 
Although this $CP$-odd contribution is expected to be small, it can be avoided
by using the decay modes $B_s\to\psi\eta^{(\prime)}$, which are pure $CP$-even.
With its excellent photon detection, BTeV is well optimized to measure the
asymmetries in these neutral modes.

\item $CP$ asymmetries in $B_s\to D_s^\pm K^\mp$: Combining the
time dependent asymmetries in these four modes allows the cleanest
determination of $\gamma-2\beta_s$. These measurements must be carried out in
the presence of the large Cabibbo allowed $B_s \to D_s \pi$ background.
Combined with the measurement (or bound) on $\beta_s$, one obtains $\gamma$,
one of the angles of the unitarity triangle. At the Tevatron, this measurement
will probably be possible only at BTeV, with an expected precision of
$\sigma(\gamma) \simeq 10^\circ$.

\item $CP$ asymmetries in $B_d\to\rho\pi$: These asymmetries seem to be the
cleanest way to measure $\alpha$, the angle at the apex of the unitarity
triangle. Once enough events are available to isolate the $\Delta I=3/2$
channel, it is possible to measure $\alpha$ without uncertainties from penguin
amplitudes, which contribute only to $\Delta I=1/2$. With its excellent
electromagnetic calorimetry, BTeV should compete well with BaBar and Belle, on
a similar time scale.

\item $CP$ asymmetry in $B_d \to\psi K_S$: These asymmetries measure the
relative phase between the amplitudes for \bbmd\ and $b\to c\bar{c}s$ decay. In
the Standard Model, this is the angle $\beta$ of the unitarity triangle, and it
is obtained with very small theoretical uncertainty. By the end of Run~IIa, the
CDF and D\O\ measurements may become competitive with BaBar and Belle. Based on
$B\to J/\psi K_S$ only, D\O\ and CDF anticipate a precision on $\sin2\beta$ in
the range 0.04--0.05, with 2\,fb$^{-1}$.

\item $CP$ asymmetries in $B_{d,s}\to h^+_1h^-_2$, $h_i=K$, $\pi$: The utility
of these modes depends on how well the uncertainty from flavor $SU(3)$ breaking
can be controlled. Data for these and other processes will tell us the range of
such effects; the resulting Standard Model constraints could be quite
stringent.  A study by CDF shows that 20\% effects from $SU(3)$ breaking lead
to an uncertainty of only $\sim 3^\circ$ on $\gamma$, which is much smaller
than CDF's expected statistical error with 2\,fb$^{-1}$ of $\sim 10^\circ$.

\item Rare semileptonic and radiative decays, such as $B\to K^*\ell^+\ell^-$ or
$B\to K^*\gamma$: Although the rate for $B\to K^*\gamma$ is higher, the
Tevatron detectors will probably not compete with BaBar and Belle. In the case
of $B\to K^*\ell^+\ell^-$, the high rate of a hadron machine is needed, and the
lepton pair provides a good trigger. In $B\to K^*\ell^+\ell^-$ the
forward-backward asymmetry is especially intriguing, because it is sensitive to
short distance physics.

\item Search for $B_{d,s}\to \ell^+\ell^-$: These flavor-changing neutral
current processes are highly suppressed in the Standard Model. Limits on the
rate constrain non-standard models, which often have other ways to mediate such
decays. Late in Run~II or at BTeV, it may be possible to observe a handful of
events at the standard model rate.

\item \CP\ asymmetries in flavor specific final states: These can be used to
measure the relative phase between the mass and width mixing amplitudes,
analogous to $\Re(\epsilon_K)$ in the kaon sector. The most often cited example
is semileptonic decays, but if this asymmetry is sufficiently large, it may be
possible to detect it using fully reconstructed modes such as $B_s\to
D_s^-\pi^+$.  In the $B$ system, the SM predicts the relative phase to be very
small, but new physics could make these asymmetries measurable without pushing
the measured value of $\sin2\beta$ outside its expected range.

\end{itemize} \vspace*{-6pt}

The most striking thing about this list is that the whole program, taken
together, is much more interesting than any single measurement. Indeed, if one
adds in measurements from the $e^+e^-$ machines, the list becomes even more
compelling. In the near future, BaBar, Belle, CDF, and D\O\ will measure
$\sin2\beta$ at the few percent level; the Tevatron measurement of $x_s$
combined with the known value of $x_d$ will determine $|V_{td}/V_{ts}|$;
BaBar's and Belle's measurements of semileptonic decays will determine
$|V_{ub}/V_{cb}|$. Apart from $\sin2\beta$, some input from hadronic physics is
needed, but the combination of these results will still test the unitarity of
the CKM matrix below the $10\%$ level.

\vspace*{8pt}
\noindent{\bf\em Tests of QCD}
\begin{itemize} \vspace*{-8pt}

\item $B_s$ width difference $\Delta\Gamma_s$: The theoretical prediction
of $\Delta\Gamma_s$ requires hadronic matrix elements of two four-quark
operators, so it relies on lattice QCD calculations to a greater extent
than $x_s$. It is still an interesting measurement, especially if it is
smaller than expected in the Standard Model.

\item $\Lambda_b$ lifetime: The measured value of the $\Lambda_b$ lifetime does
not agree with theoretical expectations. It is important to improve the
measurements with fully reconstructed hadronic decays. If the discrepancy
remains, it would presumably imply a failure of the operator product expansion
employed in inclusive decays and lifetimes, and one would have to reconcile the
failure here with success for other lifetimes and for inclusive semileptonic
decay distributions.

\item Semileptonic form factors: The $q^2$ dependence of the decay
distribution for \mbox{$\Lambda_b \to \Lambda_c \ell \bar\nu$}, $B\to
K^{(*)}\ell^+\ell^-$, etc., can be combined with theoretical information,
potentially reducing the (theoretical) error on $|V_{cb}|$, $|V_{ub}|$, and
$|V_{ts}|$.

\item Quarkonium production and polarization: Run~I data for charmonium
production disagrees at large $p_T$ with predictions from NRQCD. If the
discrepancies persist, and are confirmed through bottomonium production, they
would pose an important riddle. To make meaningful measurements, experiments
must collect higher statistics, extend to higher $p_T$, cleanly separate states
with different quantum numbers, and distinguish feed-down from direct
production.

\item $b\,\bar b$ production cross section: The cross section
$\sigma(p_T>p_T^{\rm min})$ measured in Run~I data is about twice the
prediction from perturbative QCD. With all scales ($m_b$, $\sqrt{\hat{s}}$,
$p_T$) much larger than $\lqcd$, perturbative QCD should be reliable.  A
variety of QCD effects have been studied, but do not seem to account for the
excess. If the excess is confirmed (with much higher statistics) in Run~II, one
must ask whether something in the theory has gone awry, or whether a
non-standard mechanism produces $b\,\bar b$ events.

\item Spectra of $B_c$ and other doubly heavy hadrons: CDF's $B_c$ mass and
lifetime measurements from Run~I will be improved. Fully reconstructed
nonleptonic decays are needed to obtain a good measurement of the mass, which
can be compared to calculations based on potential models or from lattice QCD.
Because there are now two heavy quarks, the lifetimes and decay widths of $B_c$
and other doubly heavy hadrons provide novel tests of the theory.

\end{itemize} \vspace*{-6pt}

Each of these is interesting for its interaction with QCD theory,
and, at least for the next several years,
each
% all except $B\to K^{(*)}\ell^+\ell^-$
can be studied only at the Tevatron.

In conclusion, one sees that $B$ physics at the Tevatron will produce a
program of many interesting and, in some cases, essential measurements.
In the case of $B$ decay measurements, the information gained is
competitive with and complementary to that gained from experiments
in $e^+e^-$ accelerators at the $\Upsilon$(4S).
Indeed, the full suite of measurements from $B_u$, $B_d$, and $B_s$
decays is much more interesting than any one or two measurements
taken in isolation.

% expand conclusions, to incorporate a wish-list?
\chapter{Common Experimental Issues}

\authors{R.~Kutschke, M. Paulini}   

\section{Introduction}

  This chapter will discuss the experimental issues
which underlie $B$ physics at CDF, D\O\ and BTeV.
Many of these issues also apply to charm physics, which will 
also be discussed.
The chapter will be painted in fairly broad strokes and 
the reader is referred to the subsequent
chapters and to the experiments' own Technical Design
Reports (TDR)~\cite{cdf_tdr}\cite{cdf_tof_l00}\cite{d0_tdr}\cite{btev_tdr}
for more details on specific experiments.


  During Run~II, the Fermilab Tevatron will collide counter-rotating
proton $p$ and anti-proton $\bar{p}$ beams at a 
center-of-mass energy of 2~TeV.  Some other design parameters of the
Tevatron for Run~II are summarized in Table~\ref{2:tab:tevatron}.
In rough terms there
are three processes which take place at this energy and which
are important to the design
of a $B$ physics experiment.  These are the production of
$b\bar{b}$ pairs, the production of $c\bar{c}$ pairs
and all of the light quark and gluon processes which
contribute to the background; the cross-sections
for these processes are summarized in 
Table~\ref{2:tab:xsecs}.  There are no known
processes which produce a single $b$ or a single $\bar{b}$
at a significant rate, only processes which produce pairs.
Despite this, one usually talks about $b$ production,
not $b\bar{b}$ production.
Similarly, there are important sources of $c\bar{c}$ production
but not of single $c$ or $\bar{c}$ production.  The
theory behind the production of heavy quark pairs
in $p\bar{p}$ collisions is discussed in 
chapter~9.
%\ref{ProductionFragmentationandSpectroscopy}.
There are, of course, many
other interesting processes
which occur, including top quark production, Higgs boson
production and perhaps even the production of supersymmetric
particles.  The cross-sections for these processes, however, are
small enough that they do not have any impact on
how one designs a $B$ physics experiment for the Tevatron.
\index{b-quark production@$b$-quark production! impact on experiments} 

  After a $b\bar{b}$ pair is produced, it hadronizes to
form pairs of $b$ hadrons including $B$ mesons, such
as $B_d$, $B_u$, $B_s$, $B_c$,
and $b$ baryons such as 
$\Lambda_b$, $\Xi_b$, $\Omega_b$, $\Xi_{bc}$, $\Omega_{cc}$ etc.
All of these states decay weakly, with a significant
lifetime and, therefore, with a significant decay length.
Excited states of these $b$ hadrons are also produced,
all of which decay strongly or electromagnetically to one
of the weakly decaying $b$ hadrons.  A similar
picture exists for the hadronization of 
$c\bar{c}$ pairs into hadrons.
Therefore the route to
all of $b$ and $c$ physics goes through the weakly decaying states.

  One shorthand which will be used in the following is,
\begin{equation}
\sigma_{BG} = \sigma_{tot}-\sigma_{c\bar{c}}-\sigma_{b\bar{b}}.
\end{equation}
This stands for the ``background'' cross-section; that is for the
total hadronic cross-section with the
$c\bar{c}$ and $b\bar{b}$ pieces excluded.  The 
$c\bar{c}$ piece is treated separately because it is interesting
to study in its own right and because it has a few critical
properties which are more like $b$'s than background.
At our current level of precision, 
$\sigma_{BG}\simeq\sigma_{tot}\simeq 75$~mb\cite{bgxsec}.
This includes elastic $p\bar{p}$ scattering, diffractive scattering
and inelastic scattering.
Because $\sigma_{BG}\simeq\sigma_{tot}$ many authors are 
are careless about distinguishing between the two.

\begin{table}[tbf]
\begin{center}
\begin{tabular}{cc} \hline
Quantity & Value \\ \hline \hline
Center of Mass Energy         & 2~TeV \\
Peak Instantaneous Luminosity & 
                        $2\times 10^{32}\; {\rm cm}^{-2}\; {\rm s}^{-1}$ 
                        \\ 
Yearly Integrated Luminosity & 2~fb$^{-1}$/year \\
Time between bunch crossings &
          \multicolumn{1}{l}{\quad\quad\quad 396~ns for $\simeq$ 2 years} \\
       &  \multicolumn{1}{l}{\quad\quad\quad 132~ns afterwards} \\
Luminous region & $(\sigma_x,\sigma_y,\sigma_z) = 
                   ( 0.003, 0.003, 30.)$~cm \\ \hline
\end{tabular}
\medskip
\caption[Tevatron parameters for Run~II]
{Tevatron parameters for Run~II. The conversion
from peak instantaneous luminosity to yearly integrated
luminosity assumes that a year consists of $10^7$ useful
seconds, as discussed in Section~\protect{\ref{2:sec:lumi}}.
\index{Tevatron parameters}
}
\label{2:tab:tevatron}
\end{center}
\end{table}

\begin{table}[tbf]
\begin{center}
\begin{tabular}{ccc} \hline
Quantity & ~~Value (mb)~~ & Comment \\ \hline \hline
$\sigma_{tot}$ & $\approx 75$ & Total hadronic cross-section including \\
                               && elastic, diffractive and inelastic 
                                  processes. \\
$\sigma_{c\bar{c}}$ & $\approx 1$  & Charm pair production cross-section. 
                                         \\
$\sigma_{b\bar{b}}$ & $\approx 0.1$ & Beauty pair production cross-section.
                                         \\
$\sigma_{BG}$ & $\approx 75$& The chapter's short-hand for
                      $\sigma_{tot}-\sigma_{c\bar{c}}-\sigma_{b\bar{b}}$.
\\ \hline
\end{tabular}
\medskip
\caption[Approximate values of cross-sections]
{Approximate values of the cross-sections which are of interest to a $B$
physics experiment using $p\bar{p}$ collisions at a center-of-mass energy of
2~TeV.  The estimate for the total cross-section is from
Ref.~\protect{\cite{bgxsec}}. The estimate of $\sigma_{b\bar{b}}$ is
discussed in Section~\ref{2:ssec:xsec}.%
\index{cross-sections!table of}%
}
\label{2:tab:xsecs}
\end{center}
\end{table}

  Throughout this chapter, the $z$ axis is defined to lie
along the beam direction and quantities such as $p_T$ are measured
with respect to this axis.  The variable $\varphi$ is the azimuth
around the $z$ axis and $\theta$ is the polar angle relative
to the $z$ axis.

\section{Separating \lowercase{$b$} and \lowercase{$c$} Hadrons from the
Backgrounds}

Inspection of Table~\ref{2:tab:xsecs} shows that
the cross-section for $b$ production is about 1.5 parts in 1000 of the 
total cross-section.  Moreover, many of the $B$ physics
processes of interest have product branching 
fractions of $10^{-6}$ or smaller.\footnote{
The product branching fraction is defined
as the product of all of the branching fractions in a decay
chain.  An example of such a decay chain is
$B_d\to J/\psi K^0$, $J/\psi\to \mu^+\mu^-$, 
$K^0\to K^0_S$, $K^0_S\to \pi^+\pi^-$.  
The branching fraction for the first decay is
about $1\times 10^{-3}$, but the product of all branching
fractions in the chain is much smaller, about $2\times 10^{-5}$.
}
Therefore one is often looking for signals
of a few parts per billion of the total cross-section!
%
% $B$   -> Psi K0s   8.9 (1.2) e-4  PDG 2000
% Psi -> mu mu     0.0588 (0.001)
% K0  -> K0s       0.5
% K0s -> pi+ pi-   0.6861 (0.0028)
% product:

  The signature which allows one to see this needle in
a haystack is the lifetime of the $b$ quark.  The $B_d$, $B_u$ and
$B_s$ mesons each have a lifetime $\tau$ of approximately
1.5~ps, or $c\tau=450$~$\mu$m.  When the momentum spectrum
of the $B$ mesons is folded in, the mean decay length of
all produced $B$ mesons is on the order of a few mm. 
Therefore almost all $B$ mesons decay inside the beam pipe.
The resolution on the decay length varies from one decay
mode to another and from one experiment to another but typical
values fall in the range of 50~$\mu$m to 100~$\mu$m;
%\marginpar{Check\\ typical\\ resolution}
therefore the $B$ decay vertices will be well resolved
and will be readily separated from the $p\bar{p}$ interaction vertex.
The $B_c$, the weakly decaying $b$ baryons and the weakly decaying 
charmed hadrons have somewhat shorter lifetimes\cite{PDG}, but
most of them have a long enough lifetime that their decay vertices too
will also be well separated from  $p\bar{p}$ interaction vertex.

The myriad background processes, with their much larger cross-sections,
do not produce particles which have this type of decay length
signature.  This brings us to the magic bullet: it is the presence 
of distinct secondary vertices which allows
the experiments to extract the $b$ and $c$ signals from the background.

About 85\% of all weakly decaying
$b$ hadrons decay into one charmed hadron plus long lived particles.
Long lived particles include pions, kaons, protons, photons,
charged leptons and neutrinos, all of which are stable enough to
escape the interaction region and leave tracks in the detector.   
Some of these particles, such as the $K^0_S$ and $\Lambda$ do decay
but their lifetimes are very long compared to the those of the $b$ 
and $c$ hadrons.\footnote{
The one exception is the $\tau$ lepton but the branching
ratio of $b\to c \tau\nu$ is small, $(2.6\pm0.4)\%$\cite{PDG}.}
About 
15\% of weakly decaying $b$ hadrons decay into 2 charmed hadrons, plus
long lived particles; the decay  $B^0\to D^{*+} D^{*-}$ is an example.
And about 1\% of all weakly decaying $b$ hadrons 
decay into only long lived particles; 
the decay $B^0\to \pi^+\pi^-$ is an example.
Therefore a typical $b\bar{b}$ event has 5
distinct vertices, all inside the beam pipe: the primary $p\bar{p}$ interaction
vertex, the two secondary $B$ decay vertices and the two
tertiary $c$ decay vertices.  On the other hand, many of the
most interesting decays involve charmless decays of the $b$ and
a typical event containing one of these
these decays has 4 vertices inside the beam pipe:
the primary $p\bar{p}$ vertex, 
the vertex from the signal charmless $b$, and the $b$ and $c$ vertices
from the other $b$ (or $\bar{b}$) produced in the $p\bar{p}$ 
interaction.

  To be complete, one more detail must be added to the description of typical
$b\bar{b}$ events.  For the running conditions anticipated  for Run~II, each
beam crossing which contains a $b\bar{b}$ interaction will also contain several
background interactions which contain no $b\bar{b}$ or $c\bar{c}$ pairs. This
is discussed in more detail in Section~\ref{2:sec:multi}.
Fig.~\ref{2:fig:cartoon} shows a cartoon of a $b\bar{b}$  event with one
charmless $b$ decay. Throughout this chapter the word ``event'' should be
understood to include all of the interactions within one beam crossing, both
the signal and background interactions.

\begin{figure}
\centerline{\epsfig{file=ch2/cartoon_crossing.eps, width=0.95\hsize}} 
\caption[A cartoon of a $b\bar{b}$ event]
{A cartoon of an interesting
$b\bar{b}$ event at the Tevatron.  In this
beam-crossing the bunches undergo three independent primary
interactions.  The one in the middle produces a $b\bar{b}$
pair plus some other hadrons while the ones to either side are 
background interactions.  In this event, one $b$ undergoes a charmless
decay while the other decays semileptonically to charm, which decays
hadronically to light hadrons.  The cartoon is meant to emphasize
the topological properties of an event: it is not to scale and does
not correctly represent the number of tracks
in a vertex or the distribution of track directions.
}
\label{2:fig:cartoon}
\index{event! cartoon of a typical}
\end{figure}

Similarly, a typical $c\bar{c}$ interaction has
three distinct vertices inside the beam pipe: the primary vertex
plus two secondary vertices, which come from the decay of the
two charmed hadrons.  A typical beam crossing which contains
a $c\bar{c}$ interaction will also contain a few background interactions.

In an event containing a $b\bar{b}$ or
a $c\bar{c}$ pair, the stable daughters
of the $b$ and $c$ hadrons usually have a large impact parameter with
respect to the primary vertex.  
%The few events without detached
%tracks include those in which all of the $b$ and $c$ hadrons decay
%with a very short decay length and those which decay mostly to
%neutrals.  In this latter class of events there can be kinematic
%accidents in which the few charged tracks do point back to
%the primary vertex.  
Because the beam spot is very narrow, roughly 30 $\mu$m in diameter, 
these tracks will also have a large $r\varphi$ impact parameter
with respect to the beam line.  A track is said to be detached
if the impact parameter, divided by its error, is large; this definition
is used both for 2D and 3D impact parameters.


  While the reconstruction of the full vertex topology
of an event is a very powerful tool to reduce backgrounds, it is often
too inefficient or too slow to be useful.  In particular present
computing technologies are too slow to allow full exploitation
of the topology at trigger time.  However a powerful
trigger can be made by looking for the presence of a few
detached tracks.   All of CDF, D\O\ and BTeV
have design triggers which make some sort of detachment requirement,
with the sophistication of that requirement changing from one
experiment to the next.   BTeV exploits detachment at all trigger
levels, including level 1, while the other experiments introduce
detachment cuts only at higher levels.  
The reader is referred to chapters~3 to~5
for further details about the triggers of each experiment.

In addition to detachment, there are other properties which can be 
used to identify events which contain $b$ quarks.  For example,
selecting events with one or two 
leptons of moderate to high $p_T$ is an excellent way to select 
events containing $b\bar{b}$ pairs while rejecting background events.  
CDF and D\O\ have successfully used single lepton and di-lepton triggers,
without any detachment requirement, to select events for their Run~I $B$ 
physics program.  
All of the Run~II detectors plan some sort of lepton triggers,
including single high $p_T$ leptons, di-leptons and 
$\psi\to\mu^+\mu^-$ triggers.
The experiments envisage some, but not all, of these triggers also
to include detachment information. For example,
when evidence of detachment is present, one can lower $p_T$
thresholds and still have an acceptable background suppression.
But when detachment information is ignored, or unavailable,
the triggers require higher $p_T$ thresholds.
Those triggers 
which do not include detachment information will provide
a useful sample for calibrating the detachment based triggers.

One of the limitations of Run~I was that the experiments
could only trigger on $b$ events with leptons in the final state.
For Run~II and beyond, both CDF and BTeV have triggers which rely only on
detachment and which are capable of triggering all hadronic final states.
The development of triggers which rely only on detachment 
is  one of the major advances since Run~I.

  Because of their topological similarities to $b\bar{b}$ events,
some $c\bar{c}$ events will also pass
these triggers.  Charm events, however, have properties which
are intermediate to the $b$ events and the background events: 
their decay lengths are shorter,
their impact parameters smaller and their stable daughters have 
both a softer momentum spectrum and a softer $p_T$ spectrum.
Therefore the cuts which reduce the background
to an acceptable level are much less efficient for $c\bar{c}$ 
events than they are for $b\bar{b}$ events.  The CDF and
D\O\ experiments do not expect that significant $c\bar{c}$ samples will
pass their trigger and have not discussed a charm physics
program.  They can, of course, do some charm physics with the
charm which is produced via $B$ decay.  BTeV, on the other hand,
expects that a significant fraction of the events which pass their
trigger, will contain $c\bar{c}$ events and they plan a
charm physics program to exploit that data.

  In summary, the long lifetimes of the weakly decaying $b$
and $c$ hadrons are the magic bullet which allow the $b$ and $c$
physics to be extracted from the background.  At trigger time
minimal cuts will be made on detachment and the offline analyses
will make more complete use of the topological information.
Various lepton based triggers, some with detachment requirements
and some without, will form a second set of triggers.

\section{Sources of Backgrounds } 
\label{2:sec:backgrounds}

The most pernicious backgrounds are those which peak in the
signal region and which can fake signals.  One example of
this is a true $B^{0}\to K^+ \pi^-$ being misreconstructed as
a $B^{0}\to \pi^+ \pi^-$ decay; this results in a peak which is
almost at the correct mass, with almost the correct width.
This sort of problem is very mode specific and will be discussed,
as needed, in the working group chapters.

  A second class of backgrounds is combinatoric background
within true $b\bar{b}$ and $c\bar{c}$ events, events which have
the correct topological properties to pass the trigger.
Suppose that one is looking for the decay, 
$B^0\to D^- \pi^+$ followed by $D^-\to K^+\pi^-\pi^-$.
All $b\bar{b}$ and $c\bar{c}$ events which produce a reconstructed 
$D^-\to K^+\pi^-\pi^-$ candidate have potential to produce background.
If another track, perhaps from the main vertex, forms a good vertex 
with the $D^-$ candidate this will be considered, incorrectly, as
a $B$ candidate.  This sort of background will not peak near the 
$B$ mass but it will produce background entries throughout the 
$D^-\pi^+$ mass plot, thereby diluting the signal.  This sort
of background can be reduced by demanding that tracks which 
participate in a $B$ candidate be inconsistent with the primary vertex.
In addition, improved vertexing precision will reduce the number of
random $D^-\pi^+$ combinations which form a vertex with an
acceptable $\chi^2$.

   There are many other background sources
of secondary vertices and detached
tracks: strange particles, interactions
of particles with the detector material, misreconstructed tracks,
multiple interactions per beam crossing, and
mis-reconstructed vertices.  While none of these backgrounds
can create fake mass peaks at the $B$ mass, they can dilute signals 
and they can overwhelm a poorly designed trigger.

At first thought one might summarily dismiss the strange hadrons as a 
source of background.  After all, they typically have lifetimes 
100 to 1000 times
longer than those of the $b$ hadrons; so only a small fraction will
decay inside the beam pipe with a decay length typical
of that for $B$ decay.  However they are produced about
a few thousand times more frequently, a few per background
interaction.  Moreover the most probable decay time of
an exponential distribution is zero, so some of the strange
hadrons will have decay lengths of a few~mm.
There is a powerful countermeasure against most of the
strange particle
background: the trigger must ignore tracks with an
impact parameter which is too large.  
One might worry that the contradictory requirements of a 
large detachment but a small impact parameter
might leave no window to accept the physics.   The answer is
clear if one recalls the definition of detachment, an impact
parameter divided by its error:  make the error
small.  In practice the detectors have sufficiently good
resolution that this background is reduced to  acceptable level.

The strange hadrons have masses much less than the those of
the $b$ hadrons;  therefore, an isolated decay of a strange particle
is unlikely to be confused for a $b$ decay.  If however,
another track, or tracks, pass close enough to the strange 
particle decay vertex that the reconstruction code incorrectly 
assigns that track to that strange particle's decay vertex,
the combination can contribute combinatorial background beneath
a $B$ signal.  This sort of background can be reduced by 
building a vertex detector with sufficiently high precision.

Another source of background comes from the
interaction of the tracks with the detector and support materials.
Photons can pair convert and hadrons can undergo
inelastic collisions.  There may be several of these secondary interactions
for each primary $p\bar{p}$ interaction.  Again, this sort of background
can be suppressed, at the trigger level, by excluding
tracks with too much detachment.  And, in the offline analysis, one can
exclude vertices which occur in the detector
material.  Having excellent resolution on vertex position
is again the secret to background reduction.

Other sorts of interaction with the detector material includes
Rutherford scattering and the tails of multiple 
Coulomb scattering.  At the trigger level, the way to deal with these 
tracks is to make sure that the detachment cuts are large enough.
At the offline reconstruction level, these tracks can often be
rejected by cutting on the confidence level of the
track fit.

Mis-reconstructed tracks are tracks which have incorrect
hit assignments.  The most direct way to deal with this problem
is to ensure a sufficiently small occupancy in the detectors.
For example, the upper limit for the long 
dimension of the BTeV pixels is set by such a study --- if the pixels 
are too long then the two track separation degrades and errors
in pattern recognition result.  This, in turn, creates false detached
tracks.  In the offline analysis, one can also reject mis-reconstructed
tracks on the basis of a bad track fit confidence level.

Multiple interactions in one beam crossing are another source of
background.   Consider the case that 
two background interactions occur in the same beam 
crossing.  In this case there are two chances that a background
interaction might trigger the detector.  But there is the
additional complication that the trigger might fire
based on some information from one vertex and some information 
from the other vertex.  This last problem can be reduced by
doing 3D vertexing in the trigger.
%At their lowest trigger levels,
%%CDF and D\O\ trigger on detachment with respect to 
%the beam spot and so, at this trigger level, the two background 
%interactions are independent.
%On the other hand, the CDF Level~3 trigger
%and all levels of the BTeV trigger look at detachment with respect
%to the primary vertex.  Their response to an event containing
%two background interactions is more complex.  If the two
%primary vertices have sufficient multiplicity, the triggers will find
%both vertices and will consider each as a candidate for the primary
%vertex of a $b\bar{b}$ interaction; it will count the
%number of detached tracks in the neighborhood of each vertex.
%This works robustly as long as the two background interactions
%are well separated in $z$.   When the two 
%vertices are close together, the trigger is no longer able to recognize
%two separate vertices.  A common failure mode is to find one false
%vertex made up of a few tracks from each of the true background vertices.
%The remaining tracks from the two background interactions
%will falsely appear to be detached tracks associated with this false
%vertex.   These false detached tracks then trigger the event.
%Again the solution is to have excellent vertex resolution.

At the Tevatron the luminous region has a length of
$\sigma_z\approx30$~cm so it is reasonable to expect the triggers
to behave acceptably with a few background interactions per beam
crossing; most of the time the interactions will be well separated.
When testing trigger algorithms it is important to measure how
the trigger degrades with an increasing number of interactions per
crossing.  The trigger performance should degrade smoothly, without
sudden drops.  

The last background class is misreconstructed vertices, which includes
both errors made when all of the tracks are well measured but also
errors made when one of the tracks suffers from one of the diseases
mentioned above.   The solution is to ensure sufficient tracking 
precision that fake track rates are small and sufficient vertex
precision that the rates for accidental vertices are small.

The above discussion has presented a number of factors  which
bound the detachment required at the trigger level from below
and which bound impact parameter cuts from above.
Using detailed simulations of their detector response,
the experiments have shown that their proposed triggers will
reduce these backgrounds to an acceptable level and that their
detectors have enough rejection power to obtain an acceptable
signal-to-background ratio during offline analysis.  The common
thread running through the discussion is that improved
vertex resolution reduces every one of these backgrounds.

\section{Basics of $b$ Production Physics}
\label{2:ssec:xsec}

 At a $p\bar{p}$ collider, it is usually most convenient
to describe particle production
in terms of three variables,
$p_T$, $y$ and $\varphi$, where $p_T$ is the transverse
momentum of the particle with respect to the beam line,
$\varphi$ is the azimuth around the beam line and where
the rapidity, $y$, is a measure of the polar angle, $\theta$, relative
to the beam line,
\begin{equation}
y = \frac{1}{2}\ln\left( \frac{E+P_{\parallel}}{E-P_{\parallel}}
       \right).
\end{equation}
\index{rapidity ($y$)!definition of}
\index{$y$ (rapidity)!definition of}
For historical reasons people sometimes work in units of pseudo-rapidity,
$\eta$, instead of $y$,
\begin{equation}
\eta = -\ln\left( \tan\theta/2\right).
\end{equation}
\index{psuedo-rapidity ($\eta$)!definition of}
\index{$\eta$ (psuedo-rapidity)!definition of}
For massless particles $\eta=y$ and for
highly relativistic particles $\eta$ approaches $y$.
The utility of the variable $\eta$ can be seen
in Fig.~\ref{2:fig:vseta}, which shows the prediction of
the {\tt PYTHIA} Monte Carlo event generator for
the production of $b$ flavored hadrons as a function of $\eta$.
The production is approximately flat in the central $\eta$
region, falling off at large $|\eta|$,  a general
feature of particle production in hadronic collisions.
The figure also shows the regions of $\eta$ which are covered by 
the three detectors.

\begin{figure}
\centerline{\epsfig{file=ch2/eta.eps, height=3.6in}} 
\caption[Production cross-section vs $\eta$ for $B$ mesons (from {\tt PYTHIA})]
{The production cross-section of $B$ mesons
vs $\eta$.  The plot is from the {\tt PYTHIA}
event generator and does not contain detector effects. 
The horizontal lines show the regions of
$\eta$ which are covered by the three detectors.
CDF and D\O\ do not cover all regions of $\eta$ with
equally quality;  the barrel region, in which they make their
best measurements covers approximately $|\eta|<1.0$.
}
\label{2:fig:vseta}
\index{psuedo-rapidity ($\eta$)!coverage of detectors}
\index{$\eta$ (psuedo-rapidity)!coverage of detectors}
\end{figure}

During Run~I, both CDF and D\O\ studied the production of $b$ quarks 
in $p\bar{p}$ collisions at a center-of-mass energy of 1.8~TeV.
Both CDF and D\O\ have studied the central rapidity region $|\eta|<1$
and D\O\ has also studied the forward region, $2.4< y < 3.2$.
The data of D\O\ are shown in Fig.~\ref{2:fig:d0xsec}.
Both CDF and D\O\ find that the
$b\bar{b}$ production cross-section in the central region is
underestimated by the Mangano, Nason and Ridolfi (MNR) next-to-leading order
QCD calculation~\cite{MNR} by
a factor of more than two.
The D\O\ data in the higher $y^{\mu}$ region is $3.6\pm 0.8$ times 
higher than the QCD calculation.

\begin{figure}[thb]
%\centerline{\epsfig{file=ch2/D0forwardxsection.eps, width=\hsize}} 
\centerline{\epsfig{file=ch2/bmu_dxdy.eps, width=\hsize}} 
\caption[$\sigma$($\mu$ from B decay) vs $y_{\mu}$]
{The cross-section for muons from $b$-decay as a function of the rapidity the
of muon, $y^{\mu}$, measured by D\O\ .  The solid curve is the prediction of
the next-to-leading order QCD calculation for a $b$-quark mass of 4.75 GeV. The
dashed curves represent the estimated theoretical 1$\sigma$ error band.}
\label{2:fig:d0xsec}
\index{cross-sections!D\O\ measurement}
\end{figure}

  When predicting their sensitivities for physics at Run~II,
CDF and D\O\ normalize their predictions to the
cross-sections which they measured in Run~I.
Not only does BTeV not 
have previous data, there are no experimental data at all 
over much of the range of the BTeV acceptance, $1.9\le|\eta|\le4.5$.
Instead BTeV uses the following procedure.  When integrated
over $\eta$ and $p_T$, the QCD predictions shown in 
Fig.~\ref{2:fig:d0xsec} predict a total $b\bar{b}$ production
cross-section of 50~$\mu$b.  Since all of the experimental data
is more than a factor of two above the theoretical calculations,
BTeV estimates the total cross-section
to be 100$\mu$b.  BTeV then uses the predictions of {\tt PYTHIA} to describe
how the cross-section is distributed over $p_T,\eta,\varphi$.
Within regions of phase space covered by CDF and D\O\, {\tt PYTHIA} has
done a good job of describing the most important experimental correlations.
 
  Other properties of $b\bar{b}$ production are illustrated
in Fig.~\ref{2:fig:bgvseta} through~\ref{2:fig:vsphi}.
Fig.~\ref{2:fig:bgvseta} shows, for $B$ mesons, the prediction
of the {\tt PYTHIA} event generator for the cross-section as
a function of $\beta\gamma$ vs $\eta$.  The figure shows that the
bulk of the cross-section is concentrated in the central region and
that forward going $B$ mesons have a much higher momentum than
do $B$ mesons produced in the central region.  This implies that
in the forward region a greater fraction of the cross-section
has long decay lengths, while in the central region there are
more events to start with.  The implications of this
tradeoff will be discussed further in Section~\ref{2:sec:centfor}.
%It turns out that once
%reasonable detachment cuts are applied, all three detectors
%CDF, D\O\ and BTeV, intercept about the same amount of $B$ meson
%cross-section.

\begin{figure}
\centerline{\epsfig{file=ch2/bg_vs_eta.eps, height=3.6in}} 
\caption[$\sigma_{b\overline{b}}$ as a function of $\beta\gamma$ vs $\eta$]
{The production cross-section for  $B$ mesons as a function
of $\beta\gamma$ and $\eta$ plane.  The plot is from the {\tt PYTHIA}
event generator and does not contain detector effects. }
\label{2:fig:bgvseta}
\index{psuedo-rapidity ($\eta$)!$\beta$ vs $\eta$}
\index{$\eta$ (psuedo-rapidity)!$\beta$ vs $\eta$}
\end{figure}

   Fig.~\ref{2:fig:thetavstheta} illustrates another of the
the properties of $b\bar{b}$ production, that $b$~hadron
and the $\bar{b}$~hadron have an RMS separation of about one unit
of $\eta$.  The figure was  made using generator level tracks from the
{\tt PYTHIA} event generator and shows
the cross-section as a function of the polar angle of one 
$B$ vs the polar angle of the other $B$.  
In a two-arm forward detector, such as BTeV,
if one $B$ is produced in a particular arm, then the other $B$
is highly likely to be produced in the same arm.
This is important for measurements which make use 
of opposite side tagging (see Section~\ref{2:sec:exp_common_tagging}).
The choice of axes for this figure, $\theta$ rather than $\eta$,
exaggerates this effect.  Some other consequences of this distribution
are discussed in Section~\ref{2:sec:centfor}.


\begin{figure}
\centerline{\epsfig{file=ch2/bbar.eps, height=3.2in}}
\vspace*{4pt}
\caption[Production angle correlations for $B\overline{B}$ pairs]
{The production angle (in degrees) for the hadron
containing a $b$ quark plotted versus the production angle for a hadron
containing $\bar{b}$ quark.  The plot is from the {\tt PYTHIA}
event generator and does not contain detector effects. 
One must be careful interpreting this plot since the natural
axes are $\eta$, not $\theta$.
}
\label{2:fig:thetavstheta}
\end{figure}

  Fig.~\ref{2:fig:vsphi} shows the azimuthal correlation
between a $b$ and its $\bar{b}$ partner.  The data are for D\O\
events in which two muons are reconstructed, both consistent
with coming from the decay of a $b$ hadron.  
%                   
\begin{figure}
\centerline{\epsfig{file=ch2/cdf_dphi.eps, height=3.2in}} 
\caption[Azimuthal angle correlation between $B\overline{B}$ pairs]
{The differential $\delta\varphi$ cross-sections for
$p^{\mu}_T> 9 $ GeV/c, $|\eta^{\mu}|<$0.6, E$^{\bar{b}}_T>$10 GeV,
$\big|\eta^{\bar{b}}\big|<1.5$ compared with theoretical predictions. The
data points have a common systematic uncertainty of $\pm$9.5\%. The uncertainty
in the theory curve arises from the error on the muonic branching fraction
and the uncertainty in the fragmentation model.
}
\label{2:fig:vsphi}
\end{figure}
%
The horizontal axis is $\delta\varphi$,
the difference in azimuth between the two muons.  Since the selection
criteria imply that the two $B$'s have a significant momentum,
the muons tend to follow the $B$ direction.  Therefore $\delta\varphi$
is a measure of the difference in azimuth between the two $b$ hadrons
in the event.  The band shows the prediction of MNR\cite{MNR}.
The $b$ hadrons are preferentially produced back to back in azimuth
and the gross shape is reproduced well by the model. 
It has already been noted that
the MNR prediction underestimates the cross-section.

  While the production of $b\bar{b}$ pairs is well described
by perturbative QCD and knowledge of the structure functions 
of the proton, the hadronization, or fragmentation, of these
quarks into the final state hadrons is described by models.
These models are usually realized as computer codes for event
generators, the most commonly used being {\tt PYTHIA}\cite{pythia},
{\tt ISAJET}\cite{isajet} and {\tt HERWIG}\cite{herwig}.  
One of the properties which must be input to the event generators is
the fraction of time that the $b$ quark fragments into each of the
allowed hadrons, $B^{-}$, $\bar{B}^0$, $\bar{B}_s$, $B_c^-$ 
or one of the $b$ baryons.  A recent measurement from
CDF\cite{cdfprodfrac} gives,
$f_u:f_d:f_s:f_{\rm baryon}$ 
$= 0.375\pm0.023:0.375\pm0.023:0.160\pm0.044:0.090\pm0.029$, 
with the assumption that $f_u=f_d$.  If they release this
assumption they obtain, $f_d/f_u=0.84\pm0.16$.
It is generally presumed that, except for threshold effects,
the fragmentation process is independent of the production 
process and is roughly independent of energy.  For comparison
the same production fractions measured at LEP and SLD 
are\cite{lepprodfrac},
$f_u:f_d:f_s:f_{\rm baryon}$ 
$=0.401\pm0.010:0.401\pm0.010:0.100\pm0.012: 0.099\pm0.017$.
In both of these measurements, the production of $B_c$
mesons is too small to be significant.
In the standard event generators
the choice of hadron species for the $b$ quark
is independent of the choice of hadron species for the $\bar{b}$
quark.  This cannot be exactly true since there is presumably some
production via the $\Upsilon(4S)$ resonance, which decays only
to $B^0\overline{B}^0$ or $B^+B^-$. Moreover the 
$B^0\overline{B}^0$ production from the $\Upsilon(4S)$ is coherent.
While it is likely that these effects do occur, they can be safely
ignored for purposes of this workshop.  If there is enough resonant
production to affect the physics results, the amount of such
production can be easily measured with the Run~II data.

The major points of this section are summarized in
Table~\ref{2:tab:bbbarprop}.

\begin{table}
\begin{enumerate}
\hrule\vspace*{2pt} \hrule
\item The mechanisms which produce heavy flavors produce
      $b\bar{b}$ pairs but not single $b$ or $\bar{b}$ quarks.
      Similarly for charm production.
\item The cross-section for $b\bar{b}$ production, integrated over
      all $\eta$ and $p_T$,  is about 100~$\mu$b
      and that for $c\bar{c}$ production is about 1~mb.
\item The $b\bar{b}$ cross-section is approximately flat in $\eta$ over the
      central region, and falls off at large $|\eta|$.
      See figure~\ref{2:fig:vseta}.
\item $b$ hadrons produced in the forward region have a higher momentum
      than those produced in the central region.
      See figure~\ref{2:fig:bgvseta}.
\item The pair of $b$ hadrons from one $b\bar{b}$ pair are approximately
      approximately back to back in $\varphi$ and have an RMS separation
      in $\eta$ of about one unit of $\eta$.
\item The production ratio of $B_u:B_d:B_s:$baryons is,
      $f_u:f_d:f_s:f_{\rm baryon}                           
       = 0.375\pm0.023:0.375\pm0.023:0.160\pm0.044:0.090\pm0.029$
      \cite{cdfprodfrac}.
\vspace*{4pt}
\hrule\vspace*{2pt} \hrule
\end{enumerate}
\caption{Summary of the important properties of $b\bar{b}$ production. }
\label{2:tab:bbbarprop}
\end{table}

\section{Production Rates and Interactions Per Crossing}
\label{2:sec:lumi}
\index{interactions!mean rates of}

The design value for the
peak instantaneous luminosity during Run~II
is $2\times 10^{32}\;{\rm cm}^{-2}\;{\rm s}^{-1}$.
This specifies the luminosity at the start of a fill, when
the beam intensities are greatest.  As a fill progresses
the instantaneous luminosity will drop.  Also there will
be shutdowns, both planned and unplanned, throughout the running
period.
The rule of thumb for converting the peak
instantaneous luminosity to the yearly integrated luminosity
is to assume that a year contains $10^7$ seconds of running
at the peak instantaneous luminosity.   This is about one third 
of the actual number of seconds in a year, which 
accounts both for the drop in luminosity as a fill
progresses and for a normal amount of down-time.
Therefore a peak instantaneous luminosity of 
$2\times 10^{32}\;{\rm cm}^{-2}\;{\rm s}^{-1}$ 
corresponds to 2~fb$^{-1}$/year.

   Given $\sigma_{b\bar{b}}=100\; \mu$b from Table~\ref{2:tab:xsecs},
the above luminosities imply $b\bar{b}$ yield of
20,000/s or $2\times 10^{11}$/year, about 3 to 4 orders of magnitude
larger than the projected yields at the $e^+e^-$ $B$ factories.
%The $c\bar{c}$ yield is expected
%to be about 10 times higher than this,
%200,000/s or $2\times 10^{12}$/year.

   Given $\sigma_{tot}=75$~mb, from Table~\ref{2:tab:xsecs},
a luminosity of $2\times 10^{32}\;{\rm cm}^{-2}\;{\rm s}^{-1}$
implies a total interaction rate of
$1.5\times 10^7$/s.  During the first few years of Run~II the
bunch structure of the Tevatron will be 396~ns between
bunch crossings.   At the design luminosity this would
correspond to about 6 interactions per crossing but it is 
not expected that the design luminosity will be achieved 
this early in the run.  After the first few years of Run~II the
bunch structure of the Tevatron will be changed to have
132~ns between bunch crossings.  The purpose of this change
is to allow an increase in luminosity while reducing the
number interactions per beam crossing.  At 132~ns between
bunches, the design luminosity corresponds to about 2 interactions per 
bunch crossing.

   The above discussion, along with the corresponding numbers
for $c\bar{c}$ production, is summarized in Table~\ref{2:tab:rates}.

The presence of multiple background interactions has 
many consequences for the design of the detector.  It was
already mentioned that the trigger must be robust against
multiple background interactions in one beam crossing.
The presence of multiple interactions must also be considered
when designing the granularity of detectors to ensure that
the occupancy is acceptably low.  



\begin{table}
\begin{center}
\begin{tabular}{lccc} \hline
Rate        & $b\bar{b}$ & $c\bar{c}$ & Total \\ \hline \hline
Interactions/s    & $2\times 10^4$ & $2\times 10^5$       &$1.5\times 10^7$ \\
Interactions/year & $2\times 10^{11}$ & $2\times 10^{12}$ &
                                                         $1.5\times 10^{14}$ \\
Interactions/crossing @ 396~ns & $0.008$ & $0.08$  & 6 \\
Interactions/crossing @ 132~ns & $0.003$ & $0.03$  & 2 \\ \hline
\end{tabular}
\medskip
\caption[Interactions per beam crossing]
{Summary of production rates for $b\bar{b}$ pairs,
$c\bar{c}$ pairs and total interactions for the design
peak luminosity of the Tevatron during Run~II,
$2\times 10^{32}\;{\rm cm}^{-2}\;{\rm s}^{-1}$
(2~fb${-1}$/year). The interactions
per bunch crossing are given twice, once for the bunch structure
planned for early in Run~II, 396~ns between bunch crossings, and once
for the bunch structure planned for later in Run~II, 132~ns
between bunch crossings.
}
\label{2:tab:rates}
\end{center}
\end{table}

\subsection{The Distribution of Interactions Per Crossing}
\label{2:sec:multi}
\index{interactions!Poisson distribution of}

To a good approximation, if there are multiple interactions
in one beam crossing, they are statistically
independent of each other. This is not 
strictly true because, once the first interaction takes place, 
there are fewer beam particles left to participate in future
interactions.  However, in the limit that the number of
particles per bunch is much larger than the number of
interactions per crossing, each interaction can be treated
as independent of all others.

Each of these independent interactions has some probability
to produce a signal interaction and some probability to produce
a background interaction.  There are two, equivalent ways of
looking at the distribution of signal and background interactions
among the multiple interactions in one event.  These equivalent
ways are related to each other by the following identity.
Given two independent
Poisson processes, signal and background for example,
the probability to observe $n_1$
interactions from the first process and $n_2$ interactions from the second
process is,
\begin{eqnarray}
P(n_1,n_2) &=& \frac{(\mu_1)^{n_1}}{n_1!}\; e^{-\mu_1}\;   
                \frac{(\mu_2)^{n_2}}{n_2!}\; e^{-\mu_2} \nonumber \\
           &=& \left[ \frac{\mu^n}{n!}
                \; e^{-\mu} \right] \;
               \left[f^{n_1} (1-f)^{n-n_1}
               \frac{n!}{n_1!(n-n_1)!} \right],
\end{eqnarray}
where $f=\mu_1/\mu$ and $\mu=\mu_1+\mu_2$.
The first factor in $[]$ is the Poisson probability to observe
\hbox{$n=n_1+n_2$} interactions in total, while the second factor in $[]$ is
the binomial probability that the $n$ interactions are split into $n_1$
from the first process and $n-n_1$ from the second process.

One can also show the general case,
that the sum of $M$ independent Poisson processes is itself a Poisson
process with a mean $\mu=\mu_1+\mu_2+\dots+\mu_M$.
In the general case, the factor multiplying the overall Poisson 
distribution will be a multinomial distribution, 
with the $M-1$ independent parameters, $\mu_1/\mu$, $\mu_2/\mu$,
\dots $\mu_{M-1}/\mu$.

The two equivalent descriptions are:
first, one can say that 
the total number of interactions within a beam crossing is
Poisson distributed with a mean of $\mu$
and that within each beam crossing the interactions are distributed
among the possible types according to a multinomial distribution.
Second, one can say that there
are $M$ independent pieces to the cross-section and that 
each piece contributes to each beam crossing
a Poisson distributed number of interactions with mean 
$\mu_M$.

This second description is less well known but it allows one
to more easily answer the following question: describe a
typical beam crossing which produces a $b\bar{b}$ pair.
For definiteness, consider the numbers summarized in 
Table~\ref{2:tab:rates} for the case of 132~ns bunch spacing;
$\mu_{b\bar{b}}=0.003$, $\mu_{c\bar{c}}=0.03$, and $\mu_{BG}=2.0$\ .
Clearly most beam crossings will contain no $b\bar{b}$ pairs.
An event which contains a typical $b\bar{b}$ pair will contain
exactly one such pair and it will be accompanied by
a Poisson distributed number of $c\bar{c}$ interactions
with a mean of 0.03 interactions per crossing and by a
Poisson distributed number of background interactions
with a mean of 2.0 interactions per crossing.

\section{Flavor Tagging}
\label{2:sec:exp_common_tagging}
\index{flavor tagging!motivation for}


One of the main $B$ physics goals of all three experimental
programs is to make precision measurements of mixing mediated CP 
violating effects, some of which are 
discussed in chapter~6 of this report.  Also, $x_s$
has yet to be measured and that
is interesting to measure in its own right.   In order to
perform any mixing related study it is
necessary to know whether a particular meson was produced as 
a $B^0(B_s)$ or as a $\bar{B}^0(\bar{B}_s)$.  Making
such a determination is called flavor tagging the $B$ 
meson.\footnote{There is another, and very different concept
called $b$ tagging.
If a lepton is part of a jet, and if the $p_T$ of the lepton with 
respect to the jet axis is sufficiently large, 
then that lepton is most probably from the 
decay of a $b$ quark within the jet.  A sample of jets containing 
such a lepton will be heavily enriched in $b$ jets.
This technique was used extensively in Run~I
to tag samples of $b$ jets which were used in the $W$ boson
and top quark physics programs.  This technique is mostly
of interest for top physics, not for $B$ physics itself.
}
%Unlike the $e^+e^-$ B-factories, which produce coherent
%$B^0\bar{B}^0$ pairs, the production of $B^0$ and
%$\bar{B}^0$ mesons via $p\bar{p}$ interactions is dominantly
%incoherent.

Every tagging method sometimes produces the wrong answer
and the effectiveness of flavor tagging is characterized by an
effective tagging efficiency $\epsilon\,D^2$,
where
$\epsilon=(N_R+N_W)/N$, 
$D=(N_R-N_W)/(N_R+N_W)$, $N$ is the number of reconstructed 
signal events before tagging, $N_R$ the number of right flavor tags,
and where $N_W$ is the number of wrong flavor tags.  
Another useful expression is $D=(1-2w)$ where $w=N_W/(N_R+N_W)$ is the
fraction of wrong sign tags; from this expression it is clear that 
the tagging power goes to
zero when the wrong sign tag fraction reaches 50\%.
Maximizing
$\epsilon D^2$ is critical to the design of every experiment.

  The quantity $D$ is known as the dilution.  This choice
of nomenclature has the anti-intuitive result
that a large dilution is good while a small dilution is bad.
Never-the-less it is the standard nomenclature.
\index{$D$(tagging dilution)}
\index{$\epsilon D^2$ (tagging power)}
\index{flavor tagging!dilution ($D$)}
\index{flavor tagging!power ($\epsilon D^2$)}

Tagging algorithms can be broken down into two classes, away
side tagging and same side tagging.  In away side tagging,
or opposite side tagging,
one looks at some property of the other $b$ hadron in the event
to determine its $b$ quantum number.   Since $b$ quarks are 
produced as $b\bar{b}$ pairs, one can infer the flavor of
the signal $B$ meson.   In same side tagging one uses
the correlations which exist between the signal $B$ meson
and the charge of nearby tracks produced either in the fragmentation
chain or in the decay of $B^{**}$ resonances.  For tagging $B^0$
mesons the correlation is with a charged pion, while for $B_s$
mesons the correlation is with a charged kaon.

\subsection{Away Side Tagging}
\index{flavor tagging!description of!away side}

The perfect away side tag would be to fully reconstruct the
other $b$ hadron in the event and to discover that it is a
$B^-$ or a $\Lambda_b$, neither of which undergoes flavor mixing.
In this case one knows that the other $b$ hadron contains
a $b$ quark and that the signal $B$ meson must have been born
with a $\bar{b}$ quark.  So the signal $B$ is tagged as being born
as a $B^0$ or as a $B_s$.  
In practice the efficiency for
reconstructing a complete $b$ hadron on the away side is much
too small to be useful.  Instead one looks for inclusive properties
of $b$ hadrons which are different from those of $\bar{b}$ hadrons.
Four such properties have been explored: lepton tagging,
kaon tagging, jet charge tagging and vertex charge tagging.


Lepton tagging exploits the sign of the lepton in the
decays $b\to X \ell^-$ compared to
$\bar{b}\to X \ell^+$, where $\ell$ is either an electron
or a muon.  The branching fractions for these decays is roughly 10\%
into each of the $e$ and $\mu$ channels.  There is some dilution
in this tag from the decay chain $b\to c \to X \ell^+ $
compared to
$\bar{b}\to \bar{c} \to X \ell^- $.   However the two different
sources of leptons have different kinematic properties and
different vertex topology properties.  So good separation between
these two sources of leptons can be achieved.  Another factor
causes further dilution.  In an ensemble of tags, the away 
side $b$ hadron will be some mixture of $B^+$, $B^0$, $B_s$ and 
several $b$ baryons.  The $B^+$ and the $b$ baryons do not mix and so
the observation of the sign of the lepton is a clear tag.  However,
17.4\% of the $B^0$ mesons will oscillate to $\bar{B}^0$ mesons
before decaying\cite{PDG} and will, therefore, give an incorrect tag.
The $B_s$ system, which is fully mixed, 
provides no tagging power at all.
%In its simplest form lepton tagging requires
%that the leptons be detached from the primary vertex.  However
%one could imagine making more sophisticated algorithms which
%ask for evidence of a secondary or tertiary vertex before
%including the lepton.
%If one were to fully reconstruct the semileptonic decay
%of a $B^0$ meson, then one could use the knowledge of the 
%proper decay time of the $B^0$ to reduce the dilution caused 
%by mixing.  However, in order to increase efficiency, 
% most of the time only a
%fragment of the $B^0$, including the lepton, will be reconstructed.  
%So the momentum is unknown and it is not possible to compute the proper 
%decay time.

  Kaon tagging exploits the charge of the kaon in the away side
decay chain,
$b\to c \to X K^-$ compared with $\bar{b}\to \bar{c} \to X K^+$.
Because of the large product branching fraction this tag has a much
higher efficiency than lepton tags but historically has had 
worse dilution.  With the improved vertexing power and
particle identification capabilities of the of the
Run~II detectors one expects significantly improved dilutions.
As with lepton tagging, there is tagging dilution from the mixing
of the away side $B^0$ and $B_s$.   It is often noted that a typical 
$B_s$ decay contains two kaons of opposite strangeness and so 
contributes no power to kaon tagging.  While this is true, one
must remember that the $B_s$ system is fully mixed and had no
tagging power to start with.  


  A method called ``jet charge tagging''
exploits the fact that the sign of the momentum weighted sum of the
particle charges of the opposite side $b$~jet is the same as the
sign of the charge of the $b$~quark producing this jet.
In a simple version, the jet charge $Q_{\rm jet}$ can be calculated as
%
\begin{equation}
  Q_{\rm jet} = \frac{ \sum_i q_i\; (\vec{p}_i \cdot\hat{a}) }
                     { \sum_i \vec{p}_i \cdot\hat{a}  },
\end{equation}
%
where $q_i$ and $\vec{p}_i$ are the charge and momentum
of track $i$ in the jet and $\hat{a}$ is
a unit~vector along the jet axis.
On average, the sign of the jet charge is the same as the sign
of the $b$~quark that produced the jet.

  Vertex charge tagging involves reconstructing the full 
vertex topology of the away side.  This does not necessarily
constitute full reconstruction of the away side since the
away side decay will usually contain $\pi^0$'s, photons,
$K^0_S$ and $K^0_L$.  However these missing particles do
not modify the charges of the remnant vertices.
If the vertices have been correctly reconstructed, and if
the away side secondary vertex has a charge of $\pm 1$, then the 
flavor of the away side $b$ is known.  If the charge of the
away side secondary vertex is zero, then there is no tagging
power.  Also, if the away side tertiary vertex has charge
$\pm 1$, one can infer the flavor of the away side $b$.

\subsection{Same Side Tagging}
\index{flavor tagging!description of!same side}

Same side tagging exploits charge correlations between the
a $B^0$, or $\bar{B}^0$, and the nearest pion in the fragmentation
chain.  Fig.~\ref{2:fig:frag} illustrates the idea behind 
the method.
 
\begin{figure}[thb]
\centerline{\epsfig{file=ch2/frag.eps, height=10.5cm}}
\medskip
\caption[Quark diagrams for same side tagging]
{Four quark diagrams for the  fragmentation of $b$ and $\bar{b}$
quarks to $B^0$ and $\bar{B}^0$ mesons.  The charged pion which
is nearest in the fragmentation chain to the $B$ meson tags the
birth flavor of the $B$ meson.  
The notation \dots indicates that the fragmentation
chain continues out of the picture.}
\label{2:fig:frag}
\end{figure}

One can think of the hadronization, or fragmentation,
processes as pulling light quark pairs from the vacuum and forming hadrons
from nearby quarks.  
In order to form a $B^0$ or a $\bar{B}^0$ meson 
the light quark pair which is nearest in the fragmentation
chain to the initial heavy quark must have been a $d\bar{d}$
pair.  This leaves a $d$ or $\bar{d}$ quark at the dangling
end of the fragmentation chain.
If the second nearest light quark pair is $u\bar{u}$ pair
then the nearest meson in the fragmentation chain
will be a $\pi^-$ or $\pi^+$, which can
be used to tag the flavor of the initial $b$ or $\bar{b}$.  
If the second nearest light quark pair is a $d\bar{d}$ pair then
the nearest meson is a $\pi^0$, which itself has no tagging power.
However the dangling end of the fragmentation chain remains 
a $d$ or $\bar{d}$ and, if the third nearest light quark pair is a $u\bar{u}$
pair, then the second nearest meson will be a $\pi^-$ or $\pi^+$
which can be used as a flavor tag.  The bottom line is that the
charge of the nearest charged pion tags the birth flavor of the
$B^0$ or $\bar{B}^0$ meson.

The question now is to discover an algorithm
that will identify the charged pion that is the
nearest charged pion in the fragmentation chain.  
CDF successfully developed such an algorithm in Run~I.
To select the same side tag pion,
all tracks within a cone 
of radius $0.7$ in $\eta\,\varphi$~space, centered
around the direction of the $B$ meson, were
considered.
Same side tag candidate tracks were required to originate from
the $B$ production point (the primary event vertex),
and were therefore required 
to satisfy $d_0/\sigma_{d_0}<3$,
where $\sigma_{d_0}$ is the uncertainty on the track
$r\,\varphi$~impact parameter $d_0$. 
This selection produced, on average, 2.2 same side tag candidate
tracks per $B$~candidate. 
String fragmentation models indicate 
that particles produced in the $b$~quark hadronization chain 
have a small momenta transverse to the direction of the $b$~quark momentum.
CDF thus selected as the tag the track that had the minimum
component of momentum, $p_T^{\rm rel}$, orthogonal to the momentum sum of
the track and the $B$~meson.

  The same fragmentation chain argument can be used to show
that the nearest charged kaon in the fragmentation chain can be used to 
tag the flavor of $B_s$ and $\bar{B}_s$ mesons. 
If the nearest kaon is a $K^+$, then the meson is a $B_s$ but if
the nearest kaon is a $K^-$, then the meson is a $\bar{B}_s$.
If, however, the nearest kaon is neutral, then there 
is no kaon
tagging power because the nearest charged neighbor will be a pion.
While there does remain a charge correlation with the nearest
charged pion, the author is not aware of any work done to exploit this.

  Compared with away side tagging methods, the same side
tagging methods have a higher efficiency but a worse dilution.
That is, they almost always find a candidate charged track but it 
is not always the correct one.  Because of its high
efficiency, same side tagging makes an important contribution to the total
tagging power of an experiment.

\subsection{Overall Tagging Strategy}
\index{flavor tagging!description of!overall strategy}

  The various methods described above have quite different
properties.  For example lepton tagging has a relatively
low efficiency but a very good dilution.  Same
side tagging and jet charge tagging, on the other hand, 
are more efficient but 
have  poorer dilutions.  At CDF in Run\,I Kaon tagging
was intermediate in both efficiency and dilution; better particle
identification capability in the CDF Run~II detector and in BTeV will 
significantly improve the dilution for this tag.  The optimal 
tagging strategy is some method which involves all of the the
tagging techniques.
Any such strategy must account for the correlations among the away
side tagging methods; same side tagging is statistically
independent of all away side methods.  
One very simple strategy is to poll each method in order 
of decreasing dilution and to accept the first method that gives
an answer.  A more powerful idea is to combine all of the 
methods into an overall likelihood ratio, a linear discriminant or 
a neural net.  The strategy employed by CDF in their Run\,I 
analysis of $\sin2\beta$ is described in reference~\cite{cdftagging}.

  For further details one should consult the chapters for 
the specific experiments and the references therein.

\section{The Measurement Error on Proper Decay Times}
\label{2:sec:properdecaytime}
\index{proper decay time}

 It is instructive to describe how one measures the proper decay time of
a $b$ hadron and to see that, for the decays of interest, the error
on the proper decay time is, to a good approximation, 
independent of momentum.

To measure the proper decay time one reconstructs the primary 
interaction vertex at which the $b$ hadron
was produced and the secondary vertex at which the $b$ hadron decayed.
The proper decay time, $t$,  is then given by $t=Lm/pc=L/\beta\gamma c$,
where $L$ is the decay length measured by the separation of the vertices,
$p$ is the measured momentum of the $b$ hadron, $m$ is the mass of
the $b$ hadron, $c$ is the speed of light, and where
$\beta$ and $\gamma$ are the usual Lorentz parameters for the $b$ hadron.
The uncertainty
on the decay length contains contributions from the error on the
primary vertex position, the error on the secondary vertex position
and the error on the momentum of the $b$.  In all three  experiments the
contributions from the position errors are much larger than
those from the error on the momentum.  And the multiplicity of 
the primary vertex is usually much higher than that of the 
secondary vertex, making the error on the primary vertex position
smaller than that of the secondary vertex;
therefore the error on the proper decay time
is dominated by the error on the position of the secondary vertex.
The relevant part of the error on the secondary vertex is the 
projection of its error ellipse onto the flight direction of
the $b$ hadron.  That is the dominant contribution to the
error is just,
\begin{equation}
\sigma_t({\rm dominant\ contribution}) = \sigma^{(2ndry)}_{L}/(\beta\gamma c),
\label{2:eq:sigt}
\end{equation}
where $\sigma^{(2ndry)}_L$ is the contribution of the secondary vertex
to the decay length.

There is one familiar exception to this rule.  When using semi-leptonic 
decays to reconstruct the $b$, the momentum carried 
by the missing neutrals is poorly known 
and the error in the proper
decay time has important contributions from the error in the momentum.

To understand how $\sigma_t$ depends on momentum, consider
two different instances of the decay $B^0\to\pi^+\pi^-$.  
In the the first case the $B^0$ has some definite momentum and
the decay takes place at a particular point in some detector.
Suppose that the momentum is large enough that both pions are boosted 
forward along the $B$ flight direction in the lab.
In the second case,  the $B^0$ decays at exactly the same space point
and with the same center-of-mass decay angles as in case 1, but
it has a larger momentum.
The decay products of this decay are then measured in the same detector.
Further suppose that the detector is sufficiently uniform that each
track from these two decays is equally well measured.
The main difference between these two cases is that
the lab frame opening angle between the two pions will be smaller in 
case 2 than in case 1.  Because the opening angle is smaller, the point
at which the tracks intersect is more poorly known.  In particular
the component of the vertex position along the $b$ flight
direction in the lab is more poorly known.  This is purely a geometric
effect.  One result of this geometric effect
is that the error on the secondary vertex position grows
like $\gamma$; that is
$\sigma^{(2ndry)}_L \propto \gamma $.  Plugging
this into Eq.~\ref{2:eq:sigt} gives $\sigma_t \propto  1/(\beta c)$.
Therefore, for $\beta\simeq1.0$, 
the error on the proper decay time is independent of momentum.
This property has been exploited by experiments such as E687, E791,
SELEX and FOCUS to make precision measurements of the charmed hadron
lifetimes.

The above analysis holds approximately for multi-body decays of $b$ 
hadrons.  It will fail for very small boosts, in which case 
the decay products travel both forwards and backwards along 
the $b$ flight direction.  It will also fail if the decay products
of the $b$ hadron are slow enough that their errors are dominated
by multiple scattering and not by the measurement errors in the
apparatus.

The above analysis is only valid if the two decays are measured
by the same detector.  It is not useful for
comparing two very different detectors; in that case there are
no short cuts and one must compute the resolution of each detector.

\section{Properties of a Good $B$ Physics Detector}
\index{detector! desired properties}

A detector for doing $b$ and $c$ physics at a hadron  
collider must have the following components, 
a high precision vertex detector,
a tracking system giving excellent momentum resolution,
excellent particle identification (ID) capability,
and a  robust trigger integrated into a high bandwidth DAQ.
And it is very desirable to have electromagnetic calorimetry so
that modes containing photons and $\pi^0$'s can be measured.

It would also be useful to have 
hadronic calorimetry which is precise enough to reconstruct a
$K^0_L$.  Because the $K^0_L$ has opposite $CP$ quantum numbers to
the $K^0_S$, many tests of the weak interactions can be made by
comparing exclusive final states which differ by substituting a $K^0_L$
for a $K^0_S$.  None of the three experiments, however, anticipate
a significant ability to reconstruct $K^0_L$ mesons.

There are many other constraints on the detector design.
For example, all of the detector elements must have sufficiently fine 
granularity to deal with the high multiplicities which occur 
in $p\bar{p}$ collisions.  
The detector must be able to deal with several such interactions 
in one beam crossing.  And it is important to design the
detector with as little mass as possible in the fiducial
volume.  

  The three experiments have approached these challenges
from different directions and with different constraints.

\subsection{Forward vs Central}
\label{2:sec:centfor}
\index{detector!central vs forward}

 At first glance the most striking difference among the three 
detectors is that CDF and D\O\ are central detectors while BTeV is a
forward detector.
But there is a much more important distinction --- BTeV
is a dedicated $B$ physics detector while CDF and D\O\ are multipurpose
detectors whose primary mission is high $p_T$ physics, including
precision top quark physics, the search for the Higgs boson and
the search for supersymmetric particles.  Some of
the constraints imposed on CDF and D\O\ are not
intrinsic limitations of the central geometry; rather they are 
consequences of their optimization for a different spectrum of physics.

But there do remain some issues for which either the forward
or central geometry has an advantage.
First, BTeV has a harder particle ID job than either CDF or
D\O\ because BTeV  must identify tracks over a much wider range of
momentum.  However the forward geometry allows for a RICH
detector which gives BTeV better overall hadronic particle ID.
Second, BTeV has a somewhat higher efficiency for reconstructing 
the decay products of the second $B$ in the event, given that the 
first $B$ has already been reconstructed.  The reasons behind
this involve the interplay of production dynamics with the
myriad constraints of detector design.  Third, the forward
geometry is more open than the central geometry, thereby simplifying
the mechanical design and maintenance.
In a central geometry, on the other hand, one unit
of $\eta$ is much more opened up in space
than in the forward region.  Since multiplicities are approximately
uniform in $\eta$, this allows a device with coarse granularity 
to have the same multi-track separation power as does a fine granularity
device in the forward region; this has advantages in channel count.
Many of the advantages discussed in this paragraph are tied
to available technologies and the situation might well change
with new developments in detector technologies.

The $B$ mesons produced in the forward region have higher momenta,
and consequently longer decay lengths, than do those produced in 
the central region.  Before drawing any conclusions about the merits of
forward produced $B$'s, one must take many other things into
account.  Not all $B$'s produced in the central region have low momentum
and
the ones which pass all analysis cuts have much higher momentum than the
average $B$ meson.  There are more $B$ mesons produced in the 
central region than in the forward region so
central detectors can tolerate a smaller efficiency for their 
topological cuts and still have a comparable event yield.
Higher momentum $B$ mesons have poorer resolution 
on their decay vertex positions (see Section~\ref{2:sec:properdecaytime});
this cuts the advantage of the highest momentum $B$ mesons in forward
detectors.
The decay products of higher momentum $B$ mesons undergo less multiple
scattering than do those of lower momentum $B$ mesons; this
helps to improve resolutions.
And the details of the detector design turn out to be the
critical.  The net result is that, after all analysis cuts, 
the early designs for the Run~II CDF detector had a significantly 
poorer resolution on proper decay time than does BTeV.  But with the 
addition of Layer-00, CDF now has a resolution on proper decay time,
after all analysis cuts, which is comparable to that of BTeV.

At trigger time a different set of priorities is present.
For example, in the lowest level of the BTeV trigger, the track 
fitting algorithms are crude and it is important that
most $B$ meson daughters are of high enough momentum that multiple
scattering is a small enough effect to treat in a crude fashion.

The most important difference which arises from of BTeV being a dedicated 
experiment, while CDF and D0 are not, is in the trigger 
and DAQ systems.  The BTeV trigger and DAQ system reconstructs
tracks and makes a detachment based trigger decision 
at the lowest trigger level.  Every beam crossing
is inspected in this way.  CDF and D\O,
on the other hand, must live within bandwidth budgets
that were established before this sort of trigger was
feasible.  Therefore they have detachment information 
available only at level 2 and higher.  For similar reasons their
triggers have a higher $p_T$ cut than does the BTeV trigger.

\subsection{A Precision Vertex Detector }
\index{vertex detector! desired properties}

   First and foremost it is necessary to have a high precision vertex 
detector.  The importance of the vertex detector to the trigger
has already been emphasized.  Also, excellent resolution on 
proper decay time, which results from excellent vertex resolution, 
is necessary to study the time dependence of mixing mediated $CP$ 
violating effects in the $B_s$ system.   

In a typical offline analysis chain, the vertex resolution 
appears to a high power.  A typical candidate-driven 
$B^0\to \pi^+\pi^-$ analysis might proceed as follows.
First one finds a $B^0$ candidate and demands that the candidate
$B$ have a well defined vertex with a good $\chi^2$.
Next, one must find the primary vertex of the $b\bar{b}$ interaction
and care must be taken to ensure that this vertex is not contaminated
by tracks from the other $b$ hadron.  
One demands that the secondary vertex be well separated from the primary
vertex.  For some analyses
it will be necessary to exclude $B$ candidates if other tracks
from the event are consistent with coming from the $B$ decay vertex.
Finally, one applies the available tagging methods.
Each of these steps exploits the vertexing power of the
experiment in a slightly different way.  With so many steps, poor
resolution has many chances to strike.

The vertex detector must have as low a mass as possible.
Less mass implies less multiple scattering and better vertex resolution. 
But a more important effect is that less mass reduces the number of
interactions of signal tracks in the detector materials.
When a signal track interacts in the detector, it is often unusable
for physics and the event is lost.  Examples include tracks which
undergo inelastic hadronic interactions before reaching the particle
ID device and photons which pair convert in the detector material.

A final consideration is the occupancy of the vertex detector.
The occupancy is defined as the fraction of channels which are 
hit during a typical beam crossing.  As the occupancy rises, the number 
of hit combinations which must be considered grows exponentially
and pattern recognition becomes more difficult.  
If the occupancy is less than a few percent, offline pattern 
recognition is straight forward and standard algorithms compute 
sufficiently quickly.  The driving
factor in behind BTeV's choice of a pixel detector, rather than
a strip detector, was to reduce the occupancy to approximately
$10^{-4}$.  With this very low occupancy, even very simple, 
pattern recognition algorithms are efficient and
produce low background levels; this allows their use at the
lowest level of the trigger.

\subsection{Tracking}
\index{tracking system!required properties}

The vertex detector must be supplemented by a tracking system with excellent
momentum resolution.  For most decay modes of interest, the mass
resolution on the $b$ hadron is dominated by the momentum resolution 
of the apparatus.  If the mass resolution can be decreased by, say,
10\%, one will get a 10\% improvement in signal-to-background
ratio without loss of signal efficiency.  In a decay chain with
several intermediate mass constraints this can add up.

 Again it is important to minimize the mass in the tracking system
and pay careful attention to the expected occupancy.

\subsection{Particle ID}
\index{particle identification!overview}

 It is important to have a excellent particle identification with the 
ability to separate, with high efficiency, all of $e,\mu,\pi,K,p$ over
a broad momentum range.  All of the detectors have triggering
modes which require lepton identification (lepton ID).  Particle ID
is also critical for reducing backgrounds
which arise when one $B$ decay mode is mistaken for another, such
as $B_s\to D_s \pi$ being mistaken for $B_s\to D_s K$.  Finally,
excellent particle ID is crucial for a 
large $\epsilon D^2$ for kaon tagging.

  All of the experiments have excellent lepton ID.  Muon ID is
done by finding tracks which penetrate a hadron 
shield.    Electron ID is done by matching tracks in the
tracking system with clusters of energy in the electromagnetic 
calorimeter.   

  BTeV is a dedicated $B$ physics experiment and one of the factors 
driving the decision to build a forward spectrometer, not a central
one, was that the forward geometry has room for a 
Ring Imaging Cherenkov counter (RICH).  This provides the 
power to separate $\pi,K,p$ from one another.
On the other hand, the CDF and D\O\ detectors were originally
optimized for high $p_T$ physics, which did not require
powerful $\pi,K,p$ separation.  Therefore the early designs for the 
CDF and D\O\ Run~II detectors did not include any device to do 
hadronic particle ID.    Since then CDF has added a time of 
flight (TOF) system to perform $\pi,K,p$ separation.  

   The reader is referred to the chapters~3 to~5 for
further details.

\subsection{Trigger and DAQ}
\index{triggering!overview of}

In order to have a broad based $B$ physics program, it is important
to have an open trigger which is able to trigger on many 
$B$ decay modes.  This must be accompanied by a 
high bandwidth DAQ system which can move the data off the detector,
move it between trigger levels and store it until a trigger
decision is made.

The job of the trigger is to 
sort through the much more copious background
interactions and extract a high purity $b$ sample to write to tape.
Ideally the trigger should be sensitive to some general property 
of $b$ events, and have a high efficiency for a wide variety
of $B$ decay modes; it is not enough that the trigger performs
well on some list of benchmark decay modes.  This allows the greatest
flexibility to explore ideas which are first thought of long after
the trigger design was frozen.  Of course one must verify that 
the trigger works well on the modes which we know now to be important. 

   A detachment based trigger meets all of these requirements; in particular
it can trigger on all hadronic decay modes, a capability which was missing
from the previous generation of experiments.  
A lepton based trigger, while missing the all hadronic modes, 
does meet many of the requirements and it will provide a 
redundant triggering method to calibrate the detachment based triggers.

  The background rejection needed by the trigger is set by several
things.  Each level of the trigger must reduce the background to
a low enough level that the bandwidth to the next level is not
saturated.  One must also consider the total amount of data which
is written to tape; if too much data is written to tape, the main 
data reconstruction pass will take too long and the production 
of physics papers will be delayed.  
The cost of archival media is also an issue.

  A final consideration is projecting results to higher 
instantaneous luminosities.  As the luminosity increases,
several limiting effects arise: one might reach the
bandwidth limit of the DAQ system; one might exceed the amount
of buffering at some level of the DAQ; the dead time might become
too large.  Once one of these limits is reached, the normal
response is to raise some trigger threshold or to prescale some 
trigger.  Typically one tries to sacrifice either the triggers which carry
the least interesting physics or the triggers with poor
signal to background ratios.  

 During an extended Run~II but it is very likely that some of the $B$ 
physics triggers will need to be modified to deal with the
increased luminosity but it is difficult to project in detail
what might need to be done. 
These decisions will depend, in part, on an understanding
of backgrounds which is not yet available.  Depending on the 
characteristics of the backgrounds, the trigger efficiency for 
some $B$ physics channels may be unaffected while the trigger 
efficiency for other $B$ physics channels may drop significantly.

  The collaborations have concentrated their $B$ physics
trigger simulations on the conditions which will be present
early in Run~II, up to a luminosity of around
$2\times 10^{32}\, {\rm cm^2 s^{-1}}$.  So they have decided not
to present projections for integrated luminosities of 10 and 
30~fb$^{-1}$.


\subsection{EM Calorimetry}
\index{EM Calorimeter!overview of}

A good electromagnetic calorimeter (EMcal) is necessary for
the reconstruction of decay modes which contain final state
photons and $\pi^0$'s.  It is also necessary for electron ID,
which can be used in triggering, in flavor tagging, in many
searches for physics beyond the standard model.

  There is one high profile decay mode for which the EMCal is
critical, the analysis of the Dalitz plot for the decay
$B^0\to \pi^+\pi^-\pi^0$, a mode which measures the CKM
angle $\alpha$.   The $\Upsilon(4S)$ machines are rate
limited and are likely not to have sufficient statistics to
make a definitive measurement of this quantity.  The Tevatron
detectors have the rate and, provided the EMCal technology
is good enough, the measurement can be done.

 A final use for an EMCal is to help sort out strong interaction
effects which are entangled with the weak interaction physics
that it the main goal.   It is most straightforward to disentangle
the strong interaction effects when all isospin permutations 
of the final state can be measured.  The classic example
of this are the decays $B^0\to \pi^+\pi^-$, 
$B^0\to \pi^0\pi^0$, and $B^+\to \pi^+\pi^0$.  While this
complete set of decay modes is probably not measurable at the
Tevatron, it illustrates the point.

  All of the detectors have electromagnetic calorimetry.
Both BTeV and CDF have discussed a $B$ physics program which exploits it.

  The EMCal system has a unique sensitivity to the issue of
track density.  As the number of tracks in the detector goes up,
the occupancy of the calorimeter goes up.  In the case of the tracking
detectors one can compensate for high occupancy by making a more
granular detector.  This works because each track usually makes
a small, localized signal in a tracking detector.  In the case of a
calorimeter, showers are extended objects which, by design,
deposit energy in neighboring crystals or cells.  Increasing
the granularity of the detector will not make the showers any smaller.
One can compensate for high occupancy by choosing
calorimeter materials in which showers are contained in a smaller 
volume; if such materials are available, have sufficient energy
resolution and radiation tolerance, and are affordable, then
they can be used to make a calorimeter which can tolerate
higher multiplicities.

\subsection{Muon Detector}
\index{muon detection!overview of}
 
The final major hardware component in a $B$ physics experiment
is a muon detector system.  This provides muon identification (ID)
for such purposes as flavor tagging, 
reconstruction of $J/\psi$ candidates,
reconstruction of semileptonic decays,
and searches for rare or forbidden decays.  These last two classes
include modes such as $B^0\to \mu^+\mu^-$ and $B^0\to e^+\mu^-$.
A second job of these system is to provide an element of the trigger
system; in some cases this is a stand alone trigger element and in
other cases it is used in conjunction with other detector components
to make a trigger decision.  

The basic design of a muon detector is an iron or
steel shield, many hadronic interaction lengths thick, which 
absorbs hadrons.  Muons which penetrate this shield are detected 
by some sort of wire chamber tracking detector, or perhaps a scintillator, 
placed behind the shield.  When possible other detector components, such 
as calorimeters and flux returns are used as part of the shielding.
In another variation, iron shielding can be magnetized to allow
measurement of the muon momentum in the muon system alone.
This gives several benefits: it allows one to design a muon
based trigger with a well defined $p_T$ cut and it allows better
matching between tracks in the muon system and tracks in the
main tracking system.


\section{Software}
\label{2:sec:software}

The software used by CDF, D\O\ and BTeV can be thought of in
four classes, event generators, the $B$ decay code, 
detector simulation tools
and reconstruction code.  The three experiments use common
tools for the first two classes of software but generally
use their own, detector specific software for the last two classes.
The one exception is the MCFast fast simulation package which was
used by both BTeV and CDF.  MCFast is described extensively
in the BTeV TDR\cite{btev_tdr}.  BTeV is a new experiment which
has tools which are quite advanced for such a young experiment
but which are still primitive on an absolute scale.  
After Run~I, CDF and D\O\ embarked on a major retooling of their software 
infrastructure, which was incomplete at the time of the workshop.
So all of the results presented here use preliminary versions of
code.  At the level of precision required for the studies performed
at this workshop, all of these tools are good enough.

  The event generators are programs which generate the physics
of a $p\bar{p}$ interaction; its output is usually just a list
of vertices and particles which come out of those vertices.
These programs are typically the intellectual property of
the theoretical physicists who developed the model which is implemented
in the program.  The generators used by CDF, D\O\ and BTeV
are {\tt PYTHIA}\cite{pythia}, 
{\tt ISAJET}\cite{isajet} and {\tt HERWIG}\cite{herwig}. Generally 
these programs are used only to predict the shapes of 
the differential cross-sections, not for the absolute cross-sections.
CDF and D\O\ have tuned the parameters of their event generators to
match their Run~I data.  BTeV, on the other hand, has no such data
against which to tune the codes.  Therefore BTeV is using the programs
as is.

   The event generators have been carefully developed to simulate
the properties of $p\bar{p}$ collisions but much less care was taken
in their model of how $b$ hadrons decay.  To circumvent this, the
$b$ and $c$ hadrons produced by the event generators are handed
to a separate code to simulate their decay.  Until recently this code
was the QQ code, which was developed and maintained by CLEO,
and which contains their integrated knowledge about the decays
of $B$'s and $D$'s.  The BaBar collaboration also has such
a program, {\tt EVTGEN} which will soon replace QQ.  The results
of the workshop were obtained using QQ.

   The next step in a typical simulation is to compute the
detector response to the simulated events.  All of the experiments
have both a fast simulation program and a detailed simulation
program.  A typical fast simulation program uses a simplified and/or
parameterized description of the detector response and directly
produces smeared 4-vectors for the tracks which were input to it.
It may also declare that a track is outside of the fiducial volume
and is not reconstructible.  The output of the fast simulation can
usually be used as is to perform the simulated analysis.

  A typical full simulation is based on the {\tt GEANT}~3 program
from CERN.  This is a program which knows how to describe a detector
by building it up from a library of known shapes.  It also has
extensive knowledge of the interactions of particles with materials.
It takes tracks from the event generator and propagates them through
the detailed description of the detector, at each step checking to
see how the track interacted with the material.  If a particle
interacts in the detector material to produce new particles, those
new particles are also propagate through the detector.  If a shower
starts in material, {\tt GEANT}~3 will follow the daughters through
each stage of the shower, and deposit the energy of the shower
in the appropriate detector cells.  The output of
this simulation is typically a list of pulse heights or arrival
times for hits in individual detector cells.   This information
is then passed to the reconstruction program and the trigger
simulation codes.

  At the time of the workshop, the experiments were still evaluating 
at the {\tt GEANT}~4 program, the C++ based successor to
{\tt GEANT}~3.  

     CDF and D\O\ have data samples from Run~I
which can be used.  Signal yields can be projected from the
Run~I signal yields by computing the ratio of efficiencies in the
old and new detectors.   This avoids the need to make assumptions
about the total cross-section, as BTeV must do.  CDF and D\O\ use
background samples from Run~I to estimate background levels in
Run~II.  BTeV must rely entirely on simulations for this purpose.

  The reconstruction code starts with raw hits, either from the
detector or from a simulation of the detector, calibrates them,
find tracks, fits them, finds showers, applies the particle
ID algorithms and so on.  The output of this step is the measured
properties of tracks and showers, which can be used directly
for physics analysis.

  The trigger simulation codes start with the same raw hits as
the reconstruction code.  In some cases the codes emulate the 
trigger hardware and produce trigger decisions which should
very closely represent the real trigger behavior.  In other
cases the simulation codes 

  For more details on the software of each experiment,
consult their 
TDRs~\cite{cdf_tdr}\cite{cdf_tof_l00}\cite{d0_tdr}\cite{btev_tdr}
and chapters~3 to~5 in this report.

\section{Comparison with \lowercase{$e^+e^-$} Machines }
\index{$e^+e^-$ machines!comparison with}

  For most of their lifetime the Tevatron experiments will
be in competition with the detectors from the
$\Upsilon(4S)$ $e^+e^-$ factories, BaBar and Belle.
The charm physics program of BTeV will also face competition from
CLEO-c.  While these programs are competition,
they also complement the Tevatron program. 
Some precision measurements will be best done in the cleaner
environment of $e^+e^-$: they have a well determined initial state, 
either pure $B^0\bar{B}^0$ or $B^+B^-$, with no additional tracks
in the event.  And the production of $B$ mesons represents about
20\% of their total cross-section, which greatly simplifies
triggering and removes many trigger biases.  Similar advantages
hold for the open charm program at CLEO-c.

  On the other hand, the Tevatron experiments have a significant
rate advantage which give it the advantage for many rare decay
modes and in those measurements which are statistically limited.  
Only the Tevatron experiments have access to the decays of the
$B_s$ and $b$ baryons, which are necessary to complete the
program of over constraining the CKM matrix.  See, for example,
the discussion of $B_s$ mixing in chapter~8.

%\section{Miscellaneous}
%   These are some things which need to fit into the above
%   hierarchy somewhere.
%   \begin{enumerate}
%     \item Reference for $\sigma(t)$ $\propto \gamma$.
%     \item Important to always quote efficiency and S/B as a pair.
%     \item  Do we want any comparison figures or tables, 
%           ie $m(\pi+\pi-)$ or $\sigma(t)$ for all three detectors?
%     \item Do not want to compare with the LHC machines since
%           they present too ephemeral a target.
%     \item When we extrapolate to 10 and 30~$fb^{-1}$ what assumptions do
%          we make --- just more running time with otherwise identical
%          performance?  For example some triggers will
%          go to hell, components will age, DAQ will have high dead times,
%          occupancies will become too high to extract anything ...
%       Also, some process which are statistics dominated at 2~$fb^{-1}$
%       will be systematics dominated at the higher luminosities and
%       the evolution of the errors could be much slower than
%       sqrt(Lumi).  Maybe this belongs elsewhere.
%
%     \item Fill in dummy references.
%     \item Check for other places which need references.
%     \item Trigger DAQ stuff.
%     \item In the section on $f_u:f_d\dots$, I talk about independence
%%           of final species.  The recent CLEO results are against this
%          but the threshold effects are worse there.

%   \end{enumerate}

\clearpage
\begin{thebibliography}{99}
\bibitem{cdf_tdr} The CDF\,II Collaboration, R.~Blair {\em et al.},
        {\it The CDF\,II Detector Technical Design Report},
        Fermilab-Pub-96/390-E.
        This Report may be found at the url:
        {\tt http://www-cdf.fnal.gov/upgrades/tdr/tdr.html}.

\bibitem{cdf_tof_l00} The CDF\,II Collaboration,
        {\em ``Update to Proposal P-909: Physics Performance of the CDF\,II
        Detector with an Inner
        Silicon Layer and a Time-of-Flight Detector''},
        submitted to the Fermilab Director and PAC, 5 Jan 1999.

\bibitem{d0_tdr} The D\O\ Collaboration, ``The D\O\ Upgrade'', submitted  to
the Fermilab PAC August 1996, Fermilab Pub-96/357-E, \\
\verb"http://higgs.physics.lsa.umich.edu/dzero/d0doc96/d0doc.html".

\bibitem{btev_tdr} The BTeV Collaboration, ``Proposal for
                   an  Experiment to Measure Mixing, CP Violation and
                   Rare Decays in Charm and Beauty Particle Decays
                   at the Fermilab Collider --- BTeV'',
                   Submitted to the Fermilab Director and PAC, May 2000.
                   This report may be found at the URL:\\
{\tt http://www-btev.fnal.gov/public\_documents/btev\_proposal/index.html}

\bibitem{bgxsec}  The CDF Collaboration, F.~Abe {\em et al.},
                  Phys.\ Rev.\ D {\bf 50}, 5550 (1994).\\
                  The E-811 Collaboration, C.~Avila {\em et al.},
                  Phys. Lett. {\bf B445} 419 (1999).

\bibitem{PDG} The Particle Data Group, D.~E.~Groom {\em et al.}, 
              The European Physical Journal {\bf C15}, 1 (2000).

\bibitem{MNR} M. Mangano, P. Nason and G. Ridolfi, 
              Nucl. Phys. {\bf B373}, 295 (1992).

\bibitem{pythia} H.~U.~Bengtsson and T.~Sjostrand,
                 Comp. Phys. Comm. {\bf 46}, 43 (1987).

\bibitem{isajet} F.~Paige and S.~Protopopescu, Brookhaven National 
                 Laboratory Report BNL-37066 (1985) (unpublished).

\bibitem{herwig} G.~Marchesini and B.R~Webber, Nucl. Phys. {\bf B310}, 461
                 (1988);
                 Nucl. Phys. {\bf B330}, 261 (1988);\\
                 I. G. Knowles, Nucl. Phys. {\bf B310}, 571 (1988);\\
                 G. Abbiendi, {\em et al.}, Comp. Phys. Comm. {\bf 67}, 
                   465 (1992).  

\bibitem{cdfprodfrac} The CDF Collaboration, T.~Affolder {\em et al.},
                      Phys. Rev. Lett. {\bf 84}, 1663 (2000).

\bibitem{lepprodfrac} D.~Abbaneo {\it et al.},
                   ``Combined results on b hadron production rates, 
                    lifetimes,  oscillations and semileptonic decays,'' 
                    .

\bibitem{cdftagging} The CDF Collaboration, T.~Affolder {\it et al.}
                     Phys.\ Rev.\ D {\bf 61}, 072005 (2000)
                     .

\bibitem{cdf_bxsec} The CDF Collaboration, F.~Abe {\em et al.},
                    Phys. Rev. Lett. {\bf 75} 1451 (1995).

\bibitem{d0_bxsec} The D\O\ Collaboration, R.~Abbott {\em et al.},
                     Phys. Rev. Lett. {\bf 74} 3548 (1995).

\bibitem{d0_bxsec1999} D.~Fein, ``Tevatron Results on $b$-Quark
                       Cross Sections and Correlations'', presented
                       at Hadron Collider Physics (HCP99), Bombay, January
                       1999.

\bibitem{d0_bxsecpub} The D\O\ Collaboration, B.~Abbott {\em et al.},
                      Phys. Rev. Lett. {\bf 84} 5478 (2000)
                      .

\bibitem{d0_bxsecnew} The D\O\ Collaboration, B.~Abbott {\em et al.},
                      ``The $b\bar{b}$ Production Cross Section and Angular
                         Correlations in $p\bar{p}$ Collisions at 
                         $\sqrt{s}$ = 1.8 TeV'', 
                      Phys. Lett. {\bf B487} 264 (2000) 
                      .
\end{thebibliography}
%
% Notes:
% 1) Forward vs central
% 2) Refs for QQ and EVTGEN.
% 3) Revist particle ID section - dE/dx in CDF/D0.
% 4) make generators into its own sub-sub section?
% 5) Comment about 10^-4 in occupancy section?
%
\chapter{The CDF Detector}

\authors{M.~Paulini, A.B.~Wicklund}

\section{Introduction}

The CDF detector has evolved over a twenty year period. 
CDF was the first experiment at the Tevatron to perform
quantitative measurements of $b$-quark production,
using single-lepton and $J/\psi$ samples in
Run 0 (1988-1989).  CDF was the first hadron collider
experiment to successfully employ a silicon vertex
detector (SVX).  The four layer, axial readout 
SVX (Run~Ia) and SVX' (Run~Ib) detectors were
used to discover the top quark through detection
of the $b$-quark decay chain.  They were also used for
a systematic program of $b$-physics studies, including
$B$-lifetimes, $B\overline{B}$ mixing, discovery
of the $B_c$, and measurement of $CP$ violation 
in the $B^0 \rightarrow J/\psi K_S^0$ mode.  In the course
of this program, CDF has developed the techniques to
identify $B$ hadron final states in $J/\psi$ and 
$\ell\nu D$ semileptonic modes, and to flavor tag
these states using away-side lepton and jet-charge
tags and toward side fragmentation correlations.
As a byproduct, CDF has developed control sample
strategies to calibrate particle identification using
relativistic rise d$E$/d$x$, to optimize flavor tagging
efficiency, and to measure material effects
(energy loss and radiation length corrections) needed
for precision mass measurements.  CDF has published over fifty
papers on $B$ physics with the Run 0 and Run~I data. In addition
to Run~I physics data, CDF recorded data on a variety
of specialized triggers in order to estimate rates and
backgrounds for the Run~II program.  Although the Run~II
$B$-physics program at CDF will be technically more
challenging than Run~I, CDF starts with the advantage
of extensive experience and benchmark data.

The upgraded CDF detector and physics program for Run~II is described in detail in
the CDF Technical Design Report~\cite{cdf_tdr}. The
additional upgrades for time-of-flight and innermost
silicon detector are described in the PAC proposal P909~\cite{cdf_p909}.
Below we summarize the Run~II detector configuration, 
with expected performance including particle identification,
issues for central solenoidal detectors, CDF $B$-physics trigger 
plans, and offline analysis issues.
 
\section{CDF Run~II Detector}
 
  The main upgrades to the CDF detector for Run~II
can be summarized as follows:
\index{CDF!detector improvements}
\begin{itemize}
\item[-]{Fully digital DAQ system designed for 132 ns bunch crossing times}
\item[-]{Vastly upgraded silicon detector}
\subitem{707,000 channels compared with 46,000 in Run~I}
\subitem{Axial, stereo, and 90$^\circ$ strip readout}
\subitem{Full coverage over the luminous region along the beam axis}
\subitem{Radial coverage from 1.35 to 28 cm over $|\eta| < 2$}
\subitem{Innermost silicon layer(``L00'') on beampipe with 6 $\mu$m axial hit resolution}
\item[-]{Outer drift chamber capable of 132 ns maximum drift}
\subitem{30,240 sense wires, 44-132 cm radius, 96 d$E$/d$x$ samples possible per track}
\item[-]{Fast scintillator-based calorimetry out to $|\eta| \simeq 3$}
\item[-]{Expanded muon coverage out to $\eta \simeq 1.5$}
\item[-]{Improved trigger capabilities}
\subitem{Drift chamber tracks with high precision at Level-1}
\subitem{Silicon tracks for detached vertex triggers at Level-2}
\item[-]{Expanded particle identification via time-of-flight and d$E$/d$x$}
\end{itemize}

\begin{figure}[t]
\centerline{\epsfxsize=13cm \epsfbox{ch3/cdfelev.ps}}
%\epsfbox{trackvol.ps}
\vspace*{.2cm}
\caption[CDF detector elevation view]{Elevation view of the Run~II CDF detector}
\label{fig:cdfelev}
\end{figure}

\begin{table}[htbp]
\begin{center}
\begin {tabular} {l c}
\hline \hline
COT                         &    \\ \hline
Radial coverage             & 44 to 132 cm \\
Number of superlayers       &  8  \\
Measurements per superlayer & 12  \\
Readout coordinates of SLs  & +2$^\circ$ 0$^\circ$ -2$^\circ$ 0$^\circ$ +2$^\circ$ 0$^\circ$ -2$^\circ$
 0$^\circ$ \\
Maximum drift distance      & 0.88 cm \\
Resolution per measurement  & 180 $\mu$m \\
Rapidity coverage           &  $|\eta| \leq 1.0$ \\
Number of channels          & 30,240 \\
%Material thickness          & 1.3\% ${\rm X_{0}}$ \\ \hline \hline
%Occupancy @ 2E32/132        & 10\% at inner superlayer  \\ \hline \hline
Layer 00                     &                        \\ \hline
Radial coverage             &1.35 to 1.65 cm \\
Resolution per measurement  & 6 $\mu$m (axial) \\
%Readout pitch               & 25 $\mu$m (r-$\phi$)\\
Number of channels          & 13,824 \\
SVX II                       &                        \\ \hline
Radial coverage             &   2.4 to 10.7 cm, staggered quadrants      \\
Number of layers            &   5          \\
Readout coordinates         &r-$\phi$ on one side of all layers \\
Stereo side                 &r-z, r-z, r-sas, r-z, r-sas (sas $\equiv
                             \pm 1.2^\circ$ stereo) \\
Readout pitch               & 60-65 $\mu$m r-$\phi$; 60-150 $\mu$m stereo\\
Resolution per measurement  & 12 $\mu$m (axial) \\
Total length                &  96.0 cm  \\
Rapidity coverage           &  $|\eta| \leq 2.0$ \\
Number of channels          & 423,900    \\
%Material thickness          &  3.5\% ${\rm X_{0}}$ \\
%Power dissipated            &  1.8 KW \\ \hline \hline
ISL                         &       \\ \hline
Radial coverage             &  20 to 28 cm \\
Number of layers            &  one for $|\eta|<1$; two for $1<|\eta|<2$  \\
Readout coordinates         &  r-$\phi$ and r-sas (sas$\equiv \pm 1.2^\circ$ stereo)
                              (all layers)\\
Readout pitch               &  110 $\mu$m (axial); 146 $\mu$m (stereo) \\
Resolution per measurement  & 16 $\mu$m (axial) \\
Total length                &  174 cm  \\
Rapidity coverage           &  $|\eta| \leq 1.9$ \\
Number of channels          & 268,800    \\
%Material thickness          &  2\% ${\rm X_{0}}$ \\ 
\hline \hline
\end{tabular}
\end{center}
\vspace*{.2cm}
\caption[CDF tracking systems]{Design parameters of the CDF tracking systems}
\label{tb_tracking}
\end{table}

The CDF detector features excellent charged particle tracking
and good electron and muon identification in the central region.
The detector is built around a 3 m diameter 5 m long superconducting
solenoid operated at 1.4 T.  The overall CDF Run~II detector 
schematic is shown in elevation view in Fig.~\ref{fig:cdfelev}.
The CDF tracking system includes a central outer drift chamber (COT),
a double-sided five layer inner silicon detector (SVX II), 
a double-sided two layer intermediate silicon tracker (ISL),
and a single layer rad-hard detector mounted on the beampipe (L00).
COT tracks above 1.5 GeV/c are available for triggering at Level-1 (XFT);
SVX layers 0-3 are combined with XFT tracks at Level-2 (SVT).
The main parameters of the CDF tracking system are summarized in
Table~\ref{tb_tracking}.

Outside the solenoid, Pb-scintillator
electromagnetic (EM) and Fe-scintillator hadronic (HAD) calorimeters cover
the range $|\eta| < 3.6$.
\index{CDF!calorimetry} 
Both the central ($|\eta| < 1.1$) and plug ($1.1 < |\eta| < 3.6$)
electromagnetic calorimeters have fine grained shower profile detectors
at electron shower maximum, and preshower pulse height detectors
at approximately $1 X_o$ depth. 
Electron identification is
accomplished using $E/p$ from the EM calorimeter and also in the shower maximum and
preshower detectors; using $HAD/EM\sim 0$; and using shower shape and position
matching in the shower max detectors.  Together with COT d$E$/d$x$, CDF gets
$\sim$10$^{-3}$ $\pi/e$ rejection in the central region.
The calorimeter cell segmentation is summarized in
Table~\ref{calseg}.  A comparison of the central and plug
calorimeters is given in Table~\ref{plugcomp}.

\begin{table}
\begin{center}
\begin {tabular} {l|c|c}
\hline \hline
$|\eta|$ Range & $\Delta\phi$ & $\Delta\eta$ \\
\hline
0. - 1.1 (1.2 h) & $15^{\circ}$ &  $\sim0.1$ \\
1.1 (1.2 h) - 1.8 & $7.5^{\circ}$ & $\sim0.1$ \\
1.8 - 2.1 & $7.5^{\circ}$ & $\sim0.16$ \\
2.1 - 3.64 & $15^{\circ}$ & 0.2 $-$ 0.6 \\
\hline \hline
\end{tabular}
\vspace*{.2cm}
\caption[CDF calorimeter segmentation]{CDF II Calorimeter Segmentation}
\label{calseg}
\end{center}
\end{table}

The calorimeter steel serves
as a filter for muon detection in the central (CMU) and
extension (CMX) muon  proportional chambers, over the
range $|\eta| < 1$, $p_T > 1.4$ GeV/c.
\index{CDF!muon systems}  Additional iron
shielding, including the magnet yoke, provides a muon filter
for the upgrade muon chambers (CMP) in the range $|\eta| < 0.6$, $p_T >2.2$ GeV/c.
The (non-energized) forward toroids from Run~I
provide muon filters for intermediate $1.0 < |\eta| < 1.5$ 
muon chambers (IMU) for $p_T > 2$ GeV/c.  Scintillators for
triggering are included in CMP, CMX, and IMU.
Muon identification is accomplished by matching track segments in the muon chambers
with COT/SVX tracks; matching is available in $R\Phi$ for all detectors and
in the $Z$ views in CMU and CMX. 
The muon systems are summarized in Table~\ref{tab:overcmu}.

\begin{table}
\begin{center}
\begin {tabular} {l|c|c}
\hline \hline
 & Central  & Plug \\
\hline
EM: & &\\
Thickness & $19 X_0, 1 \lambda$ & $21 X_0, 1 \lambda$ \\
Sample (Pb) & $0.6 X_0$ & $0.8 X_0$  \\
Sample (scint.) & 5 mm & 4.5 mm \\
WLS & sheet & fiber \\
Light yield & $160$ pe/GeV & $300$ pe/GeV \\
Sampling res. & $11.6\%/\sqrt{E_T}$ & $14\%/\sqrt{E}$ \\
Stoch. res. & $14\%/\sqrt{E_T}$ & $16\%/\sqrt{E}$ \\
Shower Max. seg. (cm)& 1.4$ \phi \times $(1.6-2.0) Z& $0.5\times0.5$ UV\\
Pre-shower seg. (cm)& $1.4 \phi\times 65$ Z & by tower \\
\hline
Hadron: &&\\
Thickness& $4.5 \lambda$  & $7 \lambda$\\
Sample (Fe)&1 to 2 in.& 2 in.\\
Sample (scint.)& 10 mm & 6 mm\\
WLS & finger & fiber\\
Light yield& $\sim40$ pe/GeV  & $39$ pe/GeV\\
\hline \hline
\end{tabular}
\vspace*{.2cm}
\caption[CDF central/plug calorimeters]{Central and Plug Upgraded Calorimeter
Comparison}
\label{plugcomp}
\end{center}
\end{table}

\begin{table}[t]
\begin{center}
\begin{tabular}{lcccc}
\hline \hline
 & CMU   & CMP   & CMX & IMU \\ \hline
Pseudo-rapidity coverage
  & $|\eta|\leq 0.6$ & $|\eta|\leq 0.6$
  & $0.6 \leq |\eta|\leq 1.0 $
  & $1.0 \leq |\eta|\leq 1.5 $ \\
Drift tube cross-section
 & 2.68 x 6.35~cm & 2.5 x 15~cm & 2.5 x 15~cm & 2.5 x 8.4~cm \\
Drift tube length   & 226~cm   & 640~cm  & 180~cm & 363~cm \\
Max drift time
 & ${800~{\rm ns}}$ & 1.4~$\mu {\rm s}$
 & 1.4~$\mu {\rm s}$ &  800~{\rm ns} \\
Total drift tubes (present)     & 2304 & 864     & 1536  & none \\
Total drift tubes (Run~II)      & 2304 & 1076    & 2208  & 1728 \\
Scintillation counter thickness &      & 2.5~cm  & 1.5~cm & 2.5~cm \\
Scintillation counter width     &      & 30~cm   & 30-40~cm & 17~cm \\
Scintillation counter length    &      &  320~cm & 180~cm & 180~cm \\
Total counters (present)        &      & 128     & 256 & none \\
Total counters (Run~II)         &      & 269     & 324 & 864 \\
Pion interaction lengths        & 5.5  & 7.8     & 6.2 & 6.2-20 \\
Minimum muon $p_T$
  & 1.4~GeV/c & 2.2 ~GeV/c & 1.4~GeV/c & 1.4-2.0~GeV/c\\
Multiple scattering resolution
  & 12~cm/$p$  & 15~cm/$p$ & 13~cm/$p$ &13-25~cm/$p$\\ \hline \hline
\end{tabular}
\vspace*{.2cm}
\caption[CDF muon detectors]{Design Parameters of the CDF II Muon Detectors.
Pion interaction lengths and multiple scattering are computed at a reference
angle of $\theta =90^{\circ}$ in CMU and CMP, at an angle of
$\theta=55^{\circ}$ in CMX, and for a range of angles for the IMU.}
\label{tab:overcmu}
\end{center}
\end{table}

 The Run~II CDF detector configuration
allows electron and muon identification with drift chamber tracking
over the range $|\eta| < 1.0$, with additional coverage out to
$|\eta| \sim 1.5$ using stand-alone silicon tracking.  Typical
thresholds are $p_T > 1$ GeV/c (electrons), $p_T > 1.5$ GeV/c (muons).
Calibration of electron and muon identification is accomplished
{\it in situ} with large samples of $J/\psi$'s and photon conversions.

 Photon identification is done using the EM calorimetry, using the preshower
and shower maximum detectors to separate $\pi^0/\gamma$.  Channels like
$B\rightarrow J/\psi\eta$ can be reconstructed with the calorimeter. Rare decays
such as $B\rightarrow K^*\gamma$ use photon conversions to obtain
precision $\gamma$ reconstruction.

Asymptotic tracking resolutions are $\sigma(p_T) \simeq 0.0007~p_T^2$ 
($|\eta < 1.1$), and $\sigma(p_T) \simeq 0.004~p_T^2$
($|\eta| < 2$ -stand-alone SVX tracking). The SVX detectors provide
typical impact parameter resolution of
15 ($R-\phi$- view) and 30 ($Z$- view) $\mu m$. For example, this results in
mass resolution of around 20 MeV for $B^0\rightarrow\pi^+\pi^-$
and proper time resolution of 45 fs for $B_s^0 \rightarrow D_s^-\pi^+$.

\begin{figure}[t]
\centerline{\epsfxsize=4.5in \epsfbox{ch3/tof_with_dedx.ps}}
%\epsfbox{trackvol.ps}
\vspace*{.2cm}
\caption[CDF particle identification]{Time difference as a function of momentum
for $\pi, K, p$ at a radius of 140 cm, expressed in ps and separation power,
assuming 100 ps resolution. The dashed line shows the $K/\pi$ separation using
$dE/dx$ in the COT.}
\label{fig:pid}
\end{figure}

Particle identification to separate pions, kaons, protons,
and electrons is provided by d$E$/d$x$ and time-of-flight
(TOF) detectors ($|\eta| < 1$):
\begin{itemize}
\item {$1/\beta^2$ d$E$/d$x$ in both COT and SVX}
\item {Relativistic rise d$E$/d$x$ in COT}
\subitem{$\simeq$1 $\sigma$ $\pi-K$ separation, $p > 2$ GeV/c}
\item {TOF bars outside the COT (1.4 m radius)}
\subitem{$\simeq$100 ps resolution}
\subitem{2$\sigma$ $K-\pi$ separation at 1.5 GeV/c}
\end{itemize}
Figure~\ref{fig:pid} shows the expected separation
as a function of momentum.  
The low momentum particle identification provides flavor
tagging with kaons, both on the away side $\overline{b}$-jet, and
on the same side as the trigger $B$ (e.g., $B^0_s$ correlated with
$K^+$, $\overline{B^0_s}$ with $K^-$); it should also provide
useful kaon separation for charm reconstruction, e.g. $B^0_s\rightarrow D_s^-\pi^+$.
Above 2 GeV/c, the COT $dE/dx$ provides a statistical
separation of pions and kaons.  This will be crucial for
determination of $CP$ asymmetries in $B^0\rightarrow\pi^+\pi^-$
and $B^0_s\rightarrow K^+K^-$.  The d$E$/d$x$ and TOF calibrations can
be determined {\it in situ} using pions and protons
from $K_S$ and $\Lambda$ decays, kaons from $\phi \rightarrow K^+K^-$,
electrons from conversion photons, and muons from $J/\psi$ decay.

 
\section[Issues for Central Solenoidal Detectors]
{Issues for Central Solenoidal Detectors at the Tevatron}

In the Run~II detector 
full charged particle tracking using the SVX and the COT 
is confined to $\eta < 1.1$.  Since all $B$-physics
triggers are track based, and rely on COT tracks in the Level-1
trigger, the $B$ triggers are basically limited to this
$\eta$ range.  Furthermore, in order to achieve a reasonable
Lorentz boost for the triggered $B$ hadrons, it is necessary
to select events with $p_T(B)$ above some minimum cutoff, typically
of order 4-6 GeV/c.  In terms of production rates, this is not
really a severe limitation.  At 10$^{32}$ luminosity, the total
$b\overline{b}$ production rate is around 10 kHz at the
Tevatron (eg. 10$^{11}$ events per fb$^{-1}$), much more than the
CDF data acquisition rate of $\simeq$75 Hz to tape. Since
$B$ production at high $p_T$ peaks at small $|y|$, the kinematic 
restriction is such that about 6\% of $B$-hadrons are produced in
the region $|y|<1$ and $p_T>6$ GeV/c (e.g., 1.2 kHz of $B$+$\overline{B}$ at
10$^{32}$.  
  
Thus, the basic trigger strategy is to select $B$ decays to 
specific final states in the central region, for example
$B^0\rightarrow J/\psi K_S$ or $B^0_s\rightarrow D^-_s\pi^+$,
with charged particle tracks confined to $|\eta| < 1$.  For mixing
and $CP$ studies, flavor tagging is accomplished using either the away-side
$\overline{B}$ or same-side fragmentation correlations.  The 
fragmentation correlations give a highly efficient flavor tag
($\simeq$ 70\% in Run~I), since the tagging tracks are guaranteed
to be in the central region.  The away side tagging would be limited
if tracking were confined to $|\eta| < 1$, but the stand-alone
SVX tracking extends the coverage in $\eta$, and so is expected to
greatly improve efficiency for jet-charge and lepton
tagging of away-side $b$-jets.  
 
There are some advantages to the central detector
configuration:
\begin{itemize}
\item{The Tevatron beamspot is small ($\sim 25 \mu$m) in
the transverse plane. This feature is exploited at the trigger level in CDF Run~II,
using the SVX detector to identify large-impact parameter tracks
from $B$ decays in the level-2 trigger. This requires the
Tevatron beam to be aligned parallel to the detector axis,
due to the long ($\sim 60$cm) luminous region in $Z$.
The $SVT$- trigger will yield a highly pure sample of inclusive
$B$ decays, approximately 10$^8$ $b\overline{b}$ events per fb$^{-1}$.}
 
\item{With the $p_T(B)$ cut, the proper time resolution
is adequate for demanding analyses such as $B_s$ mixing
(e.g., 45 fs on $B_s\rightarrow D_s^-\pi^+$)}
 
\item{The $p_T(B)$ requirement also improves the ratio
of $B$ production to QCD background.  While the total $b\overline{b}$
cross section is of order 0.2\% of the minimum bias rate, the
high-$p_T$ $b$-jet cross section is measured to be 2\% of the
QCD jet rate, in the $p_T$ range of interest for CDF.}
 
\item{In the central region, tracks are well spread out in $Z$,
and CDF has demonstrated highly efficient track reconstruction
for $b\overline{b}$ events.}
\end{itemize}

\section{Trigger Strategies}
 
The main $B$-physics goals for CDF Run~II make use of the following
final states.  Of course in each case, other final states, both
inclusive and exclusive, are used as calibration samples, and so the triggers
must be designed to be inclusive.
\begin{itemize}
\item{$B^0\rightarrow J/\psi K_S$ for $\sin{2\beta}$}
\item{$B^0_s\rightarrow J/\psi\phi$ for $CP$ and $\Delta\Gamma$}
\item{$B\rightarrow \mu^+\mu^- K^{(*)}$ rare decays}
\item{$B^0_s\rightarrow D_s^-\pi^+$ for $B_s$ mixing}
\item{$B_s\rightarrow D_s^-\ell^+\nu$ for $B_s$ mixing}
\item{$B^0_{(s)}\rightarrow K^{*0}\gamma, \phi\gamma$ using photon conversion $\gamma
\rightarrow e^+e^-$}
\item{$B^0\rightarrow \pi^+\pi^-$, $B^0_s\rightarrow K^+K^-$ for CKM
angle $\gamma$}
\end{itemize}
This is of course only a partial list.  CDF has a three-level trigger
design.  Level-1 is pipelined so as to be deadtimeless. Digitized 
data are stored in 42$\times$132 ns buffers to create and count trigger 
primitives (electrons, muons, COT tracks, and combinations of these).
Level-2 is a global trigger that allows finer grained electron and
muon track matching, and more sophisticated
combinations of trigger primitives (for example, opposite charged
lepton pairs with invariant mass requirements).  
Level-2 finds SVT-tracks using four SVX II layers; these are
matched to the XFT/COT tracks found in level-1 to define high
resolution, large impact parameter $b$-decay candidates. Level-2
also matches electrons to the central shower-maximum detector. Finally, Level-3 
provides offline quality tracking and calorimeter reconstruction.
 
Using Run~I data, CDF has carefully optimized the use of bandwidth
for Run~II triggers.  The basic trigger requirements can be
summarized as follows:
\begin{itemize}
\item{ Dimuons and dielectrons}
\subitem{Level-1: Two leptons with COT matched to muon chambers or central calorimeter}
\subitem{Level-2,3: Additional mass, charge, and $\Delta\Phi$ cuts}
\item{Single Leptons}
\subitem{Level-1: Single leptons matched to muon chambers or central calorimeter}
\subitem{Level-2,3: Additional requirement of accompanying SVT track}
\item{$B$ hadronic triggers}
\subitem{Level-1: two tracks, opposite charge, $\Delta\Phi$ cuts}
\subitem{Level-2,3: two SVX tracks, impact parameter and $ct$ cuts}
\end{itemize}
Thresholds for dilepton and two-track triggers are typically 1.5-2.0 GeV/c,
and for single leptons 3-4 GeV/c.  Level-3 can make additional cuts in order
to divide the data into manageable data sets; for example, a high mass
cut will be used to define a $B\rightarrow\pi\pi$ stream based on
the two-track triggers, while other SVX track requirements will be used
to make a $b\rightarrow c$ generic hadronic sample.
 
The total trigger bandwidth, including $B$ physics, is
designed to be 40 kHz (level-1), 300 Hz (Level-2) and 75 Hz Level-3).
Thus, the maximum rate level-3 would be 750 nb at 10$^{32}$ luminosity.
$B$ triggers are expected to require about half of that bandwidth.
Table~\ref{tab:rate} gives the expected bandwidth requirements for
the trigger streams.

\begin{table}
\begin{center}
\begin{tabular}{l|r|r|r}
\hline\hline
Trigger & L1 $\sigma$ [nb] & L2 $\sigma$ [nb] & L3 $\sigma$ [nb]\\
\hline
$B\rightarrow h^+h^-$  & 252,000 & 560 & 100 \\
$B\rightarrow \mu \mu (X)$ & 1100 & 90 & 40\\
$J/\psi \rightarrow ee$ & 18000    & 100 & 6\\
Lepton plus displaced track     & 9000     &130     & 40\\
\hline \hline
\end{tabular}
\vspace*{.2cm}
\caption[CDF $B$-physics trigger rates]{Trigger rates for the main CDF II $B$-physics triggers}
\label{tab:rate}
\end{center}
\end{table}

\section{Offline Analysis and Simulation}
CDF has estimated rates for literally hundreds of 
reconstructable decay modes that come in on the lepton or
SVT two-track triggers.  Table~\ref{tab:cdfyield} lists
some examples.

\begin{table}
\begin{center}
\begin{tabular}{l|r}
\hline\hline
Physics Channel & Event yield (2 fb$^{-1}$)\\
\hline
$b\rightarrow J/\psi X$ & 28,000,000\\
$B^0\rightarrow J/\psi K^0_S$ & 28,000\\
$B_s\rightarrow D_s\pi$ & 10,000\\
$B_s\rightarrow D_s\ell\nu$ & 30,000\\
$B\rightarrow \mu^+\mu^- K^{(*)}$ & 50\\
$B\rightarrow K^*\gamma$ & 200\\
$B\rightarrow \pi\pi,K\pi, KK$  & 30,000\\
\hline \hline
\end{tabular}
\vspace*{.2cm}
\caption[CDF $B$-physics event yields]{Event yields expected from CDF Run~II
$B$-physics triggers}
\label{tab:cdfyield}
\end{center}
\end{table}

%For two of the more important physics goals, $\sin{2\beta}$ and 
%$B_s$ mixing, Fig.~\ref{fig:lumlim} shows the respective $\sin{2\beta}$ error and
%$x_s$ 5-$\sigma$ reach as a function of integrated luminosity.
Detailed studies of the physics reach expected for CDF may be found elsewhere
in this document. We discuss briefly two issues that enter into calculations of
sensitivity, namely flavor tagging and generic simulation.

%\begin{figure}[h]
%\centerline{
%\epsfxsize=4.2in
%\epsfbox{ch3/lumlim.ps}
%}
%\epsfbox{trackvol.ps}
%\caption{Error on $\sin{2\beta}$ (left) and 5$\sigma$ reach
%for $x_s$ (right) versus CDF II integrated luminosity.}
%\label{fig:lumlim}
%\end{figure}

\subsection{Flavor Tagging}
CDF measured the flavor tagging efficiencies and dilutions for
three methods in the Run~I mixing and $\sin{2\beta}$ analyses:
``same side'' (fragmentation correlation) tagging, jet-charge
tagging, and soft electron and muon tagging.  To understand
the ``same side'' tagging at least qualitatively, CDF 
analyzed $B^{**}$ production using large samples
of semileptonic $B\rightarrow D^{(*)}\ell\nu$ decays
and applied these to tune the PYTHIA event generator.
This gave qualitative agreement with observed same-side
tagging efficiencies, including the observed differences between
$B^0$ and $B^+$ tagging caused by kaons and protons.  For jet-charge
algorithms, the tagging was optimized initially using
PYTHIA and HERWIG event generators, but actual efficiencies and
dilutions were measured with data, for example, by fitting the
$B^0\overline{B^0}$ mixing amplitude or by direct comparison
of tagging in $B^{\pm}\rightarrow J/\psi K^{\pm}$.  Soft lepton
tags were calibrated similarly.  For the $\sin{2\beta}$ analysis the
combined tagging efficiency, including correlations between tags,
was $\epsilon D^2$ = 6.3$\pm$1.7\%.
 
CDF extrapolates the Run~I efficiencies and dilutions to
Run~II conditions, taking into account the standalone silicon
tracking to $|\eta|\sim 2$, and including kaon tagging based on
TOF.  This gives estimates of $\epsilon D^2 \sim$ 9.1\% for $B^0\rightarrow J/\psi K_S$
and $\epsilon D^2 \sim$ 11.3\% for $B_s\rightarrow D_s^-\pi^+$.
The basic strategies for calibrating and optimizing the tagging
efficiencies for each method would be similar to Run~I.  For example,
the same side tags can be optimized using semileptonic decays
$B^{0,+}\rightarrow D^{(*)}\ell\nu$ and $B^0_s\rightarrow D_s^{(*)}\ell\nu$,
including the effects of fragmentation kaons.
Opposite side tags can be optimized on the same semileptonic channels,
or on inclusive samples such as $b\rightarrow J/\psi X$ (e.g., by
comparing tagging rates for prompt and long lived $J/\psi$'s).
Same-side tags for $\sin{2\beta}$ can be calibrated using
$B^{\pm}\rightarrow J/\psi K^{\pm}$ and $B^0\rightarrow J/\psi K^{*0}$.
Opposite side tags would be calibrated using $B^{\pm}\rightarrow J/\psi K^{\pm}$.
Thus, there are a variety of data channels and cross checks
that are available to understand each tagging method.  Event generator
Monte Carlo's play a role in helping to model same-side tags, including
the effects of kaons, and in understanding correlations and variations
of dilution with kinematics.

\subsection{Monte Carlo Issues}
\index{CDF!detector simulation}
CDF has adopted a GEANT based detector simulation for Run~II,
which is used for optimization of track reconstruction,
muon matching, jet energy analysis, SVT response, etc.  The calorimeter response
has to be tuned to match data from testbeam, Run~I, and eventually Run~II.
 It is obviously straightforward to understand the efficiency
for signals such as $B^0\rightarrow \pi^+\pi^-$, where the generation is
trivial, and the detector response depends only on tracking and SVT efficiency.
The main concern for $B$ triggers is backgrounds.  For lepton-based
triggers, these can be estimated from Run~I data.  For hadronic triggers
such as $B^0\rightarrow \pi^+\pi^-$, CDF can set limits on backgrounds
using Run~I data, and can use the PYTHIA generator (plus SVT simulation)
to estimate the backgrounds from real $b\overline{b}$ events.  The backgrounds
that dominate $B^0\rightarrow \pi^+\pi^-$ at the trigger level do not
appear to come from real $b\overline{b}$ events but from QCD backgrounds that
fake the impact parameter trigger; this conclusion comes from comparison
of trigger rates with $b\overline{b}$ Monte Carlo,
and also by examining Run~I data, comparing 
high mass two-track rates rates from QCD and real $b\overline{b}$ events.
Thus, while CDF can certainly estimate backgrounds from real $b\overline{b}$ events
in the SVT two-track trigger sample (these come in due to real
$b$-hadron lifetimes), it cannot reliably simulate ``QCD'' backgrounds
in this sample (these depend critically on offline SVX reconstruction).



\clearpage
\begin{thebibliography}{99}
\bibitem{cdf_tdr}The CDFII Collaboration, ``The CDF Detector Technical Design Report'',
 FERMILAB-Pub-96/390-E

\bibitem{cdf_p909}The CDFII Collaboration ``Proposal for Enhancement of the CDF II
Detector: an Inner Silicon Layer and a Time of Flight Detector'',
submitted to the PAC, October 1998

\end{thebibliography}
\chapter{The D\O\ detector}

\authors{R.~Jesik, F.~Stichelbaut, K.~Yip, A.~Zieminski}

\def\Do{\mbox{D\O}}
\def\pp{$p\bar{p}$}
\def\ptmu{$p_T^{\mu}$}
\def\pt{$p_T$}
\def\ptb{$p_T^b$}
\def\ptc{$p_T^c$}
\def\etamu{$\eta^{\mu}$}
\def\dsdptmu{\mbox{d$\sigma_b^{\mu}/\mathrm{d}p_T^{\mu}$}}
\def\dsdptb{\mbox{d$\sigma^b/\mathrm{d}p_T^b$}}
\def\bb{\mbox{$b\bar{b}$}}
\def\cc{\mbox{$c\bar{c}$}}
\def\QQ{\mbox{$Q\bar{Q}$}}
\def\ptuu{\mbox{$p_T^{\mu\mu}$}}
\def\bsdsphi{\mbox{$B_s^0 \rightarrow D_s^- + \mu^+ + \nu_{\mu}$}}
\def\bsds3pi{\mbox{$B_s^0 \rightarrow D_s^- + 3\pi^{\pm}$}}
\def\bsdslX{\mbox{$B_s^0 \rightarrow D_s^- + \ell^+ + X$}}
\def\bsdsnpi{\mbox{$B_s^0 \rightarrow D_s^- + n\pi^{\pm}$}}
\def\bks{\mbox{$B_d^0 \rightarrow K_s + J/\psi$}}
\def\jpsi{\mbox{$J/\psi$}}
\def\bkspsi{\mbox{$B_d^0 \rightarrow J/\psi K_S^0$}}
\def\bkst{$B_d^0 \rightarrow K^{*\circ} \mu^+\mu^-$}
\def\bkstee{\mbox{$B_d^0 \rightarrow K^{*\circ} e^+e^-$}}
\def\buu{\mbox{$B_s^0 \rightarrow \mu^+\mu^-$}}
\def\bsuu{ \mbox{$b \rightarrow s \mu^+\mu^-$}}
\def\dskkp{\mbox{$D_s^- \rightarrow K^+ K^- \pi^-$}}
\def\dsphi{\mbox{$D_s^- + \mu^+$}}
\def\dsmunu{\mbox{$D_s^-\, \mu^+\, \nu_{\mu}$}}
\def\ds3pi{\mbox{$D_s^- + 3\pi^{\pm}$}}
\def\kspsi{\mbox{$K_s + J/\psi$}}
\def\QQmumu{\mbox{$Q\bar{Q}\rightarrow \mu \mu$}}
\def\QQmu{\mbox{$Q\bar{Q}\rightarrow \mu$}}
\def\QQee{\mbox{$Q\bar{Q}\rightarrow e e$}}



The D\O\ Run II detector \cite{upgrade} shown in Fig.~\ref{d0det}
builds on the detector's previous strengths of excellent calorimetry
and muon detection in an extended rapidity range. The major addition to
the apparatus is a precision tracking system,
consisting of a silicon vertex detector surrounded by an eight layer central
fiber tracker. These detectors are located inside a 2 T solenoid magnet.

\section{The silicon vertex detector}

\begin{figure}[t]
\centerline{\psfig{figure=ch4/d0det.ps,height=9.5cm}}
\caption{The D\O\ detector}
\label{d0det}
\end{figure}

The silicon vertex detector\index{\Do\ silicon vertex detector}(SMT), shown in
Fig.~\ref{SMT}, is a hybrid barrel and disk design. The central detector,
covering $|z| < 32$ cm, consists of six  barrels with disks interspersed
between them.
Each barrel module consists of four radial layers of detector
ladders located at a radius of 2.7, 4.5, 6.6, and 9.5 cm,
respectively. Layers one and three have 50 $\mu$m pitch double sided
silicon microstrip detectors (with axial and $90^\circ$ strips) in
the four inner barrels, and have single sided (axial strip) detectors
in the two end barrels. Layers two and four have double sided detectors
with axial and 2$^\circ$ stereo strips. Each disk module has twelve
wedge-shaped double sided
detectors (extending radially from 2.7 to 10 cm) with 30$^\circ$ stereo angle.
The forward detector
consists of six disks of similar design (three located near each end of 
the central detector) plus four larger 24-wedge disks made of single sided
detectors, located at z = $\pm$ 110 cm and $\pm$ 120 cm. The detector
is read out using SVX-II chips and has 800,000 total channels.   
The silicon vertex detector provides tracking information out
to $|\eta|$ = 3 and gives a reconstructed vertex position resolution
of 15-40 $\mu$m in $r-\phi$ and 75-100 $\mu$m in $z$, depending on
the track multiplicity of the vertex.  

\section{The central fiber tracker}

The central fiber tracker\index{\Do\ central fiber tracker} consists of 74,000 scintillating fibers
mounted on eight concentric carbon fiber cylinders at radii from
19.5 to 51.5 cm. Each cylinder supports four layers of fibers, one
doublet in the axial direction and one doublet oriented at a 
$3^\circ$ helical stereo angle for odd numbered cylinders and at
$-3^\circ$ for even ones. The fibers are multi-clad and
have a diameter of
830 $\mu$m. Clear fiber waveguides carry the light 
for about ten meters from the scintillating fibers to the 
visible light photon counters situated in cryostats under the
calorimeter. These are silicon devices that have a
quantum efficiency of over $80\%$ and a gain of about 50,000.
They operate at a temperature of about 10 K. Digitization is
also performed with SVX-II chips and the axial fibers are used
to form fast level 1 trigger track objects.

\begin{figure}[bh]
\centerline{\psfig{figure=ch4/d0smt.eps,width=0.95\textwidth}}
\caption{The D\O\ silicon vertex detector}
\label{SMT}
\end{figure}

The combined silicon vertex detector and central fiber tracker
have excellent tracking performance. The full coverage of the
entire combined detector extends out to $|\eta|$ =  1.6.
Tracks will be reconstructed with high efficiency ($\sim 95\%$ 
in the central region) with a resolution of 
$\sigma_{p_T}/p_{T}^2 \sim 0.002$. The silicon detector disks
allow efficient tracking out to an $|\eta|$ of 3.  
     
\section{Electromagnetic pre-shower detectors}
Another important piece of the upgrade is the new pre-shower 
detectors attached to the inner surfaces of the calorimeter
cryostats. These scintillating strip detectors greatly improve
electron identification by providing a finer grain spatial match
to the inner tracker than is obtained by the calorimeter alone.
This reduces electron trigger backgrounds by a factor of 5-10.
We will be able to trigger on electrons with $p_T$ down to 1.5 
GeV/$c$, and out to an $|\eta|$ of 2.5.

The calorimeter\index{\Do\ calorimetry} itself remains unchanged from its Run I
configuration. It is a uranium-liquid argon sampling calorimeter with fine
longitudinal and transverse segmentation, $\Delta\eta \times \Delta\phi = 0.1
\times 0.1$, that allows electromagnetic showers to be distinguished from
hadronic jets. It has full coverage out to $|\eta| = 4$ with energy resolutions
of $15\%/\sqrt{E}$ for electromagnetic showers and $80\%/\sqrt{E}$ for hadronic
jets.

\section{Muon Spectrometers}

The central part of the muon system\index{\Do\ muon detector} (covering the $|\eta| < 1$ range) includes 94 proportional drift tubes (PDT), barrel scintillator counters (A-PHI) and cosmic ray veto scintillator counters. There are three PDT layers (A, B, and C) with three or four 5-cm-thick, $\pm 5$-cm-wide drift cells per layer. A $r - \phi$ magnetic field, with an average magnitude of 16.7 kGauss, is contained in  an iron toroid sandwiched between layers A and B of the PDT system. The A-PHI counters have nine segments in the $z$ direction and 80 segments in the $\phi$ direction. There are 630 A-PHI counters in total.

	The forward muon system (located at both endcaps) covers the $\eta$ [$\pm1, \pm2$] region of the \Do\ detector. It includes mini-drift tubes (MDT's with A, B, and C layers) with three or four decks of 1-cm square cells per layer. Iron toroids are located between the A and B layers of the MDT systems. The average magnitude of the field is 16.0 kGauss. Three layers of PIXEL scintillating counters, located next to corresponding MDT stations, have a $4.5^\circ$ $\phi$-segmentation and a 0.1 $\eta$-segmentation.

 The upgraded muon system offers excellent efficiency, purity, and coverage. We are 
able to trigger on muons with $p_T$ down to 1.5 GeV/$c$, and
out to an $|\eta|$ of 2.      


\section{Trigger system}

Significant upgrades to the trigger\index{\Do\ trigger system} and data
 acquisition systems are also required to operate in the high luminosity ($L = 2 \times 10^{32}$cm$^{-2}$
S$^{-1}$), high rate (7 MHz) environment of the upgraded TeVatron. The D\O\
Run II trigger consists of three levels.

\subsection{Trigger levels}

The first level (L1) trigger consists of
signals from the axial layers of the CFT, the pre-shower detectors, the 
calorimeter, and the muon scintillators and tracking chambers. The level 1
 trigger hardware for each of these systems examine the event and report
information to an array of front-end digitizing crates which have sufficient
memory to retain data from 32 crossings. Trigger decisions are made in less
than 4.2 $\mu$s at this level, providing deadtimeless operation with a maximum
accept rate of 10 kHz. Upon a level 1 trigger accept, the entire event is 
digitized and moved into a series of 16 event buffers to await a level 2
decision.

%\subsection{The level 2 trigger}
The level 2 (L2) trigger will reduce the L1 accept rate of 10 kHz by 
roughly a factor of 10 within 100 $\mu$s by correlating multi-detector
objects in an event. In the first stage, or preprocessor stage, each
detector system builds a list of trigger information. This information
is then transformed into physics objects such as energy clusters or 
tracks. The time required for the formation of these objects is about
50 $\mu$s. These objects are then sent to the level 2 global processor
where they are combined and trigger decisions are made. For example, 
spacial coorelations between tracking segments, pre-shower 
depositions, and calorimeter energy depostions may all be used to 
select electron candidates. The deadtime of the level 2 trigger is
expected to be less than $5\%$.

%\subsection{The level 3 trigger}
The third and final stage (L3) of the trigger will reconstruct events
in a farm of PC processors with a final accept rate of 50 Hz. 


\subsection{Level 1 muon triggers}
\label{sec_trigger}

D\O\ Level 1 muon trigger
\index{\Do\ muon trigger! Level 1! description} 
(L1MU)\cite{muon} identifies muon candidates by using combinatorial logic that makes use of tracks from L1 Central Fiber Tracker (L1CFT) trigger and hits from all muon detector elements: drift chambers and scintillation counters.	The central, north and south regions of the detector are divided into octants. The L1MU triggers are formed locally in each octant. For the purpose of this report the L1MU trigger terms are labelled by two digits and two letters
\index{\Do\ muon trigger! Level 1! notation}
: L1MU(i, j, A, B).
\begin{itemize}
   \item The first number refers to the number of muons requested:
   \item The second number corresponds to an approximate value of 
                    the muon \pt{} threshold in GeV/$c$. 
   The nominal values are 2,4,7 and 11 GeV/c.
   \item The first letter refers to the covered $\eta^{\mu}$ region:
   \begin{itemize}
      \item C = Central, with $\mid \eta^{\mu} \mid \leq 1$;
      \item A = All, with $\mid \eta^{\mu} \mid \leq 1.6$;
      \item X = eXtended, with $\mid \eta^{\mu} \mid \leq 2$.
   \end{itemize}
   \item Finally, the second letter describes the muon tag criterium:
   \begin{itemize}
      \item M stands for Medium tag, requiring a L1CFT track matched with the PDT or MDT centroids, and with at least one layer-scintillator confirmation.
      \item T stands for Tight tag, requiring a L1CFT track combined with the PDT or MDT centroids, and with a two layer-scintillator confirmation.
   \end{itemize}
\end{itemize}

\subsection{Level 1 muon trigger rates and efficiencies}

    Performance of various muon triggers was evaluated in Ref.~\cite{FS_note}.
For this analysis we concentrated on:
   \begin{itemize}
      \item the single muon trigger L1MU(1,4,A,T) 
      \item and the dimuon trigger L1MU(2,2,A,M). 
    \end{itemize}

The $\eta^{\mu}$
coverage of both triggers extends to $\mid \eta^{\mu} \mid \leq 1.6$ and 
effective
muon \pt{} thresholds are 2 GeV/c for the dimuon trigger and 4 GeV/c for the
single muon trigger. Muons with \pt{} $>$ 1.5 GeV/c have a chance to penetrate
the calorimeter, muons with \pt{} $>$ 3.5 GeV/c have a chance to be detected in
the entire muon detector, including B and C layers outside of the toroid
magnet.
 
    To estimate the QCD  trigger rates, we used the 
dijet 
ISAJET events generated in 6 \pt{} intervals (2--5, 5--10, 10--20, 40--80
and 80--500) with 1, 3, 5 or 7 additional minimum bias interactions
(\pt{} between 1 and 100 GeV/$c$) per event. 
The background trigger rates were computed
with the assumptions of an instantaneous luminosity of 
$2\times10^{32}$ cm$^{-2}$s$^{-1}$, a beam crossing time interval of
132 ns and a dijet total cross section of 57 mb.
    Studies of
Ref.\cite{FS_note} indicate that the expected Level 1 trigger rates at this 
instantaneous
luminosity are approximately
400 Hz and 80 Hz for the L1MU(2,2,A,T) and L1MU(1,4,A,T) triggers, 
respectively.
\index{\Do\ muon trigger! Level 1! trigger rates}%
The event samples used in these studies  contained about 1000 events each. 
Therefore some rate calculation could
suffer from large fluctuation due to the small number of selected events.
The absolute rates obtained from these samples 
could be underestimated by a factor up to two.


    For the trigger efficiency studies we have used a sample of \bks\ events processed through the latest version
of D0GEANT and the trigger simulator. Trigger efficiencies are normalized to the numbers of events with $\mid \eta^{\mu}\mid < 1.6$ and 
    kinematic cuts imposed on $p_T^{\mu}$, and $p_T^{\mu\mu}$ as
    specified in the Table~\ref{ta:trigger}.
\index{\Do\ muon trigger! Level 1! efficiency for dimuon events}%
 A large difference between efficiencies for the L1MU(1,4,A,M) and  
L1MU(1,4,A,T) triggers reflects the fact that the "tight" tag condition
requires two scintillator signals for each muon. This requirement reduces the
geometrical acceptance of the trigger due to a limited coverage of the 
scintillating
counters at the bottom of the detector.

\begin{table}
\centering
\begin{tabular}{l||c||c|c} \hline \hline
   $p_T^{\mu\mu}$ & $ > 2.0$ GeV/c & $> 5.0$ GeV/c& $> 5.0$ GeV/c \\
   $p_T^{\mu}$ & $    > 1.5$ GeV/c & $> 1.5$ GeV/c& $> 3.0$ GeV/c \\ \hline
    L1MU(2,2,A,M)  &27\% & 46\% & 57\%  \\
    L1MU(1,4,A,M)  &33\% & 69\% & 78\%  \\
    L1MU(1,4,A,T)  &15\% & 32\% & 49\%  \\
  dimuon or single (M) &41\% & 71\% & 80\%  \\
  dimuon or single (T) &32\% & 55\% & 67\%  \\
\hline \hline
\end{tabular}\vspace*{4pt}
\caption[Trigger efficiencies for dimuon events preselected with the kinematic
cuts listed~]{Trigger efficiencies for dimuon events preselected with the
kinematic cuts listed}
\label{ta:trigger}
\end{table}


 The trigger rates at the instantenous luminosity of 2 $10^{32} cm^{-2}s^{-1}$
due to dimuons from the
    genuine $Q{\bar Q}$ signal
\index{\Do\ muon trigger! Level 1! $Q{\bar Q}$ dimuon rates} 
    are $\approx$ 13 Hz ($\approx$ 4 Hz for $p_T^{\mu\mu}> $ 5 GeV/c). The 13
    Hz combines contributions from: \cc\ pair production (2.5 Hz), \bb\ pair
    production (9.5 Hz) and $b \rightarrow J/\psi +X$ decays (1.0 Hz). A
    requirement of dimuon $p_T^{\mu\mu} > 2$ GeV/$c$ reduces the rate to 9~Hz.

To estimate the contribution from $J/\psi$ production
\index{\Do\ muon trigger! Level 1! $J/\psi$ event rates} 
to the single muon 
trigger rate, we normalized the relevant Monte Carlo samples 
($b\rightarrow J/\psi + X$ and prompt $J/\psi$'s)
to the CDF measurement of the 
$J/\psi$ cross sections in the kinematic range $p_T^{J/\psi} > 5$ GeV/$c$ 
and $\mid \eta(J/\psi) \mid <0.6$~\cite{CDFpsi}.  
With this normalization we find that the irreducible L1 trigger rate 
for signal events is at least 4.0 Hz for prompt $J/\psi$'s 
and 0.8 Hz for $J/\psi$'s from $b$-quark decays
($J/\psi$ with $p_T^{J/\psi} > 2$ GeV/$c$ and $\mid \eta^{J/\psi} \mid <1.5$).   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               SVT study
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Impact of a Silicon Track Trigger }
%\label{sec_svt}

\subsection{Level 2 and Level 3 muon triggers}

	The second level of the muon trigger
\index{\Do\ muon trigger!Level 2 and 3! description} 
(L2MU) uses calibration and more precision timing information to improve the quality of muon candidates. Fast processors and a highly parallelized data pathway are the basis of the L3 muon trigger (L3MU). L3MU improves the resolution and the rejection efficiency of L2MU candidates. This is accomplished by performing local muon tracking, by adding inputs form the calorimeter and the sillicon micro-strip tracker (SMT) and by performing more analytical calculations on CFT tracks and PDT and A-PHI hits with the calibrated data.
The performance of the  L2MU and L3MU triggers has not been fully evaluated at the time of this writing.

    In addition, we expect to have the STT (Silicon Tracker Trigger)
\index{\Do\ Silicon Tracker Trigger} 
  preprocessor installed
in the middle of 2002. The STT will be part of the Level 2 trigger and will
provide an option of triggering on displaced vertices (impact parameter
significance). It will also further improve momentum resolution of muon tracks.

   The STT will allow to tag $B$ decays using displaced secondary
         vertices or tracks with large impact 
         parameters. The expected impact parameter resolution in 
the transverse plane can be parametrized as~\cite{FS_note}:
         \begin{equation}
          \sigma^2(d_0) = (12.6 \ \mu\mathrm{m})^2 +
                       \left(\frac{36.6\ \mu\mathrm{m}\, GeV}
          {p\times\sin^{3/2}\theta}\right)^2
         \end{equation}

This dependence on particle momentum and polar angle is illustrated
in Fig.~\ref{IMPACT}.
\index{\Do\ Silicon Tracker Trigger!
 impact parameter resolution} 


\begin{figure}[t]
\centerline{\psfig{figure=ch4/btag_sigmad0.eps,height=7cm}}
\caption{Expected impact parameter uncertainty dependence on track momentum and
polar angle.}
\label{IMPACT} 
\end{figure}

In Fig.~\ref{IMPACT_KST} we show the impact parameter  significance, $S =
d_0/\sigma(d_0)$,  for the \bkst{} decay products, under the condition that all
four charged particles are produced with $p_T \geq 0.5$ GeV/$c$ (muons with
$p_T \geq 1.5$ GeV/$c$). On the average 1.8   particles from the \bkst{} decay
will have an impact parameter significance greater than 2. This number
increases to 3.2 for events preselected by a request of a 400 $\mu$m
transversal separation between primary and secondary vertices.

\begin{figure}[bh]
\centerline{\psfig{figure=ch4/btag_sip.eps,height=7cm}}
\caption{Significance distribution for STT tracks reconstructed in \bkst{}
events with all four charge particles produced with $p_T \geq 0.5$ GeV/$c$.}
\label{IMPACT_KST}
\end{figure}

    Therefore, once STT becomes operational, we intend to use a simple $B$ tagging algorithm based on 
the number of tracks in the event 
         with a significance greater than $S_{min}$ to improve our Level 2
         trigger rates for selecting \bb\ events. The algorithm was tested on 
charged tracks with
        $p_T > 0.5$ GeV/$c$;
       $\mid \eta \mid < 1.6$;
	and with hits in at least 3 layers of the SVX~\cite{FS_note}. As shown
	in Fig.~\ref{BTAG_EFF}  an
        efficiency of 50\% can be achieved by requesting at least two tracks 
    per event
        with an impact parameter significance greater than 2. Similarily, requiring at least one track with an  impact parameter significance
greater than 5 permits the reduction of the background rate by a factor
10 while keeping 50\% of the signal (assuming a primary vertex smearing
of 30 $\mu$m). 
\index{\Do\ Silicon Tracker Trigger!
 efficiency} 

\begin{figure}[t]
\centerline{\psfig{figure=ch4/btag_effbks.eps,height=7cm}}
\caption{\bb{} event tagging efficiency as a function of $S_{min}$ with either
one or two tracks in the event satisfying the condition $S > S_{min}$.}
\label{BTAG_EFF}
\end{figure}

Finally, we have studied the effect of the STT trigger on the separation of the
$J/\psi$ signal due to $b$-quark decays from the prompt $J/\psi$ production.
In case we need to suppress the prompt $J/\psi$ signal, requiring at least 
one track with $\mid S_B \mid > 2.5$ provides a factor 7
reduction at the 30\% cost to the non-prompt $J/\psi$ signal.
\index{\Do\ Silicon Tracker Trigger! 
 prompt $J/\psi$ separation}

\subsection {Low \pt\ dielectron trigger}

 \Do\ has also introduced a dielectron trigger\cite{electron} aimed at
detection of soft electron pairs, primarily from $J/\psi$ dielectron decays.
\index{\Do\ dielectron trigger}%
Level 1 electron candidates are selected with a transverse energy deposit $E_T>2.0~GeV$ in the 
EM calorimeter trigger towers, and with a low $\rm p_T>1.5~GeV/c$ track coincident 
with a pre-shower cluster. The two ``electrons'' are required to have opposite signs 
and to match  within a quadrant in $\phi$ with the EM deposits. 
In the forward rapidity region ($1.5<|\eta|<2.5$) the trigger is based on EM deposits 
and preshower clusters only, since no tracking 
coverage is available.
 
 The dielectron Level 2 trigger is based on a refined spatial matching between tracks and
$\rm 0.2\times 0.2$ EM trigger towers as well as on the information on the  EM fraction of the clusters and their isolation. Finally, invariant mass and angular criteria are applied 
to select $J/\psi$ dielectron  decays. 

	Efficiencies and rates for this trigger have been estimated using {\it ISAJET} generated events 
overlaid with 2 extra interactions. Trigger rates at the nominal Run II luminosity 
are expected to be  below 1~kHz at Level 1 and about 150~Hz at Level 2.
 The expected yield of
 triggered $B_d^0~\to~J/\psi~K^0_s$ events is  
$15\times 10^3$ for an integrated luminosity of 2\,fb$^{-1}$. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                     REFERENCES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage
\begin{thebibliography}{99}

%%%%%%%%%%%%%%%%
% D0 references
%%%%%%%%%%%%%
\bibitem{upgrade} S. Abachi {\it et al. ``The D\O\ upgrade''}, \Do Note 2542, \\
\verb"http://www-d0.fnal.gov/hardware/upgrade/upgrade.html".

\bibitem{FS_note}F.~Stichelbaut, M.~Narain and A. Zieminski,
   {\it Triggers for $B$ Physics Studies in Run II}, 
   \Do{} note 3354 (December 1997).

\bibitem{muon}K. Jones {\it et al.}, 
\verb"http://hound.physics.arizona.edu/l1mu/l1mu.htm"; \\
C. Leonidopoulos (for the \Do\ Collab.) CHEP 2001 talk - in press.

\bibitem{electron} P.~ Grannis and A.~ Lucotte, {\it ``Extending the
sensitivity to sin(2$\beta$) in the $B^0/{\bar B^0}$ system using a low pt
electron $J/\psi \rightarrow e^+e^-$ trigger at the upgrade \Do\ detector''},
\Do{} Note 3596 (1999).
 
\bibitem{CDFpsi}F. Abe {\it et al.}, CDF Collaboration, Phys. Rev. Lett. {\bf
76}, 4675 (1996). 

\bibitem{d0gstar} D\O\ RunII GEANT,\\
\verb"http://www-d0.fnal.gov/newd0/d0atwork/computing/MonteCarlo/".

\end{thebibliography}
%\end{document}
\chapter{The BTeV Detector}
\authors{R. Kutschke, for the BTeV Collaboration\cite{btev_auth}}

\section{Introduction}
\index{BTeV!detector}
%%\index{detector!BTeV}

On May 15, 2000 the BTeV Collaboration submitted their
proposal\cite{btev_prop} to the Fermilab management and on 
June 27, 2000 the Fermilab director announced that 
BTeV experiment had been given Stage I approval.
The information given in this chapter is abstracted from
that proposal.   Much additional information
is available in the proposal, including
the physics case for the experiment, a detailed
description of the proposed detector, a description
of the simulation tools used to evaluate the detector design,
a summary of the physics reach, a cost
estimate and extensive appendices.

This chapter will discuss the reasons for choosing a forward
detector, followed by overviews of the detector, the
simulation tools and the analysis software.  
Some illustrative results will also be included.
As discussed in section~\ref{2:ssec:xsec} of this report, all BTeV 
event yields were computed under the presumption that the 
cross-section for $b\bar{b}$ production at the Tevatron is~100~$\mu$b.
All effieciencies and background levels
were computed for
an average 2 interactions per beam crossing, which
corresponds to a luminosity of \hbox{2$\times 10^{32}$ cm$^{-2}$s$^{-1}$}
and an interval of 132~ns between beam crossings.
\index{BTeV!presumed $b\bar{b}$ cross-section}
\index{BTeV!presumed luminosity}
\index{luminosity}

\section{Rationale for a Forward Detector at the Tevatron}
\index{BTeV!detector!forward vs central}

The BTeV detector is a double arm forward spectrometer which 
covers from 10 to 300~mrad, with respect to the colliding
beam axis.\footnote{BTeV refers to both the proton direction and the
antiproton directions as forward.}  It resembles a pair of
fixed target detectors arranged back-to-back.
In Section 2.1 of the BTeV proposal\cite{btev_prop} the reasons
for this choice are explained in detail; a summary is presented here.

According to QCD calculations of $b$ quark production, 
there is a strong correlation between the $B$ momentum and pseudorapidity,
$\eta$.  Near $\eta$ of zero, $\beta\gamma\approx 1$, while
at larger values of $|\eta |$, $\beta\gamma$ can easily reach values 
well beyond~6.  This is important because the mean decay length increases
with $\beta\gamma$ and, furthermore, the momenta of the 
decay products are larger, suppressing multiple scattering errors.
As discussed in section~\ref{2:sec:centfor} this is most important
in the trigger.

A crucially important attribute of
$b\bar{b}$ production at hadron colliders is that the $\eta$ of
the $b$ hadron and that of its companion $\bar{b}$ hadron are strongly 
correlated: 
when  the decay products of a $b$-flavored hadron are within
the acceptance of one arm 
of the spectrometer, the decay products of the accompanying 
$\bar{b}$ are usually within the acceptance of the {\em same} arm
of the detector.  This allows for reasonable levels of flavor tagging.

The long $B$ decay length, the correlated
acceptance for both $b$ hadrons and the suppression of multiple scattering
errors make the forward direction an ideal choice.

\section{Detector Description}
\label{5:sec:detector}

A sketch of the BTeV detector is shown in Fig.~\ref{5:fig:btev_det_simp}.
The geometry is complementary
to  that used in current collider experiments. 
The detector looks similar to a fixed target experiment, but
has two arms, one along the proton direction and the other along the
antiproton direction.


\begin{figure}[tbp]
\centerline{\epsfig{figure=ch5/btev_det_simp.eps,width=.95\hsize}}
\caption[A Sketch of the BTeV Detector]
{\label{5:fig:btev_det_simp}A sketch of the BTeV detector. The two 
arms are identical.
\index{BTeV!detector!sketch}
}
\end{figure} 

The key design
features of BTeV include:
\index{BTeV!detector!key features}
\begin{itemize}
\item A dipole located on the IR, which  gives BTeV an effective ``two arm''
           acceptance;
\item A precision vertex detector based on planar pixel arrays;
\item A detached vertex trigger at Level 1 that makes BTeV efficient for
   most final states, including purely hadronic modes;
\item Excellent particle identification using a Ring Imaging Cherenkov Detector
(RICH);
\item A high quality PbWO$_4$ electromagnetic calorimeter capable of
reconstructing 
final states with single photons, $\pi^o$'s, $\eta$'s or $\eta'$'s;
it can also identify electrons;
\item Precision downstream 
      tracking using straw tubes and silicon microstrip detectors,
      which provide excellent momentum and mass resolution;
\item Excellent identification of muons using a dedicated detector with the
ability to supply a dimuon trigger; and
\item A very high speed and high throughput data acquisition system which
eliminates the need to tune the experiment to specific final states.
\end{itemize}                  
%
Each of these key elements of the detector is discussed in detail
in Part~II of the proposal\cite{btev_prop} and is discussed briefly
below.

\subsection{Dipole Centered on the Interaction Region}

A large dipole magnet, bending vertically and with a 1.6 T central field, 
is centered on the
interaction region.  This is the most compact way to provide momentum 
measurements in both arms of the spectrometer.  Moreover the
pixel detector is inside the magnetic field, which
gives the Detached Vertex Trigger the capability of rejecting low
momentum tracks.  Such tracks undergo large multiple Coulomb 
scattering and might sometimes be misinterpreted as detached tracks.

\subsection{The Pixel Vertex Detector}
\index{BTeV!detector!pixel vertex detector}
%%\index{BTeV!detector!pixel vertex detector!spatial resolution}
\index{pixel detector}
\index{vertex detector! BTeV pixel}

In the center of the magnet there is a silicon pixel vertex detector.
This detector serves two functions: 
it is 
an integral part of the charged particle tracking system, providing
accurate vertex information for the offline analysis; and 
it delivers very clean, precision space points to the BTeV vertex trigger.

\begin{sloppypar}
BTeV has tested prototype pixel devices in a test beam at Fermilab. 
These devices consist
of \hbox{50 $\mu$m $\times$ 400 $\mu$m} pixel sensors
bump-bonded to custom made 
electronics chips, developed at Fermilab.  The 
position resolution achieved in the test beam 
is shown in Fig.~\ref{5:fig:pixel_res}; overlayed on that figure
is resolution function used in the Monte Carlo simulation.
The measured resolution is excellent and exceeds the requirement of 9~$\mu$m.
\end{sloppypar}

\begin{figure}[tbp]
%\vspace{-1.5cm}
\centerline{\epsfig{file=ch5/pixel_res.eps, height=3.5in}}
\vspace{0.5cm}
\caption[The Resolution of the Pixel Devices]
{\label{5:fig:pixel_res}The resolution achieved in the test beam run
using 50 $\mu$m wide pixels and an 8-bit ADC (circles) or a 2-bit ADC 
(squares), compared with the simulation (line).}
\end{figure}

The critical quantity for a $b$ experiment is $L/\sigma_L$, where $L$ is
the distance between the primary (interaction) vertex and the secondary (decay)
vertex, and $\sigma_{L}$ is its error.
%For central detectors
%the $B$'s are slower, because the mean transverse $B$ momentum is 5.3 GeV/c,
%virtually independent of the longitudinal momentum. Since they also suffer more
%multiple scattering, they have relatively poorer $L/\sigma_L$ distributions.
%LHC-b, on the other hand, does not benefit by going to higher momentum because,
%after a momentum of around 10 GeV/c (depending on the detector),
%$\sigma_L$ also increases linearly.
The efficacy of this geometry is illustrated by considering the distribution
of the resolution on the $B$ decay length, $L$, for the decay 
$B^o\to\pi^+\pi^-$.
Fig.~\ref{5:fig:das_pipi} shows
the r.m.s. error in the decay length as a function of
momentum;  it also shows the momentum distribution of the $B$'s accepted
by BTeV.
\begin{figure}[tbp]
\vspace{-1.5cm}
\centerline{\epsfig{file=ch5/pipi_decay_dist.eps, height=3.5in}}
%\vspace{-3cm}
\caption[The $B$ Momentum Distribution]
{\label{5:fig:das_pipi}The $B$ momentum distribution for
$B^o\to\pi^+\pi^-$ events (dashed) and the error in decay length
$\sigma_L$ as a function of momentum.}
\end{figure}
The following features are noteworthy:
\begin{itemize}
\item The $B$'s used by BTeV peak at $p$ = 30 GeV/c and average about 40 GeV/c.
\item The mean decay length is equal to $450~\mu$m $\times\, p/M_B$.
\item The error on the decay length is smallest near the peak of the
accepted momentum distribution. It increases at lower values of $p$, due to
multiple scattering, and increases at larger values of $p$ due to the 
smaller angles of the Lorentz-boosted decay products. 
\end{itemize}




\subsection{The Detached Vertex Trigger}
\index{BTeV!detector!detached vertex trigger}
\index{triggering!BTeV}
\index{triggering!detached vertex}

It is impossible to record data from each of the 7.5 million
beam crossings per second. A prompt decision, colloquially called a 
``trigger,'' must be made to record or discard the data from each crossing.
The main BTeV trigger is provided by the silicon pixel detector.
The Level 1 Vertex Trigger inspects every beam crossing and, using only
data from the pixel detector, reconstructs the primary vertices and
determines whether there are detached tracks which could signify a $B$
decay. Since  the $b$'s are at high momentum, 
the multiple scattering of the decay products is minimized, allowing for
triggering on detached heavy quark decay vertices. 

With outstanding pixel resolution, it is possible to trigger efficiently
at Level 1 on a variety of $b$ decays.  The trigger has been fully
simulated, including the pattern 
recognition code.   Table~\ref{5:tab:trigeff} summarizes the
results of the trigger simulations.  The trigger efficiencies
are generally above 50\% for the $b$ decay states of interest and at
the 1\% level for minimum bias background.
The efficiencies for signal channels are quoted as a percentage of 
events which pass the offline analysis cuts.  This is an
appropriate statistic because most of the $b$ decays are not
useful for doing physics: their decay products lie
outside of the fiducial volume of the detector, their decay
lengths are not long enough, their decay products can be ambiguously
assigned to several candidate vertices and so on.   
Separate Monte Carlo runs were performed to measure
overall event rates and bandwidth requirements and to
ensure that the trigger is efficient for generic $b$ and $c$ decays.

\begin{table}
\begin{center}
\begin{tabular}{lc}
\hline\hline
Process & Eff. (\%) \\
\hline\hline
Minimum bias & 1 \\
\hline
$B_s\to D_s^+ K^-$         & 74  \\
$B^0\to D^{*+} \rho^-$     & 64  \\
$B^0\to \rho^0 \pi^0$      & 56  \\
$B^0\to J/\psi K_s$        & 50  \\
$B_s\to J/\psi K^{*o}$     & 68  \\
$B^-\to D^0 K^- $          & 70  \\
$B^-\to K_s \pi^- $        & 27  \\
$B^0\to$ 2-body modes      & 63  \\
$(\pi^+\pi^-, K^+\pi^-,K^+K^-)$  & \\
\hline\hline
\end{tabular}
\end{center}
\caption[Level 1 Trigger Efficiencies in BTeV]
{Level 1 trigger efficiencies for minimum-bias events and for various
         processes of interest.
 For the minimum bias events the efficiency is quoted as a percentage
 of all events but for the signal channels the
 efficiencies are quoted as a percentage of those events which
 pass the offline analysis cuts.  
 All trigger efficiencies are determined
 for an average of two interactions per crossing.
        }
\label{5:tab:trigeff}
\end{table}

\subsection{Charged Particle Identification}
\index{BTeV!detector!RICH}
\index{BTeV!detector!charged particle identification}
%%\index{RICH detector!BTeV}
\index{particle identification!BTeV RICH}

Charged particle identification is an absolute requirement for an
experiment designed to study the decays of $b$ and $c$ quarks. 
The relatively open
forward geometry has sufficient space to install a Ring Imaging
Cherenkov detector (RICH), which provides powerful particle ID 
capabilities over a broad range of momentum.
The BTeV RICH detector must separate pions from kaons and protons in 
a momentum range from $3$ to $70$~GeV/$c$. The lower momentum limit
is determined by soft kaons useful
for flavor tagging, while the higher momentum limit
is given by two-body $B$ decays.
Separation is accomplished using a gaseous freon radiator
to generate Cherenkov light in the optical frequency range. The light is then
focused by mirrors onto Hybrid Photo-Diode (HPD) tubes. 
To separate kaons from
protons below 10 GeV/c an aerogel radiator will be used. 

As an example of the usefulness of this device,
Fig.~\ref{5:fig:dskvsdspi} shows the efficiency for
detecting the $K^-$ in the decay $B_s\to D_s^+K^-$ versus the rejection for
the $\pi^-$ in the decay $B_s\to D_s^+\pi^-$.  One sees that high 
efficiencies can be obtained with excellent rejections. 

\begin{figure}[tbp]
\vspace{-0.8cm}
\centerline{\epsfig{file=ch5/dskvsdspi.eps, height=3.6in}}
%\vspace{-3cm}
\caption[Separation of $B_s\to D_s^+K^-$ and $B_s\to D_s^+\pi^-$ with the RICH]
{The efficiency to detect the fast $K^-$ in the
reaction $B_s\to D_s^+K^-$ versus the rate to misidentify the $\pi^-$ from
$B_s\to D_s^+\pi^-$ as a $K^-$.\label{5:fig:dskvsdspi}}
\end{figure}

\subsection{Electromagnetic Calorimeter}
\index{BTeV!detector!electromagnetic calorimeter}
\index{EM Calorimeter!BTeV}

In BTeV, photons and electrons are detected when they create an electromagnetic
shower in crystals of PbWO$_4$, a dense and transparent medium that 
produces scintillation light.
The amount of light is proportional to the incident
energy. The light is sensed by photomultiplier tubes (or possibly hybrid
photodiodes). The crystals are 22 cm long and have a small transverse 
cross-section, 
26~mm~$\times$~26~mm, providing excellent segmentation.
The energy and position resolutions are exquisite,
\begin{eqnarray}
\frac{\sigma_{E}}{E} &= & \sqrt{{{(1.6\%)^2}\over{E}}+(0.55\%)^2}~, \\
\sigma_x &=& \sqrt{{{(3500~\mu m)^2}\over{E}}+(200~\mu m)^2}~,
\end{eqnarray}
where $E$ is in units of GeV. This leads to an r.m.s. $\pi^o$ mass
resolution between 2 and 5 MeV/$c^2$ over the $\pi^o$ momentum range 1 to 40 GeV/c.

The crystals are designed to point at the center of the interaction region.
They start at a radial distance of 10 cm with respect to the beam-line and
extend out to 160 cm. They cover $\sim$210 mrad. This is smaller
than the 300 mrad acceptance of the tracking detector; 
the choice was made to reduce costs. For
most final states of interest, this leads to a loss of approximately 20\%
in signal. 

At 2 interactions per crossing the calorimeter has a high rate close to the
beam pipe, where the reconstruction efficiency and resolution
is degraded by overlaps with
other tracks and photons. As one goes
out to larger radius, the acceptance becomes quite good. This can be seen
by examining the efficiency 
to reconstruct the $\gamma$ in the decay $B^o\to K^*\gamma$,
$K^*\to K^- \pi^+$.  For this study the decay products of 
the $K^*$ are required 
to reach the RICH detector. Fig.~\ref{5:fig:eff_btev_3sigma} shows the radial 
distributions of the generated
$\gamma$'s, the reconstructed $\gamma$'s and the $\gamma$ efficiency.
The shower reconstruction code, described in Chapter 
12 of the proposal, was developed from that used for the CLEO
CsI calorimeter; for reference, the efficiency of the CLEO barrel 
electromagnetic calorimeter is 89\%.

\begin{figure}[tbp]
%\vspace{-1.5cm}
\centerline{\epsfig{file=ch5/eff_btev_3sigma_prev.eps, width=.95\textwidth}}
%\vspace{-3cm}
\caption[Radial Distributions of Generated and Detected Photons]
{The radial distribution of generated and detected
photons from $B^o\to K^*\gamma$ and the resulting efficiency. The detector
response was simulated by GEANT and clusters of hit crystals were
formed by the BTeV clustering software.  This software is derived
from software used for the Crystal Ball and CLEO experiments.
The charged tracks from the $K^*$ were required to 
hit the RICH. The simulation was run at 2 interactions/crossing.}
\label{5:fig:eff_btev_3sigma}
\end{figure}


\subsection{Forward Tracking System}
\index{BTeV!detector!forward tracking system}
\index{tracking system!BTeV}

The other components of the charged-particle tracking system are straw-tube
wire proportional chambers and, near the beam where
occupancies are high, silicon microstrip detectors. These devices are used 
primarily for track momentum
measurement, $K_s$ detection and the Level 2 trigger. These detectors
measure the deflection of charged particles by the BTeV analyzing magnet and
give BTeV excellent mass and momentum resolution for charged particle
decay modes.


\subsection{Muon Detection}
\index{BTeV!detector!muon detection}
\index{muon detection!BTeV}
Muon detection is accomplished by insisting that the candidate charged track
penetrate several interaction lengths of magnetized iron and insuring that the
momentum determined from the bend in the toroid matches that 
given by the main spectrometer tracking system. The
muon system is also used to trigger on the dimuon decays of the $J/\psi$. This
is important not only to gather more signal but as a cross check on the
efficiency of the main trigger, the  Detached Vertex Trigger. 

\subsection{Data Acquisition System}
\index{BTeV!detector!data acquisition system}
%%\index{data acquisition systems!BTeV}

BTeV has a data acquisition system (DAQ) which is capable of recording
a very large number of events. The full rate of $B$'s whose decay products
are in the detector is very high, over 1 kHz. The rate from
direct charm is 
similar.  Some other experiments are forced by the limitations of their 
data acquisition system to make very harsh decisions on which $B$ events to 
accept. BTeV can record nearly all the potentially interesting $B$ and charm 
candidates in its acceptance. Therefore it can address many topics that might
be discarded by an experiment whose DAQ is more restrictive.
Since nature has a way of surprising us, the openness of the
BTeV trigger and the capability of the DAQ are genuine strengths which
permit the opportunity to learn something new and unanticipated.

\section{Simulation and Analysis Tools}
\index{BTeV!software!simulation tools}
\index{BTeV!software!analysis tools}

The physics reach of BTeV has been established by an extensive and 
sophisticated program of simulations, which is described in detail
in Part~III of the BTeV proposal~\cite{btev_prop}.
For this study $p\bar{p}\to b\bar{b}X$ events were generated using
{\tt PYTHIA}\cite{pythia}
and the $b$ hadrons were decayed using {\tt QQ}. 
These packages are discussed in chapter~2 of this report.
To model the detector response to these events,
two detector simulation packages have been used, BTeVGeant and MCFast.
BTeVGeant is a GEANT\cite{geant} based simulation of the BTeV detector 
which contains a complete description of the BTeV geometry including
the materials needed for cooling, support and readout.  GEANT
models all physical interactions of particles with material and allows 
us to see the effects of hard to calculate backgrounds.
Most of the results presented in this report were obtained using BTeVGeant.
Some of the results presented here, and all earlier BTeV results,
were obtained using MCFast\cite{MCFast}, a fast parameterized simulation
environment which allows the user to quickly change the detector
design without the need to do any coding.
BTeVGeant writes an MCFast geometry file which describes the
same detector in simplified fashion; in this way the number, size
and resolution 
of detector elements is synchronized between the two simulation tools.
MCFast models the most of the processes that GEANT does, 
energy loss, multiple scattering, pair creation, bremstrahlung,
and hadronic interactions, but in
a simplified fashion; for example some of the detector components
are described by simpler shapes, the model of multiple scattering
is purely gaussian and the model of energy deposition in the
calorimeter is parametric.  

\begin{sloppypar}
Chapter~13 of the proposal shows quantitative comparisons between
MCFast and BTeVGeant.  From these studies one sees that
MCFast is a reliable tool for computing
resolutions, efficiencies and the level of backgrounds which  
arise from real tracks; the slower, more complete, BTeVGeant is 
necessary when occupancies and event confusion are the critical issues.
\end{sloppypar}


In most circumstances the simulations are done at the hit level,
not at the digitization level.  That is, the simulation
packages produce smeared measurements, not a stream of device addresses
and
digitized pulse heights thta simulate the raw, experimental data stream.
Digitization level simulations have been done to address the
issue of required bandwidth at various levels of the trigger and
DAQ systems.

For some of the physics studies presented in this report
very high statistics Monte Carlo runs were required to reliably 
estimate the background level.
These studies were performed on a farm of
500~MHz dual Pentium~III Linux machines; over a period of 3 months
an average of about 30 machines (~60~CPUs~) per day were available.
To give one example, over a period of about 1~month, 4.5 million
generic $b\bar{b}$ events were produced to investigate backgrounds
in the channel $B^0\to\rho\pi$.

Brief descriptions of simulation and analysis software for some
specific subsystems were given in section~\ref{5:sec:detector}
of this report.  References to the relevant TDR sections were
also given.  In order to make this report a little more stand
alone, a few more details are given below.

\subsection{Tracking and Track Fitting Software}
\index{BTeV!software!tracking}

In both packages the simulation code keeps track of which tracking
hits were created by which particles.  This information is not used 
by the trigger code, which does full pattern recognition, but it is
used
to check the results of the trigger package and to debug it.
Both simulation packages smear pixel hits using resolution functions
with non-gaussian tails that were measured in the test beam.  Hits in 
the other tracking detectors are modeled with gaussian resolution 
functions.  

  In the offline analysis package no pattern recognition is 
done; instead the Monte Carlo truth table is used to collect all of 
the hits which belong on a track.  Each hit list is then Kalman 
filtered to give the track parameters and their covariance matrix
in the neighborhood of the interaction region.  Extrapolating from the 
excellent performance of the pattern recognition in the trigger,
the final pattern recognition codes are expected to be
be highly efficient and to find few false tracks; therefore the
approximation of perfect pattern recognition gives a reliable
estimate of what BTeV will achieve.

\subsection{Electromagnetic Calorimeter Software}
\index{BTeV!software!calorimeter clustering}

BTeVGeant does a complete development of electromagnetic and 
hadronic showers in all materials and follows the products of these 
showers into neighbouring detector volumes.  When a track or photon
within a shower traverses one of the PbWO$_4$ crystals, 
BTeVGeant deposits energy from that track or photon in the crystal.
The total energy deposited in crystal is summed over each
beam crossing and a parametric function is then used to convert
the total deposited energy into a measured energy plus the error on
the measured energy.  For each crystal, a record is kept of how much 
energy was deposited for which track.  This information is only
used to characterize and debug the reconstruction code --- it is not
used by the reconstruction code.

When modelling shower development, it is necessary
to stop tracing new particles when their energy drops below some
cut-off.  If these cut-off values are set too high, then the showers
are too narrow, resulting in artificially clean events and 
artificially good energy resolution.  If, on the other hand,
these cut-off values are set too low, then simulations
are prohibitively slow.  The studies done to resolve of these 
tradeoffs are discussed in detail in Section~12.1.5 of the BTeV 
proposal.  The final result is that, because adequate CPU power was
available, no significant compromises in the quality of the
simulation were necessary.

MCFast does the same calorimeter bookkeeping as BTeVGeant but 
the model of shower development is parametric, rather than a detailed
following of each generation of particles.  Both packages create identical
data structures so that the same shower reconstruction
and user analysis codes will work on 
events from both simulation packages.

\subsection{Trigger Simulation Software}
\index{BTeV!software!trigger simulation}

Chapter~9 of the proposal describes the overall plan for the trigger
and it describes in detail the algorithm for the Level~1 Detached Vertex
Trigger.   This algorithm has been coded in C and is callable from
by user code within either BTeVGeant or MCFast.  A few pieces of
the code have been ported to the target DSP's and carefully timed.
The rest is written in a high level language for ease of algorithm
development.  

Similarly a prototype for the Level~2 trigger has also been coded.
Since Level~2 will run on standard processes, the code represents
a true prototype, and is not just a clone of the algorithm.

The physics analyses presented in the BTeV Proposal~\cite{btev_prop}
and in this report have
been used as models for possible Level~3 algorithms.

The trigger results presented in Table~\ref{5:tab:trigeff}, and the
results presented in chapters~6 through~8 were obtained using this
trigger simulation software.


\subsection{RICH Software}
\index{BTeV!software!RICH simulation}

For most of the results presented in this workshop, nominal
RICH efficiencies and misidentification probabilities 
were used.  It was assumed that if a reconstructed track contained
a hit in the chambers between the RICH and the EMCal, then that
track could be identified by the RICH.  

In a few selected analyses, notably, $B_s\to D_s K$, a more detailed
simulation was used.  For this analysis tracks passing through the
RICH generated photons which were propagated through the aerogel and
chamber gas and reflected from the mirror.  They were then propagated
to the detector plan where a model of the HPD efficiency was applied.
A pattern recognition algorithm was then run to find Cherenkov
rings from the list of HPD hits.  Particle ID decisions
were made using these reconstructed rings.  The physics reach
predicted by these simulations is presented in chapter~6.

%\subsection{Cluster Finding}

%\subsection{Vertex Finding and Fitting}

\section{Flavor Tagging}
\index{BTeV!flavor tagging}
\index{flavor tagging!BTeV}

Section~\ref{2:sec:exp_common_tagging}
of this report describes the flavor tagging strategies which
are available to $B$ physics experiments at the Tevatron.
In Chapter~15 of the BTeV proposal presents a preliminary study of the
tagging power which can be achieved with the BTeV detector.
This section will summarize the results of that study.
Mixing of the opposite side $B$ meson has not yet been included in the 
results shown here.  The upcoming sections describe the algorithms
used for tagging, and tagging powers which they achieve are summarized in 
Table~\ref{5:tab:tag_effic}.

\subsection{Away Side Tagging}

Three different away side tagging methods have been studied,
lepton tagging, kaon tagging and vertex charge tagging.
The first step in all three methods is to select
tracks which are detached from all primary vertices
in the event.  In events with multiple primary vertices, detached
tracks are only considered if they are associated with the
same primary vertex as is the signal candidate.

\subsubsection{Lepton Tagging}
\index{BTeV!flavor tagging!away side lepton}

 The lepton tagging algorithm must
deal with possible wrong-sign tags which result 
from the cascade $b\to c\to\ell^+$.  Because
leptons from $b\to\ell^-$ and $b\to c\to\ell^+$ 
have quite different transverse momentum ($p_T$) distributions,
good separation can be achieved.
If there was more than one lepton tag candidate in an event,
the highest $p_T$ lepton was chosen to be the tag.

Candidates for muon tags were selected from the detached track list
if they had a momentum greater than 4.0~GeV/c.
A tagging muon with $p_T>1.0$\ GeV/c was 
considered to be from the process $b\to \ell^-$,
while one with $p_T<0.5$\ GeV/c was considered to be from 
the process $b\to c\to\ell^+$, thereby flipping the sign of the tag.

 Candidates for electron tags were selected using a parametrized
electron efficiency and hadron misidentification
probability.  The tagging lepton was required to have
$p_T>1.0$\ GeV/c and was always assumed to come from the 
process $b\to \ell^-$.  There were not enough
MC events to study electrons from $b\to c\to\ell^+$.
  
\subsubsection{Kaon Tagging}
\index{BTeV!flavor tagging!away side kaon}

Because of the large branching ratio for $b\to c\to K^- X$,
kaon tagging is the most potent tagging method at $e^+e^-$ $B$~factories.
At BTeV, in which the multiplicity of the underlying event is much
greater, excellence in both particle identification and 
vertex resolution is required to exploit kaon tagging.
Both are strong points of the forward detector geometry.  

Candidates for kaon tags were selected from the secondary track list if
they were identified as kaons in the RICH detector.  If there was more 
than one kaon tag candidate in an event, the kaon with the
largest normalized impact parameter with respect to the
primary vertex was selected.

\subsubsection{Vertex Charge Tagging}
\index{BTeV!flavor tagging! away side vertex charge}
 In this method a search was made for a detached vertex which is
consistent with being from the charged decay products of
the other $b$.  The charge of that vertex determines the charge of
the $b$.  When the opposite side $b$ hadronizes into a $\bar{B}^0$
or a $\bar{B}_s^0$, the
tagging vertex has a neutral charge and there is
no useful vertex tagging information in the event.  However this method
has the advantage that it is not affected by mixing of
the away side $b$.

\begin{sloppypar}
Tracks from the secondary list were accepted provided
they had \hbox{$p_T>100$~MeV/c} and provided they
had  $\Delta\eta<4$ with respect to the direction
of the signal $B^0$ candidate.  The tracks from the secondary
list were sorted into candidate vertices and only vertices
with a detachment of at least $1.0\, \sigma$ from the primary vertex 
were accepted.
If more than one vertex was found in an event, 
the one with the highest transverse momentum was selected;
if no secondary vertices passed the selection cuts and if
there was at least one track with $p_T>1.0$\ GeV/c, 
then the highest $p_T$ track was selected.  If the charge of the selected
vertex is non-zero, then it determines the flavor of the 
away side $b$.
\end{sloppypar}

  This tagging method is similar to jet charge tagging used by 
other experiments but BTeV has not yet investigated the possibility
of weighting the tracks by their momenta.
   
\subsubsection{Combining Away Side Tagging Methods}
\index{BTeV!flavor tagging! away side combined}
In many events, several of the same side tagging
methods may give results; moreover it can happen that two methods
will give contradictory answers.
BTeV has not yet optimized the method of combining all tagging
information but have used the following simple algorithm.  The
methods were polled in decreasing order of dilution and the first
method to give an answer was accepted.  That is, if  lepton
tagging gave a result, the result was accepted; if not, and if
kaon tagging gave a result, the kaon tag was accepted; if not,
and if the vertex charge tagging gave a result, the vertex
charge tag was accepted.

\subsection{Same Side Tagging}
\index{BTeV!flavor tagging!same side}
BTeV has studied the power of
same side kaon tagging for $B_s$ mesons.
For this study tracks were selected provided
they had a momentum greater than 3.0~GeV/c,
were identified as kaons in the RICH and
had an impact parameter with respect to the primary vertex
less than 2~$\sigma$.  It was further required that 
the system comprising the $B_s$ candidate plus the candidate
tagging track have an
invariant mass less than 7.0~GeV/c$^2$.
If more than one track passed these cuts, then the track
closest in $\phi$ to the $B_s$ direction was selected.

For same side tagging of $B_d$ events BTeV expects to use $B^{**}$ 
decays.  This study is at an early, conceptual stage.
The flag in Pythia to turn on $B^{**}$ production has not been used.
Instead BTeV has used a sample of simulated $B\to \psi K_s$ decays
and has selected events in which the $B$ and the next pion
in the generator track list have an invariant mass in the
range 5.6 - 5.8~GeV/c$^2$.   It was assumed that 30\% of these
events will come from $B^{**}$ decays and therefore be
right sign tags.  The remainder 70\% of the events were assumed to have
an equal number of right sign and wrong sign tags.
 
\subsection{Summary of Tagging}
\label{5:ssec:tagging_summary}
\index{BTeV!flavor tagging! same and away combined}

The results from the tagging study are
summarized in Table~\ref{5:tab:tag_effic}.
These results are preliminary and one should be aware that
all algorithms  have yet to exploit the full power
available to them.  In particular, the vertexing information
has yet to be fully exploited.  For example, the
$b\to\ell^-$ and $b\to c\to\ell^+$ samples differ not only in 
their $p_T$ spectra; they have distinctly different topological 
properties.  Similarly, kaon tagging can be improved if there
is evidence that the kaon comes from a tertiary vertex, indicative
of the $b\to c\to K^-$ cascade.  Finally, the vertex charge
algorithm should expect to find two vertices on the away side,
the $b$ decay vertex and the $c$ decay vertex; the charge of
both vertices provides tagging power.
Other tagging methods have yet to be studied such as using 
a $D^*$ from the decay of the opposite side $B$.
Finally, we have not yet explored the optimal use
of the correlations among all of the methods.
Therefore, the results quoted in 
Table~\ref{5:tab:tag_effic} probably underestimate the 
tagging power of BTeV, even though they do not yet incorporate
mixing on the away side. Therefore the BTeV results quoted in
this report are presented using nominal 
values of $\epsilon=0.7$ and $D=0.37$, giving $\epsilon D^2=0.1$.
The studies presented in the present chapter should be regarded as 
evidence that these nominal values lie well within the ultimate 
reach of the experiment.

\begin{table}[tb]
\begin{center}
\begin{tabular}[t]{|l|c c c|}
\hline
 Tag Type         & $\epsilon$  & $D$      &  $\epsilon{D^2}$ \\  \hline
 Muon             &  4.5\%      & 0.66        & 2.0\%              \\
 Electron         &  2.3\%      & 0.68        & 1.0\%              \\
 Kaon             &  18\%       & 0.52        & 4.9\%            \\
 Vertex Charge    &  32\%       & 0.36        & 4.1\%            \\
 Same Side Kaon   &  40\%       & 0.26        & 2.6\%            \\ 
 Same Side Pion   &  88\%       & 0.16        & 2.2\%            \\ \hline
 Total for $B_s$  &             &             & 14.6 \%    \\
 Total for $B_d$  &             &             & 14.2 \%    \\
 Total for $B_s$ with overlaps  
                  & 65\%        & 0.37        & 8.9\% \\ \hline\hline
\end{tabular}
\end{center}
\caption[Summary of Tagging Power in BTeV]
{Results of first generation studies of tagging power in BTeV.
         In the text it is discussed that these studies are incomplete
         and that they likely underestimate the tagging power which
         can be realized.
          }
\label{5:tab:tag_effic}
\end{table}

\section{Schedule}
\index{BTeV!schedule}

The BTeV program is an ambitious one.  Its goal is to begin data taking in 
2005/6. This timing is well-matched to the world $B$ physics program. 
The $e^{+}e^{-}$  $B$ factories and the Fermilab collider experiments will 
have had several years of running, the first results
in, and their significance thoroughly digested. It should be clear what the 
next set of goals is and BTeV will be guaranteed to be well-positioned to 
attack them.  This schedule also  gives BTeV a good opportunity
to have a head start in its inevitable competition with LHC-b, especially 
since BTeV can be installing and operating components of its detector in the 
collision hall well in advance of 2005. Finally, this schedule is sensibly 
related to BTeV's plan to conduct a rigorous R\&D program which includes a 
sequence of engineering runs to test the technically  challenging
systems in the BTeV design.  We believe that the scale of the BTeV construction
effort is comparable to the scale of one of the current detector upgrades.
The time scale is comparable as well.

%\section{Summary of Physics Reach}
%%
%The physics reach of BTeV is summarized in
%table~\ref{5:tab:reqmeas}.  Each of these measurements
%exploits one or more of the strengths of the BTeV detector:
%the ability to trigger efficiently
%on all hadronic events; powerful $K/\pi$ separation using
%the RICH detector; the ability to efficiently reconstruct
%photons and $\pi^{\circ}$'s; and excellent resolution on
%the proper decay time of $b$ hadrons.

%\begin{table}[tbp]
%\begin{center}
%\begin{tabular}{llcccc} \hline\hline
%Physics & Decay Mode & Hadron & $K\pi$ & $\gamma$ & Decay \\
%Quantity&            & Trigger & Sep   & Det & Time $\sigma$ \\
%\hline
%$\sin(2\alpha)$ & $B^o\to\rho\pi\to\pi^+\pi^-\pi^o$ & $\surd$ & $\surd$& $\surd$ 
%&\\
%$\cos(2\alpha)$ & $B^o\to\rho\pi\to\pi^+\pi^-\pi^o$ & $\surd$ & $\surd$& 
%$\surd$ &\\
%sign$(\sin(2\alpha))$ & $B^o\to\rho\pi$ \& $B^o\to\pi^+\pi^-$ & 
%$\surd$ & $\surd$ & $\surd$ & \\
%$\sin(\gamma)$ & $B_s\to D_s^{\pm}K^{\mp}$ & $\surd$ & $\surd$ & & $\surd$\\
%$\sin(\gamma)$ & $B^-\to \overline{D}^{0}K^{-}$ & $\surd$ & $\surd$ & & \\
%$\sin(\gamma)$ & $B^o\to\pi^+\pi^-$ \& $B_s\to K^+K^-$ & $\surd$ & $\surd$& & 
%$\surd$ \\
%$\sin(2\chi)$ & $B_s\to J/\psi\eta',$ $J/\psi\eta$ & & &$\surd$ &$\surd$\\
%$\sin(2\beta)$ & $B^o\to J/\psi K_S$ & & & & \\
%$\cos(2\beta)$ &  $B^o\to J/\psi K^o$, $K^o\to \pi\ell\nu$  & & & & \\
%$\cos(2\beta)$ &  $B^o\to J/\psi K^{*o}$ \& $B_s\to J/\psi\phi$  & & & 
%&$\surd$ \\
%$x_s$  & $B_s\to D_s^+\pi^-$ & $\surd$ & & &$\surd$\\
%$\Delta\Gamma$ for $B_s$ & $B_s\to  J/\psi\eta'$, $ D_s^+\pi^-$, $K^+K^-$ &
%$\surd$ & $\surd$ & $\surd$ & $\surd$ \\
%\hline
%\end{tabular}
%\end{center}
%\caption[Summary of BTeV Detector Capabilities]
%{Required CKM measurements for $B$ mesons and associated key detector
%characteristics.}
%\label{5:tab:reqmeas}
%\end{table}

%\begin{table}[tbp]
%\begin{center}
%\begin{tabular}{llcccc} \hline\hline
%Physics & Decay Mode & Projected \\
%Quantity&            & Sensitivity \\
%\hline
%$\sin(2\alpha)$ & $B^o\to\rho\pi\to\pi^+\pi^-\pi^o$ &  \\
%$\cos(2\alpha)$ & $B^o\to\rho\pi\to\pi^+\pi^-\pi^o$ &  \\
%sign$(\sin(2\alpha))$ & $B^o\to\rho\pi$ \& $B^o\to\pi^+\pi^-$ &  \\
%$\sin(\gamma)$ & $B_s\to D_s^{\pm}K^{\mp}$ & \\ 
%$\sin(\gamma)$ & $B^-\to \overline{D}^{0}K^{-}$ & \\
%$\sin(\gamma)$ & $B^o\to\pi^+\pi^-$ \& $B_s\to K^+K^-$ & \\
%$\sin(2\chi)$ & $B_s\to J/\psi\eta',$ $J/\psi\eta$ & \\
%$\sin(2\beta)$ & $B^o\to J/\psi K_S$ & \\
%$\cos(2\beta)$ &  $B^o\to J/\psi K^o$, $K^o\to \pi\ell\nu$  &  \\
%$\cos(2\beta)$ &  $B^o\to J/\psi K^{*o}$ \& $B_s\to J/\psi\phi$  &  \\
%$x_s$  & $B_s\to D_s^+\pi^-$ &  \\
%$\Delta\Gamma$ for $B_s$ & $B_s\to  J/\psi\eta'$, $ D_s^+\pi^-$, $K^+K^-$ & \\
%\hline
%\end{tabular}
%\end{center}
%\caption[Summary of BTeV Detector Capabilities]
%{Summary of the Physics Reach of BTeV.}
%\label{5:tab:reqmeas}
%\end{table}

\section{Conclusions}

BTeV is a powerful and precise scientific instrument capable of exquisite
tests of the Standard Model. It has great potential to discover new physics 
via rare or CP violating decays of heavy quarks.  Details of
the physics reach of BTeV and can be found in chapters~6 through~9
of this report and a summary table can be found in the
executive summary~\cite{execsum} of the BTeV Proposal.

\clearpage
\begin{thebibliography}{99}

\bibitem{btev_auth} A list of the BTeV Collaboration
                    can be found in Ref.~\cite{btev_prop}.
\bibitem{btev_prop} 
                   ``Proposal for an Experiment to Measure Mixing, CP Violation
                   and Rare Decays in Charm and Beauty Particle Decays at 
                   the Fermilab Collider --- BTeV'', 
                   Fermilab, May 2000.  This is available at the URL,\\
\verb"http://www-btev.fnal.gov/public\_documents/btev\_proposal".

\bibitem{pythia} H.~U.~Bengtsson and T.~Sjostrand,
                 Comp. Phys. Comm. {\bf 46}, 43 (1987).

\bibitem{geant}  GEANT: CERN Program Library Long Writeup W5013,\\
\verb"http://wwwinfo.cern.ch/asdoc/geant\_html3/geantall.html".

\bibitem{MCFast} P. Avery \textit{et al.},
       ``MCFast: A Fast Simulation Package for Detector
         Design Studies'', in the Proceedings of the
         International Conference on Computing in High Energy Physics,
         Berlin, (1997).  Documentation can be found at
\verb"http://www-cpd.fnal.gov/mcfast.html".

\bibitem{execsum} The executive summary is part of the BTeV Proposal,
                 Ref.~\cite{btev_prop} and can be found separately on
                 the BTeV Proposal web page.

\end{thebibliography}
\newcommand{\ra}{\rightarrow}
\newcommand{\stb} {\ifmmode \sin 2\beta \else 
                           $\sin 2\beta$\fi}
\newcommand{\jpks} {\ifmmode J/\psi K^0_S \else 
                            $J/\psi K^0_S$\fi}
\newcommand{\bjpks} {\ifmmode B^0 \ra J/\psi K^0_S \else 
                            $B^0 \ra J/\psi K^0_S$\fi}
\newcommand{\bpipi} {\ifmmode B^0 \ra \pi^+\pi^- \else 
                            $B^0 \ra \pi^+\pi^-$\fi}
\newcommand{\bkk} {\ifmmode B^0 \ra K^+ K^- \else 
                            $B^0 \ra K^+ K^-$\fi}
\newcommand{\tfb} {2~fb$^{-1}$}
\def\gevc {\ifmmode {\rm GeV}/c \else GeV$/c$\fi}
\newcommand{\eD} {\ifmmode \varepsilon{\cal D}^2 \else 
                          $\varepsilon{\cal D}^2$\fi}
\newcommand{\bskk} {\ifmmode B_s^0 \ra K^+ K^-
                        \else $B_s^0 \ra K^+ K^-$\fi}
\newcommand{\bsdsk} {\ifmmode B_s^0 \ra D_s K
                        \else $B_s^0 \ra D_s K$\fi}
\newcommand{\Pt} {\ifmmode p_T \else $p_T$\fi}
\def\bstopik {B_s^0\rightarrow \pi^+ K^-}
\def\bstokk  {B_s^0\rightarrow K^+ K^-}

\newcommand{\Bs} {\ifmmode B_{s}^{0}
                       \else $B_{s}^{0}$\fi}
\def\btopipi {B^0\rightarrow \pi^+ \pi^-}
\def\bstopik {B_s^0\rightarrow \pi^+ K^-}
\def\btokpi  {B^0\rightarrow K^+ \pi^-}
\def\mevcc {\ifmmode {\rm MeV}/c^2 \else MeV$/c^2$\fi}
\def\gevcc {\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi}
\def\pipi    {\pi^+ \pi^-}
\def\pik     {\pi^+ K^-}
\def\kpi     {K^+ \pi^-}
\def\kk      {K^+ K^-}
\newcommand{\roarrow}[1]{\stackrel{\rightharpoonup}{#1}}
\def\psietp {B_s^0 \rightarrow J\!/\psi\,\eta^\prime}
\def\psiea  {B_s^0 \rightarrow J\!/\psi\,\eta}
\def\mumu   {\mu^+\,\mu^-}

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

\input{ch6/CP_intro}

%%\newpage
\input{ch6/Bpsiks_intro}
\input{ch6/Bpsiks_CDF}
\input{ch6/Bpsiks_D0}
\input{ch6/Bpsiks_BTeV}
\input{ch6/Bpsiks_summary}

%%\newpage
\input{ch6/Bpipi_intro}
\input{ch6/Bpipi_CDF}
\input{ch6/Bpipi_D0}
\input{ch6/Bpipi_BTeV}
\input{ch6/Bpipi_summary}

%%\newpage
\input{ch6/Bdk_intro}
\input{ch6/Bdk_CDF}
\input{ch6/Bdk_BTeV}
\input{ch6/Bdk_summary}

%%\newpage
\input{ch6/B3pi_intro}
\input{ch6/B3pi_BTeV}

%%\newpage
\input{ch6/Bpsieta_intro}
\input{ch6/Bpsieta_CDF}
\input{ch6/Bpsieta_BTeV}
\input{ch6/Bpsieta_summary}

%%\newpage
\input{ch6/CP_concl}

\clearpage
\input{ch6/CP_bib}
\chapter{\protect\boldmath Rare and Semileptonic Decays}


\def\be{\begin{equation}}
\def\ee{\end{equation}}
\newcommand{\bm}[1]{\mbox{\boldmath ${#1}$}}
\def\GeV{{\rm GeV}}
\def\MeV{{\rm MeV}}
\def\ltap{\mathrel{\mathpalette\vereq<}}
\def\vereq#1#2{\lower3pt\vbox{\baselineskip1pt\lineskip1pt
     \ialign{\\$#1\hfill##\hfil\\$\crcr#2\crcr\sim\crcr}}}
\def\Do{\mbox{D0}}
\def\bentarrow{\relax\ifmmode{
     \raise 1.13ex\hbox{$\lfloor$}\kern-.4em\rightarrow\kern-.2em}
  \else{
     $\raise 1.13ex\hbox{$\lflambdaloor$}\kern-.4em\rightarrow\kern-.2em$}\fi}
\def\fm{{\rm fm}}


\authors{Christian~W.~Bauer, Gustavo~Burdman, Aida~X.~El-Khadra, 
JoAnne~Hewett, Gudrun~Hiller, Michael~Kirk, Jonathan~Lewis, 
Heather~E.~Logan, Michael~Luke, Ron~Poling, Alex~Smith, 
Ben~Speakman, Kevin~Stenson, Masa~Tanaka, Andrzej~Zieminski}


\section{Rare Decays: Theory}
\input{ch7/rare_intro}
\input{ch7/rare_inclusive}
\input{ch7/rare_exclusive}
\input{ch7/rare_beyondSM}

\section{Rare Decays: Experiment}
\input{ch7/rare_d0}
\input{ch7/rare_cdf}
\input{ch7/rare_btev}

\input{ch7/rare_summary}

\section{Semileptonic Decays: Theory}
\input{ch7/sl_theory}

\section{Semileptonic Decays: Experiment}
\input{ch7/sl_cdf}
\input{ch7/sl_btev}

\input{ch7/sl_summary}

\clearpage
\input{ch7/bibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Amol's macro:
\newcommand{\bep}{\epsilon}
%%% JoAnne's macros:
\newcommand{\ibid}{{\it ibid}.}
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\zatskip\lineskiplimit 0pt\ialign{$\matth#1\hfil##hfil$\crcr#2\crcr\sim\crcr}}}

\newcommand{\brunts}[1]{\ensuremath{\BR \, \big(\!\Bsun \rightarrow #1 \,\big)}}
\newcommand{\gqtfb}{\ensuremath{\Gamma (B_q^0(t) \rightarrow \ov{f} )}}
\newcommand{\gqbtf}{\ensuremath{\Gamma (\ov{B}{}_q^0(t) \rightarrow f )}}
\newcommand{\gqtf}{\ensuremath{\Gamma (B_q^0(t) \rightarrow f )}}
\newcommand{\gqbtfb}{\ensuremath{\Gamma (\ov{B}{}_q^0(t) \rightarrow \ov{f} )}}
\newcommand{\bbs}{$B_s$--$\ov{B}{}_s\,$}

\chapter{Mixing and Lifetimes}

%\authors{Conveners -- Neal Cason, Harry W.~K.~Cheung,
%                      Ulrich Nierste, Christoph Paus,
%                      Mikhail Voloshin}
%\authors{Theory: Amol Dighe, JoAnne Hewett}
%\authors{CDF:    Jeffrey Tseng, Matthew Jones, Masa Tanaka}
%\authors{D\O:    G.~Gutierrez, R.~Jesik, V.E.~Kuznetsov, W.~Taylor, N.~Xuan}
%\authors{BTeV:   Penny Kasper, Robert Kutschke, Gobinda Majumder,
%                 Sheldon Stone, Jianchun Wang}
%
\authors{K.~Anikeev, F.~Azfar, N.~Cason, H.W.K.~Cheung, A.~Dighe, I.~Furic,
  G.~Gutierrez, J.~Hewett, R.~Jesik, M.~Jones, P.~Kasper, J.~Kroll,
  V.E.~Kuznetsov, R.~Kutschke, G.~Majumder, M.~Martin, U.~Nierste, Ch.~Paus,
  S.~Rakitin, S.~Stone, M.~Tanaka, W.~Taylor, J.~Tseng, M.~Voloshin, J.~Wang,
  N.~Xuan}
%
%-- from experimental section
%
\newcommand{\ipb}{\ensuremath{\mathrm{pb^{-1}}}}
\newcommand{\ifb}{\ensuremath{\mathrm{fb^{-1}}}}
\newcommand{\um}{\ensuremath{\mathrm{\mu m}}}
\newcommand{\mm}{\ensuremath{\mathrm{mm}}}
%\newcommand{\eV}{\ensuremath{\mathrm{e\kern -0.1em V}}}
%\newcommand{\keV}{\ensuremath{\mathrm{ke\kern -0.1em V}}}
%\newcommand{\MeV}{\ensuremath{\mathrm{Me\kern -0.1em V}}}
%\newcommand{\GeV}{\ensuremath{\mathrm{Ge\kern -0.1em V}}}
%\newcommand{\TeV}{\ensuremath{\mathrm{Te\kern -0.1em V}}}
\newcommand{\ten}[1]{\ensuremath{\times 10^{#1}}}
%
% Next 2 needed for D0 section if not defined in chapter 4 already
%
\newcommand{\dzero}{D\O\,\,}
\newcommand{\emis}{/\!\!\!\!E}

%===============================================================================
\section{Overview} 

In hadron colliders $b$-flavored hadrons are produced with a large boost.
Therefore they are a fertile ground for measurements of decay time
distributions. The neutral $B_d^0$ and $B_s^0$ mesons mix with their
antiparticles, which leads to oscillations between the flavor eigenstates. A
measurement of the oscillation frequency allows to
determine the mass difference%
\index{B meson@$B$ meson!mass difference \dm}\index{mass difference \dm}
$\dm_q$, $q=d,s$, between the two physical mass eigenstates. The rapid
oscillations in \bbms\ %
\index{B mixing@$B$ mixing!$B_s$ mixing}% 
have not been resolved yet and their discovery has a high priority for the $B$
physics program at Run~II.  Once this has been achieved, the mass difference
$\dm_s$ will be known very precisely. By combining this information with the
already measured $\dm_d$ one will precisely
determine the length of one side of the unitarity triangle.%
\index{unitarity triangle!and \bbmd}% 
\index{unitarity triangle!and \bbms}% 
$\dm_d/\dm_s$ will have a larger impact on our knowledge of the unitarity
triangle than any previously measured quantity and even than a precisely
measured $\sin (2 \beta)$.  Accurately measured decay distributions will further
reveal the pattern of $b$-hadron lifetimes.  
\index{lifetime!$b$-flavored hadron}%
The large mass of the $b$ quark compared to the QCD scale parameter
$\Lambda_{QCD}$ allows to expand the widths in terms of $\Lambda_{QCD}/m_b$.
Differences among the total widths are dominated by terms of order $16 \pi^2
(\Lambda_{QCD}/m_b)^3$, the measurements of lifetime differences therefore probe
the heavy quark expansion at the third order in the expansion parameter. From
Run~II we expect valuable new information on the lifetimes of the $B^+$, $B_d^0$
and $B_s^0$ mesons, the width difference $\dg_s$ between the two physical $B_s$
meson eigenstates the lifetimes of the $\Lambda_b$ and eventually also of other
b-flavored baryons.  \index{lifetime!$B^+$} \index{lifetime!$B_d^0$}
\index{lifetime!$B_s^0$} \index{lifetime!$\Lambda_b$} From the current
experimental situation it is not clear whether the heavy quark expansion can be
applied to baryon lifetimes, and Run~II data will help to settle this question.

This chapter first discusses the theory predictions for the various quantities
in great detail. Where possible, we derive simple `pocket-calculator' formulae
to facilitate the analysis of the measurements.  It is described which
fundamental information can be gained from the various measurements.  Some
quantities are sensitive to physics beyond the Standard Model and we show how
they are affected by new physics. Then we summarize the experimental techniques
and present the results of the Monte Carlo simulations.

%%%% Uli's changes: \label{defxy}->\label{msh:defxy}
%===============================================================================
\section[Theory of heavy hadron lifetimes]{Theory of heavy hadron
lifetimes$\!$ \authorfootnote{Author: Mikhail Voloshin}}
\label{chmix:thlife}

The dominant weak decays of hadrons containing a heavy quark, $c$ or $b$, are
caused by the decay of the heavy quark. In the limit of a very large mass $m_Q$
of a heavy quark $Q$ the parton picture of the hadron decay should set in, where
the inclusive decay rates of hadrons, containing $Q$, mesons ($Q\bar q$) and
baryons ($Qqq$), are all the same and equal to the inclusive decay rate
$\Gamma_{parton}(Q)$ of the heavy quark. Yet, the known inclusive decay rates
\cite{pdg} are conspicuously different for different hadrons, especially for
charmed hadrons, whose lifetimes span a range of more than one order of
magnitude from the shortest $\tau(\Omega_c)=0.064 \pm 0.020$ ps to the longest
$\tau(D^+)=1.057 \pm 0.015$ ps, while the differences of lifetime among $b$
hadrons are substantially smaller. The relation between the relative lifetime
differences for charmed and $b$ hadrons reflects the fact that the dependence of
the inclusive decay rates on the light quark-gluon `environment' in a particular
hadron is a pre-asymptotic effect in the parameter $m_Q$, which effect vanishes
as an inverse power of $m_Q$ at large mass.

A theoretical framework for systematic description of the leading at $m_Q \to
\infty$ term in the inclusive decay rate $\Gamma_{parton}(Q) \propto m_Q^5$ as
well as of the terms relatively suppressed by inverse powers of $m_Q$ is
provided \cite{sv:81,sv:85,sv:86} by the operator product expansion (OPE) in
$m_Q^{-1}$. Existing theoretical predictions for inclusive weak decay rates are
in a reasonable agreement, within the expected range of uncertainty, with the
data on lifetimes of charmed particles and with the so far available data on
decays of $B$ mesons.  The only outstanding piece of present experimental data
is on the lifetime of the $\Lambda_b$ baryon: $\tau(\Lambda_b)/\tau(B_d) \approx
0.8$, for which ratio a theoretical prediction, given all the uncertainty
involved, is unlikely to produce a number lower than 0.9.  The number of
available predictions for inclusive decay rates of charmed and $b$ hadrons is
sufficiently large for future experimental studies to firmly establish the
validity status of the OPE based theory of heavy hadron decays, and, in
particular, to find out whether the present contradiction between the theory and
the data on $\tau(\Lambda_b)/\tau(B_d)$ is a temporary difficulty, or an
evidence of fundamental flaws in theoretical understanding.


It is a matter of common knowledge that application of OPE to decays of charmed
and $b$ hadrons has potentially two caveats. One is that the OPE is used in the
Minkowski kinematical domain, and therefore relies on the assumption of
quark-hadron duality at the energies involved in the corresponding decays. In
other words, it is assumed that sufficiently many exclusive hadronic channels
contribute to the inclusive rate, so that the accidentals of the low-energy
resonance structure do not affect the total rates of the inclusive processes.
Theoretical attempts at understanding the onset of the quark-hadron duality are
so far limited to model estimates \cite{cdsu:97,bsuv:99}, not yet suitable for
direct quantitative evaluation of possible deviation from duality in charm and
$b$ decays. This point presents the most fundamental uncertainty of the OPE
based approach, and presently can only be clarified by confronting theoretical
predictions with experimental data. The second possible caveat in applying the
OPE technique to inclusive charm decays is that the mass of the charm quark,
$m_c$, may be insufficiently large for significant suppression of higher terms
of the expansion in $m_c^{-1}$.  The relative lightness of the charm quark,
however, accounts for a qualitative, and even semi-quantitative, agreement of
the OPE based predictions with the observed large spread of the lifetimes of
charmed hadrons: the nonperturbative effects, formally suppressed by $m_c^{-2}$
and $m_c^{-3}$ are comparable with the `leading' parton term and describe the
hierarchy of the lifetimes.

Another uncertainty of a technical nature arises from poor knowledge of matrix
elements of certain quark operators over hadron, arising as terms in OPE. These
can be estimated within theoretical models, with inevitable ensuing model
dependence, or, where possible, extracted from the experimental data.  With
these reservations spelled out, we discuss here the OPE based description of
inclusive weak decays of charm and $b$ hadrons, with emphasis on specific
experimentally testable predictions, and on the measurements, which would less
rely on model dependence of the estimates of the matrix elements, thus allowing
to probe the OPE predictions at a fundamental level.

%-------------------------------------------------------------------------------
\subsection{OPE for inclusive weak decay rates}

The optical theorem of the scattering theory relates the total decay rate
$\Gamma_H$ of a hadron $H_Q$ containing a heavy quark $Q$ to the imaginary part
of the `forward scattering amplitude'.
\index{inclusive weak decay rates!operator product expansion}%
For the case of weak decays the latter amplitude is
described by the following effective operator
%
\begin{equation}
L_{eff}=2 \,{\rm Im} \, \left [ i \int d^4x \, e^{iqx} \, T \left \{
L_W(x),
L_W(0) \right \} \right ]\,,
\label{leff}
\end{equation}
%
in terms of which the total decay rate is given by\footnote{We use here the
  non-relativistic normalization for the {\it heavy} quark states: $\langle Q |
  Q^\dagger Q | Q \rangle =1$.}
%
\begin{equation}
  \Gamma_H=\langle H_Q | \, L_{eff} \, | H_Q \rangle\,.
  \label{lgam}
\end{equation}
%
The correlator in equation (\ref{leff}) in general is a non-local operator.
However at $q^2=m_Q^2$ the dominating space-time intervals in the integral are
of order $m_Q^{-1}$ and one can expand the correlator in $x$, thus producing an
expansion in inverse powers of $m_Q$. The leading term in this expansion
describes the parton decay rate of the quark. For instance, the term in the
nonleptonic weak Lagrangian $\sqrt{2} \, G_F \, V ({\overline q}_{1 L}
\gamma_\mu \, Q_L)({\overline q}_{2 L} \gamma_\mu \, q_{3 L})$ with $V$ being
the appropriate combination of the CKM mixing factors, generates through
Eq.~(\ref{leff}) the leading term in the effective Lagrangian
%
\begin{equation}
L^{(0)}_{eff, \, nl} = |V|^2 \, {G_F^2 \, m_Q^5 \over 64 \, \pi^3} \,
\eta_{nl} \, \left ( {\overline Q} Q \right )\,,
\label{lef0}
\end{equation}
%
where $\eta_{nl}$ is the perturbative QCD radiative correction factor.  This
expression reproduces the well known formula for the inclusive nonleptonic
decay rate of a heavy quark, associated with the underlying process $Q \to q_1
\, q_2 \, {\overline q}_3$, due to the relation $\langle H_Q | {\overline Q} Q |
H_Q \rangle \approx \langle H_Q | Q^\dagger Q | H_Q \rangle =1$, which is valid
up to corrections of order $m_Q^{-2}$. One also sees form this example, that in
order to separate individual semi-inclusive decay channels, e.g., nonleptonic
with specific flavor quantum numbers, or semileptonic, one should simply pick
up the corresponding relevant part of the weak Lagrangian $L_W$, describing the
underlying process, to include in the correlator (\ref{leff}).

The general expression for first three terms in the OPE for $L_{eff}$
has the form\index{inclusive weak decay rates!effective Lagrangian}
%
\begin{eqnarray}
L_{eff} &=&
L_{eff}^{(0)} + L_{eff}^{(2)} + L_{eff}^{(3)} \\
&=& c^{(0)} \, {G_F^2 \, m_Q^5 \over 64 \, \pi^3} \left ( {\overline
Q} Q \right ) +  c^{(2)} \,{G_F^2 \, m_Q^3 \over 64 \, \pi^3} \left
({\overline Q}\, \sigma^{\mu \nu} G_{\mu \nu} \, Q \right ) 
+ {G_F^2 \, m_Q^2 \over 4\, \pi} \, \sum_i c_i^{(3)}\, 
({\overline q}_i \Gamma_i q_i) ({\overline Q} \Gamma^\prime_i Q)\,,\nonumber 
\label{first3}
\end{eqnarray}
%
where the superscripts denote the power of $m_Q^{-1}$ in the relative
suppression of the corresponding term in the expansion with respect to the
leading one, $G_{\mu \nu}$ is the gluon field tensor, $q_i$ stand for light
quarks, $u,\, d, \, s$, and, finally, $\Gamma_i$, $\Gamma^\prime_i$ denote spin
and color structures of the four-quark operators. The coefficients $c^{(a)}$
depend on the specific part of the weak interaction Lagrangian $L_W$, describing
the relevant underlying quark process.

One can notice the absence in the expansion (\ref{first3}) of a term suppressed
by just one power of $m_Q^{-1}$, due to non-existence of operators of suitable
dimension. Thus the decay rates receive no correction of relative order
$m_Q^{-1}$ in the limit of large $m_Q$, and the first pre-asymptotic corrections
appear only in the order $m_Q^{-2}$.

\begin{figure}[bt]
\vspace*{-.25cm}
\begin{center}
\unitlength 0.9mm
\thicklines
\begin{picture}(140.00,100.00)(10,-15)
\put(10.00,70.00){\line(1,0){20.00}}
\put(30.00,70.00){\line(1,0){30.00}}
\put(60.00,70.00){\line(1,0){20.00}}
\bezier{164}(30.00,70.00)(45.00,84.00)(60.00,70.00)
\bezier{164}(30.00,70.00)(45.00,56.00)(60.00,70.00)
\put(30.00,70.00){\circle*{2.00}}
\put(60.00,70.00){\circle*{2.00}}
\put(20.00,72.00){\makebox(0,0)[cb]{$Q$}}
\put(70.00,71.00){\makebox(0,0)[cb]{$Q$}}
\put(49.00,78.00){\makebox(0,0)[cb]{$q_1$}}
\put(49.00,71.00){\makebox(0,0)[cb]{$q_2$}}
\put(49.00,62.00){\makebox(0,0)[ct]{${\overline q_3}$}}
\put(10.00,43.00){\line(1,0){20.00}}
\put(30.00,43.00){\line(1,0){30.00}}
\put(60.00,43.00){\line(1,0){20.00}}
\bezier{164}(30.00,43.00)(45.00,57.00)(60.00,43.00)
\bezier{164}(30.00,43.00)(45.00,29.00)(60.00,43.00)
\put(30.00,43.00){\circle*{2.00}}
\put(60.00,43.00){\circle*{2.00}}
\put(20.00,45.00){\makebox(0,0)[cb]{$Q$}}
\put(70.00,44.00){\makebox(0,0)[cb]{$Q$}}
\put(49.00,51.00){\makebox(0,0)[cb]{$q_1$}}
\put(49.00,44.00){\makebox(0,0)[cb]{$q_2$}}
\put(49.00,35.00){\makebox(0,0)[ct]{${\overline q_3}$}}
\put(45.00,36.00){\line(0,-1){2.00}}
\put(45.00,33.00){\line(0,-1){2.00}}
\put(45.00,30.00){\line(0,-1){2.00}}
\put(47.00,28.00){\makebox(0,0)[cc]{$g$}}
\put(10.00,12.00){\line(1,0){20.00}}
\put(30.00,12.00){\line(1,0){30.00}}
\put(60.00,12.00){\line(1,0){20.00}}
\bezier{164}(30.00,12.00)(45.00,26.00)(60.00,12.00)
\put(30.00,12.00){\circle*{2.00}}
\put(60.00,12.00){\circle*{2.00}}
\put(20.00,14.00){\makebox(0,0)[cb]{$Q$}}
\put(70.00,13.00){\makebox(0,0)[cb]{$Q$}}
\put(49.00,20.00){\makebox(0,0)[cb]{$q_1$}}
\put(49.00,13.00){\makebox(0,0)[cb]{$q_2$}}
\put(10.00,5.50){\line(3,1){20.00}}
\put(60.00,12.00){\line(4,-1){20.00}}
\put(20.00,7.00){\makebox(0,0)[ct]{$q_3$}}
\put(70.00,8.00){\makebox(0,0)[ct]{$q_3$}}
\put(87.00,70.00){\vector(1,0){11.00}}
\put(87.00,43.00){\vector(1,0){11.00}}
\put(87.00,12.00){\vector(1,0){11.00}}
\put(110.00,70.00){\line(1,0){30.00}}
\put(116.00,71.00){\makebox(0,0)[cb]{$Q$}}
\put(134.00,71.00){\makebox(0,0)[cb]{$Q$}}
\put(110.00,43.00){\line(1,0){30.00}}
\put(116.00,44.00){\makebox(0,0)[cb]{$Q$}}
\put(134.00,44.00){\makebox(0,0)[cb]{$Q$}}
\put(125.00,43.00){\line(0,-1){2.00}}
\put(125.00,40.00){\line(0,-1){2.00}}
\put(125.00,37.00){\line(0,-1){2.00}}
\put(127.00,35.00){\makebox(0,0)[cc]{$g$}}
\put(110.00,12.00){\line(1,0){30.00}}
\put(116.00,13.00){\makebox(0,0)[cb]{$Q$}}
\put(134.00,13.00){\makebox(0,0)[cb]{$Q$}}
\put(110.00,7.00){\line(3,1){15.00}}
\put(125.00,12.00){\line(3,-1){15.00}}
\put(117.00,9.00){\makebox(0,0)[lt]{$q$}}
\put(132.00,9.00){\makebox(0,0)[rt]{$q$}}
\put(125.00,62.00){\makebox(0,0)[cc]{ $m_Q^5 \, ({\overline Q} Q)$}}
\put(125.00,27.00){\makebox(0,0)[cc]{ $ m_Q^3 \, \left ({\overline Q}
(\vec \sigma \cdot \vec B) Q \right )$}}
\put(125.00,0.00){\makebox(0,0)[cc]{ $m_Q^2 \, ({\overline q} \Gamma
q)({\overline Q} \Gamma^\prime Q)$}}
\put(123.50,70.00){ \circle*{2.83}}
\put(123.50,43.00){ \circle*{2.83}}
\put(123.50,12.00){ \circle*{2.83}}
\end{picture}
\vspace*{-.75cm}
\caption{Graphs for three first terms in OPE for inclusive decay rates:
  the parton term, the chromomagnetic interaction, and the four-quark term.}
\end{center}
\label{fig:ope}
\end{figure}

The mechanisms giving rise to the three discussed terms in OPE are shown in
Figure~8.1. The first, leading term corresponds to the parton decay, and does
not depend on the light quark and gluon `environment' of the heavy quark in a
hadron. The second term describes the effect on the decay rate of the gluon
field that a heavy quark `sees' in a hadron. This term in fact is sensitive only
to the chromomagnetic part of the gluon field, and contains the operator of the
interaction of heavy quark chromomagnetic moment with the chromomagnetic field.
Thus this term depends on the spin of the heavy quark, but does not depend on
the flavors of the light quarks or antiquarks. Therefore this effect does not
split the inclusive decay rates within flavor $SU(3)$ multiplets of heavy hadrons,
but generally gives difference of the rates, say, between mesons and baryons.
The dependence on the light quark flavor arises from the third term in the
expansion (\ref{first3}) which explicitly contains light quark fields.
Historically, this part is interpreted in terms of two mechanisms
\cite{sv:81,gnpr:79,bgt:84}: the weak scattering (WS) and the Pauli interference
(PI).\index{inclusive weak decay rates!Pauli interference}
\index{inclusive weak decay rates!weak scattering}
The WS corresponds to a cross-channel of the
underlying decay, generically $Q \to q_1 \, q_2 \, {\overline q}_3$, where
either the quark $q_3$ is a spectator in a baryon and can undergo a weak
scattering off the heavy quark: $q_3 \, Q \to q_1 \, q_2$, or an antiquark in
meson, say ${\overline q}_1$, weak-scatters (annihilates) in the process
${\overline q}_1 \, Q \to q_2 \, {\overline q}_3$. The Pauli interference effect
arises when one of the final (anti)quarks in the decay of $Q$ is identical to
the spectator (anti)quark in the hadron, so that an interference of identical
particles should be taken into account. The latter interference can be either
constructive or destructive, depending on the relative spin-color arrangement of
the (anti)quark produced in the decay and of the spectator one, thus the sign of
the PI effect is found only as a result of specific dynamical calculation. In
specific calculations, however, WS and PI arise from the same terms in OPE,
depending on the hadron discussed, and technically there is no need to resort to
the traditional terminology of WS and PI.

In what follows we discuss separately the effects of the three terms in the
expansion (\ref{first3}) and their interpretation within the existing and future
data.

%-------------------------------------------------------------------------------
\subsection{The parton decay rate}

The leading term in the OPE amounts to the perturbative expression for the decay
rate of a heavy quark. In $b$ hadrons the contribution of the subsequent terms
in OPE is at the level of few percent, so that the perturbative part can be
confronted with the data in its own right. In particular, for the $B_d$ meson
the higher terms in OPE contribute only about 1\% of the total nonleptonic as
well as of the semileptonic decay rate.  Thus the data on these rates can be
directly compared with the leading perturbative term in OPE.

\index{B decay@$B$ decay!semileptonic} The principal theoretical topic,
associated with this term is the calculation of QCD radiative corrections, i.e.
of the factor $\eta_{nl}$ in Eq.~(\ref{lef0}) and of a similar factor,
$\eta_{nl}$, for semileptonic decays. It should be noted, that even at this,
perturbative, level there is a known long-standing problem between the existing
data and the theory in that the current world average for the semileptonic
branching ratio for the $B$ mesons, $B_{sl}(B)= 10.45 \pm 0.21 \%$, is somewhat
lower than the value $B_{sl}(B) \ge 11.5$ preferred from the present knowledge
of theoretical QCD radiative corrections to the ratio of nonleptonic to
semileptonic decay rates (see, e.g., \cite{bbsv:94}). However, this apparent
discrepancy may in fact be due to insufficient `depth' of perturbative QCD
calculation of the ratio $\eta_{nl}/\eta_{sl}$. In order to briefly elaborate on
this point, we notice that the standard way of analyzing the perturbative
radiative corrections in the nonleptonic decays is through the renormalization
group (RG) summation of the leading log terms and the first next-to-leading
terms \cite{ap:91,bbbg:95} in the parameter $L \equiv \ln(m_W/m_b)$. For the
semileptonic decays the logarithmic dependence on $m_W/m_b$ is absent in all
orders due to the weak current conservation at momenta larger than $m_b$, thus
the correction is calculated by the standard perturbative technique, and a
complete expression in the first order in $\alpha_s$ is available both for the
total rate \cite{hp:83,nir:89} and for the lepton spectrum \cite{cj:94}.  In
reality however the parameter $L \approx 2.8$ is not large, and non-logarithmic
terms may well compete with the logarithmic ones. This behavior is already seen
from the known expression for the logarithmic terms: when expanded up to the
order $\alpha_s^2$ the result of Ref.\cite{bbbg:94} for the rate of decays with
single final charmed quark takes the form
%
\begin{equation}
{{\Gamma(b \to c \bar u d) +
\Gamma(b \to c \bar u s)} \over {3 \, \Gamma(b \to c e
\bar \nu )}}=
 1+ {\alpha_s \over \pi} + {\alpha_s^2 \over \pi^2} \left [ 4 \, L^2
+ \left
( {7 \over 6} + { 2 \over 3} \, c(m_c^2/m_b^2) \right ) L \right ]\,,
\label{as2l}
\end{equation}
%
where, in terms of notation of Ref.\cite{bbbg:94}, $c(a)=c_{22}(a)-c_{12}(a)$.
The behavior of the function $c(a)$ is known explicitly \cite{bbbg:94} and is
quite weak: $c(0)=19/2$, $c(1) = 6$, and $c(m_c^2/m_b^2) \approx 9.0$ for the
realistic mass ratio $m_c/m_b \approx 0.3$. One can see that the term with the
single logarithm $L$ contributes about two thirds of that with $L^2$ in the term
quadratic in $\alpha_s$. Under such circumstances the RG summation of the terms
with powers of $L$ does not look satisfactory for numerical estimates of the QCD
effects, at least at the so far considered level of the first next-to-leading
order terms, and the next-to-next-to-leading terms can be equally important as
the two known ones, which would eliminate the existing impasse between the
theory and the data on $B_{sl}(B)$. One can present some arguments \cite{mv:96}
that this is indeed the case for the $b$ quark decay, although a complete
calculation of these corrections is still unavailable.

%-------------------------------------------------------------------------------
\subsection{Chromomagnetic and time dilation effects in decay rates}

The corrections suppressed by two powers of $m_Q^{-1}$ to inclusive decay rates
arise from two sources \cite{buv:92}: the $O(m_Q^{-2})$ corrections to the
matrix element of the leading operator, $({\overline Q} Q$, and the second term
in OPE (\ref{first3}) containing the chromomagnetic interaction. The expression
for the matrix element of the leading operator with the correction included is
written in the form
%
\begin{equation}
\langle H_Q| {\overline Q} Q | H_Q \rangle = 1-
{\mu_\pi^2(H_Q)-\mu_g^2(H_Q) \over 2 \, m_Q^2} + \ldots ,
\label{qbarq}
\end{equation}
%
where $\mu_\pi^2$ and $\mu_g^2$ are defined as
%
\begin{eqnarray}
&&\mu_\pi^2=\langle H_Q| {\overline Q}\, (i {\vec D})^2 \, Q | H_Q
\rangle \,, \nonumber \\
&&\mu_g^2=\langle H_Q| {\overline Q}\, {1 \over 2} \sigma^{\mu \nu}
G_{\mu \nu} Q | H_Q \rangle\,,
\label{mus}
\end{eqnarray}
%
with $D$ being the QCD covariant derivative. The correction in equation
(\ref{qbarq}) in fact corresponds to the time dilation factor $m_Q/E_Q$, for the
heavy quark decaying inside a hadron, where it has energy $E_Q$, which energy is
contributed by the kinetic part ($\propto \mu_\pi^2$) and the chromomagnetic
part ($\propto \mu_g^2$). The second term in OPE describes the effect of the
chromomagnetic interaction in the decay process, and is also expressed through
$\mu_g^2$.\index{kinetic energy of a heavy quark}\index{chromomagnetic energy of
  a heavy quark}

The explicit formulas for the decay rates, including the effects up to the order
$m_Q^{-2}$ are found in \cite{buv:92} and for decays of the $b$ hadrons read as
follows. For the semileptonic decay rate
%
\begin{equation}
\Gamma_{sl}(H_b)={|V_{cb}|^2 \, G_F^2 \, m_b^5 \over 192 \, \pi^3}
\,\langle H_b | {\overline b} b | H_b \rangle \,\left [1+
 {{\mu_g^2} \over m_b^2} \left ( {x \over 2}
\,{ d \over
{dx}} -2\right)  \right ] \,\eta_{sl} \, I_0 (x,\,0,\,0)\,,
\label{gamsl1}
\end{equation}
%
and for the nonleptonic decay rate
%
\begin{equation}
\Gamma_{nl}(H_b)={|V_{cb}|^2 \, G_F^2 \, m_b^5 \over 64 \, \pi^3} \,
\langle H_b | {\overline b} b | H_b \rangle \, \left \{\left [1+
 {{\mu_g^2} \over m_b^2} \left ( {x \over 2}
\,{ d \over
{dx}} -2\right)  \right ] \,\eta_{nl} \, I (x)-8 \eta_2 \, {{\mu_g^2}
\over m_b^2} \, I_2(x) \right \}\,.
\label{gamnl1}
\end{equation}
%
These formulas take into account only the dominant CKM mixing $V_{cb}$ and
neglect the small one, $V_{ub}$. The following notation is also used:
$x=m_c/m_b$, $I_0(x,y,z)$ stands for the kinematical suppression factor in a
three-body weak decay due to masses of the final fermions.  In
particular,\index{inclusive weak decay rates!kinematical integrals}
%
\begin{eqnarray}
&&I_0(x,0,0)=(1-x^4)(1-8\, x^2+ x^4)-24 \, x^4 \, \ln x\,, \\ \nonumber
&&I_0(x,x,0)=(1-14\, x^2-2 \, x^4 -12 x^6)\sqrt{1-4\, x^2} + 24 \,
(1-x^4) \, \ln {1+ \sqrt{1-4\, x^2} \over 1- \sqrt{1-4\, x^2}}\,.
\label{kints}
\end{eqnarray}
%
Furthermore, $I(x)=I_0(x,0,0)+I_0(x,x,0)$, and
$$
I_2(x)=(1-x^2)^3+\left ( 1+{1 \over
2} x^2+ 3 x^4 \right ) \, \sqrt{1-4\, x^2}-3x^2 \, (1-2x^4)
\ln{{1+\sqrt{1-4\, x^2}} \over {1-\sqrt{1-4\, x^2}}}\,.
$$
Finally, the QCD radiative correction factor $\eta_2$ in Eq.~(\ref{gamnl1}) is
known in the leading logarithmic approximation and is expressed in terms of the
well known coefficients $C_+$ and $C_-$ in the renormalization of the
nonleptonic weak interaction: $\eta_2=(C_+^2(m_b)-C_-^2(m_b))/6$
with\index{nonleptonic weak interaction!QCD renormalization}
%
\begin{equation}
C_-(\mu)=C_+^{-2}(\mu)=\left [ {\alpha_s(\mu) \over \alpha_s(m_W)}
\right ]^{4/b}\,,
\label{cpm}
\end{equation}
%
and $b$ is the coefficient in the QCD beta function. The value of $b$ relevant
to $b$ decays is $b=23/3$.


Numerically, for $x \approx 0.3$, the expressions for the decay rates can be
written as
%
\begin{eqnarray}
&&\Gamma_{sl}(H_b)=\Gamma_{sl}^{parton} \, \left ( 1-
{\mu_\pi^2(H_b)-\mu_g^2(H_b) \over 2 \, m_b^2}-2.6 {\mu_g^2(H_b) \over
m_b^2} \right )\,, \nonumber \\
&&\Gamma_{nl}(H_b)=\Gamma_{nl}^{parton} \, \left ( 1-
{\mu_\pi^2(H_b)-\mu_g^2(H_b) \over 2 \, m_b^2}-1.0 {\mu_g^2(H_b) \over
m_b^2} \right )\,,
\label{gamnum}
\end{eqnarray}
%
where $\Gamma^{parton}$ is the perturbation theory value of the corresponding
decay rate of $b$ quark.

The matrix elements $\mu_\pi^2$ and $\mu_g^2$ are related to the spectroscopic
formula for a heavy hadron mass $M$,
%
\begin{equation}
M(H_Q)=m_Q + {\overline \Lambda}(H_Q)+{\mu_\pi^2(H_Q)-\mu_g^2(H_Q) \over
2 \, m_Q} + \ldots
\label{massf}
\end{equation}
%
Being combined with the spin counting for pseudoscalar and vector mesons, this
formula allows to find the value of $\mu_g^2$ in pseudoscalar mesons from the
mass splitting:
%
\begin{equation}
\mu_g^2(B)={3 \over 4} \left ( M_{B^*}^2-M_B^2 \right ) \approx 0.36 \,
GeV^2\,.
\label{mugn}
\end{equation}
%
The value of $\mu_\pi^2$ for $B$ mesons is less certain. It is constrained by
the inequality \cite{mv:95}, $\mu_\pi^2(H_Q) \ge \mu_g^2(H_Q)$, and there are
theoretical estimates from the QCD sum rules \cite{bb:94}: $\mu_\pi^2(B) = 0.54
\pm 0.12 \, GeV^2$ and from an analysis of spectroscopy of heavy hadrons
\cite{neubert:00}: $\mu_\pi^2(B) = 0.3 \pm 0.2 \, GeV^2$. In any event, the
discussed corrections are rather small for $b$ hadrons, given that
$\mu_g^2/m_b^2 \approx 0.015$. The largest, in relative terms, effect of these
corrections in $B$ meson decays is on the semileptonic decay rate, where it
amounts to 4 -- 5 \% suppression of the rate, which rate however is only a
moderate fraction of the total width. In the dominant nonleptonic decay rate
the effect is smaller, and, according to the formula (\ref{gamnum}) amounts to
about 1.5 -- 2 \%.

The effect of the $m_Q^{-2}$ corrections can be evaluated with a somewhat better
certainty for the ratio of the decay rates of $\Lambda_b$ and $B$ mesons. This
is due to the fact that $\mu_g^2(\Lambda_b)=0$, since there is no correlation of
the spin of the heavy quark in $\Lambda_b$ with the light component, having
overall quantum numbers $J^P=0^+$. Then, applying the formula (\ref{gamnum}) to
$B$ and $\Lambda_b$, we find for the ratio of the (dominant) nonleptonic decay
rates:
%
\begin{equation}
{\Gamma_{nl}(\Lambda_b) \over \Gamma_{nl}(B)} = 1
-{\mu_\pi^2(\Lambda_b)-\mu_\pi^2(B) \over 2 \, m_b^2}+0.5 {\mu_g^2(B)
\over m_b^2}\,.
\label{lambdab}
\end{equation}
%
The difference of the kinetic terms, $\mu_\pi^2(\Lambda_b)-\mu_\pi^2(B)$, can be
estimated from the mass formula:
%
\begin{equation}
\mu_\pi^2(\Lambda_b)-\mu_\pi^2(B)={2 \, m_b \, m_c \over m_b-m_c} \left
[ {\overline M}(B)-{\overline M}(D)-M(\Lambda_b)+ M(\Lambda_c) \right] =
0 \pm 0.04 \, GeV^2\,,
\label{difmu}
\end{equation}
%
where ${\overline M}$ is the spin-averaged mass of the mesons, e.g., ${\overline
  M}(B)=(M(B)+3 M(B^*))/4$. The estimated difference of the kinetic terms is
remarkably small. Thus the effect in the ratio of the decay rates essentially
reduces to the chromomagnetic term, which is also rather small and accounts for
less than 1\% difference of the rates. For the ratio of the semileptonic decay
rates the chromomagnetic term is approximately four times larger, but then the
contribution of the semileptonic rates to the total width is rather small. Thus
one concludes that the terms of order $m_b^{-2}$ in the OPE expansion for the
decay rates can account only for about 1\% difference of the lifetimes of
$\Lambda_b$ and the $B$ mesons.

The significance of the $m_Q^{-2}$ terms is substantially different for the
decay rates of charmed hadrons, where these effects suppress the inclusive
decays of the $D$ mesons by about 40\% with respect to those of the charmed
hyperons in a reasonable agreement with the observed pattern of the lifetimes.

It should be emphasized once again that the $m_Q^{-2}$ effects do not depend on
the flavors of the spectator quarks or antiquarks. Thus the explanation of the
variety of the inclusive decay rates within the flavor $SU(3)$ multiplets,
observed for charmed hadrons and expected for the $b$ ones, has to be sought
among the $m_Q^{-3}$ terms.

%-------------------------------------------------------------------------------
\subsection{$L_{eff}^{(3)}$ Coefficients and operators}

Although the third term in the expansion (\ref{first3}) is formally suppressed
by an extra power of $m_Q^{-1}$, its effects are comparable to, or even larger
than the effects of the second term. This is due to the fact that the diagrams
determining the third term (see Fig.~8.1) contain a two-body phase space, while
the first two terms involve a three-body phase space. This brings in a numerical
enhancement factor, typically $4 \pi^2$. The enhanced numerical significance of
the third term in OPE, generally, does not signal a poor convergence of the
expansion in inverse heavy quark mass for decays of $b$, and even charmed,
hadrons the numerical enhancement factor is a one time occurrence in the series,
and there is no reason for similar `anomalous' enhancement among the higher
terms in the expansion.

Here we first present the expressions for the relevant parts of $L_{eff}^{(3)}$
for decays of $b$ and $c$ hadrons in the form of four-quark operators and then
proceed to a discussion of hadronic matrix elements and the effects in specific
inclusive decay rates. The consideration of the effects in decays of charmed
hadrons is interesting in its own right, and leads to new predictions to be
tested experimentally, and is also important for understanding the magnitude of
the involved matrix elements using the existing data on charm
decays.\index{inclusive weak decay rates!spectator quark effects}

We start with considering the term $L_{eff}^{(3)}$ in $b$ hadron nonleptonic
decays, $L_{eff, nl}^{(3,b)}$, induced by the underlying processes $b \to c \,
{\overline u} \, d$, $b \to c \, {\overline c} \, s$, $b \to c \, {\overline u}
\, s$, and $b \to c \, {\overline c} \, d$. Unlike the case of three-body decay,
the kinematical difference between the two-body states $c \overline c$ and $c
\overline u$, involved in calculation of $L_{eff, nl}^{(3,b)}$ is of the order
of $m_c^2/m_b^2 \approx 0.1$ and is rather small. At present level of accuracy
in discussing this term in OPE, one can safely neglect the effect of finite
charmed quark mass\footnote{The full expression for a finite charmed quark mass
  can be found in \cite{ns:97}}. In this approximation the expression for
$L_{eff, nl}^{(3,b)}$ reads as \cite{sv:86}
%
\begin{eqnarray}
\label{l3nlb}
L_{eff, \, nl}^{(3, b)} &=&  |V_{cb}|^2 \,{G_F^2 \, m_b^2 \over 4
\pi} \,
\bigg\{
{\tilde C}_1 \, (\overline b \Gamma_\mu b)(\overline u \Gamma_\mu u) +
{\tilde C}_2  \,
(\overline b \Gamma_\mu u) (\overline u \Gamma_\mu b) \nonumber\\
&&{} + {\tilde C}_5 \, (\overline  b \Gamma_\mu b +
{2 \over 3}\overline b \gamma_\mu \gamma_5 b) (\overline q \Gamma_\mu
q)+ {\tilde C}_6 \, (\overline  b_i \Gamma_\mu b_k +
{2 \over 3}\overline b_i \gamma_\mu \gamma_5 b_k)
(\overline q_k \Gamma_\mu q_i) \nonumber\\ 
&&{} + {1 \over 3} \,
{\tilde \kappa}^{1/2} \, ({\tilde \kappa}^{-2/9}-1) \, \bigg[ 2 \,
({\tilde C}_+^2 - {\tilde C}_-^2) \,
 (\overline b \Gamma_\mu t^a b) \,
j_\mu^a  \nonumber\\ 
&&{} -
(5{\tilde C}_+^2+{\tilde C}_-^2 - 6 \, {\tilde C}_+ \, {\tilde C}_-)
(\overline  b \Gamma_\mu t^a b +
{2 \over 3}\overline b \gamma_\mu \gamma_5 t^a b) j_\mu^a \bigg]
\bigg\}\,,
\end{eqnarray}
%
where the notation $(\overline q \, \Gamma \, q)= (\overline d \, \Gamma \, d) +
(\overline s \, \Gamma \, s)$ is used, the indices $i, \, k$ are the color
triplet ones, $\Gamma_\mu=\gamma_\mu \, (1-\gamma_5)$, and $j_\mu^a=\overline u
\gamma_\mu t^a u + \overline d \gamma_\mu t^a d + \overline s \gamma_\mu t^a s$
is the color current of the light quarks with $t^a = \lambda^a /2$ being the
generators of the color $SU(3)$. The notation ${\tilde C}_\pm$, is used as
shorthand for the short-distance renormalization coefficients $C_\pm(\mu)$ at
$\mu=m_b$: ${\tilde C}_\pm \equiv C_\pm(m_b)$. The expression (\ref{l3nlb}) is
written in the leading logarithmic approximation for the QCD radiative effects
in a low normalization point $\mu$ such that $\mu \ll m_b$ (but still, at least
formally, $\mu \gg \Lambda_{QCD}$). For such $\mu$ there arises so called
`hybrid' renormalization \cite{sv:87}, depending on the factor ${\tilde
  \kappa}=\alpha_s(\mu)/\alpha_s(m_b)$. The coefficients ${\tilde C}_A$ with
$A=1, \ldots , 6$ in Eq.~(\ref{l3nlb}) have the following explicit expressions in
terms of ${\tilde C}_\pm$ and ${\tilde \kappa}$:
%
\begin{eqnarray}
&&{\tilde C}_1= {\tilde C}_+^2+{\tilde C}_-^2 + {1 \over 3} (1 -
\kappa^{1/2}) ({\tilde C}_+^2-{\tilde C}_-^2)\,,
\nonumber \\
&&{\tilde C}_2= \kappa^{1/2} \, ({\tilde C}_+^2-{\tilde C}_-^2)\,,
\nonumber \\
&&{\tilde C}_3=- {1 \over 4} \, \left [ ({\tilde C}_+-{\tilde C}_-)^2 +
{1 \over 3}
(1-\kappa^{1/2})
(5{\tilde C}_+^2+{\tilde C}_-^2+6{\tilde C}_+{\tilde C}_-) \right] \,,
\nonumber \\
&&{\tilde C}_4=-{1 \over 4} \, \kappa^{1/2} \, (5{\tilde C}_+^2+{\tilde
C}_-^2+6{\tilde C}_+{\tilde C}_-)\,, \nonumber
\\
&&{\tilde C}_5=-{1 \over 4} \, \left [ ({\tilde C}_++{\tilde C}_-)^2 +
{1 \over 3} (1-\kappa^{1/2})
(5{\tilde C}_+^2+{\tilde C}_-^2-6{\tilde C}_+{\tilde C}_-) \right]\,,
\nonumber \\
&&{\tilde C}_6=-{1 \over 4} \, \kappa^{1/2} \, (5{\tilde C}_+^2+{\tilde
C}_-^2-6{\tilde C}_+{\tilde C}_-)\,.
\label{coefs}
\end{eqnarray}

The expression for the CKM dominant semileptonic decays of $b$ hadrons,
associated with the elementary process $b \to c \, \ell \, \nu$ does not look to
be of an immediate interest. The reason is that this process is intrinsically
symmetric under the flavor $SU(3)$, and one expects no significant splitting of
the semileptonic decay rates within $SU(3)$ multiplets of the $b$ hadrons. The
only possible effect of this term, arising through a penguin-like mechanism can
be in a small overall shift of semileptonic decay rates between $B$ mesons and
baryons. However, these effects are quite suppressed and are believed to be even
smaller than the ones arising form the discussed $m_b^{-2}$ terms.

For charm decays there is a larger, than for $b$ hadrons, variety of effects
associated with $L_{eff}^{(3)}$ that can be studied experimentally, and we
present here the relevant parts of the effective Lagrangian. For the CKM
dominant nonleptonic decays of charm, originating from the quark process $c \to
s \, u \, {\overline d}$, the discussed term in OPE has the form
%
\begin{eqnarray}
\label{l3nl}
&&L_{eff, nl }^{(3, \, \Delta C = \Delta S)}= \cos^4\theta_c \,{G_F^2 \,
m_c^2 \over 4 \pi} \,
\bigg\{
C_1 \, (\overline c \Gamma_\mu c)(\overline d \Gamma_\mu d) + C_2  \,
(\overline c \Gamma_\mu d) (\overline d \Gamma_\mu c)
\nonumber \\
&&{} + C_3 \, (\overline  c \Gamma_\mu c +
{2 \over 3}\overline c \gamma_\mu \gamma_5 c) (\overline s \Gamma_\mu
s)+ C_4 \, (\overline  c_i \Gamma_\mu c_k +
{2 \over 3}\overline c_i \gamma_\mu \gamma_5 c_k)
(\overline s_k \Gamma_\mu s_i)
\\ \nonumber
&&{} + C_5 \, (\overline  c \Gamma_\mu c +
{2 \over 3}\overline c \gamma_\mu \gamma_5 c) (\overline u \Gamma_\mu
u)+ C_6 \, (\overline  c_i \Gamma_\mu c_k +
{2 \over 3}\overline c_i \gamma_\mu \gamma_5 c_k)
(\overline u_k \Gamma_\mu u_i) \\ \nonumber
&&{} + {1 \over 3} \,
\kappa^{1/2} \, (\kappa^{-2/9}-1) \, \bigg[ 2 \, (C_+^2 - C_-^2) \,
 (\overline c \Gamma_\mu t^a c) \,
j_\mu^a - (5C_+^2+C_-^2)
(\overline  c \Gamma_\mu t^a c +
{2 \over 3}\overline c \gamma_\mu \gamma_5 t^a c) j_\mu^a \bigg]
\bigg\}\,,
\end{eqnarray}
%
where, $\theta_c$ is the Cabibbo angle, and the coefficients without the tilde
are given by the same expressions as above for the $b$ decays (i.e., those with
tilde) with the replacement $m_b \to m_c$. The part of the notation in the
superscript $\Delta C = \Delta S$ points to the selection rule for the dominant
CKM unsuppressed nonleptonic decays.  One can rather realistically envisage
however a future study of inclusive rates for the once CKM suppressed decays of
charmed hadrons\footnote{Even if the inclusive rate of these decays is not to be
  separated experimentally, they contribute about 10\% of the total decay rate,
  and it is worthwhile to include their contribution in the balance of the total
  width.}, satisfying the selection rule $\Delta S=0$ and associated with the
quark processes $c \to d \, u \, {\overline s}$ and $c \to d \, u \, {\overline
  d}\,$. The corresponding part of the effective Lagrangian for these processes
reads as
%
\begin{eqnarray}
&&L_{eff, \, nl}^{(3, \Delta S=0)}= \cos^2 \theta_c \, \sin^2 \theta_c
\,{G_F^2 \, m_c^2 \over 4 \pi} \,
\bigg\{
C_1 \, (\overline c \Gamma_\mu c)(\overline q \Gamma_\mu q) + C_2  \,
(\overline c_i \Gamma_\mu c_k) (\overline q_k \Gamma_\mu q_i)
\nonumber \\*
&&{} + C_3 \, (\overline  c \Gamma_\mu c +
{2 \over 3}\overline c \gamma_\mu \gamma_5 c) (\overline q \Gamma_\mu
q)+ C_4 \, (\overline  c_i \Gamma_\mu c_k +
{2 \over 3}\overline c_i \gamma_\mu \gamma_5 c_k)
(\overline q_k \Gamma_\mu q_i) \\*
&&{} + 2 \, C_5 \, (\overline  c \Gamma_\mu c +
{2 \over 3}\overline c \gamma_\mu \gamma_5 c) (\overline u \Gamma_\mu
u)+ 2 \, C_6 \, (\overline  c_i \Gamma_\mu c_k +
{2 \over 3}\overline c_i \gamma_\mu \gamma_5 c_k)
(\overline u_k \Gamma_\mu u_i)  \nonumber\\
&&{} + {2 \over 3} \,
\kappa^{1/2} \, (\kappa^{-2/9}-1) \, \bigg[ 2 \, (C_+^2 - C_-^2) \,
 (\overline c \Gamma_\mu t^a c) \,
j_\mu^a - (5C_+^2+C_-^2)
(\overline  c \Gamma_\mu t^a c +
{2 \over 3}\overline c \gamma_\mu \gamma_5 t^a c) j_\mu^a \bigg]
\bigg\}\,, \nonumber
\label{l3nl1}
\end{eqnarray}
%
where again the notation $(\overline q \, \Gamma \, q)= (\overline d \, \Gamma
\, d) + (\overline s \, \Gamma \, s)$ is used.

The semileptonic decays of charm, the CKM dominant, associated with $c \to s \,
\ell \, \nu$, and the CKM suppressed, originating from $c \to s \, \ell \, \nu$,
contribute to the semileptonic decay rate, which certainly can be measured
experimentally. The expression for the part of the effective Lagrangian,
describing the $m_Q^{-3}$ terms in these decays is
\cite{mv:96,cheng:97,gm:98}
\index{inclusive weak decay rates!semileptonic decay of charm}%
%
\begin{eqnarray}
&&L_{eff, \, sl}^{(3)}= 
{G_F^2 \, m_c^2 \over 12 \pi} \, \bigg\{ \cos^2 \theta_c \left[
L_1 \, (\overline  c \Gamma_\mu c +
{2 \over 3}\overline c \gamma_\mu \gamma_5 c) (\overline s \Gamma_\mu
s)+ L_2 \, (\overline  c_i \Gamma_\mu c_k +  {2 \over 3}\overline c_i
\gamma_\mu \gamma_5 c_k)
(\overline s_k \Gamma_\mu s_i) \right ] \nonumber \\
&&{} + \sin^2 \theta_c \left [
L_1\, (\overline  c \Gamma_\mu c +
{2 \over 3}\overline c \gamma_\mu \gamma_5 c) (\overline d \Gamma_\mu
d)+L_2 \, (\overline  c_i \Gamma_\mu c_k +
{2 \over 3}\overline c_i \gamma_\mu \gamma_5 c_k)
(\overline d_k \Gamma_\mu d_i) \right ]  \nonumber \\
&&{} - 2 \, \kappa^{1/2} \, (\kappa^{-2/9}-1) \,
(\overline  c \Gamma_\mu t^a c +
{2 \over 3}\overline c \gamma_\mu \gamma_5 t^a c) j_\mu^a
\bigg\} \,,
\label{l3sl}
\end{eqnarray}
%
with the coefficients $L_1$ and $L_2$ found as
%
\begin{equation}
L_1=(\kappa^{1/2}-1), ~~~~~
L_2 = -   3\, \kappa^{1/2}\,.
\label{coefl}
\end{equation}

%-------------------------------------------------------------------------------
\subsection{Effects of $L_{eff}^{(3)}$ in mesons}

The expressions for the terms in $L_{eff}^{(3)}$ still leave us with the problem
of evaluating the matrix elements of the four-quark operators over heavy hadrons
in order to calculate the effects in the decay rates according to the formula
(\ref{lgam}). In doing so only few conclusions can be drawn in a reasonably
model independent way, i.e., without resorting to evaluation of the matrix
elements using specific ideas about the dynamics of quarks inside hadrons. The
most straightforward prediction can in fact be found for $b$ hadrons. Namely,
one can notice that the operator (\ref{l3nlb}) is symmetric under the flavor U
spin (an SU(2) subgroup of the flavor $SU(3)$, which mixes $s$ and $d$ quarks).
This is a direct consequence of neglecting the small kinematical effect of the
charmed quark mass. However the usual (in)accuracy of the flavor $SU(3)$ symmetry
is likely to be a more limiting factor for the accuracy of this symmetry, than
the corrections of order $m_c^2/m_b^2$. Modulo this reservation the immediate
prediction from this symmetry is the degeneracy of inclusive decay rates within
U spin doublets:
%
\begin{equation}
\Gamma(B_d)=\Gamma(B_s)\,,~~~~\Gamma(\Lambda_b)=\Gamma(\Xi_b^0)\,,
\label{upred}
\end{equation}
%
where $\Gamma(B_s)$ stands for the average rate over the two eigenstates of the
$B_s - {\overline B}_s$ oscillations. The data on decay rates of the cascade
hyperon $\Xi_b^0$ are not yet available, while the currently measured lifetimes
of $B_d$ and $B_s$ are within less than 2\% from one another. Theoretically, the
difference of the lifetimes, associated with possible violation of the $SU(3)$
symmetry and with breaking of the U symmetry of the effective Lagrangian
(\ref{l3nlb}), is expected to not exceed about 1\%.

For the non-vanishing matrix elements of four-quark operators over pseudoscalar
mesons one traditionally starts with the factorization formula and parameterizes
possible deviation from factorization in terms of `bag constants'. Within the
normalization convention adopted here the relations used in this parameterization
read as\index{Bag constants}
%
\begin{eqnarray}
&&\langle P_{Q \overline q} | (\overline Q \Gamma_\mu q) \, (\overline q
\Gamma_\mu Q) | P_{Q \overline q} \rangle = {1 \over 2} \, f_P^2 \, M_P
\,B \,, \nonumber \\
&&\langle P_{Q \overline q} | (\overline Q \Gamma_\mu Q) \, (\overline q
\Gamma_\mu q) | P_{Q \overline q} \rangle = {1 \over 6} \, f_P^2 \, M_P
\,{\tilde B}\,,
\label{bagc}
\end{eqnarray}
%
where $P_{Q \overline q}$ stands for pseudoscalar meson made of $Q$ and
${\overline q}$, $f_P$ is the annihilation constant for the meson, and $B$ and
${\tilde B}$ are bag constants. The parameters $B$ and ${\tilde B}$ generally
depend on the normalization point $\mu$ for the operators, and this dependence
is compensated by the $\mu$ dependence of the coefficients in $L_{eff}^{(3)}$,
so that the results for the physical decay rate difference do not depend on
$\mu$. If the normalization point $\mu$ is chosen at the heavy quark mass (i.e.
$\mu=m_b$ for $B$ mesons, and $\mu=m_c$ for $D$ mesons) the predictions for the
difference of total decay rates take a simple form in terms of the corresponding
bag constants (generally different between $B$ and $D$):
%
\begin{eqnarray}
\Gamma(B^\pm) - \Gamma(B^0) &=& |V_{cb}|^2 \,{G_F^2 \, m_b^3 \, f_B^2
\over 8
\pi} \left [ ({\tilde C}_+^2-{\tilde C}_-^2) \, B(m_b) + {1 \over 3}\,
({\tilde C}_+^2+{\tilde C}_-^2) \,  {\tilde B}(m_b) \right] \nonumber \\
&\approx& - 0.025 \, \left ( {f_B \over 200 \, MeV } \right )^2 \,
ps^{-1}\,, \label{bpred} \\[6pt]
\Gamma(D^\pm) - \Gamma(D^0) &=&  \cos^4 \theta_c \,{G_F^2 \, m_c^3 \,
f_D^2 \over 8
\pi} \left [ (C_+^2-C_-^2) \,  B(m_c) + {1 \over 3}\, (C_+^2+C_-^2) \,
{\tilde B}(m_c) \right] \nonumber \\
&\sim& - 0.8 \, \left ( {f_D \over 200 \, MeV } \right )^2 \, ps^{-1}\,,
\label{dpred}
\end{eqnarray}
%
where the numerical values are written in the approximation of exact
factorization: $B=1$, and ${\tilde B}=1$. It is seen from the numerical
estimates that, even given all the theoretical uncertainties, the presented
approach is in reasonable agreement with the data on the lifetimes of $D$ and
$B$ mesons. In particular, this approach describes, at least qualitatively, the
strong suppression of the decay rate of $D^\pm$ mesons relative to $D^0$, the
experimental observation of which has in fact triggered in early 80-s the
theoretical study of preasymptotic in heavy quark mass effects in inclusive
decay rates.  For the $B$ mesons the estimate (\ref{bpred}) is also in a
reasonable agreement with the current data for the discussed difference ($-0.043
\pm 0.017 \, ps^{-1}$).

%-------------------------------------------------------------------------------
\subsection{Effects of $L_{eff}^{(3)}$ in baryons}

The weakly decaying heavy hyperons, containing either $c$ or $b$ quark are:
$\Lambda_Q \sim Qud,~ \Xi_Q^{(u)} \sim Qus,~ \Xi_Q^{(d)} \sim Qds$, and
$\Omega_Q \sim Qss$. The first three baryons form an $SU(3)$ (anti)triplet. The
light diquark in all three is in the state with quantum numbers $J^P = 0^+$, so
that there is no correlation of the spin of the heavy quark with the light
component of the baryon.\index{heavy baryons!$SU(3)$ antitriplet} On the contrary,
in $\Omega_Q$ the two strange quarks form a $J^P = 1^+$ state, and a correlation
between the spins of heavy and light quarks is present. The absence of spin
correlation for the heavy quark in the triplet of hyperons somewhat reduces the
number of independent four-quark operators, having nonvanishing diagonal matrix
elements over these baryons. Indeed, the operators entering $L_{eff}^{(3)}$
contain both vector and axial bilinear forms for the heavy quarks. However the
axial part requires a correlation of the heavy quark spin with that of a light
quark, and is thus vanishing for the hyperons in the triplet.  Therefore only
the structures with vector currents are relevant for these hyperons. These
structures are of the type $(\overline c \, \gamma_\mu \, c) (\overline q \,
\gamma_\mu \, q)$ and $(\overline c_i \, \gamma_\mu \, c_k) (\overline q_k \,
\gamma_\mu \, q_i)$ with $q$ being $d$, $s$ or $u$.  The flavor $SU(3)$ symmetry
then allows to express, for each of the two color combinations, the matrix
elements of three different operators, corresponding to three flavors of $q$,
over the baryons in the triplet in terms of only two combinations: flavor octet
and flavor singlet. Thus all effects of $L_{eff}^{(3)}$ in the triplet of the
baryons can be expressed in terms of four independent combinations of matrix
elements.  These can be chosen in the following way:
%
\begin{eqnarray}
\label{msh:defxy}
x&=&\left \langle  {1 \over 2} \, (\overline Q \, \gamma_\mu \, Q) \left
[
(\overline u \, \gamma_\mu u) - (\overline s \, \gamma_\mu s) \right]
\right \rangle_{\Xi_Q^{(d)}-\Lambda_Q}  \nonumber\\
&=& \left \langle  {1 \over 2} \,
(\overline Q \, \gamma_\mu \, Q) \left [ (\overline s \, \gamma_\mu s) -
(\overline d \, \gamma_\mu d) \right] \right \rangle_{\Lambda_Q -
\Xi_Q^{(u)}}\,,   \nonumber\\
y&=&\left \langle  {1 \over 2} \, (\overline Q_i \, \gamma_\mu \, Q_k)
\left [ (\overline u_k \, \gamma_\mu u_i) - (\overline s_k \, \gamma_\mu
s_i) \right ] \right \rangle_{\Xi_Q^{(d)}-\Lambda_Q}  \nonumber\\
&=& \left \langle  {1
\over 2} \, (\overline Q_i \, \gamma_\mu \, Q_k) \left [ (\overline s_k
\, \gamma_\mu s_i) - (\overline d_k \, \gamma_\mu d_i) \right] \right
\rangle_{\Lambda_Q - \Xi_Q^{(u)}}\,,
\end{eqnarray}
%
with the notation for the differences of the matrix elements: $\langle {\cal O}
\rangle_{A-B}= \langle A | {\cal O} | A \rangle - \langle B | {\cal O} | B
\rangle$, for the flavor octet part and the matrix elements:
%
\begin{eqnarray}
&&x_s = {1 \over 3} \, \langle H_Q | ({\overline Q} \, \gamma_\mu \, Q)
\left ( ({\overline u} \, \gamma_\mu \, u)+({\overline d} \, \gamma_\mu
\, d)+ ({\overline s} \, \gamma_\mu \, s) \right ) | H_Q \rangle
\nonumber \\
&&y_s = {1 \over 3} \, \langle H_Q | ({\overline Q}_i \, \gamma_\mu \,
Q_k) \left ( ({\overline u}_k \, \gamma_\mu \, u_i)+({\overline d}_k \,
\gamma_\mu \, d_i)+ ({\overline s}_k \, \gamma_\mu \, s_i) \right ) |
H_Q \rangle
\label{xys}
\end{eqnarray}
%
for the flavor singlet part, where $H_Q$ stands for any heavy hyperon in the
(anti)triplet.

The initial, very approximate, theoretical estimates of the matrix elements
\cite{sv:86} were essentially based on a non-relativistic constituent quark
model, where these matrix elements are proportional to the density of a light
quark at the location of the heavy one, i.e., in terms of the wave function,
proportional to $|\psi(0)|^2$. Using then the same picture for the matrix
elements over pseudoscalar mesons, relating the quantity $|\psi(0)|^2$ to the
annihilation constant $f_P$, and assuming that $|\psi(0)|^2$ is approximately
the same in baryons as in mesons, one arrived at the estimate
\index{heavy baryons!matrix elements}%
%
\begin{equation}
  y=-x=x_s=-y_s \approx {f_D^2 \, M_D \over 12} \approx 0.006 \, GeV^2\,,
\label{xyold}
\end{equation}
%
where the sign relation between $x$ and $y$ is inferred from the color
antisymmetry of the constituent quark wave function for baryons. Since the
constituent picture was believed to be valid at distances of the order of the
hadron size, the estimate (\ref{xyold}) was applied to the matrix elements in a
low normalization point where $\alpha_s(\mu) \approx 1$. For the matrix elements
of the operators, containing $s$ quarks over the $\Omega_Q$ hyperon, this
picture predicts an enhancement factor due to the spin correlation:
%
\begin{equation}
\langle \Omega_Q | ({\overline Q} \, \Gamma_\mu \, Q)
 ({\overline s} \, \Gamma_\mu \, s)  |\Omega_Q \rangle = - \langle
\Omega_Q | ({\overline Q}_i \, \Gamma_\mu \, Q_k)
 ({\overline s}_k \, \Gamma_\mu \, s_i)  |\Omega_Q \rangle = {10 \over
3}\, y
\label{omegaold}
\end{equation}
%
Although these simple estimates allowed to correctly predict \cite{sv:86} the
hierarchy of lifetimes of charmed hadrons prior to establishing this hierarchy
experimentally, they fail to quantitatively predict the differences of lifetimes
of charmed baryons. We shall see that the available data indicate that the color
antisymmetry relation is badly broken, and the absolute value of the matrix
elements is larger, than the naive estimate (\ref{xyold}), especially for the
quantity $x$.


It should be emphasized that in the heavy quark limit the matrix elements
(\ref{msh:defxy}) and (\ref{xys}) do not depend on the flavor of the heavy
quark, provided that the same normalization point $\mu$ is used. Therefore,
applying the OPE formulas to both charmed and $b$ baryons, one can extract the
values for the matrix elements from available data on charmed hadrons, and then
make predictions for $b$ baryons, as well as for other inclusive decay rates,
e.g., semileptonic, for charmed hyperons.

The only data available so far, which would allow to extract the matrix
elements, are on the lifetimes of charmed hyperons. Therefore, one has to take
into account several essential types of inclusive decay, at least those that
contribute to the total decay rate at the level of about 10\%. Here we first
concentrate on the differences of the decay rates within the $SU(3)$ triplet of
the hyperons, which will allow us to extract the non-singlet quantities $x$ and
$y$, and then discuss the $SU(3)$ singlet shifts of the rates.

The differences of the dominant Cabibbo unsuppressed nonleptonic decay rates
are given by
%
\begin{eqnarray}
\delta_1^{nl, \,0} \equiv \Gamma^{nl}_{\Delta S =
\Delta C}(\Xi_c^0)-\Gamma^{nl}_{\Delta S = \Delta C}(\Lambda_c) = \cos^4
\theta_c \, {G_F^2 \,
m_c^2 \over 4 \pi} \left [ (C_5 - C_3) \, x + (C_6 - C_4) \, y \right
]\,, \nonumber \\
\delta_2^{nl, \,0} \equiv \Gamma^{nl}_{\Delta S =
\Delta C}(\Lambda_c)-\Gamma^{nl}_{\Delta S =\Delta C}(\Xi_c^+) = \cos^4
\theta_c \, {G_F^2 \,
m_c^2 \over 4 \pi} \left [ (C_3 - C_1) \, x + (C_4 - C_2) \, y \right
]\,.
\label{dnl0}
\end{eqnarray}
%
The once Cabibbo suppressed decay rates of $\Lambda_c$ and $\Xi_c^+$ are equal,
due to the $\Delta U =0$ property of the corresponding effective Lagrangian
$L_{eff, nl}^{(3,1)}$ (Eq.~(\ref{l3nl1})). Thus the only difference for these
decays in the baryon triplet is
%
\begin{eqnarray}
\delta^{nl,1} & \equiv & \Gamma^{nl}_{\Delta S =0}
(\Xi_c^0)-\Gamma^{nl}_{\Delta S = 0 }(\Lambda_c) \nonumber \\
& = & \cos^2 \theta_c \,
\sin^2 \theta_c \, {G_F^2
\, m_c^2 \over 4 \pi} \left [ (2\, C_5 - C_1 - C_3) \, x + (2 \, C_6 -
C_2 - C_4) \, y \right ]\,.
\label{dnl1}
\end{eqnarray}
%
The dominant semileptonic decay rates are equal among the two $\Xi_c$ baryons
due to the isotopic spin property $\Delta I =0$ of the corresponding interaction
Lagrangian, thus there is only one non-trivial splitting for these decays:
%
\begin{equation}
\delta^{sl,0} \equiv \Gamma^{sl}_{\Delta S = -1}(\Xi_c) -
\Gamma^{sl}_{\Delta S = -1} (\Lambda_c) = - \cos^2 \theta_c {G_F^2 \,
m_c^2 \over 12
\pi} \left [ L_1 \, x + L_2 \, y \right ]\,.
\label{dsl0}
\end{equation}
%
Finally, the Cabibbo suppressed semileptonic decay rates are equal for
$\Lambda_c$ and $\Xi_c^0$, due to the $\Delta V =0$ property of the
corresponding interaction. Thus the only difference for these is
%
\begin{equation}
\delta^{sl,1} \equiv \Gamma^{sl}_{\Delta S = 0}(\Lambda_c) -
\Gamma^{sl}_{\Delta S = 0}(\Xi_c^+) = - \sin^2 \theta_c {G_F^2 \, m_c^2
\over 12
\pi} \left [ L_1 \, x + L_2 \, y \right ]\,.
\label{dsl1}
\end{equation}

Using the relations (\ref{dnl0})--(\ref{dsl1}) on can find expressions for two
differences of the measured total decay rates, $\Delta_1 = \Gamma (\Xi_c^0) -
\Gamma (\Lambda_c)$ and $\Delta_2= \Gamma (\Lambda_c) - \Gamma (\Xi_c^+)$, in
terms of the quantities $x$ and $y$:
%
\begin{eqnarray}
\Delta_1 &=& \delta_1^{nl, \,0} + \delta^{nl,1} + 2 \, \delta^{sl,0}
\nonumber \\
&=& {G_F^2 \, m_c^2 \over 4 \pi} \, \cos^2 \theta \, \bigg\{ x
 \left [ \cos^2 \theta \, (C_5-C_3) + \sin^2 \theta \, (2 \, C_5 - C_1
-C_3)- {2 \over 3} \, L_1 \right ] \nonumber \\
&&{} + y \left[ \cos^2 \theta \,
(C_6-C_4) + \sin^2 \theta \, (2 \, C_6 - C_2 -C_4)- {2 \over 3} \,
L_2\right ] \bigg\}\,,
\label{d1}
\end{eqnarray}
%
and
%
\begin{eqnarray}
\Delta_2 &=& \delta_2^{nl, \,0} - 2 \, \delta^{sl,0} + 2 \, \delta^{sl,1}
\nonumber \\
&=& {G_F^2 \, m_c^2 \over 4 \pi}\, \bigg\{ x \left[ \cos^4
\theta \, (C_3-C_1) + {2 \over 3} \, (\cos^2 \theta - \sin^2 \theta) \,
L_1 \right] \nonumber \\
&&{} + y \left[ \cos^4 \theta \, (C_4-C_2) + {2 \over 3} \,
(\cos^2 \theta - \sin^2 \theta) \, L_2 \right] \bigg\}\,.
\label{d2}
\end{eqnarray}

By comparing these relations with the data, one can extract the values of $x$
and $y$. Using the current data for the total decay rates:
$\Gamma(\Lambda_c)=4.85 \pm 0.28 \, ps^{-1}$, $\Gamma(\Xi_c^0)= 10.2 \pm 2 \,
ps^{-1}$, and the updated value \cite{pdg1} $\Gamma(\Xi_c^+)=3.0 \pm 0.45 \,
ps^{-1}$, we find for the $\mu$ independent matrix element $x$
%
\begin{equation}
x = -(0.04 \pm 0.01) \, GeV^3 \, \left ( 1.4 \, GeV \over m_c \right
)^2\,,
\label{resx}
\end{equation}
%
while the dependence of the thus extracted matrix element $y$ on the
normalization point $\mu$ is shown in Fig.~8.2.\footnote{It should be noted
  that the curves at large values of $\kappa$, $\kappa > \, \sim 3$, are shown
  only for illustrative purpose. The coefficients in the OPE, leading to the
  equations (\ref{d1},\ref{d2}), are purely perturbative.  Thus, formally, they
  correspond to $\alpha_s(\mu) \ll 1$, i.e., to $\kappa \ll 1/\alpha_s(m_c) \sim
  (3 - 4)$.}

\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \epsfbox{ch8/th_hbaryons1.eps}
    \vspace*{6pt}
\caption[The values of the extracted matrix elements $x$ and $y$ in $GeV^3$
vs.\ the normalization point parameter
$\kappa=\alpha_s(\mu)/\alpha_s(m_c)$.]{The values of the extracted matrix
elements $x$ and $y$ in $GeV^3$ vs.\ the normalization point parameter
$\kappa=\alpha_s(\mu)/\alpha_s(m_c)$. The thick lines correspond to the central
value of the data on lifetimes of charmed baryons, and the thin lines show the
error corridors. The extracted values of $x$ and $y$ scale as $m_c^{-2}$ with
the assumed mass of the charmed quark, and the plots are shown for $m_c=1.4 \,
GeV$.}
\label{fig:xy}
  \end{center}
\end{figure}

Notably, the extracted values of $x$ and $y$ are in a drastic variance with the
simplistic constituent model: the color antisymmetry relation, $x=-y$, does not
hold at any reasonable $\mu$, and the absolute value of $x$ is substantially
enhanced\footnote{A similar, although with a smaller enhancement, behavior of
  the matrix elements was observed in a recent preliminary lattice study
  \cite{psm:99}.}

Once the non-singlet matrix elements are determined, they can be used for
predicting differences of other inclusive decay rates within the triplet of
charmed hyperons as well as for the $b$ baryons.
\index{inclusive weak decay rates!splitting for baryons}%
Due to correlation of errors in $x$ and $y$ it
makes more sense to express the predictions directly in terms of the total decay
rates of the charmed hyperons. The thus arising relations between the rates do
not depend on the normalization parameter $\mu$. In this way one finds
\cite{mv:99-1} for the difference of the Cabibbo dominant semileptonic decay
rates between either of the $\Xi_c$ hyperons and $\Lambda_c$:
%
\begin{equation}
\Gamma_{sl}(\Xi_c)-\Gamma_{sl}(\Lambda_c) \approx \delta^{sl, 0}= 0.13
\, \Delta_1 - 0.065 \, \Delta_2 \approx 0.59 \pm 0.32 \, ps^{-1}\,.
\label{dslr}
\end{equation}
%
When compared with the data on the total semileptonic decay rate of
$\Lambda_c$, $\Gamma_{sl}(\Lambda_c)=0.22 \pm 0.08 \, ps^{-1}$, this prediction
implies that the semileptonic decay rate of the charmed cascade hyperons can be
2--3 times larger than that of $\Lambda_c$.

The predictions found in a similar way for the inclusive Cabibbo suppressed
decay rates are \cite{mv:99-1}: for nonleptonic decays
%
\begin{equation}
\delta^{nl,1}=0.082 \, \Delta_1 + 0.054 \, \Delta_2 \approx 0.55 \pm
0.22 \, ps^{-1}
\label{dnl1r}
\end{equation}
%
and for the semileptonic ones
%
\begin{equation}
\delta^{sl,1}=\tan^2 \theta_c \, \delta^{sl, 0} \approx 0.030 \pm 0.016
\, ps^{-1}\,.
\label{dsl1r}
\end{equation}

For the only difference of the inclusive rates in the triplet of $b$ baryons,
$\Gamma(\Lambda_b)-\Gamma(\Xi_b^-)$, one finds an expression in terms of $x$ and
$y$, or alternatively, in terms of the differences $\Delta_1$ and $\Delta_2$
between the charmed hyperons,
%
\begin{eqnarray}
\Gamma(\Lambda_b)-\Gamma(\Xi_b^-) &=& \cos^2 \theta_c \, |V_{cb}|^2
\,{G_F^2 \, m_b^2 \over 4 \pi} \, \left[ (\tilde C_5 - \tilde C_1) \, x
+ (\tilde C_6 - \tilde C_2) \, y \right] \\
&\approx& |V_{cb}|^2  \, {m_b^2 \over m_c^2} \, (0.85 \,
\Delta_1 + 0.91 \, \Delta_2) \approx 0.015 \, \Delta_1 + 0.016 \,
\Delta_2 \approx 0.11 \pm 0.03 \, ps^{-1}\,.\nonumber 
\label{dbres}
\end{eqnarray}
%
When compared with the data on the total decay rate of $\Lambda_b$ this result
predicts about 14\% longer lifetime of $\Xi_b^-$ than that of $\Lambda_b$.

The singlet matrix elements $x_s$ and $y_s$ (cf. Eq.~(\ref{xys})) are related to
the shift of the average decay rate of the hyperons in the triplet:
\index{heavy baryons!average decay rate}%
%
\begin{equation}
{\overline \Gamma}_Q= {1 \over 3} \, \left ( \Gamma(\Lambda_Q) +
\Gamma(\Xi_Q^1)+\Gamma(\Xi_Q^2) \right )\,.
\label{avgam}
\end{equation}
%
For the charmed baryons the shift of the dominant nonleptonic decay rate is
given by \cite{mv:99-2}
\begin{equation}
\delta_{nl}^{(3,0)} {\overline \Gamma}_c= \cos^4 \theta \, {G_F^2 \,
m_c^2 \over 8 \pi} (C_+^2 + C_-^2)\, \kappa^{5/18} \, (x_s-3 \, y_s)\,,
\label{d3gc}
\end{equation}
%
while for the $b$ baryons the corresponding expression reads as
%
\begin{equation}
\delta^{(3)} {\overline \Gamma}_b= |V_{cb}|^2 \, {G_F^2 \, m_b^2 \over 8
\pi} ({\tilde C}_+ - {\tilde C}_-)^2\, {\tilde \kappa}^{5/18} \, (x_s-3
\, y_s)\,.
\label{d3gb}
\end{equation}

The combination $x_s-3 \, y_s$ of the $SU(3)$ singlet matrix elements cancels in
the ratio of the shifts for $b$ hyperons and the charmed ones:
%
\begin{equation}
\delta^{(3)} {\overline \Gamma}_b= {|V_{cb}|^2 \over \cos^4 \theta} \,
{m_b^2 \over m_c^2} \,
{({\tilde C}_+ - {\tilde C}_-)^2 \over C_+^2 + C_-^2 } \,  \left [
{\alpha_s (m_c) \over \alpha_s(m_b)} \right ]^{5/18} \,
\delta_{nl}^{(3,0)} {\overline \Gamma}_c \approx 0.0025 \,
\delta_{nl}^{(3,0)} {\overline \Gamma}_c\,.
\label{rbc}
\end{equation}
%
(One can observe, with satisfaction, that the dependence on the unphysical
parameter $\mu$ cancels out, as it should.)  This equation shows that relatively
to the charmed baryons the shift of the decay rates in the $b$ baryon triplet is
greatly suppressed by the ratio $({\tilde C}_+ - {\tilde C}_-)^2 / (C_+^2 +
C_-^2)$, which parametrically is of the second order in $\alpha_s$, and
numerically is only about 0.12.

An estimate of $\delta^{(3)} {\overline \Gamma}_b$ from Eq.~(\ref{rbc}) in
absolute terms depends on evaluating the average shift $\delta_{nl}^{(3,0)}
{\overline \Gamma}_c$ for charmed baryons. The latter shift can be
conservatively bounded from above by the average total decay rate of those
baryons: $\delta_{nl}^{(3,0)} {\overline \Gamma}_c < {\overline \Gamma_c} = 6.0
\pm 0.7 \, ps^{-1}$, which then yields, using Eq.~(\ref{rbc}), an upper bound
$\delta^{(3)} {\overline \Gamma}_b < 0.015 \pm 0.002 \, ps^{-1}$. More
realistically, one should subtract from the total average width ${\overline
  \Gamma_c}$ the contribution of the `parton' term, which can be estimated from
the decay rate of $D_0$ with account of the $O(m_c^{-2})$ effects, as amounting
to about $3 \, ps^{-1}$. (One should also take into account the semileptonic
contribution to the total decay rates, which however is quite small at this
level of accuracy). Thus a realistic evaluation of $\delta^{(3)} {\overline
  \Gamma}_b$ does not exceed $0.01 \, ps^{-1}$, which constitutes only about 1\%
of the total decay rate of $\Lambda_b$.  Thus the shift of the total decay rate
of $\Lambda_b$ due to the effects of $L_{eff}^{(3)}$ is dominantly associated
with the $SU(3)$ non-singlet difference (\ref{dbres}). The shift of the
$\Lambda_b$ decay rate with respect to the average width ${\overline \Gamma}_b$
due to the non-singlet operators is one third of the splitting (\ref{dbres}),
i.e., about 5\%. Adding to this the 1\% shift of the average width and another
1\% difference from the meson decays due to the suppression of the latter by the
$m_b^{-2}$ chromomagnetic effects, one concludes that at the present level of
theoretical understanding it looks impossible to explain a more than 10\%
enhancement of the total decay rate of $\Lambda_b$ relative to $B_d$, where an
ample 3\% margin is added for the uncertainties of higher order terms in OPE as
well as for higher order QCD radiative effects in the discussed corrections. In
other words, the expected pattern of the lifetimes of the $b$ hyperons in the
triplet, relative to $B_d$, is
%
\begin{equation}
\tau(\Xi_b^0) \approx \tau(\Lambda_b) < \tau(B_d) < \tau(\Xi_b^-)\,,
\label{pattern}
\end{equation}
%
with the ``best" theoretical estimate of the differences to be about 7\% for
each step of the inequality.

For the double strange hyperons $\Omega_c$ and $\Omega_b$ there is presently no
better approach to evaluating the four-quark matrix elements, than the use of
simplistic relations, like (\ref{omegaold}) based on constituent quark model.
Such relations imply that the effects of the strange quark, WS and PI, in the
$\Omega_Q$ baryons are significantly enhanced over the same effects in the
cascade hyperons. In charmed baryons a presence of strange spectator quark
enhances the decay through positive interference with the quark emerging from
the $c \to s$ transition in the decay. For $\Omega_c$ this implies a significant
enhancement of the total decay rate \cite{sv:86}, which is in perfect agreement
with the data on the $\Omega_c$ lifetime. Also a similar enhancement is expected
for the semileptonic decay rate of $\Omega_c$.  In $b$ baryons, on the
contrary, the interference effect for a spectator strange quark is negative.
Thus the nonleptonic decay rate of $\Omega_b$ is expected to be suppressed,
leaving $\Omega_b$ most probably the longest-living particle among the $b$
baryons.

%-------------------------------------------------------------------------------
\subsection{Relation between spectator effects in baryons and \\ the decays
  $\Xi_Q \to \Lambda_Q \, \pi$}

Rather unexpectedly, the problem of four-quark matrix elements over heavy
hyperons is related to decays of the type $\Xi_Q \to \Lambda_Q \, \pi$.  The
mass difference between the charmed cascade hyperons $\Xi_c$ and $\Lambda_c$ is
about 180 MeV. The expected analogous mass splitting for the $b$ hyperons should
be very close to this number, since in the heavy quark limit
%
\begin{equation}
M(\Xi_b)-M(\Lambda_b)=M(\Xi_c)-M(\Lambda_c) + O(m_c^{-2}-m_b^{-2}).
\label{massdiff}
\end{equation}
%
Therefore in both cases are kinematically possible decays of the type $\Xi_Q \to
\Lambda_Q \, \pi$, in which the heavy quark is not destroyed, and which are
quite similar to decays of ordinary `light' hyperons.  Surprisingly, the rate of
these decays for both $\Xi_c$ and $\Xi_b$ is not insignificantly small, but
rather their branching fraction can reach a level of few per mill for $\Xi_c$
and of one percent or more for $\Xi_b$ \cite{mv:00}.

The transitions $\Xi_Q \to \Lambda_Q \, \pi$ are induced by two underlying weak
processes: the `spectator' decay of a strange quark, $s \to u \, \overline u \,
d$, which does not involve the heavy quark, and the `non-spectator' weak
scattering (WS)
%
\begin{equation}
s \, c \to c \, d
\label{ws}
\end{equation}
%
trough the weak interaction of the $c \to d$ and $s \to c$ currents.  One can
also readily see that the WS mechanism contributes only to the decays $\Xi_c \to
\Lambda_c \, \pi$ and is not present in the decays of the $b$ cascade hyperons.
An important starting point in considering these transitions is that in the
heavy quark limit the spin of the heavy quark completely decouples from the spin
of the light component of the baryon, and that the latter light component in
both the initial an the final baryons forms a $J^P=0^+$ state with quantum
numbers of a diquark.  Since the momentum transfer in the considered decays is
small in comparison with the mass of the heavy quark the spin of the amplitudes
with spin flip of the heavy quark, and thus of the baryon, are suppressed by
$m_Q^{-1}$. In terms of the two possible partial waves in the decay $\Xi_Q \to
\Lambda_Q \, \pi$, the $S$ and $P$, this implies that the $P$ wave is strongly
suppressed and the decays are dominated by the $S$ wave.

According to the well known current algebra technique, the $S$ wave amplitudes
of pion emission can be considered in the chiral limit at zero four-momentum of
the pion, where they are described by the PCAC reduction formula (pole terms are
absent in these processes):
%
\begin{equation}
\langle \Lambda_Q \, \pi_i (p=0) \,| H_W |\, \Xi_Q \rangle = {\sqrt{2}
\over f_\pi} \, \langle \Lambda_Q \, |\left[Q^5_i, \, H_W \right ] |\,
\Xi_Q \rangle\,,
\label{pcac}
\end{equation}
%
where $\pi_i$ is the pion triplet in the Cartesian notation, and $Q^5_i$ is the
corresponding isotopic triplet of axial charges. The constant $f_\pi \approx 130
\, MeV$, normalized by the charged pion decay, is used here, hence the
coefficient $\sqrt{2}$ in Eq.~(\ref{pcac}). The Hamiltonian $H_W$ in
Eq.~(\ref{pcac}) is the nonleptonic strangeness-changing Hamiltonian:
%
\begin{eqnarray}
H_W = &&\sqrt{2} \, G_F \cos \theta_c \sin \theta_c \left \{ \left
( C_+ + C_-
\right ) \left [ (\overline u_L \, \gamma_\mu \, s_L)\, (\overline
d_L \, \gamma_\mu \, u_L) - (\overline c_L \, \gamma_\mu \, s_L)\,
(\overline d_L \, \gamma_\mu \, c_L) \right ] \right . \nonumber \\
&&{} + \left. \left ( C_+ - C_- \right ) \left [ (\overline d_L \,
\gamma_\mu \, s_L)\, (\overline u_L \, \gamma_\mu \, u_L) - (\overline
d_L \, \gamma_\mu \, s_L)\, (\overline c_L \, \gamma_\mu \, c_L) \right
] \right \}\,.
\label{hw}
\end{eqnarray}
%
In this formula the weak Hamiltonian is assumed to be normalized (in LLO) at
$\mu = m_c$.  The terms in the Hamiltonian (\ref{hw}) without the charmed quark
fields describe the `spectator' nonleptonic decay of the strange quark, while
those with the $c$ quark correspond to the WS process (\ref{ws}).

It is straightforward to see from Eq.~(\ref{pcac}) that in the PCAC limit the
discussed decays should obey the $\Delta I =1/2$ rule. Indeed, the commutator of
the weak Hamiltonian with the axial charges transforms under the isotopic SU(2)
in the same way as the Hamiltonian itself. In other words, the $\Delta I=1/2$
part of $H_W$ after the commutation gives an $\Delta I=1/2$ operator, while the
$\Delta I = 3/2$ part after the commutation gives an $\Delta I = 3/2$ operator.
The latter operator however cannot have a non vanishing matrix element between
an isotopic singlet, $\Lambda_Q$, and an isotopic doublet, $\Xi_Q$. Thus the
$\Delta I=3/2$ part of $H_W$ gives no contribution to the $S$ wave amplitudes in
the PCAC limit.

Once the isotopic properties of the decay amplitudes are fixed, one can
concentrate on specific charge decay channels, e.g., $\Xi_b^- \to \Lambda_b \,
\pi^-$ and $\Xi_c^0 \to \Lambda_c \, \pi^-$. An application of the PCAC relation
(\ref{pcac}) with the Hamiltonian from Eq.~(\ref{hw}) to these decays, gives the
expressions for the amplitudes at $p=0$ in terms of baryonic matrix elements of
four-quark operators:
%
\begin{eqnarray}
&&\hspace*{-.5cm}{} \langle \Lambda_b \, \pi^- (p=0) \,| H_W |\, \Xi_b^- \rangle \\
&=& {\sqrt{2} \over f_\pi} \, G_F \cos \theta_c \sin \theta_c \,
\langle \Lambda_b
\,| \left ( C_+ + C_- \right ) \left [ (\overline u_L \, \gamma_\mu
\, s_L)\, (\overline d_L \, \gamma_\mu \, d_L) - (\overline u_L \,
\gamma_\mu \, s_L)\, (\overline u_L \, \gamma_\mu \, u_L) \right ]
\nonumber \\
&&{} + \left ( C_+ - C_- \right ) \left [ (\overline d_L \, \gamma_\mu \,
s_L)\, (\overline u_L \, \gamma_\mu \, d_L) - (\overline u_L \,
\gamma_\mu \, s_L)\, (\overline u_L \, \gamma_\mu \, u_L) \right] | \,
\Xi_b^- \rangle \nonumber \\
&=& {\sqrt{2} \over f_\pi} \, G_F \cos \theta_c \sin \theta_c \,
\langle \Lambda_b
\,|\, C_- \left [ (\overline u_L \, \gamma_\mu \, s_L)\, (\overline d_L
\, \gamma_\mu \, d_L) - (\overline d_L \, \gamma_\mu \, s_L)\,
(\overline u_L \, \gamma_\mu \, d_L) \right ] \nonumber \\
&&{} - {C_+ \over 3} \left [ (\overline u_L \, \gamma_\mu \, s_L)\,
(\overline d_L \, \gamma_\mu \, d_L) + (\overline d_L \, \gamma_\mu \,
s_L)\, (\overline u_L \, \gamma_\mu \, d_L) +2 \, (\overline u_L \,
\gamma_\mu \, s_L)\, (\overline u_L \, \gamma_\mu \, u_L) \right ] | \,
\Xi_b^- \rangle\,, \nonumber
\label{xib}
\end{eqnarray}
%
where in the last transition the operator structure with $\Delta I =3/2$ giving
a vanishing contribution is removed and only the structures with explicitly
$\Delta I =1/2$ are retained, and
%
\begin{eqnarray}
&& \langle \Lambda_c \, \pi^- (p=0) \,| H_W |\, \Xi_c^0 \rangle = \langle
\Lambda_b \, \pi^- (p=0) \,| H_W |\, \Xi_b^- \rangle + 
{\sqrt{2} \over f_\pi} \, G_F \cos \theta_c \sin \theta_c \\
&&{}\qquad \times\langle \Lambda_c \,
| \left ( C_+ + C_- \right ) (\overline c_L \, \gamma_\mu \, s_L)\,
(\overline u_L \, \gamma_\mu \, c_L) + 
\left ( C_+ - C_- \right ) (\overline u_L \, \gamma_\mu \, s_L)\,
(\overline c_L \, \gamma_\mu \, c_L) |\, \Xi_c^0 \rangle\,.\nonumber 
\label{xic}
\end{eqnarray}
%
In the latter formula the first term on the r.h.s. expresses the fact that in
the heavy quark limit the `spectator' amplitudes do not depend on the flavor or
the mass of the heavy quark. The rest of the expression (\ref{xic}) describes
the `non-spectator' contribution to the amplitude of the charmed hyperon decay.
Using the flavor $SU(3)$ symmetry the latter contribution can be related to the
non-singlet matrix elements (\ref{msh:defxy}) (normalized at $\mu=m_c$) as
%
\begin{eqnarray}
\Delta A &\equiv& \langle \Lambda_c \, \pi^- (p=0) \,| H_W |\, \Xi_c^0
\rangle - \langle \Lambda_b \, \pi^- (p=0) \,| H_W |\, \Xi_b^- \rangle
\nonumber \\
&=& {G_F \cos \theta_c \sin \theta_c \over 2 \, \sqrt{2} \, f_\pi} \,
\left[ \left ( C_- -
C_+ \right ) \, x - \left ( C_+ + C_- \right ) \, y \right ]\,.
\label{das}
\end{eqnarray}
%
Furthermore, with the help of the equations (\ref{d1}) and (\ref{d2}) relating
the matrix elements $x$ and $y$ to the differences of the total decay widths
within the triplet of charmed hyperons, one can eliminate $x$ and $y$ in favor
of the measured width differences. The resulting expression has the form
%
\begin{eqnarray}
\Delta A &\approx& - {\sqrt{2} \, \pi \cos \theta_c \sin \theta_c  \over 
G_F \, m_c^2 \, f_\pi} \left [ 0.45 \left (
\Gamma(\Xi_c^0)-\Gamma(\Lambda_c) \right ) + 0.04 \left (
\Gamma(\Lambda_c)-\Gamma(\Xi_c^+) \right ) \right ] \nonumber \\
&=& -10^{-7} \left [
0.97 \left ( \Gamma(\Xi_c^0)-\Gamma(\Lambda_c) \right ) + 0.09 \left (
\Gamma(\Lambda_c)-\Gamma(\Xi_c^+) \right ) \right ] \left ( {1.4 \,
GeV \over m_c} \right )^2 \, ps \,,
\label{dasm}
\end{eqnarray}
%
where, clearly, in the latter form the widths are assumed to be expressed in
$ps^{-1}$, and $m_c=1.4 \, GeV$ is used as a `reference' value for the charmed
quark mass. It is seen from Eq.~(\ref{dasm}) that the evaluation of the
difference of the amplitudes within the discussed approach is mostly sensitive
to the difference of the decay rates of $\Xi_c^0$ and $\Lambda_c$, with only
very little sensitivity to the total decay width of $\Xi_c^+$. Using the current
data the difference $\Delta A$ is estimated as
%
\begin{equation}
  \Delta A= -(5.4 \pm 2) \times 10^{-7}\,,
\label{dasn}
\end{equation}
with the uncertainty being dominated by the experimental error in the
lifetime of $\Xi_c^0$. An  amplitude $A$ of the magnitude,
given by the central value in Eq.~(\ref{dasn}) would produce a decay rate
$\Gamma (\Xi_Q \to \Lambda_Q \, \pi) = |A|^2 \, p_\pi/(2 \pi) \approx
0.9 \times 10^{10} \, s^{-1}$, which result can also be written in a
form of triangle inequality
\begin{equation}
\sqrt{\Gamma(\Xi_b^- \to \Lambda_b \, \pi^-)} + \sqrt{\Gamma(\Xi_c^0 \to
\Lambda_c \, \pi^-)} \ge \sqrt{0.9 \times 10^{10} \, s^{-1}}\,.
\label{tri}
\end{equation}

Although at present it is not possible to evaluate in a reasonably model
independent way the matrix element in Eq.~(\ref{xib}) for the `spectator' decay
amplitude, the inequality (\ref{tri}) shows that at least some of the discussed
pion transitions should go at the level of $0.01 \, ps^{-1}$, similar to the
rates of analogous decays of `light' hyperons.

%-------------------------------------------------------------------------------
\subsection{Summary on predictions for lifetimes}

We summarize here specific predictions for the inclusive decay rates, which can
be argued with a certain degree of theoretical reliability, and which can be
possibly experimentally tested in the nearest future.
\index{inclusive weak decay rates!theoretical predictions summary}%

$B$ mesons:
\begin{equation}
\tau(B_d)/\tau(B_s)=1 \pm 0.01\,.
\end{equation}

Charmed hyperons:
\begin{eqnarray}
& \Gamma_{sl}(\Xi_c) = (2 - 3) \, \Gamma_{sl}(\Lambda_c)\, \qquad
\Gamma_{sl}(\Omega_c) > \Gamma_{sl}(\Xi_c)\,,& \nonumber\\[4pt]
& \Gamma^{nl}_{\Delta S =
-1}(\Xi_c^+) \approx \Gamma^{nl}_{\Delta S = -1}(\Lambda_c) \,,& \\[4pt]
&\Gamma^{nl}_{\Delta S =
-1}(\Xi_c^0)-\Gamma^{nl}_{\Delta S = -1}(\Lambda_c) \approx 0.55 \pm
0.22 \, ps^{-1}\,.& \nonumber
\end{eqnarray}

$b$ hyperons:
\begin{eqnarray}
& \tau(\Xi_b^0) \approx \tau(\Lambda_b) < \tau(B_d) <
\tau(\Xi_b^-)<\tau(\Omega_b)\,, &  \nonumber\\[4pt]
& \Gamma(\Lambda_b) - \Gamma(\Xi_b^-) \approx 0.11 \pm 0.03 \,
ps^{-1}\,, & \\[4pt]
& 0.9 < \displaystyle {\tau(\Lambda_b) \over \tau(B_d)} < 1\,.&  \nonumber
\end{eqnarray}

Strangeness decays $\Xi_Q \to \Lambda_Q \, \pi$:
The $\Delta I =1/2$ rule should hold in these decays, so that
$\Gamma(\Xi_Q^{(d)} \to \Lambda_Q \pi^-)  = 2 \, \Gamma(\Xi_Q^{(u)} \to
\Lambda_Q \pi^0)$. The rates are constrained by the triangle inequality
(\ref{tri}).

%===============================================================================
\boldmath
\section[Theory of \bbm]{Theory of \bbm$\!$ 
\authorfootnote{Author: Ulrich~Nierste}}
\unboldmath
\label{chmix:sec:thbmix}

In Sect.~\ref{subs:mix} the time evolution of the $B^0\!-\!\ov{B}{}^0\,$ system
has been discussed.  \bbm\ involves three physical, rephasing-invariant
quantities: $|M_{12}|$, $|\Gamma_{12}|$ and the phase $\phi$ defined in
\eq{defphi}.  In the following we will discuss how they are related to physical
observables. The discussed quantities are summarized in
Table~\ref{chmix:tab:obs}.  \index{B mixing@$B$ mixing}

%-------------------------------------------------------------------------------
\subsection{Mass difference}
\label{chmix:sub:mass}

\index{B meson@$B$ meson!mass difference \dm}\index{mass difference \dm} The
mass difference \dm\ can be measured from the tagged time evolutions in
(\ref{gtfres}-\ref{gbtfbres}) from any decay $B^0\to f$, unless $\lambda_f=\pm
1$, in which case the oscillatory terms vanish.  The time evolution is
especially simple for flavor-specific decays, which are characterized by
$\lambda_f=0$. The corresponding formulae can be found in \eq{gtfs} and
\eq{defa0}. Interesting flavor-specific \index{flavor-specific!decay modes}
modes are tabulated in Table~\ref{chmix:tab:fsd}. Time integrated measurements
determine $x_q=\dm_q/\Gamma_q=\dm_q \tau_{B_q}$, $q=d,s$, defined in \eq{defxy}.
While unfortunately it is common practice to quote measurements of $\dm_q$ in
terms of $x_q$, it should be clear that the measured oscillation frequencies
determine $\dm_d$ and $\dm_s$ and not $x_d$ and $x_s$. Fundamental physics
quantities like CKM elements are related to $\dm_d$ and $\dm_s$, so that the
errors of the lifetimes entering $x_q$ are irrelevant.

\index{B mixing@$B$ mixing!$B_d$ mixing} In order to predict the mass
difference $\dm_q$ within the Standard Model or one of its extensions, one must
first calculate the \dbt\ transition amplitude, which triggers \bbm. The lowest
order contribution to this amplitude in the Standard Model is the box diagram in
Fig.~\ref{ch1:fig:box}. Then one must match the result to an effective field
theory, in which the interactions mediated by heavy particles are described by
local operators represented by pointlike vertices. In the Standard Model only
one operator, $Q$ in \eq{ch1:uli:defq}, emerges.\index{operator!$Q$} This
procedure separates short- and long-distance physics and is described in
Sect.~\ref{1:Heff}. It results in the effective Hamiltonian in \eq{ch1:uli:h2}.
The interesting short-distance physics is contained in the Wilson coefficient
$C$ in \eq{ch1:uli:c2}. New physics can modify $C$ and can introduce new
operators in addition to $Q$ in \eq{ch1:uli:defq}.  \index{Wilson coefficient}
The Standard Model prediction is readily obtained from \eq{ch1:uli:h2} to
\eq{defbb}:
%
\begin{equation}
\dm_q = 2 |M_{12}^q| \, = \, 
   \frac{|\bra{B_q^0} H^{|\Delta B|=2}  \ket{\ov{B}{}^0_q}| }{m_{B_q}}  
=  \frac{G_F^2}{6 \pi^2}\, \eta_B\, m_{B_q} \, \widehat{B}_{B_q} f_{B_q}^2 
    M_W^2\, S \bigg( \frac{m_t^2}{M_W^2} \bigg) 
    \left| V_{tb} V_{tq}^* \right|^2 . 
    \label{chmix:dmb}
\end{equation}
where $q=s$ or $d$.
\index{B meson@$B$ meson!mass difference \dm}\index{mass difference \dm}%

\begin{table}
\begin{center}
\begin{tabular}{l@{~~~~}l@{~~~~}l@{~~~~}l} \hline
observable & defined in & 
                          \multicolumn{2}{l}{SM prediction for } \\
& & $B_d$ in & $B_s$ in 
                          \\\hline &&&\\[-3mm]
$\dm \simeq 2 |M_{12}| $ & \eq{mgsol:a} & 
 \eq{chmix:dmb},\eq{chmix:dmbrat} & 
 \eq{chmix:dmb},\eq{chmix:dmsnum},\eq{chmix:dmbrat} \\
$\phi$ & \eq{defphi} & \eq{chmix:phinum} & \eq{chmix:phinum} \\
$\dg \simeq 2 |\Gamma_{12}| \cos\phi $ & 
  \eq{mgsol:b} &
  \eq{chmix:dgdm},\eq{chmix:dgdsm}   & 
   \eq{chmix:dgnum2},\eq{chmix:dgdm}  \\
$\dg_{\rm CP} \simeq 2 |\Gamma_{12}|   $ & \eq{chmix:dgcp}
        &   &  \\
$\dg_{\rm CP}^\prime \simeq 2 |\Gamma_{12}| \cos^2\phi$ 
  & \eq{chmix:dgp}  & &  \\
$\dg_{\rm CP}^{cc} =  2 |\xi_c^{d\,2}\Gamma_{12}^{cc}|$ & 
    \eq{chmix:dgcp:d} & \eq{chmix:dgcc}   &  \\
$\ega =  \lt| \frac{\Gamma_{12}}{M_{12}} \rt| \sin \phi  $ & 
   \eq{defepsg} & \eq{chmix:aqnum} & \eq{chmix:aqnum}  \\[4pt] \hline
\end{tabular}
\end{center}
\caption{Observables related to $|M_{12}|$, $|\Gamma_{12}|$ and $\phi$
          discussed in Sect.~\ref{chmix:sec:thbmix}.}
\label{chmix:tab:obs}
\end{table}

Next we discuss the phenomenology of $\dm_d$ in the Standard Model.  We first
insert the numerical values of those quantities which are well-known into
\eq{chmix:dmb}. The QCD factor $\eta_B=0.55$ \cite{bjw} corresponds to the
$\ov{\rm MS}$ scheme for $m_t$. The $\ov{\rm MS}$ value of $m_t=167\,$GeV is
numerically smaller than the pole mass measured at the Tevatron by roughly 7
GeV. Solving
\eq{chmix:dmb} for $|V_{td}|$ one finds:%
\index{B meson@$B$ meson!mass difference \dm!$\dm_d$}%
\index{mass difference \dm!$\dm_d$}
\index{unitarity triangle!and \bbmd}%
\index{Cabibbo-Kobayashi-Maskawa matrix!$V_{td}$}%
%
\begin{equation}
|V_{td}| = 0.0078 \, \sqrt{\frac{\dm_d}{0.49\, \mbox{ps}^{-1}}} \, 
             \frac{200 \, \mbox{MeV}}{f_{B_d}}\,
             \sqrt{\frac{1.3}{\widehat{B}_{B_d}}} \,.
\end{equation}
%
The relation of $|V_{td}|$ to the improved Wolfenstein parameters is%
\index{Wolfenstein parameters!for $V_{td}$}%
%
\begin{equation}
|V_{td}| = A \lambda^3 R_t \lt( 1+ {\cal O} (\lambda^4) \rt)
\, = \, |V_{cb}| \, \lambda \,R_t \lt( 1+ {\cal O} (\lambda^4) \rt)
\end{equation}
%
and 
%
\begin{equation}
R_t = \sqrt{ (1- \ov{\rho})^2 + \ov{\eta}^2} \label{chmix:defrt}
\end{equation}
%
is the length of one side of the unitarity triangle. Hence the measurement of
$\dm_d$ defines a circle in the $(\ov{\rho},\ov{\eta})$ plane centered around
$(1,0)$. Yet the hadronic uncertainties associated with
$f_{B_d}\sqrt{\widehat{B}_{B_d}}$ obscure a clean extraction of $|V_{td}|$ and
$R_t$ from the well-measured $\dm_d$.  The summer 2000 world averages from
lattice calculations are $f_{B_d} = (200 \pm 30)\,$MeV and
$\widehat{B}_{B_d}=1.30 \pm 0.17$ \cite{hash}. This gives $|V_{td}|=0.0078 \pm
0.0013$ and, with $|V_{cb}|= (40.4 \pm 1.8)\times 10^{-3}$, $R_t= 0.88 \pm 0.15$.
\index{hadronic parameter!$f_{B_d}$} \index{hadronic parameter!$\widehat{B}_B$}

\index{B mixing@$B$ mixing!$B_s$ mixing} For the discussion of $\dm_s$ we
first note that the corresponding CKM
element in \eq{chmix:dmb} is fixed from CKM unitarity:%
\index{Cabibbo-Kobayashi-Maskawa matrix!$V_{ts}$}%
\index{Wolfenstein parameters!for $V_{ts}$}%
\index{B meson@$B$ meson!mass difference \dm!$\dm_s$}%
\index{mass difference \dm!$\dm_s$}
%
\begin{equation}
|V_{ts}| = |V_{cb}| \lt[ 1 + \lambda^2 \lt( \ov{\rho} - \frac{1}{2} \rt) 
               + {\cal O} (\lambda^4) \rt] .
\end{equation}
%
$|V_{ts}|$ is smaller than $|V_{cb}|$ by roughly 1\%. Hence within the Standard
Model a measurement of $\dm_s$ directly probes the calculation of the hadronic
matrix element. \eq{chmix:dmb} specifies to
%
\begin{equation}
\dm_s = 17.2 \, \mbox{ps}^{-1} \lt( \frac{|V_{ts}|}{0.04} \, 
          \frac{f_{B_s}}{230 \, \mbox{MeV}} \rt)^2 
             \frac{\widehat{B}_{B_s}}{1.3}
             \label{chmix:dmsnum} .
\end{equation}
%
This can be rewritten as 
%
\begin{equation}
f_{B_s}\sqrt{\widehat{B}_{B_s}} =
     \sqrt{\frac{\dm_s}{14.9\, \mbox{ps}^{-1}}} \frac{0.04}{|V_{cb}|} 
     \, 247\, \mbox{MeV} . \label{chmix:dmsbd}
\end{equation}
%
\index{hadronic parameter!$f_{B_s}$}%
The present 95\% C.L.\ limit of
$\dm_s\geq 14.9\,$ps$^{-1}$ \cite{os} implies a lower bound on
$f_{B_s}\sqrt{\widehat{B}_{B_s}}$ which is only marginally consistent
with some of the quenched lattice calculations. Hence global fits of
the unitarity triangle (using $\dm_d/\dm_s $ to constrain $R_t$) which
use too small values of $f_{B_s}\sqrt{\widehat{B}_{B_s}}$ confine the
apex of the triangle to a too small region of the
$(\ov{\rho},\ov{\eta})$ plane or may even be in conflict with the
measured lower bound on $\dm_s$.

The determination of $R_t$ profits enormously from a measurement of $\dm_s$,
because the ratio of the hadronic matrix elements entering $\dm_d/\dm_s$ can be
calculated with a much higher accuracy than the individual matrix elements:
%
\begin{equation}  
\xi = \frac{f_{B_s}\sqrt{\widehat{B}_{B_s}}}{f_{B_d}\sqrt{\widehat{B}_{B_d}}}
\label{chmix:defxi}
\end{equation} 
%
\index{hadronic parameter!and $SU(3)_F$ symmetry}%
\index{hadronic parameter!$\xi$}%
is equal to 1 in the limit of exact $SU(3)_F$ symmetry. Hence the
theorists' task is reduced to the calculation of the deviation from 1. The
current world average from lattice calculations is \cite{hash}
%
\begin{equation}
\xi = 1.16 \pm 0.05 \label{chmix:xilatt} .
\end{equation}
Further $|V_{cb}|$ drops out from the ratio 
\begin{equation} 
 \frac{\dm_d}{\dm_s} = \lambda^2 R_t^2 \lt( 1 +  
 \lambda^2 (1 - 2 \ov{\rho} ) + {\cal O} (\lambda^4)  \rt) \, 
 \frac{m_{B_d}}{m_{B_s}}\, \frac{1}{\xi^2}  . \label{chmix:dmbrat}
\end{equation} 

With the expected experimental accuracy of $\dm_{d,s}$ and the anticipation of
progress in the determination of $\xi$ in \eq{chmix:xilatt} a determination of
$R_t$ at the level of 1-3\% is possible.  Then, eventually, even the uncertainty
in $\lambda$ cannot be neglected anymore. Keeping the overall factor of
$\lambda^2$ while inserting $\lambda=0.22$ in the subleading terms one finds
from \eq{chmix:dmbrat}:
%
\begin{equation} 
R_t = 0.880 \, \sqrt{\frac{\dm_d}{0.49 \,\mbox{ps}^{-1}}} \, 
        \sqrt{\frac{17 \,\mbox{ps}^{-1} }{\dm_s}} \,
        \frac{0.22}{\lambda} \,
        \frac{\xi}{1.16} \, 
        \lt( 1 \, + \,  0.05 \, \ov{\rho} \rt). \label{chmix:rtapp}  
\end{equation} 
%
Here the omission of ${\cal O}(\lambda^4)$ terms induces a negligible error of
less than 0.1\%. At present $R_t$ is obtained from a global fit of the unitarity
triangle and \eq{chmix:dmbrat} is used to predict
$\dm_s = 17.3\epm{1.5}{0.7}$ \cite{cckm}.% 
\index{B meson@$B$ meson!mass difference \dm!SM prediction for $\dm_s$}%
\index{mass difference \dm!$\dm_s$!SM prediction} One should be aware that some
of the quantities entering the global fit, especially $\epsilon_K$, are
sensitive to new physics. Hence the measurement of a $\dm_s$ well above the
quoted range would be very exciting. In a large class of extensions of the
Standard model $\dm_d$ and $\dm_s$ change, while their ratio does not. In these
models $\dm_s$ could be in conflict with \eq{chmix:dmsbd} without affecting
$R_t$ in \eq{chmix:rtapp}.  Therefore it is desirable to gain additional
experimental information on $f_{B_s}$, so that the dependence on
non-perturbative methods is reduced. This information can be obtained from the
$B_s$ width difference discussed in Sect.~\ref{chmix:sub:width}.

%-------------------------------------------------------------------------------
\subsection{Width difference}
\label{chmix:sub:width}

%...............................................................................
\subsubsection{Calculation}\label{chmix:subsub:width:calc}

\index{B meson@$B$ meson!width difference $\dg$}%
\index{width difference $\dg$} The two mass eigenstates $B_L$ and $B_H$ in
\eq{defpq} differ not only in their masses but also in their widths. The
prediction of the width difference $\dg = \Gamma_L - \Gamma_H \simeq 2\,
|\Gamma_{12}| \cos \phi $ in \eq{mgsol:b} requires the calculation of
$|\Gamma_{12}|$ and $\phi$. $\Gamma_{12}$ is determined from the absorptive part
of the \dbt\ transition amplitude. It receives contributions from all final
states which are common to $B^0$ and $\ov{B}{}^0$ as shown in the first line of
\eq{1:delgam}. The leading order (LO) diagrams contributing to $\Gamma_{12}$ in
the $B_s$ system are shown in Fig.~\ref{chmix:fig:dga}.
%
\begin{figure}[tb]
  \centerline{\epsfxsize=0.85\textwidth \epsffile{ch8/th_dga_uli.eps}}
  \vspace*{6pt}
\caption[Leading order diagrams determining $\Gamma_{12}^s$. Only the 
CKM-favored contribution $\xi_c^{s*2} \Gamma_{12}^{cc}$ is shown.]{Leading
order diagrams determining $\Gamma_{12}^s$. Only the  CKM-favored contribution
$\xi_c^{s*2} \Gamma_{12}^{cc}$ is shown.  The left diagram is the \emph{weak
annihilation diagram} and the right one is the \emph{spectator interference
diagram}. The dashed line indicates the cut through the final state.}
\label{chmix:fig:dga}
\end{figure}
%
The dominant contribution comes from the spectator interference diagram, the
weak annihilation diagram is color-suppressed.  We write
%
\begin{equation}
\Gamma_{12}^q \, = \, -  \lt[  
\xi_c^{q*2} \Gamma_{12,q}^{cc} \, + \, 
               2 \xi_u^{q*} \xi_c^{q*} 
               \Gamma_{12,q}^{uc} \, +  \, \xi_u^{q*2} \Gamma_{12,q}^{uu}
               \rt] , 
               \qquad q=d,s, 
\label{chmix:ga12dec} 
\end{equation}
%
where the three terms denote the contributions from the diagram with
(anti-)quarks $i$ and $j$ in the final state.
% The mass $M_{B_q}}$ and the square of the decay constant $f_{B_q}$ are
% factored out.
The $\xi_i^{q}$'s denote the corresponding CKM factors:
%
\begin{equation}  
\xi_u^{q} \,=\, V_{uq} V_{ub}^*, \qquad  
\xi_c^{q} \,=\, V_{cq} V_{cb}^*, \qquad  
\xi_t^{q} \,=\, V_{tq} V_{tb}^*. \label{chmix:xiq}   
\end{equation}  
%
They satisfy $\xi_u^{q}+\xi_c^{q}+\xi_t^{q}=0$ from CKM unitarity and
read in terms of the improved Wolfenstein parameters \cite{wblo}:
\index{Wolfenstein parameters!for $V_{ud}V_{ub}^*$}
\index{Wolfenstein parameters!for $V_{us}V_{ub}^*$}
\index{Wolfenstein parameters!for $V_{cd}V_{cb}^*$}
\index{Wolfenstein parameters!for $V_{cs}V_{cb}^*$}
\index{Wolfenstein parameters!for $V_{td}V_{tb}^*$}
\index{Wolfenstein parameters!for $V_{ts}V_{tb}^*$}
%
\begin{eqnarray}
\xi_u^{d} \, = \, A \lambda^3 \lt( \ov{\rho}+ i \ov{\eta} \rt) + 
              {\cal O} (\lambda^5), && \qquad
\xi_c^{d} \, = \,  - A \lambda^3 +{\cal O} (\lambda^5), \nn 
&& \qquad
   \xi_t^{d} \, = \,  A \lambda^3 \lt( 1 - \ov{\rho}- i \ov{\eta} \rt)
           +{\cal O} (\lambda^5), \no \\[4pt] 
\xi_u^{s} \, = \, A \lambda^4 ( 1+\frac{\lambda^2}{2} ) 
              \lt( \ov{\rho}+ i \ov{\eta} \rt) + 
              {\cal O} (\lambda^8), 
&& \qquad
\xi_c^{s} \, = \, A \lambda^2 ( 1 - \frac{\lambda^2}{2} ) 
               +{\cal O} (\lambda^6), \nn 
&& \qquad
   \xi_t^{s} =   -A \lambda^2 \lt[ 1- \lambda^2 ( \frac{1}{2}
              - \ov{\rho} - i \ov{\eta}   )   \rt] 
              +{\cal O} (\lambda^6),  \label{chmix:xis}
\end{eqnarray}
%
$-\xi_u^{d}/\xi_c^{d}$ and $-\xi_t^{d}/\xi_c^{d}$ define two sides
of the standard unitarity triangle depicted in Fig.~\ref{1:fig:UT}. One has 
%
\begin{equation}
-\frac{\xi_u^{d}}{\xi_c^{d}} \,=\, R_b e^{i \gamma},\qquad\quad
-\frac{\xi_t^{d}}{\xi_c^{d}} \, = \, R_t e^{-i\beta},
\end{equation}
\index{unitarity triangle!angle $\gamma$}%
\index{unitarity triangle!angle $\beta$}%
\index{unitarity triangle!angle $\beta_s$}%
%
where $R_b=\sqrt{\ov{\rho}^2+\ov{\eta}^2}$ and $R_t$ is defined in
\eq{chmix:defrt}. The ratio $\xi_t^{s}/\xi_c^{s}$ involves the phase $\beta_s$
measured from the mixing-induced $CP$ asymmetry in $B_s \to D_s^+ D_s^-$:
\index{decay!$B_s \rightarrow D_s^+ D_s^-$}
%
\begin{equation}
-\frac{\xi_t^{s}}{\xi_c^{s}} \, = \, [1+ \ov{\rho} \lambda^2 
                     +{\cal O}(\lambda^4)] \, 
 e^{i\beta_s}\qquad \mbox{with }\ 
 \beta_s =
  \ov{\eta}\lambda^2 [1+(1-\ov{\rho}) \lambda^2] +{\cal O} (\lambda^6) 
 . \label{chmix:betas}
\end{equation}

The coefficients $\Gamma_{12}^{ij}$ in \eq{chmix:ga12dec} are positive.  They
are inclusive quantities and can be calculated using the heavy quark expansion
described for the heavy hadron lifetimes in Sect.~\ref{chmix:thlife}.  The
leading term of the power expansion in $\Lambda_{QCD}/m_b$ contains two Dirac
structures, each of which can be factorized into a short-distance Wilson
coefficient and the matrix element of a local operator:
%
\begin{equation}
\Gamma_{12,q}^{ij} = \frac{G_F^2 m_b^2}{6 \pi } f_{B_q}^2 M_{B_q}   
    \lt[ F^{ij}(z) \frac{2}{3} B_{B_q} - 
          F_S^{ij} (z) \frac{5}{12} B_{B_{q}}^{S\prime} \rt] 
          \lt[ 1 + {\cal O} \lt( \frac{\Lambda_{QCD}}{m_b} \rt)\rt].
 \label{chmix:ga12F}
\end{equation}
%
Here $z= m_c^2/m_b^2$. The operator $Q$, which we
already encountered in the discussion of \dm, was defined in 
\eq{ch1:uli:defq}. When $Q$ occurs in the context of \bbms, 
we understand that the $d$ quarks in \eq{ch1:uli:defq} are
appropriately replaced by $s$ quarks. In $\Gamma_{12,q}$ a second 
operator occurs:
%
\begin{equation}
 Q_S = \ov{q}_L b_R \, \ov{q}_L b_R, \qquad\quad q=d \mbox{ or } s.
\end{equation}
%
$B_{B_q}$ and $B_{B_{q}}^{S\prime}$ (or, equivalently, $B_{B_{q}}^{S}$)
are `bag' parameters quantifying the hadronic matrix elements of 
$Q$ and $Q_S$:%
\index{Bag constants!$B_{B_q}$}% 
\index{hadronic parameter!$B_{B_q}$}%
\index{Bag constants!$B_{B_q}^S$}% 
\index{hadronic parameter!$B_{B_q}^S$}
%
\begin{eqnarray}
\bra{B_q^0} Q (\mu) \ket{\ov{B}{}_q^0} &=& 
        \frac{2}{3} f_{B_q}^2 m_{B_q}^2 B_{B_q}(\mu)\,,\\
\bra{B_q^0} Q_S (\mu) \ket{\ov{B}{}_q^0} &=& 
        - \frac{5}{12} f_{B_q}^2 m_{B_q}^2 B_{B_q}^{S\prime} (\mu) 
        \, = \, - \frac{5}{12} f_{B_q}^2 m_{B_q}^2 
        \frac{m_{B_q}^2}{\lt[ m_b(\mu)+m_q(\mu) \rt]^2} B_{B_q}^S (\mu) . \no
\label{chmix:Bdef}
\end{eqnarray}
%
$B_{B_{q}}^S$ and $B_{B_{q}}^{S\prime}$ simply differ by the factor $m_{B_q}^2/[
m_b(\mu)+m_q(\mu)]^2$. While lattice results are usually quoted for
$B_{B_{q}}^S$, the forthcoming formulae are shorter when expressed in terms of
$B_{B_{q}}^{S\prime}$.  These `bag' factors depend on the renormalization scale
$\mu$ and the renormalization scheme. In the literature on \dg\ it is customary
to use $B_{B_q}$ and $B_{B_q}^S$ in the $\ov{\rm MS}$ scheme as defined in
\cite{bbgln}.  Numerical values obtained in lattice calculations are usually
quoted for $\mu=m_b$. The invariant bag factor $\widehat{B}_{B_q}$ defined in
\eq{defbb} is related to $B_{B_q}(\mu)$ by
%
\begin{eqnarray}
\widehat{B}_{B_q} &=& B_{B_q}(\mu) \, b_B (\mu) \\
 b_B (\mu) &=& \lt[ \alpha_s (\mu ) \rt]^{-6/23} \, 
     \lt[ 1+ \frac{\alpha_s(\mu)}{4 \pi} \frac{5165}{3174}   \rt], 
     \qquad \quad  b_B(m_b) \, = \, 1.52 \pm 0.03 . 
    \no
\end{eqnarray}
\index{hadronic parameter!$\widehat{B}_B$!relation to $B_{B_q}$}%
%
A recent preliminary lattice calculation with two dynamical flavors found
$B_{B_s} (m_b) = 0.83 \pm 0.08$ and $B_{B_s}^S(m_b) = 0.84\pm 0.08$ \cite{hy}. 
No deviation of $B_{B_s}/B_{B_d}$ and $B_{B_s}^S/B_{B_d}^S$ from 1 is seen. 
The quoted value for $B_{B_q} (m_b)$ corresponds to $\widehat{B}_{B_q}=1.26\pm
0.12$.  The short distance physics entering $\Gamma_{12}^q$ in \eq{chmix:ga12F}
is contained in $F(z)$ and $F_S (z)$. To leading order in $\alpha_s$ they read:
%
\begin{eqnarray}
F^{cc} (z)   &=& \sqrt{1-4 z} \lt[ (1-z) K_1 + \frac{1- 4z}{2} K_2 \rt] \nn
F_S^{cc} (z) &=& \sqrt{1-4 z}\, (1+2 z) \lt( K_1 - K_2 \rt)\nn
F^{uc} (z)   &=& (1-z)^2 \lt[ \frac{2+z}{2} K_1 + \frac{1-z}{2} K_2
                 \rt] \nn  
F_S^{uc} (z)   &=& (1-z)^2\, (1+2 z) \lt( K_1 - K_2 \rt)\nn
F^{uu} (z)   &=& F^{cc} (0)   \, =\, F^{uc} (0) \, = \, 
                 K_1 + \frac{1}{2} K_2 \nn
F_S^{uu} (z)   &=& F_S^{cc} (0)   \, =\, F_S^{uc} (0) \, = \, 
                 K_1 - K_2 . \label{chmix:theFs}
\end{eqnarray}
%
$K_1$ and $K_2$ are combinations of the Wilson coefficients $C_1$ and $C_2$,
which are tabulated in Table~\ref{tab}:
%
\begin{equation}
K_1 \,=\, 3 C_1^2 + 2 C_1 C_2, \qquad \quad 
K_2 \, = \, C_2^2.
\end{equation}
%
The scale at which the Wilson coefficients are evaluated must equal the scale
used in $B_{B_q} (\mu) $ and $B_{B_q}^S(\mu)$. In \eq{chmix:ga12F} and
\eq{chmix:theFs} we have neglected the small contributions from penguin
coefficients, which can be found in \cite{bbgln}.

It is instructive to eliminate $\xi_c^q=-\xi_t^q-\xi_u^q$ in favor of $\xi_u^q$
and $\xi_t^q$ in \eq{chmix:ga12dec}:
%
\begin{equation}
\Gamma_{12}^q \, = \, - 
               \xi_t^{q*2} \, \lt[ \Gamma_{12,q}^{cc} \, + \, 
               2\, \frac{\xi_u^{q*}}{\xi_t^{q*}} 
               \lt( \Gamma_{12,q}^{cc} - 
               \Gamma_{12,q}^{uc} \rt) \, +  \, 
               \frac{\xi_u^{q*2}}{\xi_t^{q*2}} 
               \lt( \Gamma_{12,q}^{cc} - 2 \Gamma_{12,q}^{uc} + 
               \Gamma_{12,q}^{uu} \rt) \rt].
\label{chmix:ga12dec2} 
\end{equation}
%
In the limit $z=0$ all the $\Gamma_{12}^{ij}$'s become equal and $\arg
M_{12}=\arg(-\Gamma_{12}) = \arg (\xi_t^{q*2})$, so that $\phi$ vanishes in this
limit.  With \eq{chmix:xis} one verifies that $\phi_d={\cal O} (\ov{\eta} z)$
and $\phi_s={\cal O} (\lambda^2 \ov{\eta} z)$. This GIM suppression is lifted,
if new physics contributes to $\arg M_{12}^q$ spoiling the cancellation between
$\arg M_{12}$ and $\arg(-\Gamma_{12}^q)$. Therefore $\phi$ is extremely
sensitive to new physics.  \index{phase!of $CP$ violation in mixing, $\phi$}
%%\index{phase!of $CP$ violation in mixing, $\phi$!SM prediction for $\phi_s$}
\index{phase!of $CP$ violation in mixing, $\phi$!SM prediction for $\phi_d$ 
and $\phi_s$}

Combining \eq{chmix:ga12dec2} and \eq{chmix:theFs} we find the Standard Model
predictions for $\phi_d=\phi(B_d)$ and $\phi_s=\phi(B_s)$:
%
\begin{eqnarray}
\phi_d & = & 
   - \frac{2\ov{\eta}}{R_t^2} \, 
     \frac{\Gamma_{12,d}^{uc} -
   \Gamma_{12,d}^{cc}}{\Gamma_{12,d}^{cc}} 
   \lt[ 1+ {\cal O} \lt( \ov{\eta} z^2, \ov{\eta} \lambda^4\rt) \rt]
      \approx - \frac{24}{5} \, \frac{\ov{\eta}}{R_t^2} 
         \, \frac{B_{B_d}}{B_{B_d}^{S\prime}} \, 
         \frac{K_1+K_2}{K_2-K_1} \, z \, \approx -0.1 \, = \, 
          - 5^\circ \,, \no\\[3mm]
\phi_s & = & 
   2 \lambda^2 \ov{\eta} \, 
     \frac{\Gamma_{12,s}^{uc} -
   \Gamma_{12,s}^{cc}}{\Gamma_{12,s}^{cc}} 
   \lt[ 1+ {\cal O} \lt( \ov{\eta} z^2, \ov{\eta} \lambda^4\rt)
   \rt]
     \approx \frac{24}{5} \, \ov{\eta} \, \lambda^2 
         \, \frac{B_{B_s}}{B_{B_s}^{S\prime}} \, 
         \frac{K_1+K_2}{K_2-K_1} \, z  \, \approx \, 3 \times 10^{-3} 
         \, = \, 0.2^\circ \,. \nonumber\\
         \label{chmix:phinum}
\end{eqnarray}
%
That is, in the Standard Model, we can safely neglect the factor of $\cos
\phi_q$ in the relation $\dg_q = \ 2 |\Gamma_{12}^q| \cos \phi_q$ (see
\eq{mgsol:b}). For $\Gamma_{12}^s$ the CKM-suppressed contributions
$\Gamma_{12,s}^{uc}$ and $\Gamma_{12,s}^{uu}$ are completely irrelevant.
$\Gamma_{12}^d$ is also dominated by the double-charm contribution
$\Gamma_{12,d}^{cc}$, with up to ${\cal O}(5\% )$ corrections from the second
term in \eq{chmix:ga12dec2}:
%
\begin{eqnarray} 
\dg_s^{\rm SM} &=& 2 |\Gamma_{12}^s| \, = \, 
        |\xi_t^s|^2 \, 2 |\Gamma_{12,s}^{cc}| \nn 
\dg_d^{\rm SM} &=& 2 |\Gamma_{12}^d| \, = \, 
        |\xi_t^d|^2 \, 2 | \Gamma_{12,d}^{cc} | 
        \, \bigg| 1 + 2 \frac{R_t^2 + \ov{\rho} - 1}{R_t^2} 
        \frac{\Gamma_{12,d}^{uc}-\Gamma_{12,d}^{cc}}{\Gamma_{12,d}^{cc}} 
            + {\cal O} \bigg( \frac{R_B^4}{R_t^4}\, z^2 \bigg) \bigg| \,. 
  \label{chmix:dgsm}
\end{eqnarray} 
%
\index{B meson@$B$ meson!width difference $\dg$!SM prediction for $\dg_s$}%
\index{width difference $\dg$!SM prediction for $\dg_s$}%
$\Gamma_{12,s}^{cc}$ has been calculated in the next-to-leading order of
$\alpha_s$ \cite{bbgln} and $\Lambda_{QCD}/m_b$ \cite{bbd}. The result gives
the following prediction for $\dg_s^{\rm SM}$:
%
\begin{equation} 
\frac{\dg_s^{\rm SM}}{\Gamma} \, =\, \frac{2 |\Gamma_{12}^s|}{\Gamma} 
        \, = \, 
        \left( \frac{f_{B_s}}{245~{\rm MeV}} \right)^2 \,
        \left[ \, (0.234\pm 0.035)\, {B_{B_s}^S} - 
        0.080 \pm 0.020 \, \right] .
\label{chmix:dgnum} 
\end{equation} 
%
Here the coefficient of $B_{B_s}^S$ has been updated to
$m_b(m_b)+m_s(m_b)=4.3\,$GeV (in the $\ov{\rm MS}$ scheme) compared to
\cite{bbgln}. Since the coefficient $F^{cc}(z)$ of $B_{B_s}$ in \eq{chmix:ga12F}
is very small, $B_{B_s}$ in \eq{chmix:dgnum} has been fixed to $B_{B_s} (m_b) =
0.85\pm 0.05$ obtained in quenched lattice QCD \cite{hy}. Recently the
KEK--Hiroshima group succeeded in calculating $f_{B_s}$ in an unquenched lattice
QCD calculation with two dynamical fermions \cite{fbs}. The result is
$f_{B_s}=(245\pm 30)\,$MeV.  With this number and $B_{B_s}^S (m_b) = 0.87\pm
0.09$ from quenched lattice QCD \cite{hy,hioy} one finds from \eq{chmix:dgnum}:
%
\begin{equation} 
\frac{\dg_s^{\rm SM}}{\Gamma} \,=\, 0.12\pm 0.06 
        . \label{chmix:dgnum2}
\end{equation}
%
Here we have conservatively added the errors from the two lattice quantities 
linearly. $f_{B_s}$ drops out from the ratio
%
\begin{equation}  
\frac{\dg_s^{\rm SM}}{\dm_s^{\rm SM}} \,\simeq\, \frac{5 \pi}{6} 
        \frac{m_b^2}{M_W^2\, \eta_B b_B  S (m_t^2/M_W^2) } \, |F_S (z)| 
       \, \frac{B_{B_s}^{S\prime}}{B_{B_s}} 
       \lt[ 1 + {\cal O} \lt( \frac{\Lambda_{QCD}}{m_b} \rt) \rt] .   
\label{chmix:dgdm}
\end{equation}
%
The full result with next-to-leading order corrections in $\alpha_s$ and
$\Lambda_{QCD}/m_b$ can be found in \cite{bbgln}.  Including these corrections
one finds \cite{bbgln}:
%
\begin{equation} 
\frac{\dg_s^{\rm SM}}{\dm_s^{\rm SM}} \,=\, \lt( 3.7 \epm{0.8}{1.5} \rt) \times 10^{-3}.
\label{chmix:dgdmnum}
\end{equation}
%
The uncertainty in \eq{chmix:dgdmnum} is dominated by the renormalization scale
dependence. Its reduction requires a painful three-loop calculation. The
numerical value in \eq{chmix:dgdmnum} is obtained with $B_{B_s}^S/B_{B_s}=1.0\pm
0.1$ \cite{hy}, which is larger than the one used in \cite{bbgln}.

Next we consider the width difference in the $B_d$ meson system: since the
second term in \eq{chmix:dgsm} is negligible in view of the other uncertainties,
\eq{chmix:dgdm} also holds for $\dg_d^{\rm SM}/\dm_d^{\rm SM}$ with the replacement
$B_{B_s}^{S\prime}/B_{B_s} \to B_{B_d}^{S\prime}/B_{B_d}$. The $SU(3)_F$
breaking in these `bag' factors and in the $\Lambda_{QCD}/m_b$ corrections can
safely be neglected, so that the numerical range \eq{chmix:dgdmnum} also holds
for $\dg_d^{\rm SM}/\dm_d^{\rm SM}$. With $\dm_d =0.49\,$ps$^{-1}$ and
$\tau_{B_d}=1.5\,$ps one finds $\dg_d^{\rm SM}\approx 3\times 10^{-3}
\Gamma_d$.%
\index{B meson@$B$ meson!width difference $\dg$!SM prediction for $\dg_d$}%
\index{width difference $\dg$!SM prediction for $\dg_d$} Since $\dg_d$ and $\dm_d$
are affected by new physics in different ways, it is instructive to consider the
ratio of the two width differences: from \eq{chmix:dgsm} and \eq{chmix:dgnum}
one finds
%
\begin{equation}
\frac{\dg_d^{\rm SM}}{\dg_s^{\rm SM}} \,=\, \frac{|\Gamma_{12}^d|}{|\Gamma_{12}^s|}
      \, \simeq \, \frac{f_{B_d}^2 B_{B_d}^S}{f_{B_s}^2 B_{B_s}^S} 
              \, \lt| \frac{\xi_t^d}{\xi_t^s} \rt|^2 
\, \simeq \,  0.04 \, R_t^2  . \label{chmix:dgdsm}     
\end{equation}
%
In the last line we have used $f_{B_s}/f_{B_d}=1.16\pm 0.05$ and
$B_{B_s}^S=B_{B_d}^S$. The numerical predictions for $\dg_d$ from
\eq{chmix:dgdmnum} and \eq{chmix:dgdsm} are consistent with each other. Since
$\dg_d$ stems from the CKM-suppressed decay modes, it can be substantially
enhanced in models of new physics.

%...............................................................................
\boldmath
\subsubsection{Phenomenology of $\dg_s$}\label{chmix:subsub:width:phen}
\unboldmath
\index{B meson@$B$ meson!width difference $\dg$!measurement of $\dg_s$}
\index{width difference $\dg$!measurement of $\dg_s$}

\paragraph{Time Evolution:}
%==========================
The width difference $\dg_s$ can be measured from the time evolution of an
untagged $B_s$ sample, as shown in sect.~\ref{subs:unt}. In general the decay
$\Bsun \to f$ is governed by the two-exponential formula in \eq{guntfeig}. With
\eq{defpq} and \eq{deflaf} we can calculate $\langle f \ket{B_{L,H}}$ and find
from \eq{guntfeig}:
%
\begin{equation}
\guntf \,=\,  {\cal N}_f \, \frac{|A_f|^2}{2} \lt( 1+ |\lambda_f|^2 \rt) 
            \lt[ \lt( 1- \adg{f} \rt) e^{- \Gamma_L t} + 
                    \lt( 1+ \adg{f} \rt) e^{- \Gamma_H t} \rt]
\label{chmix:utadg} 
\end{equation}
%
\index{untagged $B$!two-exponential decay}% 
\index{$A_{\Delta \Gamma}$!and untagged decay}%
with $\adg{f}$ defined in \eq{adeg}.  Throughout this
Sect.~\ref{chmix:subsub:width:phen} we neglect small terms of order $\ega$.  The
time-independent prefactor of the square bracket can be eliminated in favor of
the branching ratio $Br \, \big( \!\Bun \rightarrow f \, \big)$ using
\eq{untnorm}.  In principle one could measure $\dg_s=\Gamma_L-\Gamma_H$ by
fitting the decay distribution of any decay with $|\adg{f}|\neq1$ to $\guntf$ in
\eq{chmix:utadg}. In practice, however, one will at best be able to measure the
deviation from a single exponential up to terms linear in $\dg_s t$. In
\eq{guntf} $\guntf$ is expressed in terms of $\Gamma_s=(\Gamma_L+\Gamma_H)/2$
and $\dg_s$. With \eq{untnorm} one finds
%
\begin{equation}
\guntf \,=\,
2\, \brunts{f} \,
        \Gamma_s 
        \, e^{- \Gamma_s t} \lt[ 
        1 +  \frac{\dg_s}{2} \, \adg{f} \, 
        \lt( t - \frac{1}{ \Gamma_s } \rt) 
        \rt] 
        + {\cal O} \lt(\lt( \dg_s\, t \rt)^2 \rt) . \label{chmix:guntf3} 
\end{equation} 
%
That is, unless one is able to resolve quadratic ${\cal O} \lt( ( \dg_s)^2 \rt)
$ terms, one can only determine the product $\adg{f} \dg_s $ from the time
evolution. A flavor-specific decay mode like $B_s \to
D_s^- \pi^+$%
\index{decay!$B_s \rightarrow D_s^- \pi^+$} is characterized by $\lambda_f=0$
and therefore has $\adg{f}=0$. In these decay modes the term involving $\dg_s $
in \eq{chmix:guntf3} vanishes. Flavor-specific decays therefore determine
$\Gamma_s$ up to corrections of order $( \dg_s)^2 $. For those decays \guntf\ is
insensitive to new physics in $M_{12}$, because $\lambda_f=0$. In order to gain
information on $\dg_s$ from \eq{chmix:guntf3}, one must consider decays with
$\lambda_f\neq 0$.  But $\lambda_f$ and \adg{f}\ depend on the mixing phase
$\phi_M$ (see \eq{defphm}) and therefore change in the presence of new physics
in $M_{12}$. In the Standard Model we can calculate $\phi_M$ and then extract
$\dg_s$ from the measured $\adg{f} \dg_s $. In the presence of new physics,
however, one needs additional information.  We therefore discuss these two cases
independently below.

Lifetimes are conventionally measured by fitting the decay distribution to a
single exponential.  We now write the two-exponential formula of
\eq{chmix:utadg} as \index{untagged $B$!two-exponential decay}
%
\begin{eqnarray}
\guntf &=& A \, e^{-\Gamma_L t} \, + \, B \, e^{-\Gamma_H t} \nn
       & = & e^{-\Gamma_s t} \lt[ 
        \lt( A+B \rt) \cosh \frac{\dg_s t}{2} + 
        \lt( B-A \rt) \sinh \frac{\dg_s t}{2}   \rt] \! , 
   \label{chmix:twoex2}
\end{eqnarray}
%
where $A=A(f)$ and $B=B(f)$ can be read off from \eq{chmix:utadg}.  If one uses
a maximum likelihood fit of \eq{chmix:twoex2} to a single exponential,
%
\begin{equation}
 F\lt[ f, t \rt] \,=\,
        { \Gamma_f} \,   e^{-\Gamma_f \, t}, \label{chmix:singex} 
\end{equation}
%
it will yield the following result \cite{hm}:
%
\begin{equation}
\Gamma_f \,=\,
        \frac{A/\Gamma_L + B/\Gamma_H}{A/\Gamma_L^2 + B/\Gamma_H^2}     
 . \label{fitex}
\end{equation}
%
We expand this to second order in $\dg_s$:
%
\begin{equation}
     \Gamma_f \,=\, \Gamma_s \, + \,  
\frac{A-B}{A+B} \, \frac{\dg_s}{2} \, 
                - \frac{2 \, A B}{(A+B)^2}  \frac{(\dg_s)^2}{\Gamma_s} \,  
        + \, {\cal O} \lt( \frac{(\dg_s)^3}{\Gamma_s^2} \rt)
 . \label{chmix:fit}
\end{equation}
%
In flavor-specific decays we have $A=B$ (see \eq{guntf}). We see from
\eq{chmix:fit} that here a single-exponential fit determines
%
\begin{equation}
 \Gamma_{\rm fs} \,=\, \Gamma_s \, - \, \frac{(\dg_s)^2}{2\, \Gamma_s} 
             \, + \,  {\cal O} \lt( \frac{(\dg_s)^3}{\Gamma_s^2} \rt)
             . \label{chmix:gafs}
\end{equation}
%
Heavy quark symmetry predicts that the average \emph{widths}\ $\Gamma_s$ and
$\Gamma_d$ are equal up to corrections of less than one percent
\cite{bbsuv,bbd}. From \eq{chmix:gafs} we then realize that the average $B_s$
\emph{lifetime}\ (defined as $1/\Gamma_f$) can exceed the $B_d$ lifetime by more
than one percent, if $\dg_s$ is sizable.

\paragraph{\boldmath $CP$ Properties and Branching Ratios:}
%==============================================
In the $B_s$ system \dg\ is dominated by $\Gamma_{12}^{s,cc}$. In the following
we will neglect the Cabibbo-suppressed contributions from $\Gamma_{12}^{s,uc}$
and $\Gamma_{12}^{s,uu}$. We also specify to the PDG phase convention for the
CKM matrix, in which $\arg (V_{cb} V_{cs}^*) = {\cal O} (\lambda^6)$, see
\eq{chmix:betas}. For the discussion in the forthcoming paragraphs it will be
useful to define the $CP$ eigenstates
%
\begin{equation}
\ket{B_s^{\textrm{\scriptsize even}}} \,=\,
 \frac{1}{\sqrt{2}} \lt( \ket{B_s} - \ket{\ov{B}{}_s} \rt), \qquad 
 \mbox{and} \qquad
\ket{B_s^{\textrm{\scriptsize odd}}} \,=\, 
 \frac{1}{\sqrt{2}} \lt( \ket{B_s} + \ket{\ov{B}{}_s} \rt)
 .\label{chmix:cpe}  
\end{equation}
%
Here we have used the standard convention for the $CP$ transformation:%
\index{CP transformation@\CP\ transformation!B meson@$B$ meson} $CP \ket{B_s} =
- \ket{\ov{B}{}_s}$.

\index{B meson@$B$ meson!width difference $\dg$!$\dg_s$ and branching ratios}%
\index{width difference $\dg$!measurement of $\dg_s$!branching ratios}
Interestingly, one can measure $\dg_s$ from branching ratios, without
information from lifetime fits.  We define
%
\begin{equation}
\dg_{\rm CP}^s \,\equiv\, 2 |\Gamma_{12}^s| \, = \, 
2 \sum_{f  \in X_{c\ov{c}}} \,  \lt[
\Gamma ( B_s \to f_{\rm CP+} ) \, - \,
        \Gamma ( B_s \to f_{\rm CP-}) \rt]
        . 
        \label{chmix:dgcp}
\end{equation}
%
Here $X_{c\ov {c}}$ represents the final states containing a
$(c,\ov{c})$ pair, which constitute the dominant contribution to
$\dg_{\rm CP}^s$ stemming from the decay $b \to c\ov{c} s$.  In
\eq{chmix:dgcp} we have decomposed any final state $f$ into its
$CP$-even and $CP$-odd component, $\ket{f}=\ket{f_{\rm CP+}} 
  + \ket{f_{\rm CP-}}$\footnote{The factor of 2 in \eq{chmix:dgcp} is an
  artifact of our normalization of $\ket{f_{\rm CP\pm}}$.} and defined
%
\begin{equation}
\Gamma ( B_s \to f_{\rm CP\pm} ) \,=\, 
        {\cal N}_f \, | \bra{f_{\rm CP\pm}} B_s \rangle |^2 
  \, = \, 
     \frac{ | \bra{f_{\rm CP\pm}} B_s \rangle |^2}{ | \bra{f} B_s \rangle
     |^2} \, 
        \Gamma ( B_s \to f ) . 
\end{equation}
%
${\cal N}_f$ is the usual normalization factor originating from the phase-space
integration.  $\dg_{\rm CP}^s$ equals $\dg_s$ in the Standard Model, but these
quantities differ by a factor of $\cos \phi_s $ in models of new physics, see
\eq{mgsol:b}. We will later exploit this feature to probe the Standard Model and
to determine $|\cos \phi_s|$.

We now prove the second equality in \eq{chmix:dgcp} and subsequently discuss how
$\Gamma ( B_s \to f_{\rm CP\pm})$ can be measured.  Start from the definition of
$\Gamma_{12}^s$:
%
\begin{equation}
\Gamma_{12}^s \,=\, \sum_f {\cal N}_f \, \bra{B_s} f \rangle 
                \bra{f} \ov{B}_s \rangle 
        \, = \, \frac{1}{2} \sum_f {\cal N}_f  
        \lt[ \, \bra{B_s} f \rangle \bra{f} \ov{B}_s \rangle + 
        \bra{B_s} \ov{f} \rangle \bra{{ \ov{f}}} \ov{B}_s \rangle \, \rt]
 .\label{defg12} 
\end{equation}
%
In the second equation we have paired the final state $\ket{f}$ with its $CP$
conjugate $\ket{\ov{f}}=-CP\ket{f}$.
In the next step we 
trade $f$ for $f_{\rm CP+}$ and $f_{\rm CP-}$ and use the $CP$ transformation% 
\index{CP transformation@\CP\ transformation!B meson@$B$ meson}
%
\begin{equation}
 \bra{f_{\rm CP\pm}} \ov{B}_s \rangle \,=\,
        \mp \, \bra{f_{\rm CP\pm}} B_s \rangle 
\end{equation}
%
in our phase convention with $\arg (V_{cb} V_{cs}^*) =0$. Then \eq{defg12}
becomes
%
\begin{eqnarray}
- \, \Gamma_{12}^s &=& 
        \sum_{f\in X_{c\ov{c}}} {\cal N}_f \,
        \lt[ | \bra{f_{\rm CP+}} B_s \rangle |^2 - 
             | \bra{f_{\rm CP-}} B_s \rangle |^2 \rt] \nn
& = & \sum_{f\in X_{c\ov{c}}}
\lt[ \Gamma ( B_s \to f_{\rm CP+} ) \, - \,
        \Gamma ( B_s \to f_{\rm CP-}) \rt] .
\label{chmix:g12oe}
\end{eqnarray}
%
Interference terms involving both $ \bra{f_{\rm CP+}} B_s \rangle$ and
$\bra{f_{\rm CP-}} B_s \rangle $ drop out when summing the two terms $\bra{B_s}
f \rangle \bra{f} \ov{B}_s \rangle$ and $\bra{B_s} \ov{f} \rangle \bra{{
    \ov{f}}} \ov{B}_s \rangle $.  An explicit calculation of $\Gamma_{12}^s$
reveals that the overall sign of the LHS of \eq{chmix:g12oe} is positive, which
completes the proof of \eq{chmix:dgcp}.

Loosely speaking, $\dg_{\rm CP}^s$ is measured by counting the $CP$-even and
$CP$-odd double-charm final states in $B_s$ decays.  Our formulae become more
transparent if we use the $CP$-eigenstates defined in \eq{chmix:cpe}.  With
$\ket{B_s}= (\ket{B_s^{\textrm{\scriptsize even}}}+\ket{B_s^{\textrm{\scriptsize
      odd}}})/\sqrt{2}$ one easily finds from \eq{chmix:g12oe}:
%
\begin{equation}
\dg_{\rm CP}^s \,=\, 2 |\Gamma_{12}^s| \, = \, 
\Gamma \lt( B_s^{\textrm{\scriptsize even}} \rt) - 
        \Gamma ( B_s^{\textrm{\scriptsize odd}} ) .
        \label{chmix:dgcp2}
\end{equation}
%
Here the RHS refers to the total widths of the $CP$-even and $CP$-odd $B_s$
eigenstates.  We stress that the possibility to relate $|\Gamma_{12}^s|$ to a
measurable quantity in \eq{chmix:dgcp} crucially depends on the fact that
$\Gamma_{12}^s$ is dominated by a single weak phase. For instance, the final
state $K^+ K^-$ is triggered by $b \to u \ov{u} s$ and involves a weak phase
different from $b \to c\ov{c} s$.  Although $K^+ K^-$ is $CP$-even, the decay
$B_s^{\textrm{\scriptsize odd}}\to K^+ K^- $ is possible.% 
\index{decay!$B_s \rightarrow K^+ K^-$}
An inclusion of such CKM-suppressed modes into \eq{chmix:g12oe}
would add interference terms that spoil the relation to measured
quantities.  The omission of these contributions to $\Gamma_{12}^s$
induces a theoretical uncertainty of order 3--5\% on \eq{chmix:dgcp2}.

A measurement of $\dg_{\rm CP}^s$ has been performed by the ALEPH
collaboration \cite{aleph}.  ALEPH has measured
\index{decay!$B_s \rightarrow D_s^+ D_s^-$}
\index{decay!$B_s \rightarrow D_s^{*+} D_s^-$}
\index{decay!$B_s \rightarrow D_s^+ D_s^{*-}$}
\index{decay!$B_s \rightarrow D_s^{*+} D_s^{*-}$}
%
\begin{equation}
 2\, \brunts{D_s^{(*)}{}^+ D_s^{(*)}{}^-} \,=\,
        0.26 \epm{0.30}{0.15} \label{chmix:alexp}
\end{equation}
%
and related it to $\dg_{\rm CP}^s$. For this the following theoretical input has
been used \cite{ayopr}:
\begin{itemize} 
\item[i)] In the heavy quark limit $m_c \to \infty $ and neglecting certain
  terms of order $1/N_c$ (where $N_c=3$ is the number of colors) the decay
  $B_s^{\textrm{\scriptsize odd}} \to D_s^\pm D_s^{*}{}^\mp$ is forbidden. Hence
  in this limit the final state in $\Bsun \to D_s^\pm D_s^{*}{}^\mp$ is $CP$-even.
  Further in $\Bsun \to D_s^{*}{}^+ D_s^{*}{}^-$ the final state is in an
  S-wave.
\item[ii)] In the small velocity limit when $m_c \to \infty $ with $m_b - 2
  m_c$ fixed \cite{sv}, $\dg_{\rm CP}^s$ is saturated by $\Gamma ( \Bsun \to
  D_s^{(*)}{}^+ D_s^{(*)}{}^- )$.  With i) this implies that in the considered
  limit the width of $B_s^{\textrm{\scriptsize odd}}$ vanishes.  For $N_c \to
  \infty$ and in the SV limit, ${ \Gamma ( \Bsun \to D_s^{(*)}{}^+ D_s^{(*)}{}^-
    )}$ further equals the parton model result for $\dg_{\rm CP}^s$
  (quark-hadron duality).
\end{itemize}
%
Identifying $\Gamma ( B_s^{\textrm{\scriptsize even}} \to D_s^{(*)}{}^+
D_s^{(*)}{}^- ) \simeq \dg_{\rm CP}^s$ and $\Gamma ( B_s^{\textrm{\scriptsize
    odd}} \to D_s^{(*)}{}^+ D_s^{(*)}{}^- ) \simeq 0$ we can integrate
$\gunt{D_s^{(*)}{}^+ D_s^{(*)}} = \Gamma ( B_s^{\textrm{\scriptsize even}}\to
D_s^{(*)}{}^+ D_s^{(*)}{}^- )\exp (-\Gamma_L t) $ over $t$ to find:
%
\begin{equation}
2\, \brunts{ D_s^{(*)}{}^+ D_s^{(*)}{}^-} \,\simeq\, 
         \, \frac{\dg_{\rm CP}^s}{\Gamma_L}   
. \label{chmix:bdg}
\end{equation}
%
Thus the measurement in \eq{chmix:alexp} is compatible with the theoretical
prediction in \eq{chmix:dgnum2}.

When using \eq{chmix:bdg} one should be aware that the corrections to the limits
i) and ii) adopted in \cite{ayopr} can be numerically sizeable. For instance, in
the SV limit there are no multibody final states like $D_s^{(*)} \ov{D} X_s$,
which can modify \eq{chmix:bdg}.  As serious would be the presence of a sizeable
$CP$-odd component of the $D_s^{(*)}{}^+ D_s^{(*)}{}^- $ final state, since it
would be added with the wrong sign to $\dg_{\rm CP}^s$ in \eq{chmix:bdg}. A
method to control the corrections to the SV limit experimentally is proposed
below in the paragraph on new physics. One feature of the SV limit is the
absence of $CP$-odd double-charm final states. (Indeed there are only very few
$CP$-odd final states in Table~\ref{chmix:tab:cpe}.) This has the consequence that
$\dg_{\rm CP}^s$ cannot be too small, because for $\Gamma
(B_s^{\textrm{\scriptsize odd}} \to X_{\ov{c} c})$ the spectator contributions
and non-spectator diagrams like those in Fig.~\ref{chmix:fig:dga} must sum to
zero. This favors values of $\dg_{\rm CP}^s = \dg_s^{\rm SM}$ in the upper range of
\eq{chmix:dgnum2}.


\paragraph{Standard Model:}
%==========================
In the Standard Model the \bbms\ phase $\phi_M^s=-2\beta_s$ can be safely
neglected for the discussion of $\dg_s$. Then the mass eigenstates coincide with
the $CP$ eigenstates defined in \eq{chmix:cpe} with
$\ket{B_L}=\ket{B_s^{\textrm{\scriptsize even}}}$ and
$\ket{B_H}=\ket{B_s^{\textrm{\scriptsize odd}}}$. Any $b\to c \ov{c} s$ decay
into a $CP$-even final state like $D_s^+D_s^-$ stems solely from the $\ket{B_L}$
component in the untagged $B_s$ sample. A lifetime fit to this decay therefore
determines $\Gamma_L$.  Conversely, the $b\to c \ov{c} s$ decay into a $CP$-odd
eigenstate determines $\Gamma_H$. We can easily verify this from
\eq{chmix:utadg} by calculating $\adg{f}$: $q/p$ in \eq{qpsol} equals $-1$ and
$\ov{A}_f/A_f=-\eta_f$, where $\eta_f$ is the $CP$ parity of the final state. Then
\eq{deflaf} yields $\lambda_f=\eta_f$, so that $\adg{f} =-\eta_f$. Hence for any
$b\to c \ov{c} s$ decay the coefficient of $\exp(-\Gamma_H t)$ in
\eq{chmix:utadg} vanishes for a $CP$-even final state, while the $\exp(-\Gamma_L
t)$ term vanishes for a $CP$-odd final state. In practice one will encounter much
more statistics in $CP$-even final states, so that the best determination of
$\dg_s$ will combine
$\Gamma_L$ with $\Gamma_{\rm fs}$ measured in a flavor-specific decay.%
\index{B meson@$B$ meson!width difference $\dg$!$\dg_s$ and lifetimes}%
\index{B meson@$B$ meson!width difference $\dg$!$\dg_s$ and branching ratios}%
\index{width difference $\dg$!measurement of $\dg_s$!lifetimes} From
\eq{chmix:gafs} and $\Gamma_L=\Gamma_s + \dg_s/2$ one finds
%
\begin{equation}
\dg_s \,=\, 2 \lt( \Gamma_L - \Gamma_{\rm fs} \rt)
             \lt( 1 - 
        2 \frac{\Gamma_L - \Gamma_{\rm fs}}{\Gamma_{\rm fs}} \rt) 
             \, + \,  {\cal O} \lt( \frac{(\dg_s)^3}{\Gamma_s^2}\rt)
\label{chmix:dgressm} .
\end{equation}
%
Here we have expanded to second order in $\dg_s$, which should be sufficient for
realistic values of $\dg_s$.

It should be stressed that every $b\to c \ov{c} s$ decay encodes the same
information on $\dg_s$, once its $CP$ parity is known. This is also true for $b\to
c \ov{u} d$ decays into $CP$ eigenstates, because the decay amplitude carries the
same phase as the one in $b\to c \ov{c} s$.  Therefore the extracted values for
$\dg_s$ in these decays can be combined to gain statistics. Interesting decay
modes are summarized in Table~\ref{chmix:tab:cpe}. Many of the listed modes, like
$\Bsun \to \psi \phi$, require an angular analysis to separate the $CP$-even from
the $CP$-odd component. This procedure is described in detail in
Sect.~\ref{chmix:twocp}.%

\index{decay!$B_s \rightarrow K^+ K^-$} It is tempting to use $\Bsun \to K^+
K^-$ to measure $\dg_s$ because of its nice experimental signature. But such
CKM-suppressed decay modes cannot be used, because the weak phase of the decay
amplitude is not known. If $B_s \to K^+ K^-$ were dominated by penguin loops and
new physics were absent from these loops, $\lambda_{K^+K^-}$ would indeed be
equal to $+1$ and the coefficient of $\exp(-\Gamma_H t)$ in \eq{chmix:utadg}
would vanish. In practice, however, the tree-level amplitude $b\to u \ov{u} s$
is expected to give a non-negligible contribution. Since this amplitude carries
a different phase, $2 \arg(V_{ub})=-2\gamma$, $\lambda_{K^+K^-}$ deviates from
$\pm 1$ and both exponentials in \eq{chmix:utadg} contribute.


\paragraph{New Physics:}
%=======================
\index{B meson@$B$ meson!width difference $\dg$!new physics}%
\index{width difference $\dg$!new physics}%
In the presence of new physics the
CP-violating phase $\phi$ in \eq{defphi} and \eq{mgsol:b} can be large. Since
various observables in untagged $B_s$ decays depend on $\cos \phi_s$ in
different ways, one can reveal new physics and determine $|\cos \phi_s|$ by
combining different measurements. We have already seen above that $\dg_{\rm
  CP}^s$ in \eq{chmix:dgcp} does not depend on $\phi_s$ at all, while $\dg_s$ is
diminished in the presence of new physics:
%
\begin{equation} 
\dg_s \,=\, \dg_{\rm CP}^s \cos \phi_s .
\end{equation}
%
On the other hand $\sin \phi_s$ can be obtained from $CP$ asymmetries in $B_s$
decays like $B_s \to \psi \phi$.  Therefore measurements of \dg\ are
complementary to the study of $CP$ asymmetries, which require tagging and the
resolution of the rapid \bbs\ oscillation and come with a loss in statistics,
efficiency and purity.  Both avenues should be pursued and their results
combined, because they measure the same fundamental quantities. A detailed
analysis of both tagged and untagged decays can be found in \cite{dfn}.

In our phase convention $\arg (V_{cb} V_{cs}^*) =0$ we simply have 
%
\begin{equation}
\arg(M_{12}) \,=\, \phi_s .  \label{chmix:dgcph}  
\end{equation}
%
The mass eigenstates can be expressed as
%
\begin{eqnarray}
\ket{B_L} & = & \phantom{- \,}
                \frac{1+e^{i\phi}}{2} \, 
                  \ket{B_s^{\textrm{\scriptsize even}}} \, { -} \, 
                  \frac{1-e^{i\phi}}{2} \, 
                  \ket{B_s^{\textrm{\scriptsize odd}}} 
                  , \nn
\ket{B_H} & = & { -} \, \frac{1-e^{i\phi}}{2} \, 
                  \ket{B_s^{\textrm{\scriptsize even}}} \, + \, 
                  \frac{1+e^{i\phi}}{2} \, 
                  \ket{B_s^{\textrm{\scriptsize odd}}} 
         \, . 
        \label{chmix:rot}
\end{eqnarray} 
%
Whenever we use $B_s^{\textrm{\scriptsize even}}$ and $B_s^{\textrm{\scriptsize
    odd}}$ we implicitly refer to our phase conventions for the CKM matrix and
the $CP$ transformation.  If formulae involving $B_s^{\textrm{\scriptsize even}}$
and $B_s^{\textrm{\scriptsize odd}}$ are used to constrain models with an
extended quark sector, the phase convention used for the enlarged CKM matrix
must likewise be chosen such that $\arg (V_{cb} V_{cs}^*) \simeq 0$.

We next consider the time evolution of a $b\to c\ov{c}s$ decay into a CP
eigenstate with $CP$ parity $\eta_f$. \adg{f}\ reads
%
\begin{equation}
\adg{f} \,=\, -\eta_f \cos \phi_s . \label{acp2}
\end{equation}
%
In the Standard Model, where $\phi_s\simeq 0$, \guntf\ simplifies to a
single-exponential law, which can be verified from \eq{chmix:guntf3} or by
inserting \eq{chmix:rot} into \eq{guntfeig}.

Since $\dg_{\rm CP}^s$ is unaffected by new physics and $\dg_{\rm CP}^s > 0$,
several facts hold beyond the Standard Model:
\begin{itemize}
\item[i)] There are more $CP$-even than $CP$-odd final states in $B_s$ decays. 
\item[ii)] The shorter-lived mass eigenstate is always the one with
  the larger $CP$-even component in \eq{chmix:rot}. Its branching ratio into a
  $CP$-even final state $f_{\rm CP+}$ exceeds the branching ratio of the
  longer-lived mass eigenstate into $f_{\rm CP+}$, if the weak phase
  of the decay amplitude is close to $ \arg V_{cb} V_{cs}^*$.
\item[iii)] For $ \cos \phi_s > 0$ $B_L$ has a shorter lifetime than
  $B_H$, while for $ \cos \phi_s < 0$ the situation is the opposite
  \cite{g}.
\end{itemize}
Allowing for a new physics phase $\phi_s$ the result in \eq{chmix:bdg}
is changed. In the SV limit one now predicts:
%
\begin{eqnarray}
2\, \brunts{ D_s^{(*)}{}^+ D_s^{(*)}{}^-} & \simeq & 
        \dg_{\rm CP}^s \, \lt[ \, \frac{1+\cos \phi_s}{2\, \Gamma_L}  + 
                            \frac{1-\cos \phi_s}{2\, \Gamma_H} \, \rt] 
         \nn
& =&  \,  \frac{\dg_{\rm CP}^s}{\Gamma_s} \lt[ 
  1 + {\cal O} \lt( \frac{\dg_s}{\Gamma_s} \rt) \rt] . \label{chmix:bdg2}
\end{eqnarray}
%
\index{decay!$B_s \rightarrow D_s^+ D_s^-$}
\index{decay!$B_s \rightarrow D_s^{*+} D_s^-$}
\index{decay!$B_s \rightarrow D_s^+ D_s^{*-}$}
\index{decay!$B_s \rightarrow D_s^{*+} D_s^{*-}$}
The term in square brackets accounts for the fact that in general the $CP$-even
eigenstate $\ket{B_s^{\textrm{\scriptsize even}}}$ is a superposition of
$\ket{B_L}$ and $\ket{B_H}$. It is straightforward to obtain \eq{chmix:bdg}:
inserting \eq{chmix:rot} into \eq{guntfeig} expresses \guntf\ in terms of
$\Gamma ( B_s^{\textrm{\scriptsize even}} \to f )$ and $\Gamma (
B_s^{\textrm{\scriptsize odd}} \to f )$. After integrating over time the
coefficient of $\Gamma ( B_s^{\textrm{\scriptsize even}} \to f )$ is just the
term in square brackets in \eq{chmix:bdg2}.  We verify from \eq{chmix:bdg2} that
the measurement of $\brunts{D_s^{(*)}{}^+ D_s^{(*)}{}^-}$ determines $\dg_{\rm
  CP}^s$.  Its sensitivity to the new physics phase $\phi_s$ is suppressed by
another factor of $\dg_s/\Gamma_s$ and is irrelevant in view of the theoretical
uncertainties.

Next we discuss the determination of $\dg_s$ and $|\cos \phi_s|$.  There are two
generic ways to obtain information on $\dg_s$ and $\phi_s$ :
\begin{itemize}
\item[i)] The measurement of the $B_s$ lifetime in two decay modes
  $\Bsun \to f_1$ and  $\Bsun \to f_2$ with\\
  $\adg{f_1} \neq \adg{f_2}$.
\item[ii)] The fit of the decay distribution of $\Bsun \to f$ to the
  two-exponential formula in \eq{guntf}.
\end{itemize}
As first observed in \cite{g}, the two methods are differently affected by a new
physics phase $\phi_s \neq 0$. Thus by combining the results of methods i) and
ii) one can gain information on $\phi_s$.  In this paragraph we consider two
classes of decays:
\begin{itemize}
\item flavor-specific decays, which are characterized by $\ov{A}_f=0$ {
    implying} $\adg{f}=0$.  Examples are $B_s \to D_s^- \pi^+$ and $B_s \to X
  \ell^+ \nu_{\ell} $,
\index{decay!$B_s \rightarrow D_s^- \pi^+$}%
\index{decay!$B_s \rightarrow X \ell^+ \nu$}
\item the CP-specific decays of Table~\ref{chmix:tab:cpe}, with $\adg{f} = -
  \eta_f \cos \phi_s$.
\end{itemize}
In both cases the time evolution of the untagged sample in \eq{guntf} is not
sensitive to the sign of $\dg_s$ {(or, equivalently, of $\cos \phi_s$)}. For the
CP-specific decays of Table~\ref{chmix:tab:cpe} this can be seen by noticing that
%
\begin{equation}
\adg{f}\, \sinh \frac{\dg_s \, t}{2} =
        - \, \eta_f \, |\cos \phi_s| \, \sinh \frac{|\dg_s| \, t}{2} .\no
\end{equation}
%
Here we have used the fact that $\dg_s$ and $\cos \phi_s $ always have the same
sign, because $\dg_{\rm CP}^s>0$.  Hence our untagged studies can only determine
$|\cos \phi_s|$ and therefore lead to a four-fold ambiguity in $\phi_s$.  The
sign ambiguity in $\cos \phi_s$ reflects the fact that from the untagged time
evolution in \eq{guntf} one cannot distinguish, whether the heavier or the
lighter eigenstate has the shorter lifetime.  (Methods to resolve the discrete
ambiguity can be found in \cite{dfn}.)

In order to experimentally establish a non-zero $\dg_s$ from the time evolution
in \eq{guntf} one needs sufficient statistics to resolve the deviation from a
single-exponential decay law, see \eq{chmix:guntf3}.  As long as we are only
sensitive to terms linear in $\dg_s\, t$ and $\dg_s/\Gamma_s$, we can only
determine $\adg{f}\, \dg_s$ from \eq{chmix:guntf3}. $\adg{f}\, \dg_s$ vanishes
for flavor-specific decays and equals $-\eta_f \dg_s\, \cos \phi_s$ for
CP-specific final states.  Hence from the time evolution alone one can only
determine the product $\dg_s\, \cos \phi_s $ in the first experimental stage.


\boldmath
\subparagraph{Determination of $\Gamma_s$ and $\dg_s \cos \phi_s$:}
% \label{sec:dgcos}
\unboldmath

In Eqs.~(\ref{chmix:twoex2}) -- (\ref{chmix:fit}) we have related the width
found in a single-exponential fit to the parameters $A(f)$, $B(f)$, $\Gamma_s$
and $\dg_s$ of the two-exponential formula.  \index{untagged $B$!two-exponential
  decay}

In \eq{chmix:gafs} we found that a single-exponential fit in flavor-specific
decays (which have $A=B$) determines $\Gamma_s$ up to corrections of order
$(\dg_s)^2/\Gamma_s^2$.

With \eq{guntf} and \eq{chmix:twoex2} we can read off $A$ and $B$ for the
CP-specific decays of Table~\ref{chmix:tab:cpe} and find $A(f_{\rm CP+})/B(f_{\rm
  CP+}) = (1+\cos \phi)/(1-\cos \phi)$ and $A(f_{\rm CP-})/B(f_{\rm CP-}) =
(1-\cos \phi)/(1+\cos \phi)$ for $CP$-even and $CP$-odd final states, respectively.
Our key quantity for the discussion of $CP$-specific decays 
$\Bsun \to f_{\rm CP}$ is
%
\begin{equation}
\dg^{s\,\prime}_{\rm CP} \,\equiv\, - \eta_f \adg{f} \, \dg_s \, 
                \, = \, 
              \dg_s \, \cos \phi_s 
          \, = \, 
          \dg_{\rm CP}^s \, \cos^2 \phi_s  
        .\label{chmix:dgp}
\end{equation}   
%
With this definition \eq{chmix:fit} reads for the decay rate $\Gamma_{\rm CP,
  \eta_f}$ measured in $\Bsun \to f_{\rm CP}$ \cite{dfn}:
%
\begin{equation} 
     \Gamma_{\rm CP,\eta_f} \,=\, \Gamma_s \, + \,  
        \eta_f \, \frac{\dg^{s\,\prime}_{\rm CP}}{2} 
        { \, - \, \sin^2 \phi_s \, \frac{(\dg_s)^2}{2 \Gamma_s} }
        \, + \, {\cal O} \lt( \frac{(\dg_s)^3}{\Gamma_s^2} \rt) . 
   \label{chmix:gacp} 
\end{equation}
%
That is, to first order in $\dg_s$, comparing the $\Bsun$ lifetimes measured in
a flavor-specific and a CP-specific final state determines $\dg^{s\,\prime}_{\rm
  CP}$. The first term in \eq{chmix:gacp} agrees with the result in \cite{g},
which has been found by expanding the time evolution in \eq{chmix:twoex2} and
\eq{chmix:singex} for small $\dg_s \, t$.

From \eq{chmix:gafs} and \eq{chmix:gacp} one finds
%
\begin{equation}
\Gamma_{\rm CP,\eta_f}  \, - \, \Gamma_{\rm fs}  \,=\,
        { \frac{\dg^{s\,\prime}_{\rm CP}}{2}} \lt( \eta_f \, + \, 
                \frac{\dg^{s\,\prime}_{\rm CP}}{\Gamma} \rt) \, + \, 
        {\cal O} \lt( \frac{(\dg)^3}{\Gamma^2}  \rt)  
        . \label{dgp2}
\end{equation}
%
Hence for a $CP$-even ($CP$-odd) final state the quadratic corrections enlarge
(diminish) the difference between the two measured widths.  A measurement of
$\dg^{s\,\prime}_{\rm CP}$ has a high priority at Run~II of the Tevatron.  The
LHC experiments ATLAS, CMS and LHCb expect to measure $\dg^{s\,\prime}_{\rm
  CP}/\Gamma_s$ with absolute errors between 0.012 and 0.018 for
$\dg^{s\,\prime}_{\rm CP}/\Gamma_s =0.15$ \cite{cern}.  An upper bound on
$\dg^{s\,\prime}_{\rm CP}$ would be especially interesting. If the lattice
calculations entering \eq{chmix:dgnum2} mature and the theoretical uncertainty
decreases, an upper bound on $|\dg^{s\,\prime}_{\rm CP}|$ may { show that}
$\phi_s \neq 0,\pi$ through
%
\begin{equation}
\frac{\dg^{s\,\prime}_{\rm CP}}{\dg_{\rm CP}^s} \,=\, \cos^2 \phi_s  .
 \label{chmix:cp2}
\end{equation}  
%
Note that conversely the experimental establishment of a non-zero
$\dg^{s\,\prime}_{\rm CP}$ immediately helps to constrain models of new physics,
because it excludes values of $\phi_s$ around $\pi/2$.

The described method to obtain $\dg_{\rm CP}^{s\,\prime}$ can also be used, if
the sample contains a known ratio of $CP$-even and $CP$-odd components.  This
situation occurs e.g.\ in decays to $J/\psi \phi$, if no angular analysis is
performed or in final states, which are neither flavor-specific nor CP
eigenstates. We discuss this case below for $\Bsun \to D_s^\pm D_s^{(*)}{}^\mp
$.  Further note that the comparison of the lifetimes measured in $CP$-even and
$CP$-odd final states determines $\dg^{s\,\prime}_{\rm CP}$ up to corrections of
order $(\dg_s/\Gamma_s)^3$.

The theoretical uncertainty in \eq{chmix:dgnum2} dilutes the extraction of
$|\cos \phi_s|$ from a measurement of $\dg_{\rm CP}^{s\,\prime}$ alone.  One can
bypass the theory prediction in \eq{chmix:dgnum2} altogether by measuring both
$\dg_{\rm CP}^{s\,\prime}$ and $|\dg_s|$ and determine $|\cos \phi_s|$ through
%
\begin{equation}
\frac{\dg^{s\,\prime}_{\rm CP}}{\lt| \dg_s \rt|} \,=\, |\cos \phi_s | . 
 \label{chmix:cp3}
\end{equation}
%
To obtain additional information on $\dg_s$ and $\phi_s$ from the time evolution
in \eq{guntf} requires more statistics: the coefficient of $t$ in
\eq{chmix:guntf3}, $\dg_s\, \adg{f}/2$, vanishes in flavor-specific decays and
is equal to $-\eta_f \dg_{\rm CP}^{s\,\prime}/2 $ in the CP-specific decays of
Table~\ref{chmix:tab:cpe}. Therefore the data sample must be large enough to be
sensitive to the terms of order $(\dg_s \, t)^2$ in order to get new information
on $\dg_s$ and $\phi_s$.  We now list three methods to determine $|\dg_s|$ and
$|\cos \phi_s|$ separately \cite{dfn}. The theoretical uncertainty decreases and
the required experimental statistics increases from method 1 to method 3.  Hence
as the collected data sample grows, one can work off our list downwards. The
first method exploits information from branching ratios and needs no information
from the quadratic $(\dg_s \,t)^2$ terms.

\subparagraph{Method 1:} We assume that $\dg_{\rm CP}^{s\,\prime}$ has been
measured as described on page~\pageref{chmix:dgp}. The method presented now is a
measurement of $\dg_{\rm CP}^s$ using the information from branching ratios.
With \eq{chmix:cp2} one can then find $|\cos \phi_s|$ and subsequently $|\dg_s|$
from \eq{chmix:cp3}.  In the SV limit the branching ratio $\brunts{D_s^{(*)}{}^+
  D_s^{(*)}{}^-}$ equals $\dg_{\rm CP}^s/(2 \Gamma_s)$ up to corrections of
order $\dg_s/\Gamma_s$, as discussed above \cite{ayopr}.  Corrections to the SV
limit, however, can be sizeable. Yet we stress that one can control the
corrections to this limit experimentally, successively arriving at a result
which does not rely on the validity of the SV limit.  For this it is of prime
importance to determine the $CP$-odd component of the final states $D_s^{\pm}
D_s^{*\mp}$ and $D_s^{*+} D_s^{*-}$. We now explain how the $CP$-odd and $CP$-even
component of any decay $\Bsun \to f$ corresponding to the quark level transition
$b \to c\ov{c} s$ can be obtained. This simply requires a fit of the time
evolution of the decay to a single exponential, as in \eq{chmix:singex}.  Define
the contributions of the $CP$-odd and $CP$-even eigenstate to $B_s \to f$:
%
\begin{equation}
\Gamma (B_s^{\textrm{\scriptsize odd}} \to f) \, \equiv \, {\cal N}_f 
 \, |  \langle f
         \ket{B_s^{\textrm{\scriptsize odd}}} |^2,  \qquad 
\Gamma (B_s^{\textrm{\scriptsize even}} \to f) \, \equiv \, {\cal N}_f 
        \, | \langle f
         \ket{B_s^{\textrm{\scriptsize even}}} |^2 . \label{chmix:defoe}
\end{equation}
%
It is useful to define the $CP$-odd fraction $x_f$ by
%
\begin{equation}
\frac{  \Gamma (B_s^{\textrm{\scriptsize odd}}  
                \to f) }{
        \Gamma ( B_s^{\textrm{\scriptsize even}} 
                \to f) }
\,=\, \frac{  |  \langle f
         \ket{B_s^{\textrm{\scriptsize odd}}} |^2 }{
        | \langle f
         \ket{B_s^{\textrm{\scriptsize even}}} |^2 } \, =\, 
\frac{  |  \langle \ov{f}
         \ket{B_s^{\textrm{\scriptsize odd}}} |^2 }{
        | \langle \ov{f}
         \ket{B_s^{\textrm{\scriptsize even}}} |^2 } \, =\,
  \frac{x_f}{1-x_f}  . \label{chmix:x1x}
\end{equation}
%
The time evolution $(\guntf +\guntfb)/2$ of the CP-averaged untagged decay
$\Bsun \to f,\ov{f}$ is governed by a two-exponential formula:
\begin{equation}
\frac{\guntf +\guntfb }{2} \,=\, 
        A(f) \, e^{-\Gamma_L t} + B(f) \, e^{-\Gamma_H t} .
\label{chmix:twoex3} 
\end{equation}
%
With \eq{chmix:rot} and \eq{guntfeig} one finds  
%
\begin{eqnarray}
A(f) & = & 
 \frac{{\cal N}_f}{2} \,  | \langle  f \ket{B_L} |^2 \, +  
 \frac{{\cal N}_f}{2} \,        | \langle  \ov{f} \ket{B_L} |^2 \nn
  & = & \frac{1+ \cos \phi}{2} \, 
        \Gamma ( B_s^{\textrm{\scriptsize even}} \to f) \, + \,
        \frac{1- \cos \phi}{2} \, 
        \Gamma ( B_s^{\textrm{\scriptsize odd}} \to f) , \nn
B(f) & = & \frac{{\cal N}_f}{2} \,
         | \langle f  \ket{B_H } |^2 \, + \, 
        \frac{{\cal N}_f}{2} \, | \langle \ov{f}  \ket{B_H } |^2 \nn
  & = & \frac{ 1- \cos \phi }{2} \, 
        \Gamma ( B_s^{\textrm{\scriptsize even}} \to f) \, + \, 
          \frac{ 1+ \cos \phi }{2} \, 
        \Gamma ( B_s^{\textrm{\scriptsize odd}} \to f) 
        .\label{chmix:abg}
\end{eqnarray}
%
With \eq{chmix:x1x} we arrive at
%
\begin{equation}
\frac{A(f)}{B(f)} =  
 \frac{ (1+ \cos \phi) 
        \Gamma ( B_s^{\textrm{\scriptsize even}} \to f) + 
        (1- \cos \phi) 
        \Gamma ( B_s^{\textrm{\scriptsize odd}} \to f)}{
        (1- \cos \phi) 
        \Gamma ( B_s^{\textrm{\scriptsize even}} \to f) + 
        (1+ \cos \phi) 
        \Gamma ( B_s^{\textrm{\scriptsize odd}} \to f) 
        } 
        \, = \, 
        \frac{1+(1- 2 x_f) \cos \phi}{1-(1- 2 x_f) \cos \phi}
\,.\label{chmix:ab}
\end{equation}
%
In \eq{chmix:abg} and \eq{chmix:ab} it is crucial that we average the decay
rates for $\Bsun \to f$ and the CP-conjugate process $\Bsun \to \ov{f}$. This
eliminates the interference term $\bra{B_s^{\textrm{\scriptsize odd}}} f\rangle
\bra{f} B_s^{\textrm{\scriptsize even}} \rangle $, so that $A(f)/B(f)$ only
depends on $x_f$.  The single-exponential fit with \eq{chmix:singex} determines
$\Gamma_f$. Equations \eq{chmix:fit} and \eq{chmix:ab} combine to give
%
\begin{equation}
2\, ( \Gamma_f - \Gamma_s ) =
        (1- 2 x_f ) \, \dg_s \, \cos \phi_s 
  \, = \, (1- 2 x_f ) \, \dg_{\rm CP}^s \, \cos^2 \phi_s
  \, = \, (1- 2 x_f ) \, \dg_{\rm CP}^{s\,\prime} \,,
 \label{chmix:dgmx} 
\end{equation}
%
up to corrections of order $(\dg_s)^2/\Gamma_s$. In order to determine $x_f$
from \eq{chmix:dgmx} we need $\dg_{\rm CP}^{s\,\prime}$ from the
lifetime measurement in a CP-specific final state like $D_s^+ D_s^-$%
\index{decay!$B_s \rightarrow D_s^+ D_s^-$} or from the angular separation of
the $CP$ components in $\Bsun \to \psi
\phi$.%
\index{decay!$B_s \rightarrow \psi \phi$} 
The corrections of order $(\dg_s)^2/\Gamma_s$ to \eq{chmix:dgmx} can be read off
from \eq{chmix:fit} with \eq{chmix:ab} as well.  Expressing the result in terms
of $\Gamma_f$ and the rate $\Gamma_{\rm fs}$ measured in flavor-specific decays,
we find
%
\begin{equation}
1- 2\, x_f \,=\, 2\, 
        \frac{\Gamma_f-\Gamma_{\rm fs}}{\dg_{\rm CP}^{s\,\prime}} 
        \, \lt[ 1 \, -  \,  
        2\, \frac{\Gamma_f-\Gamma_{\rm fs}}{\Gamma_s} \rt] \, 
        + {\cal O} \lt( \frac{(\dg_s)^2}{\Gamma_s^2} \rt)
        . \label{chmix:xfres} 
\end{equation} 
%
In order to solve for $\Gamma (B_s^{\textrm{\scriptsize even}} \to f) $ and
$\Gamma (B_s^{\textrm{\scriptsize odd}} \to f) $ we also need the branching
ratio $\brunts{f}+\brunts{\ov{f}}$. Recalling \eq{untnorm} one finds from
\eq{chmix:twoex3} and \eq{chmix:abg}:
%
\begin{eqnarray}
\brunts{f}+\brunts{\ov{f}} 
&=& \phantom{ + }
\Gamma (B_s^{\textrm{\scriptsize even}}  \to f)
  \lt[    \frac{1+ \cos \phi_s }{2 \Gamma_L}  + 
         \frac{1- \cos \phi_s }{2 \Gamma_H} \rt]  \nn
&&  + \, 
\Gamma (B_s^{\textrm{\scriptsize odd}}  \to f)
  \lt[    \frac{1- \cos \phi_s }{2 \Gamma_L}  + 
         \frac{1+ \cos \phi_s }{2 \Gamma_H} \rt] 
. \label{chmix:boe}
\end{eqnarray}
%
By combining \eq{chmix:x1x} and \eq{chmix:boe} we can solve for the two CP
components:
%
\begin{eqnarray}
\Gamma (B_s^{\textrm{\scriptsize even}} \to f)
&=& 
        \lt[ \Gamma_s^2- \lt( \dg_s/2 \rt)^2 \rt]
        \lt( \brunts{ f} + \brunts{\ov{f}} \rt)    
   \frac{1- x_f}{ 2 \Gamma_s \, - \, \Gamma_f } \nn
& = &   (1- x_f) \lt( \brunts{f} + \brunts{\ov{f}} \rt)
        \Gamma_s \,+\, {\cal O} \lt( \dg_s  \rt) \nn  
\Gamma (B_s^{\textrm{\scriptsize odd}}  \to f)
&=&  
        \lt[ \Gamma_s^2- \lt( \dg_s/2 \rt)^2 \rt]
        \lt( \brunts{f} + \brunts{\ov{f}} \rt)
   \frac{x_f}{ 2 \Gamma_s \, - \, \Gamma_f } \nn
& = &   x_f  \lt( \brunts{ f} + \brunts{\ov{f}} \rt)   \,
         \Gamma_s \,  
        +\, {\cal O} \lt( \dg_s \rt) .
\end{eqnarray}
%
From \eq{chmix:dgcp2} we now find the desired quantity by summing over all final
states $f$:
%
\begin{eqnarray} 
\dg_{\rm CP}^s & = & 
\Gamma \lt( B_s^{\textrm{\scriptsize even}} \rt) - 
        \Gamma \lt( B_s^{\textrm{\scriptsize odd}} \rt) 
\, = \,
        2 \lt[ \Gamma_s^2- \lt( \dg_s/2 \rt)^2 \rt]
\sum_{f \in X_{c\ov{c}}}  \brunts{ f}    \,
   \frac{1- 2\, x_f}{ 2 \Gamma_s \, - \, \Gamma_f} 
  \nn 
&=& 2\,  \Gamma_s \sum_{f \in X_{c\ov{c}}}  \brunts{ f} 
        \, 
        (1- 2\, x_f) \lt[ 1 \, + \, {\cal O} \lt( \frac{\dg_s}{\Gamma_s}
        \rt) \rt].
\label{chmix:dgcpres} 
\end{eqnarray}
%
\index{decay!$B_s \rightarrow D_s^+ D_s^-$}%
\index{decay!$B_s \rightarrow D_s^{*+} D_s^-$!$CP$-odd fraction}%
\index{decay!$B_s \rightarrow D_s^+ D_s^{*-}$!$CP$-odd fraction}%
\index{decay!$B_s \rightarrow D_s^{*+} D_s^{*-}$!$CP$-odd fraction}%
It is easy to find $\dg_{\rm CP}^s$: first determine $1-2x_f$ from
\eq{chmix:xfres} for each studied decay mode, then insert the result into
\eq{chmix:dgcpres}. The small quadratic term $( \dg_s/2 )^2=\dg_{\rm CP}^s
\dg_{\rm CP}^{s\,\prime}/4$ is negligible. 
This procedure can be performed
for $\brunts{D_s^{\pm} D_s^{*}{}^{\mp}}$ and $\brunts{D_s^{*}{}^+ D_s^{*}{}^-}$
to determine the corrections to the SV limit. In principle the $CP$-odd P-wave
component of $\brunts{D_s^{*}{}^+ D_s^{*}{}^-}$ (which vanishes in the SV limit)
could also be obtained by an angular analysis, but this is difficult in
first-generation experiments at hadron colliders, because the photon from
$D_s^* \to D_s \gamma$ cannot be detected.  We emphasize
that it is not necessary to separate the $D_s^{(*)}{}^+D_s^{(*)}{}^-$ final
states; our method can also be applied to the semi-inclusive $D_s^{(*)}{}^\pm
D_s^{(*)}{}^\mp $ sample, using $\dg_{\rm CP}^{s\,\prime}$ obtained from an
angular separation of the CP
components in $\Bsun \to \psi \phi$.%
\index{decay!$B_s \rightarrow \psi \phi$} Further one can successively include
those double-charm final states which vanish in the SV limit into
\eq{chmix:dgcpres}.  If we were able to reconstruct all $b \to c \ov{c} s$ final
states, we could determine $\dg_{\rm CP}^s$ without invoking the SV limit.  In
practice a portion of these final states will be missed, but the induced error
can be estimated from the corrections to the SV limit in the measured decay
modes. By comparing $\dg_{\rm CP}^s$ and $\dg_{\rm CP}^{s\,\prime}$ one finds
$|\cos \phi_s|$ from \eq{chmix:cp2}.  The irreducible theoretical error of
method 1 stems from the omission of CKM-suppressed decays and is of order $2
|V_{ub} V_{us}/(V_{cb} V_{cs})| \sim 3-5\%$.

Method 1 is experimentally simple: at the first stage (relying on the SV limit)
it amounts to counting the $\Bsun$ decays into $ D_s^{(*)}{}^+ D_s^{(*)}{}^-$.
The corrections to the SV limit are obtained by one-parameter fits to the time
evolution of the collected double-charm data samples.  This sample may include
final states from decay modes which vanish in the SV limit, such as
multiparticle final states.  No sensitivity to $(\dg_s \, t)^2$ is needed.  A
further advantage is that $\dg_{\rm CP}^s$ is not diminished by the presence of
new physics.
\index{B meson@$B$ meson!width difference $\dg$!$\dg_s$ and branching ratios}%
\index{width difference $\dg$!measurement of $\dg_s$!branching ratios}

\subparagraph{Method 2:} \index{untagged $B$!two-exponential decay} In the
Standard Model the decay into a $CP$ eigenstate $f_{\rm CP}$ is governed by a
single exponential. If a second exponential is found in the time evolution of a
CKM-favored decay $\Bsun \to f_{\rm CP}$, this will be clear evidence of new
physics \cite{dun}. To this end we must resolve the time evolution in \eq{guntf}
up to order $(\dg_s \, t)^2$. At first glance this seems to require a
three-parameter fit to the data, because \guntf\ in \eq{guntf} depends on
$\Gamma_s$, $\dg_s$ and (through $\adg{f}$, see \eq{acp2}) on $\phi_s$. It is
possible, however, to choose these parameters in such a way that one of them
enters \gunt{f_{\rm CP}}\ at order $(\dg_s )^3$, with negligible impact.  The
fit parameters are $\Gamma^\prime$ and $Y$.  They are chosen such that
%
\begin{equation} 
\guntfcpp \,=\,  2\, \brunts{f_{\rm CP+} } \,
        \Gamma^\prime e^{-\Gamma^\prime t}
        \lt[ 1 + Y \, \Gamma^\prime \, t \, 
        \lt( -1 
                +\frac{\Gamma^\prime t}{2} \rt)
                + {\cal O} \lt( (\dg_s )^3 \rt)
         \rt] 
        . \label{chmix:t2}
\end{equation}
%
Here we have considered a $CP$-even final state, for which a lot more data are
expected than for $CP$-odd states.  With \eq{chmix:t2} we have generalized the
lifetime fit method described in \eq{chmix:guntf3} -- \eq{chmix:gafs} to the
order $(\dg\, t)^2$.  A non-zero $Y$ signals the presence of new physics.  The
fitted rate $\Gamma^\prime$ and $Y$ are related to $\Gamma_s$, $\dg_s$ and
$\phi_s$ by
%
\begin{equation} 
Y \, = \, \frac{(\dg_s)^2}{4 \Gamma^{\prime 2}} \sin^2 \phi_s 
        ,  \qquad \qquad
\Gamma^\prime \,=\, \Gamma_s (1-Y) + \frac{\cos \phi_s}{2} \dg_s
      \label{chmix:defgpy} .
\end{equation} 
%
Note that for $|\cos \phi_s|=1$ the rate $\Gamma^\prime$ equals the rate of the
shorter-lived mass eigenstate and the expansion in \eq{chmix:t2} becomes the
exact single-exponential formula.  After determining $\Gamma^\prime$ and $Y$ we
can solve \eq{chmix:defgpy} for $\Gamma_s$, $\dg_s$ and $\phi_s$. To this end we
need the width $\Gamma_{\rm fs}$ measured in flavor-specific decays. We find
%
\begin{eqnarray}
|\dg_s| &\,=\,& 2 \sqrt{(\Gamma^\prime-\Gamma_{\rm fs})^2 +\Gamma_{\rm fs}^2} 
        \lt[ 1+ {\cal O} \lt( \frac{\dg_s}{\Gamma_s}  \rt) \rt], \nn
\Gamma_s\, &=& \, \Gamma_{\rm fs} + \frac{(\dg_s)^2}{2 \Gamma_s} + 
                {\cal O} \lt( \lt(\frac{\dg_s}{\Gamma_s}\rt)^3  \rt) \nn 
\dg_{\rm CP}^{s\,\prime} \, &=&\, 
  2 \lt[ \Gamma^\prime - \Gamma_s \lt( 1- Y \rt) \rt] 
        \lt[ 1 + {\cal O} \lt( \lt(\frac{\dg_s}{\Gamma_s}\rt)^2 \rt) \rt], \nn
|\sin \phi_s| \, &=& \, \frac{2\Gamma_s \sqrt{Y}}{|\dg_s|} \, 
        \lt[ 1+ {\cal O} \lt( \frac{\dg_s}{\Gamma_s}  \rt) \rt] . 
\label{chmix:gpyres}
\end{eqnarray} 
%
The quantity $\dg_{\rm CP}^{s\,\prime}$, which we could already determine from
single-exponential fits, is now found beyond the leading order in
$\dg_s/\Gamma_s$. By contrast, $\dg_s$ and $|\sin \phi_s|$ in \eq{chmix:gpyres}
are only determined to the first non-vanishing order in $\dg_s/\Gamma_s$.

In conclusion, method 2 involves a two-parameter fit and needs
sensitivity to the qua\-dratic term in the time evolution. The presence
of new physics can be invoked from $Y\neq 0$ and does not require to
combine lifetime measurements in different decay modes. 

\subparagraph{Method 3:} \index{untagged $B$!two-exponential decay} Originally
the following method has been proposed to determine $|\dg_s|$ \cite{dun,g}: The
time evolution of a $\Bsun$ decay into a flavor-specific final state is fitted
to two exponentials. This amounts to resolving the deviation of $\cosh (\dg_s\,
t/2)$ from 1 in \eq{guntf} in a two-parameter fit for $\Gamma_s$ and $|\dg_s|$.
If one adopts the same parameterization as in \eq{chmix:t2}, $\Gamma^\prime$ and
$Y$ are obtained from \eq{chmix:defgpy} by replacing $\phi_s$ with $\pi/2$.  The
best suited flavor-specific decay modes at hadron colliders are $\Bsun \to
D_s^{(*)\pm} \pi^\mp $, $\Bsun \to D_s^{(*)\pm} \pi^\mp
\pi^+ \pi^- $ and $\Bsun \to X \ell^{\mp} \nu$.%
\index{decay!$B_s \rightarrow D_s^- \pi^+$}%
\index{decay!$B_s \rightarrow D_s^- \pi^+ \pi^- \pi^+$}%
\index{decay!$B_s \rightarrow D_s^{*-} \pi^+$}%
\index{decay!$B_s \rightarrow D_s^{*-} \pi^+ \pi^- \pi^+$}%
\index{decay!$B_s \rightarrow X \ell^+ \nu$}%
Depending on the event rate in these modes, method 3 could be superior to method
2 in terms of statistics. On the other hand, to find the ``smoking gun'' of new
physics, the $|\dg_s|$ obtained must be compared to $\dg_{\rm CP}^{s\,\prime}$
from CP-specific decays to prove $|\cos \phi_s|\neq 1$ through \eq{chmix:cp3}.
Since the two measurements are differently affected by systematic errors, this
can be a difficult task. First upper bounds on $|\dg_s|$ using method 3 have
been obtained in \cite{semi}.

The L3 collaboration has determined an upper bound $|\dg_s|/\Gamma_s\leq 0.67$
by fitting the time evolution of fully inclusive decays to two exponentials
\cite{l3}. This method is quadratic in $\dg_s$ as well. The corresponding
formula for the time evolution can be simply obtained from \eq{chmix:twoex2}
with $A=\Gamma_L$ and $B=\Gamma_H$.

%...............................................................................
\boldmath
\subsubsection{Phenomenology of $\dg_d$}
\unboldmath
\label{chmix:subsub:width:phen:d}

\index{B meson@$B$ meson!width difference $\dg$!measurement of $\dg_d$}%
\index{width difference $\dg$!measurement of $\dg_d$} The Standard Model
value $\dg_d^{\rm SM}/\Gamma_d \approx 3\times 10^{-3}$ derived before
\eq{chmix:dgdsm} is presumably too small to be measured from lifetime
fits. In extension of the Standard Model, however, $\dg_d/\Gamma_d$
can be large, up to a few percent. The expected high statistics for
the decay $B_d \to \psi K_S$ can be used to measure the lifetime
$1/\Gamma_{B_d\to \psi K_S}$ in this channel with 
\index{decay!$B_d \to \psi K_S$}
%
\begin{equation}    
\Gamma_{B_d \to \psi K_S} =
                \Gamma_d - \frac{\dg^d}{2} \cos (2 \beta_{\psi K_S})
\, = \,  \Gamma_d - 
        |\Gamma_{12}^d | \cos (2 \beta_{\psi K_S})   
        \cos  \phi_d 
        .\label{chmix:dgp:d}
\end{equation}
%
$\sin (2 \beta_{\psi K_S})$ is the quantity characterizing the
mixing-induced $CP$ asymmetry measured from tagged $B_d \to \psi K_S$
decays.  $\Gamma_d$ is obtained from a lifetime measurements in
flavor-specific decay modes.  We stress that this measurement of
$\Gamma_{B_d\to \psi K_S}$ can be done from the \emph{untagged}\ 
$\Bdun \to \psi K_S$ data sample. If $\Gamma_{12}^d$ is dominated by
new physics, its phase and therefore also $\phi_d$ is unknown.  If one
neglects the small SM contribution in \eq{chmix:phinum} to $\phi_d$,
\eq{chmix:dgp:d} reads 
\begin{equation}
\Gamma_{B_d \to \psi K_S} \simeq
   \Gamma_d - |\Gamma_{12}^d | \,\cos (2 \beta_{\psi K_S}) 
              \cos (2 \beta_{\psi K_S} - 2 \beta)  \, = \, 
   \Gamma_d - |\Gamma_{12}^d | \cos (\phi_d + 2 \beta) 
              \cos \phi_d  , 
\end{equation} 
where $\beta$ is the true angle of the unitarity triangle as defined
in \eq{1:eq:beta}. Note that in the presence of new physics $\beta$ is
unknown.  When combined with the $CP$ asymmetry in flavor-specific
decays discussed in Sect.~\ref{chmix:sub:afs} one can determine
$|\Gamma_{12}^d|$ and $\sin \phi_d$.  Then up to discrete ambiguities
also $\beta=\beta_{\psi K_S}-\phi_d/2$ can be determined.  Depending
on whether the enhancement of $\Gamma_{12}^d$ is due to $b\to \ov{c}c
d$, $b\to \ov{u}c d$ or $b\to \ov{u}u d$, transitions, $CP$ asymmetries
in these channels can also help to disentangle $|\Gamma_{12}^d|$ and
$\phi_d$.

Interestingly, one can isolate the contribution to 
$|\Gamma_{12}^d| $ from $b\to
\ov{c} c d$ decays. Define
%
\begin{equation}
\dg_{\rm CP}^{d,cc} \equiv 2\, | \xi_c^{d\,2} \Gamma_{12}^{d,cc}| \, = \, 
2 \sum_{f  \in X_{c\ov{c}}} \lt[
\Gamma ( B_d \to f_{\rm CP+} ) \, - \,
        \Gamma ( B_d \to f_{\rm CP-}) \rt]
        . 
        \label{chmix:dgcp:d}
\end{equation}
%
in analogy to \eq{chmix:dgcp}. In the Standard Model $\dg_{\rm CP}^{d,cc} $ is
slightly larger than $2 |\Gamma_{12}^d|$, by a factor of $1/R_t^2$.  From
\eq{chmix:dgdsm} one finds
%
\begin{equation} 
\frac{\dg_{\rm CP}^{d,cc}}{\dg^s_{\rm CP}} \simeq 
  \frac{f_{B_d}^2 B_{B_d}^S}{f_{B_s}^2 B_{B_s}^S} 
              \, \lt| \frac{\xi_c^d}{\xi_t^s} \rt|^2 \, \simeq \,
0.04, \label{chmix:dgcc}
\end{equation} 
%
i.e.\ $\dg_{\rm CP}^{d,cc}/\Gamma_d \simeq 5\times 10^{-3}$. Now
$\dg_{\rm CP}^{d,cc} $ can be measured by counting the $CP$-even and
$CP$-odd final states in $b\to c \ov{c} d$ decays, just as described in
Sect.~\ref{chmix:subsub:width:phen} for $b\to c \ov{c} s$ decays of
$B_s$ mesons. Again, in the SV limit the inclusive decay $\Bdun \to
X_{c\ov{c}d}$ is exhausted by $\Bdun \to D^{(*)+} D^{(*)-}$, which is
purely $CP$-even in this limit.  With \eq{chmix:dgcpres} one can find
$\dg_{\rm CP}^{d,cc}$.  That is, in the SV limit one just has to
measure $Br \big(\Bdun \to D^{(*)+} D^{(*)-}\big)$, which equals
$\dg_{\rm CP}^{d,cc}/(2 \Gamma_d)$.  However, the lifetime method
described in `Method 1' above cannot be used to determine the
corrections to the SV limit, because $\dg_d$ is too small. Yet in the
limit of exact U-spin symmetry ($m_d=m_s$) the $CP$-odd components of
$D^{(*)+} D^{(*)-}$ from $B_d$ decay and $D_s^{(*)+} D_s^{(*)-}$ from
$B_s$ decay are the same.  Finally in $b\to c \ov{c} d$ transitions CP
violation in decay could be relevant. It results from penguin loops
involving top- or up-quarks and spoils the relation \eq{chmix:dgcpres}
between branching ratios and $\dg_{\rm CP}^{d,cc} $.  This effect,
however, is calculable for inclusive decays like $\Bdun \to
X_{c\ov{c}d}$. In the Standard Model $CP$ violation in this inclusive
decay is of order $1\%$ and therefore negligible \cite{lno}.
CP-violation from non-standard sources can be revealed by comparing
CP-asymmetries in $b\to c \ov{c} d$ decays with those in $b\to c\ov{c}
s$ decays (namely $\sin 2 \beta$ from $B_d\to \psi K_S$).  Since
\eq{chmix:dgcc} depends on no CKM elements and the hadronic factor is
known exactly in the $SU(3)_F$ limit, a combined measurement of
$\dg_{\rm CP}^{d,cc}$ and $\dg_{\rm CP}^s$ provides an excellent probe
of new physics in $b\to c \ov{c} d$ transitions.

%-------------------------------------------------------------------------------
\subsection[$CP$ Asymmetry in Flavor-specific Decays]
{\boldmath $CP$ Asymmetry in Flavor-specific Decays}
\label{chmix:sub:afs}

\index{CP asymmetry@\CP\ asymmetry!in flavor-specific decay} In the preceding
sections we have set the small parameter $a_q\equiv a(B_q)$, $q=d,s$, defined in
\eq{defepsg} to zero. In order to study $CP$ violation in mixing we must keep
terms of order $a_q$. The corresponding ``wrong-sign'' $CP$ asymmetry is measured
in flavor-specific decays $B_q \to f$ and equals
%
\begin{equation}
a^q_{\rm fs} 
     = \frac{\gqbtf - \gqtfb}{\gqbtf + \gqtfb} 
     = \imag \frac{\Gamma_{12}^q}{M_{12}^q} = a_q \,,
        \qquad  \mbox{for~ }  \ov{A}_f  =  0 
         \mbox{ ~and~ } |A_f| = |\ov{A}_{\ov{f}}|
 . \label{chmix:defafs}
\end{equation}
%
\index{CP asymmetry@\CP\ asymmetry!semileptonic}%
A special case of $a^q_{\rm fs}$ is the semileptonic asymmetry, where $f=X
\ell^+ \nu$, introduced in Sect.~\ref{ch1:sect:ty}.  A determination of $a_q$
gives additional information on the three rephasing-invariant quantities
$|M_{12}^q|$, $|\Gamma_{12}^q|$ and $\phi_q$ characterizing \bbm.

Observe that $a^q_{\rm fs}$ in \eq{chmix:defafs} is time-independent. While both
numerator and denominator depend on $t$, this dependence drops out from the
ratio.  The ``right-sign'' asymmetry, vanishes:
%
\begin{equation}
    \gqtf - \gqbtfb \, = \, 0\,,
        \qquad \mbox{for~ } \ov{A}_f  =  0 
        \mbox{ ~and~ }  |A_f| = |\ov{A}_{\ov{f}}|
 . \label{chmix:unm}
\end{equation}
%
This implies that one can measure $a^q_{\rm fs}$ from \emph{untagged} decays
\cite{y,dfn}. It is easily verified from the sum of \eq{gtfres} and \eq{gbtfres}
that to order $a_q$ the time evolution of untagged decays exhibits oscillations
governed by $\dm_q$.  Since $a$ is small, a small production asymmetry $\epsilon
= N_{\ov{B}}/N_B -1$, which also leads to oscillations in the untagged sample,
could introduce an experimental bias. To first order in the small parameters
$a_q$, and $\epsilon$ one finds
%
\begin{equation}
a_{\rm fs}^{q,unt} =
   \frac{\guntf - \guntfb}{\guntf + \guntfb}  =
        \frac{a_q}{2} - \frac{a_q+\epsilon}{2} \,  
        \frac{\cos (\dm_q\, t)}{\cosh (\dg_q t/2) }\,,
        \qquad \mbox{for~ } \ov{A}_f  =  0 
        \mbox{ ~and~ } |A_f| = |\ov{A}_{\ov{f}}| 
        . \,  \label{chmix:fsun} 
\end{equation}
%
Note that the production asymmetry between $B_q^0$ and $\ov{B}{}_q^0$ cannot
completely fake the effect of a non-zero $a_q$ in \eq{chmix:fsun}: while both
$a_q\neq 0$ and $\epsilon\neq 0$ lead to oscillations, the offset from the
constant term indicates $a_q\neq 0$.

The Standard Model predictions for $a_d$ and $a_s$ are 
\index{CP asymmetry@\CP\ asymmetry!in flavor-specific decay!SM prediction}%
\begin{eqnarray}
a_d & \approx & - \frac{\ov{\eta}}{R_t^2} 
   \frac{4 \pi \, \lt( K_1+K_2 \rt) m_c^2}{M_W^2\, \eta_B b_B  
  S (m_t^2/M_W^2) } \, \approx \, - 8 \times 10^{-4} \nn 
a_s & \approx & \ov{\eta} \lambda^2 \, 
   \frac{4 \pi \, \lt( K_1+K_2 \rt) m_c^2}{M_W^2\, \eta_B b_B  
  S (m_t^2/M_W^2) } \, \approx \, 5 \times 10^{-5}. \label{chmix:aqnum}
\end{eqnarray}
%
The huge GIM suppression factor $m_b^2/M_W^2 \sin \phi_d \propto m_c^2/M_W^2$
leads to these tiny predictions for $CP$ violation in mixing.  $a_d$ plays a
preeminent role in the search for new physics :
%
\begin{itemize}
\item its sensitivity to new physics is enormous, it can be enhanced
  by two orders of magnitude, 
\item it is affected by a wide range of possible new physics effects: 
      new $CP$ violating effects in $\phi_d$ relax the GIM-suppression
      $ \propto m_b^2/M_W^2 \sin \phi_d$, because $\phi_d$ is no more 
      proportional to $z=m_c^2/m_b^2$. New physics contributions to 
      \emph{any}\ of the CKM-suppressed decay modes $b\to \ov{c}c d$, $b\to
      \ov{u}c d$ or $b\to \ov{u}u d$ can significantly enhance 
      $|\Gamma_{12}^d|$ and thereby $a_d$. 
\end{itemize}
%
Of course new physics contributions to $\arg M_{12}^d$ will not only affect
$\phi_d$, but also the $CP$ asymmetry in $B_d^0 \to \psi K_S$.\index{decay!$B_d
  \rightarrow \psi K_S$} But from this measurement alone one cannot extract the
new physics contribution, because one will know the true value of $\beta= \arg
(-\xi_{t}^{d*}/\xi_{c}^{d*})$ only poorly, once new physics affects the standard
analysis of the unitarity triangle. For the discussion of new physics it helps
to write
%
\begin{equation}
a_q  \,=\, \frac{2 |\Gamma_{12}^q|}{\dm_q} \, \sin \phi_q  
\, = \,
\frac{|\dg_q|}{\dm_q} \, \frac{\sin \phi_q}{|\cos \phi_q|}. 
\label{chmix:aqnp}
\end{equation}
%
If both $|\dg_d|$ and $a_d$ are measured, one can determine both
$|\Gamma_{12}^d|$ and $\sin \phi_d$.

$a_s$ is less interesting than $a_d$, because $\Gamma_{12}^s$ stems from
CKM-favored decays and is not very sensitive to new physics.  The ratio
$\dg_{\rm CP}^s/\Gamma_s \leq 0.2$ from \eq{chmix:dgnum2} and the current
experimental limit $\dm_s \geq 14.9\,$ps${}^{-1}$ \cite{os} imply that $|a_s|
\leq 0.01$. New physics can affect $\phi_s$ only through $\arg M_{12}^s$, but
this new physics can be detected most easily through $CP$ asymmetries in $B_s\to
\psi \phi$ or
\index{decay!$B_s \rightarrow \psi \phi$}
\index{decay!$B_s \rightarrow D_s^+ D_s^-$}
\index{decay!$B_s \rightarrow D_s^{*+} D_s^-$!$CP$-odd fraction}
\index{decay!$B_s \rightarrow D_s^+ D_s^{*-}$!$CP$-odd fraction}
\index{decay!$B_s \rightarrow D_s^{*+} D_s^{*-}$!$CP$-odd fraction}
$B_s \to D_s^{(*)+} D_s^{(*)-}$ decays. Since the Standard Model predictions for
these asymmetries are essentially zero, there is no problem here to disentangle
standard from non-standard physics.  Note that the measurement of $\mbox{sgn}\,
\sin \phi_s$ reduces the four-fold ambiguity in $\phi_s$ from the measurement of
$|\cos \phi_s| $ to a two-fold one.  The unambiguous determination of $\phi_s$ is
discussed in detail in \cite{dfn}.  \index{discrete ambiguity of $CP$ phase}


%-------------------------------------------------------------------------------
\subsection[Angular analysis to separate the $CP$ components]{\boldmath Angular analysis to
separate the $CP$ components$\!$ \authorfootnote{Author: Amol Dighe}}

\subsection[$CP$-odd and $CP$-even components in $B_s \to J/\psi\phi$]
{\boldmath $CP$-odd and $CP$-even components in $B_s \to J/\psi \phi$} 
\label{chmix:twocp}

The most general amplitude for $B_s \to J/\psi \phi$ can be written in terms of
the polarization states of the two vector mesons as \cite{rosner,ddlr}
%
\beq
A(B_s(t) \to J/\psi\, \phi) = \frac{A_0(t)}{x}\, {\bep}^{*L}_{J/\psi}\,
{\bep}^{*L}_{\phi} - A_{\|}(t)\, {\bep}^{*T}_{J/\psi} \cdot {\bep}^{*T}_{\phi} /
\sqrt{2} - i A_{\perp}(t)\, {\bep}^*_{J/\psi} \times {\bep}^*_{\phi} \cdot
\hat{\bf p}_{\phi} / \sqrt{2}\,,
\label{chmix:ampl}
\eeq
%
where $x\equiv p_{J/\psi}\cdot p_{\phi}/(m_{J/\psi} m_{\phi})$ and $\hat{\bf
  p}_{\phi}$ is the unit vector along the direction of motion of $\phi$ in the
rest frame of $J/\psi$.

Since the ``$CP$ violation in decay'' of $B_s \to J/\psi \phi$ is vanishing,
%
\beq
\overline{A}_0(0) = A_0(0) \,, \qquad
\overline{A}_{\|}(0) = A_{\|}(0) \,,\qquad
\overline{A}_{\perp}(0) = -A_{\perp}(0)\,.
\label{chmix:cpstates}
\eeq
%
The final state is thus an admixture of different $CP$ eigenstates: $A_0$ and
$A_\|$ are $CP$-even amplitudes whereas $A_\perp$ is $CP$-odd. The decay rate is
given by \beq \Gamma (t) \propto |A_0(t)|^2 + |A_{\|}(t)|^2 +
|A_{\perp}(t)|^2\,,
\label{chmix:no-angle}
\eeq
where the time evolutions of the individual terms are
\cite{ddf1} 
\beqa
|A_{0, \|}(t)|^2 & =  & |A_{0, \|}(0)|^2 \left[e^{-\Gamma_L t} -
e^{-\overline{\Gamma}t}
\sin(\Delta m_s t)\delta\phi\right], \nonumber \\
%|A_{\|}(t)|^2 & =  & |A_{\|}(0)|^2 \left[e^{-\Gamma_L t} -
%e^{-\overline{\Gamma}t}
%\sin(\Delta m_s t)\delta\phi\right] \nonumber \\
|A_{\perp}(t)|^2 & =  & |A_{\perp}(0)|^2 \left[e^{-\Gamma_H t} +
e^{-\overline{\Gamma}t}
\sin(\Delta m_s t)\delta\phi\right] \,.
\label{chmix:time-one}
\eeqa
Here, $\bar{\Gamma} \equiv \Gamma_s = (\Gamma_L + \Gamma_H) /2$. Note that
this is {\it not} the average lifetime of $B_s$ as measured
through its semileptonic decays \cite{paulini}.

The value of 
\beq
\delta \phi \equiv 2 \beta_s \approx 2 \lambda^2 \eta \approx 0.03
\label{chmix:deltaphi}
\eeq is small in the standard model\footnote{Generalizations of the
  formulae to the case of new physics can be found in \cite{dfn}.}, so
that the terms proportional to $\delta \phi$ in (\ref{chmix:time-one})
can be neglected in the first approximation.  The time evolution of
(\ref{chmix:no-angle}) is then a sum of two exponential decays with
lifetimes $1/\Gamma_H$ and $1/\Gamma_L$.

In principle, a fit to the time dependence of
the total decay rate (\ref{chmix:no-angle})
can give the values of $\Gamma_H$ and $\Gamma_L$ separately,
but $\Delta \Gamma_s / \bar{\Gamma}$ is expected to be less than 20\%,
and it is not easy to separate two closely spaced lifetimes.
The inclusion of angular 
information will increase the accuracy in the measurement of 
$\Delta \Gamma_s$ multi-fold, as we'll see in the section
\ref{chmix:transversity} below. 


%-------------------------------------------------------------------------------
\subsection{The transversity angle distribution}
\label{chmix:transversity}

Since there are four particles in the final state,
the directions of their momenta can define three 
independent physical angles. Our
convention for the definitions of angles \cite{ddlr,ddf1}
is as shown in Fig.~\ref{chmix:ang-conv}.
The $x$-axis is the direction of $\phi$ in  
the $J/\psi$ rest frame, the $z$  axis is perpendicular to 
the decay plane of $\phi \to K^+ K^-$, and $p_y(K^+)
\geq 0$. The coordinates $(\theta, \varphi)$ describe the 
direction of $l^+$ in the $J/ \psi$ rest frame and $\psi$ is 
the angle made by $\vec p(K^+)$ with the $x$ axis in the $\phi$ 
rest frame. With this convention,
\beqa
& {\bf x} = {\bf p}_{\phi} , \qquad
{\bf y} = \displaystyle \frac{ {\bf p}_{K^+} - {\bf p}_{\phi} ( {\bf p}_{\phi}
	\cdot {\bf p}_{K^+} ) }
	{ | {\bf p}_{K^+} - {\bf p}_{\phi} ( {\bf p}_{\phi}
	\cdot {\bf p}_{K^+} ) | } , \qquad
{\bf z} = {\bf x} \times {\bf y}, & \nonumber \\[4pt]
&\sin \theta \cos \varphi =   {\bf p}_{\ell^+} \cdot  {\bf x}, \qquad
\sin \theta \sin \varphi = {\bf p}_{\ell^+} \cdot  {\bf y}, \qquad
\cos \theta =  {\bf p}_{\ell^+} \cdot {\bf z}\,.&
\label{chmix:angdef}
\eeqa
Here, the bold-face characters represent {\it  unit} 3-vectors and
everything is measured in the rest frame of $J/\psi$.
Also
\beq
\cos \psi = - {\bf p}'_{K^+}
                        \cdot {\bf p}'_{J/\psi},
\label{chmix:psidef}
\eeq
where the primed quantities are {\it  unit vectors}
measured in the rest frame of $\phi$.

\begin{figure}[t]
\begin{center}
\epsfig{file=ch8/th_angles.eps,height=3.0in}
\caption{The definitions of angles $\theta, \varphi, \psi$.
Here $\theta$ is the ``transversity'' angle.}
\label{chmix:ang-conv}
\end{center}
\end{figure}

The $\theta$ defined here is the {\it transversity} angle
\cite{dqstl}, which separates out the $CP$-even and
$CP$-odd components. The angular distribution in terms of
$\theta$ is given by \cite{ddlr}:
\beq
\frac{d \Gamma (t)}{d \cos \theta} \propto
   (|A_0(t)|^2 + |A_{\|}(t)|^2)\, \frac{3}{8}\, (1 + \cos ^2 \theta)
                + |A_{\perp}(t)|^2\, \frac{3}{4} \sin^2 \theta \, ,
\label{chmix:one-angle}
\eeq
where the time evolutions of the terms are as given in
(\ref{chmix:time-one}).

The $CP$-even and $CP$-odd components are now separated by not only
their different lifetimes (which are very close) but also by their
decay angular distributions (which are distinctly different). The
study of information content about the value of $\Delta \Gamma_s$ in
the time and angular measurements \cite{sen} suggests that, in order
to get the same degree of accuracy in $\Delta \Gamma_s$ with only time
measurements, one would need about two orders of magnitude more number
of events than if both the time and angular measurements were used
(see Fig.~3 in \cite{sen}).  This indicates that the strategy of
selecting one decay mode ({\it e.g.} $J/\psi \phi$) and studying its
angular distribution will turn out to be more fruitful than trying to
combine all $CP$ eigenstate decay modes and determine $\Gamma_H$ and
$\Gamma_L$ solely from their time evolutions.  Note that, in the
limiting case of $\Gamma_H = \Gamma_L$, the time evolution by itself
{\it cannot} separate the $CP$ even and odd components, whereas the
angular measurements can.

A fit to the transversity angle distribution (\ref{chmix:one-angle}) with its
complete time evolution (\ref{chmix:time-one}) also gives the value of $\delta
\phi$ and $\Delta m_s$, though a better measurement of the latter may be obtained
through other decay channels.

The transversity angle distribution (\ref{chmix:one-angle}) is valid for any
$B_s \to J/\psi(\to \ell^+ \ell^-) C_1 C_2$ decay, where $C_1$ and $C_2$ are (a)
self-conjugate particles, or (b) scalars and $CP$ conjugates of each other
\cite{dqstl}.  The particles $C_1$ and $C_2$ need not be the products of any
resonance, and their total angular momentum is irrelevant. So the time and
transversity angle measurements from all the resonant and non-resonant decays of
this form may be combined to gain statistics. Here the values of $(|A_0(0)|^2 +
|A_\|(0)|^2)$ and $|A_\perp(0)|^2$ are just some {\it effective} average values,
but the decay widths $\Gamma_H$ and $\Gamma_L$ are the same for all such decay
modes, and hence for the whole data sample.
 

%-------------------------------------------------------------------------------
\subsection{Three-angle distribution in
$B_s\to J/\psi(\to l^+l^-)\,\phi(\to K^+K^-)$}

While the one-angle distribution (\ref{chmix:one-angle}) is in principle
sufficient to determine the values of $\Gamma_H$, $\Gamma_L$, $\delta \phi$ and
$\Delta m_s$, using the information present in all the angles $(\theta, \phi,
\psi)$ will improve the measurements. In addition, one also gets access to the
magnitudes of all the three amplitudes $A_0(0), A_\|(0), A_\perp(0)$ and the
strong phases between them, which was not possible with the one-angle
distribution \cite{ddf1}. A method to combine the three-angle distributions of
$B_s \to J/\psi \phi$ and $B_d \to J/\psi K^*$ to resolve a discrete ambiguity
in the CKM angle $\beta$ has also been proposed \cite{ddf2}.


The three angle distribution of an initially present (i.e.\ tagged) $B_s$ meson
is \cite{ddlr,ddf1}
\beqa
&& \frac{d^3 \Gamma [B_s(t) \to J/\psi (\to l^+ l^-) \phi (\to K^+K^-)]}
{d \cos \theta\, d \varphi\, d \cos \psi}
\propto \frac{9}{32 \pi} \bigg\{ 2 |A_0(t)|^2 \cos^2 \psi (1 - \sin^2
\theta
\cos^2 \varphi) \nonumber\\
&&{} + \sin^2 \psi\, \Big[ |A_\parallel(t)|^2  (1 - \sin^2 \theta \sin^2
\varphi)
+ |A_\perp(t)|^2 \sin^2 \theta - \mbox{Im}\,(A_\parallel^*(t)
A_\perp(t))
\sin 2 \theta \sin \varphi \Big] \nonumber\\
&&{} +\frac{1}{\sqrt{2}}\,\sin2\psi\, \Big[\mbox{Re}\,(A_0^*(t) A_\parallel(t))
\sin^2 \theta \sin2\varphi 
+ \mbox{Im}\,(A_0^*(t)A_\perp(t))\sin2\theta\cos\varphi \Big] \bigg\}\,.
\label{chmix:three-angle}
\eeqa 
Note that the same angular distribution (\ref{chmix:three-angle}) is also
valid for $B_d \to J/\psi(\to \ell^+ \ell^-) K^*(\to K^\pm \pi^\mp)$.  The
angular distribution for the $CP$ conjugate decay is obtained simply by
replacing all $A$'s by $\bar{A}$'s \cite{ddf1}.


\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|l|} \hline
Observable & Time evolution \\
\hline
$|A_0(t)|^2$  & $|A_0(0)|^2 \left[e^{-\Gamma_L t} -
e^{-\overline{\Gamma}t}
\sin(\Delta m_s t)\delta\phi\right]$\\
$|A_{\|}(t)|^2$ &$ |A_{\|}(0)|^2 \left[e^{-\Gamma_L t} -
e^{-\overline{\Gamma}t}\sin(\Delta m_s t)\delta\phi\right]$\\
$|A_{\perp}(t)|^2$ & $|A_{\perp}(0)|^2 \left[e^{-\Gamma_H t} +
e^{-\overline{\Gamma}t}\sin(\Delta m_s t)\delta\phi\right]$\\
\hline
Re$(A_0^*(t) A_{\|}(t))$ &  $|A_0(0)||A_{\|}(0)|\cos(\delta_2 -
\delta_1)\left[e^{-\Gamma_L t} - e^{-\overline{\Gamma}t}
\sin(\Delta m_s t)\delta\phi\right]$\\
Im$(A_{\|}^*(t)A_{\perp}(t))$ & $|A_{\|}(0)||A_{\perp}(0)|\left[
e^{-\overline{\Gamma}t}\sin (\delta_1-\Delta m_s t)+\frac{1}{2}\left(
e^{-\Gamma_H t}-e^{-\Gamma_L
t}\right)\cos(\delta_1)\delta\phi\right]$\\
Im$(A_0^*(t)A_{\perp}(t))$ & $|A_0(0)||A_{\perp}(0)|\left[
e^{-\overline{\Gamma}t}\sin (\delta_2-\Delta m_s t)+\frac{1}{2}\left(
e^{-\Gamma_H t}-e^{-\Gamma_L
t}\right)\cos(\delta_2)\delta\phi\right]$\\
\hline
\end{tabular}
\end{center}
\caption{Time evolution of the decay $B_s\to J/\psi(\to l^+l^-)
\phi(\to K^+K^-)$ of an initially (i.e.\ at $t=0$) pure $B_s$ meson.
Here $\bar{\Gamma} \equiv (\Gamma_H + \Gamma_L)/2$.}
\label{chmix:timetable_bs}
\end{table}


\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|l|}
\hline
Observable & Time evolution \\  \hline
$|\overline{A}_0(t)|^2$  & $|A_0(0)|^2 \left[e^{-\Gamma_L t} +
e^{-\overline{\Gamma}t}\sin(\Delta m_s t)\delta\phi\right]$\\
$|\overline{A}_{\|}(t)|^2$ &$ |A_{\|}(0)|^2 \left[e^{-\Gamma_L t} +
e^{-\overline{\Gamma}t}\sin(\Delta m_s t)\delta\phi\right]$\\
$|\overline{A}_{\perp}(t)|^2$ & $|A_{\perp}(0)|^2 \left[e^{-\Gamma_H
t} -
e^{-\overline{\Gamma}t}\sin(\Delta m_s t)\delta\phi\right]$\\
\hline
Re$(\overline{A}_0^*(t)\overline{A}_{\|}(t))$ &
$|A_0(0)||A_{\|}(0)|\cos(\delta_2 -
\delta_1)\left[e^{-\Gamma_L t} + e^{-\overline{\Gamma}t}
\sin(\Delta m_s t)\delta\phi\right]$\\
Im$(\overline{A}_{\|}^*(t)\overline{A}_{\perp}(t))$ &
$-|A_{\|}(0)||A_{\perp}(0)|\left[
e^{-\overline{\Gamma}t}\sin (\delta_1-\Delta m_s t)-\frac{1}{2}\left(
e^{-\Gamma_H t}-e^{-\Gamma_L
t}\right)\cos(\delta_1)\delta\phi\right]$\\
Im$(\overline{A}_0^*(t)\overline{A}_{\perp}(t))$ &
$-|A_0(0)||A_{\perp}(0)|\left[
e^{-\overline{\Gamma}t}\sin (\delta_2-\Delta m_s t)-\frac{1}{2}\left(
e^{-\Gamma_H t}-e^{-\Gamma_L
t}\right)\cos(\delta_2)\delta\phi\right]$\\
\hline
\end{tabular}
\end{center}
\caption{Time evolution of the decay $\overline{B_s}\to J/\psi
(\to l^+ l^-)\phi(\to K^+ K^-)$
of an initially (i.e.\ at $t=0$) pure $\overline{B_s}$ meson.
Here $\bar{\Gamma} \equiv (\Gamma_H + \Gamma_L)/2$.}
\label{chmix:timetable_bsbar}
\end{table}


The time evolution of the observables in the angular distribution
(the coefficients of the angular terms in (\ref{chmix:three-angle})
and its $CP$ conjugate mode)
are given in Table \ref{chmix:timetable_bs} and
\ref{chmix:timetable_bsbar} respectively. 
Here $\delta_1 \equiv {\rm Arg}(A_\perp^* A_\|)$ and
$\delta_2 \equiv {\rm Arg}(A_0^* A_\|)$.
A finite lifetime difference $\Delta \Gamma$ implies that
the $CP$ violating terms proportional to
\beq
\left(e^{-\Gamma_H t} - e^{-\Gamma_L t}\right) \cos(\delta_{1(2)})\,
\delta \phi
\label{chmix:untagged}
\eeq
survive even when the $B_s$ is untagged \cite{df,ddf1}.
An experimental feasibility study for extracting the parameters 
from the time dependent three angle distribution has been performed
for the LHC in \cite{lhcreport}.

%-------------------------------------------------------------------------------
\subsection{Angular moments method}

The likelihood fit to the complete angular distribution
(\ref{chmix:three-angle}) -- including the time evolution of
the observables (Tables \ref{chmix:timetable_bs} and
\ref{chmix:timetable_bsbar}) -- is a difficult task due to the
large number of parameters involved. The method of angular
moments proposed in \cite{ddf1} can disentangle
the angular dependences and split up the likelihood fit
into a number of likelihood fits with a smaller number of
parameters.

The angular distributions ((\ref{chmix:one-angle}) or
(\ref{chmix:three-angle})) are of the form
\beq
f( \Theta, {\cal P}; t) = \sum b^{(k)} ( {\cal P}; t)\, g^{(k)}
        (\Theta) \,,
\label{chmix:f=bg}
\eeq
where ${\cal P}$ represents the parameters, 
and $\Theta$ denotes the angles.
If we can find {\it weighting functions} $w^{(i)}$ such that
\beq
\int [{\cal D} \Theta]\, w^{(i)}(\Theta)\, g^{(k)} (\Theta) = 
\delta_{ik}\,,
\label{chmix:wg=delta}
\eeq
then
\beq
b^{(i)} \approx  \sum_{events} w^{(i)}(\Theta)\,, 
\label{chmix:b=sum-w}
\eeq
and the observables are determined directly from the data.
It can be shown \cite{ddf1} that such a set of weighting
functions exists for any angular distribution of the
form (\ref{chmix:f=bg}) and such a set can be determined
without any {\it a priori} knowledge of the values of
the observables $b^{(i)}$. A likelihood fit can then be
performed on each observable $b^{(i)}$ independently in
order to determine the parameters.

The angular moments (AM) method is more transparent and easier to
implement than the complete likelihood fit method. 
Although the AM method in its naive form involves some
loss of information, the extent of this loss of information
in the case of the transversity angle distribution has been
found to be less than 10\% in the parameter range of interest
(see Fig.~5 of \cite{sen}). In its full form, the
AM method can determine the values of parameters almost as well as
the likelihood fit method (see, {\it e.g.} \cite{cleo-drho}).

To conclude, the angular analysis of $B_s \to J/\psi \phi$ decays
can separate the $CP$ even and odd components in the final state, 
and it is perhaps the best way to determine the lifetime difference 
between $B_s^H$ and $B_s^L$. As a byproduct, it also helps  the
measurement of $CP$ odd and even components and their relative 
strong phases, and with enough statistics, the determination of
$\Delta m_s$ and $\delta \phi$. The angular analysis, possibly
employing the angular moments method if the likelihood fit
is inadequate, is highly recommended.


%-------------------------------------------------------------------------------
\boldmath
\subsection[\ddm]{\ddm$\!$ \authorfootnote{Author: Ulrich~Nierste}}
\label{chmix:sec:d}
\unboldmath
\index{D mixing@$D$ mixing}%
\index{D meson@$D$ meson!mass difference $\dm_D$}%
\index{D meson@$D$ meson!width difference $\dg_D$}%
We define the mass eigenstates in \ddm\ as 
\begin{eqnarray}
\ket{D_1} &=& 
        p \ket{D^0} + q \ket{\ov{D}{}^0} \,,  \nn
\ket{D_2} &=& 
        p \ket{D^0} - q \ket{\ov{D}{}^0} \,,
\qquad \mbox{with } \lt|p\rt|^2+\lt|q\rt|^2 = 1\,. \label{chmix:d:defpq}
\end{eqnarray}
$q$ and $p$ are obtained from the solution of the eigenvalue problem
for $M - i \Gamma/2$. In \eq{mgqp:c} $q/p$ is determined in terms of 
$M_{12}$ and $\Gamma_{12}$. We define 
\begin{eqnarray}
\dm_D &=& m_2-m_1, \qquad\qquad 
\dg_D \, = \, \Gamma_1-\Gamma_2 \label{chmix:d:dmdg} \\
x_D &=& \frac{\dm_D}{\Gamma} \qquad\qquad
y_D \, = \, - \frac{\dg_D}{2\Gamma} 
\, = \, \frac{\Gamma_2-\Gamma_1}{2\Gamma} \label{chmix:d:xy} .
\end{eqnarray}
The definitions in \eq{chmix:d:defpq} and \eq{chmix:d:dmdg} comply
with the conventions of Sect.~\ref{ch1:sect:gen} for \bbm. In particular
the time evolution formulae of \eq{tgg} - \eq{gpgms} are also valid
for \ddm\ with the replacement $B^0 \to D^0$.  Unlike in the case of
\bbm\ we cannot expand in $\dg_D/\dm_D$. We also refrain from
expanding in $\ega$ defined in \eq{defepsg}. Then \eq{gtfres} - \eq{gbtfbres} 
can be used for $D^0$ mesons, if $1+\ega$ in \eq{gbtfres} is replaced 
by $|p/q|^2$ and $1-\ega$ in \eq{gtfbres} is replaced by $|q/p|^2$.
Note that the definition of $y_D$ in \eq{chmix:d:xy} is opposite in
sign to the one of $y$ in the $B$ meson system in \eq{defxy}. In 
\eq{chmix:d:xy} we have used the sign convention which is usually used
in \ddm. Since \ddm\ is very small, one can expand in $\dm_D t$ and
$\dg_D t$. Using $x_D$ and $y_D$ from \eq{chmix:d:xy} the small $t$
expansion of \eq{gtfres} - \eq{gbtfbres} gives%
\index{time evolution!decay rate!$D$ meson}
\begin{eqnarray} 
\Gamma (D^0(t) \rightarrow f ) &\,=\,& 
 {\cal N}_f \lt| A_f \rt|^2 e^{-\Gamma_D t}\, 
  \Bigg[ 1 + \lt( -\imag \lambda_f \, x_D \, + \, \real \lambda_f \, 
                y_D  \rt) \Gamma_D t 
\phantom{\frac{1}{2}} \nn
&& \phantom{ {\cal N}_f \lt|  A_{f} \rt|^2 e^{-\Gamma_D t}\,} 
  \lt.
        + 
                \lt( \frac{|\lambda_f|^2+1}{4}\, y_D^2 + 
                     \frac{|\lambda_f|^2-1}{4}\, x_D^2  \rt) 
                     \!\lt( \Gamma_D t \rt)^2 \rt] ,
 \label{chmix:d:df} \\[2mm] 
\Gamma (\ov{D}{}^0(t) \rightarrow f ) &\,=\,& 
 {\cal N}_f \lt| A_{f} \rt|^2 e^{-\Gamma_D t}
 \lt| \frac{p}{q} \rt| ^2 \, 
  \Bigg[ 
 |\lambda_f|^2  + \lt( \imag \lambda_f \, x_D \,  + 
                             \real \lambda_f \, y_D \rt) \Gamma_D t \nn
&& \phantom{ {\cal N}_f \lt| \ov{A}_{\ov{f}} \rt|^2 e^{-\Gamma_D t}
 \lt| \frac{p}{q} \rt| ^2 } 
  \lt.
       +
        \lt( \frac{|\lambda_f|^2+1}{4}\, y_D^2 - 
                     \frac{|\lambda_f|^2-1}{4}\, x_D^2  \rt) 
                     \!\lt( \Gamma_D t \rt)^2 \rt] ,
 \label{chmix:d:dbf} \\[2mm] 
\Gamma (D^0(t) \rightarrow \ov{f} ) &\,=\,& 
 {\cal N}_f \lt| \ov{A}_{\ov{f}} \rt|^2 e^{-\Gamma_D t}
  \lt| \frac{q}{p} \rt|^2
   \lt[ |\lambda_{\ov{f}}|^{-2} + 
     \lt( \imag \frac{1}{\lambda_{\ov{f}}} \, x_D + 
          \real \frac{1}{\lambda_{\ov{f}}} \, y_D \rt) \Gamma_D t \rt. \nn
&& \phantom{ {\cal N}_f \lt| \ov{A}_{\ov{f}} \rt|^2 e^{-\Gamma_D t}
 \lt| \frac{p}{q} \rt| ^2} 
  \lt.
       +
    \lt( \frac{|\lambda_{\ov{f}}|^{-2}+1}{4}\, y_D^2 -  
                     \frac{|\lambda_{\ov{f}}|^{-2}-1}{4}\, x_D^2  \rt) 
                     \!\lt( \Gamma_D t \rt)^2 \rt] ,
 \label{chmix:d:dfb} \\[2mm] 
\Gamma (\ov{D}{}^0(t) \rightarrow \ov{f} ) &\,=\,& 
   {\cal N}_f \lt| \ov{A}_{\ov{f}} \rt|^2 e^{-\Gamma_D t}
   \lt[ 1 + 
     \lt( - \imag \frac{1}{\lambda_{\ov{f}}} \, x_D + 
          \real \frac{1}{\lambda_{\ov{f}}} \, y_D \rt) \Gamma_D t \rt. \nn
&& \phantom{ {\cal N}_f \lt| \ov{A}_{\ov{f}} \rt|^2 e^{-\Gamma_D t}} 
  \lt.
       +
    \lt( \frac{|\lambda_{\ov{f}}|^{-2}+1}{4}\, y_D^2 +  
                     \frac{|\lambda_{\ov{f}}|^{-2}-1}{4}\, x_D^2 \rt) 
                     \!\lt( \Gamma_D t \rt)^2 \rt] ,
  \label{chmix:d:dbfb}
\end{eqnarray}
with $\Gamma_D=(\Gamma_1+\Gamma_2)/2$. $\dm_D$ and $\dg_D$ are 
very small, because they are GIM-suppressed by a factor of
$m_s^2/M_W^2$. For this reason they are also difficult to calculate,
because at scales of order $m_s$ non-perturbative effects become
important. A recent calculation, which incorporates non-perturbative
effects with the help of the quark condensate, can be found in \cite{bu}.

%-------------------------------------------------------------------------------
\subsection[New Physics Effects in Meson Mixing]{New Physics Effects in Meson
Mixing$\!$ \authorfootnote{Author: JoAnne Hewett}}

The existence of new physics may modify the low-energy effective Hamiltonian
governing $B$ and $D$ physics in several ways: (i) via contributions to the
Wilson coefficients of the Standard Model operators, (ii) by generating new
operators, or (iii) through the presence of new $CP$ violating phases.  These
effects may originate from new interactions in tree-level meson decays or from
the virtual exchange of new physics in loop-mediated processes.  The scale of
new physics is expected to be large compared to $M_W$, and hence it is generally
anticipated that additional tree-level contributions to meson decays are
suppressed\cite{howe}.  However, large new contributions may be present in loop
processes, making meson mixing a fertile ground to reveal the influence of new
interactions.  All three above classes of new physics contributions may play a
role in meson mixing.  Such effects may be discovered in observables which are
suppressed in the Standard Model, such as the asymmetry $a_{\rm fs}^q$ measuring
$CP$ violation in $B_{d,s}$ mixing or in $D^0$ meson mixing, they may modify the
mixing-induced $CP$ asymmetries in $B\to\psi K_S$ and $B\to\pi\pi$ decays, or they
may alter the precisely calculated SM value of the ratio of $B_s$ to $B_d$
mixing.  We will discuss each of these observables in this section.  The effects
of new physics on meson width differences is described in
Sect.~\ref{chmix:subsub:width:phen}.

%-------------------------------------------------------------------------------
\subsubsection{$B_d$ Mixing}
\index{B mixing@$B$ mixing!$B_d$ mixing}%
\index{B meson@$B$ meson!mass difference \dm!$\dm_d$}%
\index{B meson@$B$ meson!mass difference \dm!new physics}%
\index{mass difference \dm!$\dm_d$}%
\index{mass difference \dm!$\dm_d$!new physics}%
It is well-known that new physics can play a large role in $B_d$
mixing.  One important consequence of this is that the constraints
in the $\rho-\eta$ plane from $\Delta m_d$
can be altered, resulting in a
significant shift\cite{indirect_rev} of the allowed region in this plane
from its Standard Model range.  This in turn modifies the expected values for
$\sin 2\beta$ and $\sin 2\alpha$, even if new sources of $CP$ violation 
are not present.  In fact, this comprises the most significant effect from
new physics on the $CP$ asymmetries in $B\to\psi K_S$ and $B\to\pi\pi$
decays in a large class of models.%
\index{decay!$B_d \rightarrow \psi K_S$}%
\index{decay!$B_d \to \pi \pi $ }%
\index{unitarity triangle!angle $\beta$}%
\index{unitarity triangle!angle $\alpha$}%

A model independent determination of such effects has been presented
in Ref.~\cite{yossi}.  New physics contributions to $B_d$ mixing can be
parameterized in a model independent fashion by considering the ratio
\begin{equation}
{\langle B_d^0|{\cal H}^{\rm full}|\bar B^0_d\rangle\over
 \langle B_d^0|{\cal H}^{\rm SM}|\bar B^0_d\rangle} 
   = \Big( r_d\, e^{i\theta_d} \Big)^2\,,
\end{equation}
where ${\cal H}^{\rm full(SM)}$ represents the Hamiltonian responsible
for $B_d$ mixing in the case of the Standard Model plus new physics
(just Standard Model), and $r_d(\theta_d)$ represents the new physics
contribution to the magnitude (phase) of $B_d$ mixing.  In the
Standard Model, the unitarity triangle is constrained by measurements
of $\sin 2\beta$, $\sin 2\alpha$, the ratio of semileptonic decays $
{\Gamma(b\to u\ell\nu)/\Gamma(b\to c\ell\nu)}$, and $x_d=C_tR_t^2$,
where $R_t$ is defined in Section~\ref{chmix:sub:mass}.%  
\index{B decay@$B$ decay!semileptonic}%
\index{decay!$B_d \rightarrow X \ell^+ \nu$}%
\index{unitarity triangle!and Vub@and $V_{ub}$}%
\index{Cabibbo-Kobayashi-Maskawa matrix!$V_{ub}$}%
These quantities are then modified in the presence of new interactions
via $a_{\psi K_S}=\sin (2\beta+2\theta_d)$,
$a_{\pi\pi}=\sin(2\alpha-2\theta_d)$,\ and $x_d=C_tR_t^2r_d^2$. Note
that the new phase contributions in $a_{\psi K_S}$ and $a_{\pi\pi}$
conspire to cancel in the triangle constraint and the relation
$\alpha+\beta+\gamma=\pi$ is retained.%
\index{unitarity triangle!angle $\alpha$}%  
Measurement of these four quantities allows one to disentangle the new
physics effects and fully reconstruct the true unitarity triangle
(\ie, find the true values of $\alpha$, $\beta$, and $R_t$) as well as
determine the values of $r_d$ and $\theta_d$ in a geometrical fashion.
This is depicted in Fig.~\ref{mix_modelindep}.  While this technique
is effective in principle, in practice it is limited by theoretical
uncertainties in $x_d\,, \alpha$, and the ratio of semileptonic
decays, as well as discrete ambiguities.

\begin{figure}[thb]
\centerline{
\epsfig{file=ch8/th_gnwfig1.eps,width=12cm}}
\caption[The model independent analysis in the $\rho-\eta$ plane]{The model
independent analysis in the $\rho-\eta$ plane: (i) the $a_{\psi K_S}$ ray, (ii)
the $a_{\pi\pi}$ circle, (iii) the $x_d$ circle, (iv) the semileptonic decay
ratio circle.  The $\gamma$ ray is given by the dashed line and the true
$\beta$ ray corresponds to the dotted line.  The true vertex of the unitarity
triangle is at $(\rho,\eta)$, while the point $(\rho',\eta')$ serves to
determine $r_d$ and $\theta_d$.}
\label{mix_modelindep}
\end{figure}
    
Model independent bounds on new physics contributions to $B_d$ mixing
can also be directly placed from measurements of $\Delta m_d$ and $\sin
2\beta$.  In the class of models
which respect $3\times 3$ CKM unitarity, where tree-level $B$
decays (in particular their phase) are dominated by the SM, and where
$\Gamma_{12}\simeq\Gamma_{12}^{\rm SM}$, the new physics effects in $B_d$
mixing can be isolated. The modification to $M_{12}$ can then be
described as above in terms of $r_d\,, \theta_d$.
The direct determination of $\Delta m_d$ provides a
bound on the magnitude of new physics contributions, $r_d$, while 
measurement of $\sin 2\beta$ constrains the new
phase $\theta_d$.  Taking into account the uncertainties on the values
of the relevant CKM factors and the hadronic matrix elements, present
data constrains $0.5\leq r_d\leq 1.8$ and $\sin 2\theta_d\geq -0.53$
at $95\%$ C.L.  It is clear that large contributions to $B_d$ mixing from new
interactions are still allowed, and may hence admit for an exciting discovery
as future measurements improve.

We note that another useful parameterization of new physics contributions
which is common in the literature is given by
\begin{equation}
M_{12}^{\rm NP}=h\, e^{i\sigma}M_{12}^{\rm SM}\,,
\end{equation}
where these variables are related to the previous ones via
\begin{equation}
r_d^2\, e^{2i\theta_d}=1+h\, e^{i\sigma}\,.
\end{equation}
Constraints on this set of parameters from current data is presented in
\cite{niretal} in various classes of models.

These parameters can be related to new physics contributions in other
observables.  For instance, the $CP$ asymmetry in flavor-specific $B_d$
decays (see Sect.~\ref{chmix:sub:afs}), which is given by%
\index{CP asymmetry@\CP\ asymmetry!in flavor-specific decay}%
\begin{equation}
a^d_{\rm fs}  = \imag {\Gamma^d_{12}\over M^d_{12}}\,,
\end{equation}
and has a value of $-8 \times 10^{-4}$ in the Standard Model (see
\eq{chmix:aqnum}), provides a good opportunity to probe the existence
of new interactions.  Since $\Gamma^d_{12}/M^d_{12}$ is essentially
real in the Standard Model, the contributions of new interactions to
the flavor-specific $CP$ asymmetry can be written as
\begin{equation}
a^d_{\rm fs}=-\left( {\Gamma_{12}^d\over M^d_{12}} \right)_{\rm SM} 
               {\sin 2 \theta_d \over r_d^2}\,.
\end{equation}
The above bounds from present data on the new physics contributions to
$B_d$ mixing restrict
\begin{equation}
-2.1\leq {a_{\rm fs}^d\over (\Gamma^d_{12}/M^d_{12})_{\rm SM}} \leq 4.0\,.
\end{equation}
It is important to note that $a_{\rm fs}^d$ can lie outside this
range, if new new physics enhances $\Gamma_{12}^d$, which is composed
of CKM-suppressed decay modes.

There are a plethora of new physics scenarios which can yield
substantial contributions to $B_d$ mixing; some examples are briefly
cataloged here.  Models which respect the structure of the $3\times 3$
CKM matrix contribute simply to the Wilson coefficient of the Standard
Model operator.  This is best illustrated by the virtual exchange in
the box diagram of charged Higgs bosons which are present in flavor
conserving two-Higgs-doublet models\cite{chHiggs} and by the
contributions of supersymmetric particles\cite{mix_susy} when a
Standard Model-like super-CKM structure is assumed.%
\index{supersymmetry}%
\index{two-Higgs doublet model}%
\index{new physics!supersymmetry}%
If the super-CKM angles ($\tilde V$) are allowed to be arbitrary, the
structure of the Wilson coefficients are altered.  In this case, the
supersymmetric amplitude relative to that of the Standard Model is
roughly given by $\sim (M_W/\tilde m)^n (\tilde V_{td}\tilde
V_{tb}/V_{td}V_{tb})$ and can constitute a flavor problem for
Supersymmetry if the sparticle masses, $\tilde m$, are near the weak
scale.  The existence of a fourth generation would also modify the CKM
structure of the Wilson coefficients.  New $|\Delta B|=2$ operators
are generated in scenarios\cite{newops} such as Left-Right Symmetric
models, theories of strong dynamics, as well as in Supersymmetry.
\index{new physics!left-right symmetry}%
\index{new physics!strong dynamics}%
Tree-level contributions\cite{mix_tree} are manifest in flavor
changing two-Higgs-doublet models, in scenarios with a flavor changing
extra neutral gauge boson, and in Supersymmetry with R-parity
violation.%
\index{two-Higgs doublet model!flavor-changing Higgs}%
\index{new physics!two-Higgs doublet model!flavor-changing Higgs}%
Most of these examples also contain new phases which may
be present in $B_d$ mixing.  It is interesting to note that various
forms of Supersymmetry may affect $B_d$ mixing in all possible
manners!

While it is possible to obtain large effects in $B_d$ mixing in all of
these scenarios, it is difficult to use $\Delta m_d$ at present to
tightly restrict these contributions and constrain the parameter space
in the models, due to the current sizable errors on the Standard Model
theoretical prediction arising from the imprecisely determined values
of the CKM factors and the hadronic matrix elements.  Frequently,
other flavor changing neutral current processes, such as $b\to
s\gamma$, provide more stringent constraints.

%-------------------------------------------------------------------------------
\subsubsection{$B_s$ Mixing}
\index{B mixing@$B$ mixing!$B_s$ mixing}
\index{B meson@$B$ meson!mass difference \dm!$\dm_s$}%
\index{B meson@$B$ meson!mass difference \dm!new physics}%
\index{mass difference \dm!$\dm_s$}%
\index{mass difference \dm!$\dm_s$!new physics}%
A similar analysis as employed above may be used to constrain new physics
contributions to $B_s$ mixing.  In the class of models which respect the
$3\times 3$ CKM unitarity and where tree-level decays are dominated by the
Standard Model contributions, we can again parameterize potential new
contributions to $\Delta m_s$ via $M_{12}=(r_se^{i\theta_s})^2M_{12}^{\rm SM}$.
This gives
%
\begin{equation}
{\Delta m_d\over\Delta m_s} = {\lambda^2R_t^2\over\xi^2}\, 
  {m_{B_d}\over m_{B_s}}\, {r^2_d\over r^2_s}\,,
\end{equation}
%
which reduces to the Standard Model expression when $r_d=r_s$.  The parameter
$\theta_s$ can be constrained once $CP$ asymmetries in the $B_s$ system are
measured or, if $\theta_s$ is large, from measurements of $\dg_s$ as described
in Sect.~\ref{chmix:subsub:width:phen}.

As discussed above, the ratio $\Delta m_d/\Delta m_s$ yields a good
determination of the CKM ratio $|V_{td}/V_{ts}|$ within the Standard Model,
since the ratio of hadronic matrix elements is accurately calculated in lattice
gauge theory.  Remarkably, this remains true in many scenarios beyond the
Standard Model.  In a large class of models which retain the $3\times 3$ CKM
structure, the virtual exchange of new particles in the box diagram alters the
Inami-Lim function, but not the remaining factors in the expression for $\Delta
m_{d,s}$.  The effects of new physics thus cancel in the ratio.  As an explicit
example, consider charged Higgs exchange in the box diagram within the context
of two-Higgs-doublet models.  The expression for the mass difference in $B_s$
mixing in this case is
\begin{eqnarray}
\Delta m_s & = & {G_F^2M_W^2\over 6\pi^2}\,f_{B_s}^2B_{B_s}\eta_{B_s}
m_{B_s}\, |V_{tb}V^*_{ts}|^2\, 
\Big[S(m_t^2/M_W^2) + F(m_t^2/m^2_{H^\pm},\tan\beta)\Big]
\nonumber\\
& = & \Delta m_d\, \xi^2\, {m_{B_s}\over m_{B_d}}\, 
{|V_{ts}|^2\over |V_{td}|^2} \,,
\end{eqnarray}
where $m_{H^\pm}$ represents the charge Higgs mass and $\tan\beta$ is
the ratio of vevs.  Here,
we see that the charged Higgs contribution is the same for $B_d$
and $B_s$ mixing (neglecting $d$- and $s$-quark masses)
and thus cancels exactly in the ratio.  This cancellation also occurs
in several other classes of models, including minimal 
Supersymmetry with flavor conservation.
Notable exceptions to this
are found in models which (i) change the structure of the 
CKM matrix, such as the addition of a fourth generation, or extra singlet
quarks, or in Left-Right symmetric models, (ii) have sizable Yukawa couplings 
to the light fermions, such as leptoquarks or Higgs models with
flavor changing couplings, 
or (iii) have generational dependent couplings, including supersymmetry with
R-parity violation.

%-------------------------------------------------------------------------------
\subsubsection{Mixing in the Charm Sector}
\index{D mixing@$D$ mixing}%
\index{D meson@$D$ meson!mass difference $\dm_D$!new physics}%

The short distance Standard Model contribution to $\Delta m_D$ proceeds through
a $W$ box diagram with internal $d,s,b$-quarks.  In this case the external
momentum, which is of order $m_c$, is communicated to the light quarks in the
loop and can not be neglected.  The effective Hamiltonian is
%
\begin{equation}
{\cal H}^{\Delta c=2}_{eff} = {G_F\alpha\over 8\sqrt 2x_w}\, \Big[
|V_{cs}V^*_{us}|^2\, (I_1^s\,{\cal O} - m_c^2 I_2^s\,{\cal O}') 
+ |V_{cb}^*V_{ub}|^2\, (I_3^b\, {\cal O} - m_c^2 I^b_4\, {\cal O}') \Big]\,,
\end{equation}
%
where the $I_j^q$ represent integrals\cite{dmixsm} that are functions of
$m_q^2/M_W^2$ and $m_q^2/m_c^2$, and ${\cal O}=[\bar u\gamma_\mu(1-
\gamma_5)c]^2$ is the usual mixing operator, while ${\cal O}'=[\bar u
\gamma_\mu(1+\gamma_5)c]^2$ arises in the case of non-vanishing external
momentum.  The numerical value of the short distance contribution is $\Delta
m_D\sim 5\times 10^{-18}$ GeV (taking $f_D=200$ MeV).  The long distance
contributions have been computed via an intermediate state dispersive approach
and in heavy quark effective theory, yielding values\cite{dmixld} in the range
$\Delta m_D\sim 10^{-17} - 10^{-16}$ GeV.  The Standard Model predictions are
clearly quite small and allow for a large window for the observation of new
physics effects.

\begin{figure}[t]
\vspace*{-.5cm}
\centerline{\hspace*{20mm}
\epsfig{file=ch8/th_slac6821_fig2a.ps,
        height=0.52\textwidth,angle=-90}
\hspace*{1.5cm}
\raisebox{7pt}{\epsfig{file=ch8/th_slac6821_fig2b.ps,
        height=0.5\textwidth,angle=90}}}
\vspace*{-2cm}
\centerline{
  \psfig{file=ch8/th_slac6821_fig2c.ps,width=6cm,angle=-90}}
\vspace*{-.5cm}
\caption[$\Delta m_D$ in the (a) four generation Standard Model; (b)
two-Higgs-doublet model~II; and (c) flavor-changing Higgs model with
tree-level contributions.]{$\Delta m_D$ in (a) the four generation Standard
Model as a function of the appropriate $4\times 4$ CKM elements, taking the
$b'$-quark mass to be $100,200,300,400$ GeV from top to bottom, (b) the
two-Higgs-doublet model II as a function of $\tan\beta=v_2/v_1$ with
$m_{H^\pm}=50,100,250, 500, 1000$ GeV from top to bottom, and (c) the
flavor-changing Higgs model with tree-level contributions as a function of the
mixing factor, with $m_{h^0}=50,100,250,500,1000$ GeV from top to bottom.}
\label{charm_mix}
\end{figure}

Since the Standard Model expectation is so small, large enhancements in $\Delta
m_D$ are naturally induced by new interactions.  A compilation of such effects
in various models and list of references can be found in \cite{harry}.  This
article shows that the present experimental bound on $D$-mixing already
constrains the parameter space in many scenarios, and an order of magnitude
improvement would exclude (or discover) some models.  Here, for purposes of
illustration, we present the potential enhancements that can occur in three
scenarios\cite{indirect_rev}.  We examine (i) the case of a fourth generation to
demonstrate the effect of heavy fermions participating in the box diagram, (ii)
the contributions from charged Higgs exchange in flavor conserving
two-Higgs-doublet models, which is often used as a benchmark in studying new
physics, and (iii) the tree-level contributions in flavor-changing Higgs models,
where the flavor changing couplings are taken to be $\lambda_{h^0f_if_j}\sim
(\sqrt 2 G_F)^{1/2} \sqrt{m_im_j}\Delta_{ij}$ with $\Delta_{ij}$ being
a combination of mixing angles.%  
\index{new physics!two-Higgs doublet model}%
\index{new physics!two-Higgs doublet model!flavor-changing Higgs}%
\index{new physics!fourth generation}%
\index{fourth generation}%
\index{two-Higgs doublet model!flavor-changing Higgs}%
\index{two-Higgs doublet model}%
The mass difference as a function of the model parameters is shown in
Fig.~\ref{charm_mix} for each case.  We see that in each case the
parameter space is already restricted by the current experimental
value, and that an improvement in the bound would provide a sensitive
probe of these models.
  
%===============================================================================
\section[Interesting decay modes]{Interesting decay modes$\!$ \authorfootnote{Author: Ulrich~Nierste}}

In this section we list the decay modes which are useful for the determination
of $\dm_q$, $\dg_q$ and $B$ meson lifetimes.  Flavor-specific decay modes are
summarized in Table~\ref{chmix:tab:fsd}, CKM-favored decays into $CP$ eigenstates
are listed in Table~\ref{chmix:tab:cpe} and decays which are neither
flavor-specific nor $CP$-specific can be found in Table~\ref{chmix:tab:cpne}.

\begin{table}[thb]
\begin{center}
\begin{tabular}{@{}r|p{0.45\textwidth}|p{0.35\textwidth}} \hline
quark decay & hadronic decay & remarks \\\hline\hline  
$\ov{b} \rightarrow \ov{c} \ell^+  \nu_\ell $ & 
  $B_{d,s}\rightarrow D_{(s)}{}^- \ell^+ \nu_\ell$ & 
\index{decay!$B_s \rightarrow D_s^- \ell^+ \nu$}
\index{decay!$B_d \rightarrow D^- \ell^+ \nu$}
\\
& $B_{d,s}\rightarrow X \ell^+ \nu_\ell $ & 
\index{decay!$B_s \rightarrow X \ell^+ \nu$}
\index{decay!$B_d \rightarrow X \ell^+ \nu$}
\\\hline
$\ov{b} \rightarrow \ov{c} u \ov{d}$ & 
  $B_d \rightarrow D^{(*)}{}^- \pi^+$ & 
\index{decay!$B_d \rightarrow D^- \pi^+$}
\index{decay!$B_d \rightarrow D^{*-} \pi^+$}
\\
& $B_d \rightarrow D^{(*)}{}^- \pi^+ \pi^+ \pi^-$ & 
\index{decay!$B_d \rightarrow D^- \pi^+ \pi^- \pi^+$}
\index{decay!$B_d \rightarrow D^{*-} \pi^+ \pi^- \pi^+$}
\\
& $B_d \rightarrow D^{*}{}^- \pi^+ \pi^+ \pi^- \pi^0$ &
\index{decay!$B_d \rightarrow D^{*-} \pi^+ \pi^+ \pi^- \pi^0$}
\\
& $B_d \rightarrow \ov{D}{}^{(*)0} \rho^0$
\index{decay!$B_d \rightarrow \ov{D}{}^0 \rho^0$}
\index{decay!$B_d \rightarrow \ov{D}{}^{*0} \rho^0$}
  [$\rightarrow  K^+ \pi^-$ or $K^+ \pi^+ \pi^- \pi^-$ 
   \mbox{\hspace{15ex}or} 
   $K^{(*)}{}^+ \ell^- \ov{\nu}_{\ell}$ etc.] &
   $\BR(\rho^0 \rightarrow \pi^+ \pi^-) \approx 100\%$. \hfill\break
   The $\ov{D}{}^{(*)0}$ must be detected in a final state $f$
         such that $D^{(*)0} \rightarrow f$ is forbidden or suppressed. \\
& $B_s \rightarrow D_s^{(*)}{}^- \pi^+$ & 
\index{decay!$B_s \rightarrow D_s^- \pi^+$}
\index{decay!$B_s \rightarrow D_s^{*-} \pi^+$}
\\ 
& $B_s \rightarrow D_s^{(*)}{}^- \pi^+ \pi^+ \pi^-$ & 
\index{decay!$B_s \rightarrow D_s^- \pi^+ \pi^+ \pi^-$}
\index{decay!$B_s \rightarrow D_s^{*-} \pi^+ \pi^+ \pi^-$}
\\
& $B_s \rightarrow D_s^{*}{}^- \pi^+ \pi^+ \pi^- \pi^0$ &
\index{decay!$B_s \rightarrow D_s^- \pi^+ \pi^+ \pi^- \pi^0$}
\index{decay!$B_s \rightarrow D_s^{*-} \pi^+ \pi^+ \pi^- \pi^0$}
\\
& $B_s \rightarrow \ov{D}{}^{(*)0} K_S$
  [$\rightarrow  K^+ \pi^-$ or $K^+ \pi^+ \pi^- \pi^-$ 
   \mbox{\hspace{15ex}or} 
   $K^{(*)}{}^+ \ell^- \ov{\nu}_{\ell}$ etc.] &
\index{decay!$B_s \rightarrow \ov{D}{}^0 K_S $}
\index{decay!$B_s \rightarrow \ov{D}{}^{*0} K_S $}
\\\hline
$\ov{b} \rightarrow \ov{c} c \ov{s}$ & 
   $B_d \rightarrow \psi K^+ \pi^-$ & 
        mainly \mbox{$B_d \rightarrow \psi K^{(*)}$ 
        [$\rightarrow  K^+ \pi^-$]}  
\index{decay!$B_d \rightarrow \psi K^+ \pi^-$}
\\
&  $B_d \rightarrow D^{(*)+} D_s^{(*)-}$ & 
\index{decay!$B_d \rightarrow D^+ D^-$}
\index{decay!$B_d \rightarrow D^{*+} D^-$}
\index{decay!$B_d \rightarrow D^+ D^{*-}$}
\index{decay!$B_d \rightarrow D^{*+} D^{*-}$}
\\\hline
$\ov{b} \rightarrow \ov{c} c \ov{d}$ & 
$B_s \rightarrow \psi K^- \pi^+ $  & 
        mainly \mbox{$B_s \rightarrow \psi \ov{K}{}^{(*)}$ 
        [$\rightarrow  K^- \pi^+$]}  
\index{decay!$B_s \rightarrow \psi K^- \pi^+$}
\\
&  $B_s \rightarrow D_s^{(*)}{}^- D^{(*)}{}^+$
& 
\index{decay!$B_s \rightarrow D_s^+ D_s^-$}
\index{decay!$B_s \rightarrow D_s^{*+} D_s^-$}
\index{decay!$B_s \rightarrow D_s^+ D_s^{*-}$}
\index{decay!$B_s \rightarrow D_s^{*+} D_s^{*-}$}
\\\hline
$\ov{b} \rightarrow \ov{c} X $ 
& $B_d \rightarrow D^{(*)}{}^- X$  & 
        small contamination from \hfill\break 
	$\ov{b} \rightarrow \ov{c} c \ov{d}$ 
\index{decay!$B_d \rightarrow D^- X$}
\index{decay!$B_d \rightarrow D^{*-} X$}
\\\hline  
\end{tabular}
\end{center}
\caption{Interesting flavor-specific decays.}
\label{chmix:tab:fsd}
\end{table}


\begin{table}[thbp]
\begin{center}
\begin{tabular}{@{}r|p{0.49\textwidth}|p{0.31\textwidth}} \hline
quark decay & hadronic decay & remarks \\\hline\hline  
$\ov{b} \rightarrow \ov{c} u \ov{d}$ 
& $B_d \rightarrow \ov{D}{}^{(*)0}  \rho^0 $
  [$\rightarrow  \rho_0 K_S$ or $K \ov{K}$ or $\pi^+ \pi^- $] 
  &   The $\ov{D}{}^{(*)0}$ must be detected in a $CP$-specific final state $f$
      (hence $D^{(*)}{}^0\rightarrow f$ is equally allowed). 
\index{decay!$B_d \rightarrow \ov{D}{}^0  \rho^0 $}
\index{decay!$B_d \rightarrow \ov{D}{}^{*0}  \rho^0 $}
\\
& $B_d \rightarrow \ov{D}{}^{(*)0} \pi^+ \pi^- $
  [$\rightarrow  \rho_0 K_S$ or $K \ov{K}$ or $\pi^+ \pi^- $] 
  & This decay mode has color-unsuppressed contributions.  
\index{decay!$B_d \rightarrow \ov{D}{}^{0} \pi^+ \pi^- $}
\index{decay!$B_d \rightarrow \ov{D}{}^{*0} \pi^+ \pi^- $}
\\
& $B_s \rightarrow \ov{D}{}^{(*)0} K_S$
  [$\rightarrow  \rho_0 K_S$ or $  K \ov{K}$ or $\pi^+ \pi^- $] &  
\index{decay!$B_s \rightarrow \ov{D}{}^0 K_S $}
\index{decay!$B_s \rightarrow \ov{D}{}^{*0} K_S $}
\\\hline
$\ov{b} \rightarrow \ov{c} c \ov{s}$ 
& $B_d \rightarrow \psi K_S$ &
\index{decay!$B_s \rightarrow \psi K_S$}
\\
& $B_d \rightarrow \psi K_S \rho^0$ & 
The $\ov{B}_d$ can decay into the
same final state $K_S \rho^0$. \hfill\break 
Angular analysis separates $CP$-eigenstates.
\index{decay!$B_d \rightarrow \psi K_S \rho^0$}
\\
& $B_d \rightarrow \psi \phi K_S$ or $\psi K_S \rho^0$ & 
                        Angular analysis required.
\index{decay!$B_d \rightarrow \psi \phi K_S$}
\index{decay!$B_d \rightarrow \psi K_S \rho^0$}
\\
& $B_d \rightarrow \psi \phi K^*$ [$\rightarrow K_S \pi^0$] &
                        Angular analysis required, \hfill\break
			$\pi^0$ is problematic. 
\index{decay!$B_d \rightarrow \psi \phi K^*$}
\\
&  $B_d \rightarrow D_s^{(*)}{}^+ D_s^{(*)}{}^- K_S$ &
                        Angular analysis required.
\index{decay!$B_d \rightarrow D_s^+ D_s^- K_S$}
\index{decay!$B_d \rightarrow D_s^{*+} D_s^{*-} K_S$}
\\
& $B_s \rightarrow \psi \phi$ & 
                        Angular analysis required. 
\index{decay!$B_s \rightarrow \psi \phi$}
\\
& $B_s \rightarrow \psi K \ov{K}$ or $\psi K^* \ov{K}{}^*$ &
                        Same remark 
\index{decay!$B_s \rightarrow \psi K \ov{K}$}
\index{decay!$B_s \rightarrow \psi K^* \ov{K}{}^*$}
\\
& $B_s \rightarrow \psi \phi \phi$ &
                        Same remark. 
\index{decay!$B_s \rightarrow \psi \phi \phi$}
\\
& $B_s \rightarrow \psi \eta $& 
\index{decay!$B_s \rightarrow \psi \eta $}
\\
& $B_s \rightarrow \psi \eta^\prime$ & 
\index{decay!$B_s \rightarrow \psi \eta^\prime$}
\\
& $B_s \rightarrow D_s^+ D_s^-$ &
\index{decay!$B_s \rightarrow D_s^+ D_s^-$}
\\
& $B_s \rightarrow D_s^{*}{}^+ D_s^{*}{}^-$ & 
                        Angular analysis required.
\index{decay!$B_s \rightarrow D_s^+ D_s^-$}
\index{decay!$B_s \rightarrow D_s^{*+} D_s^-$}
\index{decay!$B_s \rightarrow D_s^+ D_s^{*-}$}
\index{decay!$B_s \rightarrow D_s^{*+} D_s^{*-}$}
\\
& $B_s \rightarrow D^{(*)+} D^{(*)-}$ or $D^{(*)0}\ov{D}{}^{(*)0}$ &
                        Non-spectator decays.
\index{decay!$B_s \rightarrow D^+ D^-$}
\index{decay!$B_s \rightarrow D^{*+} D^-$}
\index{decay!$B_s \rightarrow D^+ D^{*-}$}
\index{decay!$B_s \rightarrow D^{*+} D^{*-}$}
\\ 
& $B_s \rightarrow \psi f_0 $ &
                        $CP$-odd final state.
\index{decay!$B_s \rightarrow \psi f_0$}
\\
& $B_s \rightarrow \chi_{c0} \phi $ &
                        $CP$-odd final state.
\index{decay!$B_s \rightarrow \chi_{c0} \phi $}
\\\hline 
$\ov{b} \rightarrow \ov{c} c \ov{d}$ 
& $B_d \rightarrow D^+ D^-$ &
                        $CP$-even final state.
\index{decay!$B_d \rightarrow D^+ D^-$}
\index{decay!$B_d \rightarrow D^{*+} D^-$}
\index{decay!$B_d \rightarrow D^+ D^{*-}$}
\index{decay!$B_d \rightarrow D^{*+} D^{*-}$}
\\
& $B_d \rightarrow D^{*}{}^+ D^{*}{}^-$ & 
\\
& $B_d \rightarrow \psi \rho^0$ & 
\index{decay!$B_d \rightarrow \psi \rho^0$}
\\
& $B_s \rightarrow \psi K_S$ &
\index{decay!$B_s \rightarrow \psi K_S$}
\\
& $B_s \rightarrow \psi K_S \pi^0$ &         
    Mainly \hfill\break $B_s \rightarrow \psi \ov{K}{}^{(*)}$ 
                [$\rightarrow  K_S \pi^0$].\hfill
\index{decay!$B_s \rightarrow \psi K_S \pi^0$}%
\\\hline
\end{tabular}
\end{center}
\caption{Interesting CKM-favored decays into $CP$-eigenstates.}
\label{chmix:tab:cpe}
\end{table}

\begin{table}[hbtp]
\begin{center}
\begin{tabular}{@{}r|p{0.49\textwidth}|p{0.31\textwidth}} \hline
quark decay & hadronic decay & remarks \\\hline\hline  
$\ov{b} \rightarrow \ov{c} c \ov{s}$ 
& $B_s \rightarrow \psi K \ov{K}{}^*$ combined with 
        $\psi  K^* \ov{K}$ & angular analysis plus \hfill\break analysis
analogous to \hfill\break $B_s \rightarrow D_s^{\pm} K^{\mp}$ required.
\index{decay!$B_s \rightarrow \psi K \ov{K}{}^*$}
\index{decay!$B_s \rightarrow \psi K^* \ov{K}$}
 \\ \hline
\end{tabular}
\end{center}
\caption{Interesting CKM-favored decays into $CP$ non-eigenstates 
         accessible to $B$ and $\ov{B}$.}
\label{chmix:tab:cpne}
\end{table}

%abwabwabwabwabwabwabwabwabwabwababwabwabwabwabwabw$$$$$$$$$$$$$$$$$$$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%===============================================================================
\section{Introduction and Physics Input for CDF}

%This is an introduction to the following CDF sections discussing technical
%issues and physics inputs relevant for the chapter on mixing and lifetimes.  In
%the following there is a list of the most important improvements of the CDF
This section summarizes technical issues and physics inputs that are relevant to
mixing and lifetime measurements at CDF. The CDF detector improvements that are
critical for these measurements are the following
\index{CDF!detector improvements}:
%\index{CDF!detector improvements} detector relevant for this chapter.
%
\begin{enumerate}
\item The Level-1 and Level-2 trigger systems have been upgraded to allow
  triggers on high momentum tracks at Level-1, using the central drift chamber (``XFT''),
  and on large impact parameter
  tracks at Level-2, using the silicon vertex detector (``SVT''). This in turn allows
  CDF to trigger on all-hadronic decays of $b$ hadrons, such as
  $B_s\to D_s\pi$. Extensive simulations, together with run~I data, have
  been used to estimate trigger rates and event yields for the analyses
  discussed below.
\item The silicon vertex detector has been upgraded for improved silicon tracking and
  extended fiducial coverage.  This upgrade is most relevant to analyses which
  depend critically on vertex position resolution, in particular the measurement of
  $B_s$ mixing.
\item The CDF muon system has been upgraded to allow extended fiducial coverage
 and lower trigger thresholds. The increase of the fiducial volume
  is treated as a simple scale factor for all analyses using muons; the change
  in trigger threshold applies to the analysis of central di-muons only.

 % taken in Run~I cannot be used directly to make projections.  Therefore
 % extensive simulations have been performed. The event yields for the relevant
 %  analyses are going to be discussed in detail.
\end{enumerate}
%
The basic information needed to make projections for Tevatron Run~II is
the event yield for the $b$ hadron decay channels in question. Projections
are relatively simple for channels obtained from lepton triggers
using extrapolations from the Run~I data. An example of
this scaling is given in~\ref{sec:cdf-sin2beta}.
The projections for channels which are triggered by the newly implemented
displaced track trigger system, often referred to as the hadronic trigger, are more
difficult and are based primarily on simulations.

The CDF trigger system is organized in three levels of which only the first two
are simulated for the following studies, since the third level should not
reject good signal events.  In addition to the trigger simulation,
physics inputs are needed for the total $B$ cross sections and production
and branching fractions. Both the physics inputs and the description of the
trigger simulation are given below. Some of the issues are already partially
covered in the $CP$ violation chapter in Section~\ref{sec:cdf-sin2beta}. Here
the emphasis is mostly on the hadronic trigger.
%...............................................................................
\subsection{Physics Input}

The Monte Carlo program Bgenerator~\cite{Ref:BGEN}
\index{CDF!Bgenerator}%
is used to generate $b$ hadrons; it parameterizes the $p_T$ and $y$ distributions for
$b$ quarks according to next-to-leading-order calculations~\cite{Ref:NDE}. The $b$
quarks are fragmented into $b$ hadrons using the
Peterson~\cite{Ref:PETERSON} function with a fragmentation parameter value of
$\epsilon_b=0.006$. The CLEO Monte Carlo Program QQ~\cite{Ref:CLEOMC} is used to
decay the $b$ hadrons.  Events generated with Bgenerator contain
particles only from the decay of the $b$ hadrons, and do not include
particles produced in the $b$ quark fragmentation or the underlying event from the $p\overline{p}$ collision.
%$p\bar{p}$ collision are not available.

The overall production cross section is normalized using the CDF
measurement for $B^0$ production with $\rm p_T > 6 ~\GeV, ~|y| < 1$
~\cite{Ref:CDF-XS-BD}.
The $B_s$ and $\Lambda_b$ production fractions in $p\bar{p}$ collisions
are based on the CDF measurement of $f_s / (f_u + f_d)$~\cite{Ref:CDF-FSFUFD}
and the world average value for $f_{\Lambda_b}$, respectively.
  Assuming the $b$ hadron production spectra follow the
distributions from~\cite{Ref:NDE}, and using the CDF measurement
from Reference~\cite{Ref:CDF-XS-BD}, a total production cross section
for $B^0$ mesons of 50.1 $\rm \mu b$ is obtained.
Table~\ref{tab:br-prod} lists the measured rates and production fractions
assumed in the CDF analyses, together with the relevant hadronic decay
branching fractions that are known. In addition, we have estimated
the branching fractions for $b$-hadron decays that have not been
measured directly, using various symmetry assumptions as
described below.
%Since certain decays are forced in the generation process, the branching
%fractions for these decay modes are needed when calculating the total event
%yields. Branching fractions which are measured are listed in
%Table~\ref{tab:br-prod}, while some other branching fractions are not known. In
%the following section it is explained how they are estimated. In
%Table~\ref{tab:br-prod} all parameters discussed in this section are summarized
%together with the appropriate references.

\begin{table}
\begin{center}
\begin{tabular}{ l  c  l }
\hline %-----------------------------------------------------------
\hline %-----------------------------------------------------------
Quantity                  & Value                     & Reference\\
\hline %-----------------------------------------------------------
{\cal B}($D_s^+ \to \phi \pi^+$)    &$( 3.6  \pm 0.9  )\ten{-2}$& \cite{Ref:PDG}\\
{\cal B}($D_s^+ \to K^+\overline{K^{*0}}$)       &$( 3.3  \pm 0.9  )\ten{-2}$& \cite{Ref:PDG}\\
{\cal B}($D_s^+ \to \pi^+\pi^+\pi^-$)   &$( 1.0  \pm 0.4  )\ten{-2}$& \cite{Ref:PDG}\\
\hline %-----------------------------------------------------------
{\cal B}($D^0\to K^-\pi^+$)     &$( 3.85 \pm 0.09 )\ten{-2}$& \cite{Ref:PDG}\\
{\cal B}($D^0\to K^-\pi^+\pi^+\pi^-$)     &$( 7.6  \pm 0.4  )\ten{-2}$& \cite{Ref:PDG}\\
\hline %-----------------------------------------------------------
{\cal B}($\Lambda_c^+ \to p K^-\pi^+$)
                                &$( 5.0  \pm 1.3  )\ten{-2}$& \cite{Ref:PDG}\\
{\cal B}($\Lambda_c^+ \to \Lambda\pi^+\pi^-\pi^+$)
                                &$( 3.3  \pm 1.0  )\ten{-2}$& \cite{Ref:PDG}\\
\hline %------------------------------------------------------------
{\cal B}($\Lambda \to p \pi^-$) &$(63.9  \pm 0.5  )\ten{-2}$& \cite{Ref:PDG}\\
\hline %------------------------------------------------------------
{\cal B}($\phi(1020) \to K^+K^-$)        &$(49.1  \pm 0.8  )\ten{-2}$& \cite{Ref:PDG}\\
{\cal B}($K^*(892) \to K \pi$)       &$ 1                       $& \cite{Ref:PDG}\\
\hline %-----------------------------------------------------------
$f_{\Lambda_b}$                 &$(0.116 \pm 0.020)\ten{-2}$& \cite{Ref:PDG}\\
$f_s/(f_u + f_d)$               &$(0.213 \pm 0.038)\ten{-2}$& \cite{Ref:CDF-FSFUFD}\\
$\mathrm{\sigma_B^0}(\mathrm{p_T(B^0)>6~\GeV;~|y|<1})$
                                &$(3.52  \pm 0.61 )\mu b   $& \cite{Ref:CDF-XS-BD}\\
\hline %-----------------------------------------------------------
\hline %-----------------------------------------------------------
\end{tabular}
\end{center}
\caption{Physics input used for event yield estimates.}
\label{tab:br-prod}
\end{table}

\paragraph{Branching Fraction Estimates}
%=======================================
Since many of the hadronic decay channels have so far not been measured or even
observed, certain branching ratios have to be estimated.
This is relatively simple for $B_s$ decays, using  related $B^0$
decay modes. These estimates are summarized in
Table~\ref{tab:bs-branchings}.  The related uncertainties should be small, on the
order of roughly 10\% in the form factors or 20\% in the event
yields.

\begin{table}
\begin{center}
\begin{tabular}{ l  c  l }
\hline %-----------------------------------------------------------------------
\hline %-----------------------------------------------------------------------
Quantity                        & Value               & Reference\\
\hline %-----------------------------------------------------------------------
{\cal B}($B_s\to D_s \pi$)      &$(3.0\pm0.4)\ten{-3}$& from $B^0$~\cite{Ref:PDG}\\
{\cal B}($B_s\to D_s \pi\pi\pi$)&$(8.0\pm2.5)\ten{-3}$& from $B^0$~\cite{Ref:PDG}\\
{\cal B}($B_s\to D_s D_s$)      &$(8.0\pm3.0)\ten{-3}$& from $B^0$~\cite{Ref:PDG}\\
{\cal B}($B_s\to D_s^* D_s$)    &$(2.0\pm0.6)\ten{-2}$& from $B^0$~\cite{Ref:PDG}\\
{\cal B}($B_s\to D_s^* D_s^*$)  &$(2.0\pm0.7)\ten{-2}$& from $B^0$~\cite{Ref:PDG}\\
\hline %-----------------------------------------------------------------------
\hline %-----------------------------------------------------------------------
\end{tabular}
\end{center}
\caption{Branching fraction estimates for $B_s$ decays.}
\label{tab:bs-branchings}
\end{table}
\index{decay!$B_s \rightarrow D_s^- \pi^+$}
\index{decay!$B_s \rightarrow D_s^- \pi^+ \pi^- \pi^+$}
\index{decay!$B_s \rightarrow D_s^+ D_s^-$}
\index{decay!$B_s \rightarrow D_s^{*+} D_s^-$}
\index{decay!$B_s \rightarrow D_s^+ D_s^{*-}$}
\index{decay!$B_s \rightarrow D_s^{*+} D_s^{*-}$}

%{\tt For $B_c$ mesons ... Vaia's estimations and her Table.}

For $\Lambda_b$ baryons the situation is
more complicated. Several patterns arise when comparing bottom and charm
branching fractions, as well as meson and baryon branching fractions. The most
important is the fact that the branching fractions of $B$ mesons are often quite small
compared to those of the corresponding $D$ decays. For instance
%
\begin{eqnarray}
  {\cal B}(B^0\to D^-\pi^+) & = & (3.0 \pm 0.4) \ten{-3}\,,\nonumber\\
  {\cal B}(D^0\to K^-\pi^+) & = & (3.83\pm 0.09)\ten{-2}\,.
\end{eqnarray}
%
Comparing the two gives a $B$ to $D$ ratio of $0.08$.  One might suppose,
neglecting that $c \to s$ involves a light quark, that the widths of these modes
would be similar, but the total widths reflected in the mean lifetimes are
different. Moreover, the $b$ sector involves considerably more decay channels.
The semileptonic decays, however, remain qualitatively different, and do not
scale in the same way.

% Discuss with Jeff?! not clear to me what is meant.
%
% One might suppose,... that the width of these modes would be
% similar...'' is not well put:  B -> D pi has CKM-element V_cb
% und D -> K pi has V_cs, i.e. the naive factor between the decay width
% is |V_cb|^2 / |V_cs|^2  = 0.04^2 = 0.0016  -- (V_cs roughly 1)

A similar pattern for the hadronic decay modes may reasonably be expected for
baryons.  Indeed, in the one hadronic branching fraction measured for the
$\Lambda_b$, we have:
%
\begin{eqnarray}
  {\cal B}(\Lambda_b\to J/\psi\Lambda) & = & (4.7\pm 2.8) \ten{-4}\,,\nonumber \\
  {\cal B}(\Lambda_c^+\to p K^{*0})    & = & (1.6\pm 0.5) \ten{-2}\,,\nonumber \\
  {\cal B}(\Lambda_c^+\to p \phi)      & = & (1.2\pm 0.5) \ten{-3}\,.
\end{eqnarray}
\index{decay!$\Lambda_b \rightarrow J\psi\Lambda$}
%
In comparing the $\Lambda_c^+$ branching fraction to the $\Lambda_b$ branching
fraction, it is assumed that virtual $W^- \to \overline{c}s$ occurs about as
frequently as $W^- \to \overline{u}d$.  The $\Lambda_b$ to $\Lambda_c^+$ ratio is about 0.03 when comparing
to $\Lambda_c^+ \to pK^{*0}$.  A similar comparison can be made with the second
$\Lambda_c^+$ decay mode, which, aside from the $|V_{us}/V_{ud}|^2$ factor, is
most similar to $\Lambda_b \to J/\psi \Lambda$; the ratio is about $0.02$.
However, since applying this factor to a color-suppressed mode is
problematic, the first ratio is preferred
%
\begin{equation}
  g_{bc} = 0.03
\end{equation}
%
to multiply $\Lambda_c^+$ hadronic branching fractions to obtain estimates of
corresponding $\Lambda_b$ fractions.

Another difference between charm and bottom decays is that the bottom hadrons
can avail themselves of the virtual $W^- \to \overline{c}s$ transition which is
disallowed for charm decays. The ratio of branching fractions for an external
$W^- \to \overline{c}s$ to $W^- \to \overline{u}d$ is similar to the ratio of
the square of the decay constants
%
\begin{equation}
  g_{D_s}^2 = \left(\frac{f_{D_s^+}}{f_{\pi^+}}\right)^2
            = \left(\frac{280\;{\rm MeV}}{130.7\;{\rm MeV}}\right)^2
          = 4.58 \,,
\end{equation}
%
the effect of which is seen in comparing branching fractions of $B^0 \to
D^-D_s^+$ and $D^-\pi^+$, with the usual caveats.

Adding a $\pi^+\pi^-$ to the final state of a decay mode tends to result in a
larger branching fraction. This effect is observed in the mesons, but the ratio
calculated among $\Lambda_c^+$ modes is preferred because of the different
baryon structure:
%
\begin{equation}
  g_{\pi\pi} = \frac{{\cal B}(\Lambda_c^+\to\Lambda\pi^+\pi^+\pi^-)}
                    {{\cal B}(\Lambda_c^+\to\Lambda\pi^+)}
             = \frac{3.3}{0.9} = 3.7 \,.
\end{equation}

\begin{table}
\begin{center}
\begin{tabular}{ l  l  c  c  c  }
\hline\hline
$\Lambda_b$ decay & $\Lambda_c$ decay &    ${\cal B}(\Lambda_c^+)$ &     estimate & ${\cal B}(\Lambda_b)$ estimate \\
\hline
$\Lambda_c^+\pi^-$&
            $\Lambda \pi^+$           & $b_1=(9.0\pm 2.8)\ten{-3}$ & $b_1 g_{bc}$ & $2.6\ten{-4}$ \\
$\Lambda_c^+\pi^-\pi^+\pi^-$ &
            $\Lambda \pi^+\pi^-\pi^+$ & $b_2=(3.3\pm 1.0)\ten{-2}$ & $b_2 g_{bc}$ & $9.7\ten{-4}$ \\
\hline
$pD^0\pi^-$(nr)   & $p K^- \pi^+$(nr) & $b_3=(2.8\pm 0.8)\ten{-2}$ & $b_3 g_{bc}$ & $8.2\ten{-4}$ \\
\hline
$p K^-$           &                   &                            & $b_1 g_{bc} g_{bu} g_{\pi K}$ & $8.1\ten{-6}$ \\
\hline
$p\pi^-$          &                   &                            & $b_1 g_{bc} g_{bu}$ & $2.0\ten{-6}$ \\
$p\pi^-\pi^+\pi^-$&                   &                            & $b_2 g_{bc} g_{bu}$ & $7.4\ten{-6}$ \\
\hline\hline
\end{tabular}
\end{center}
\caption{
  Branching ratio estimates for $\Lambda_b$ decays using scale factors
  described in the text.}
\label{tab:lb-branchings}
\end{table}
\index{decay!$\Lambda_b \rightarrow \Lambda_c^+\pi^-$}
\index{decay!$\Lambda_b \rightarrow \Lambda_c^+\pi^-\pi^+\pi^-$}
\index{decay!$\Lambda_b \rightarrow pD^0\pi^-$}

Another factor is used to estimate the branching fractions where the external
$W$ yields a $D_s^{*+}$ rather than a $D_s^+$.  Here the $B^0$ branching
fractions are used
%
\begin{equation}
  g_{*} = \frac{{\cal B}(B^0\to D^{(*)-}D_s^{*+})}
               {{\cal B}(B^0\to D^{(*)-}D_s^+)}
        = \frac{1.0 + 2.0}{0.80 + 0.96}
        = 1.7.
\end{equation}
%
A similar factor is obtained when comparing analogous decays with $\rho^+$ and
$\pi^+$ final states of $B^0$ decays.  Decay modes such as $\Lambda_c^+ \to
\Lambda\rho^+$ have not been observed.

Once certain $b \to c$ branching fractions have been estimated, they are scaled
by
%
\begin{equation}
  g_{bu} = |V_{ub}/V_{cb}|^2 \sim 0.0077
\end{equation}
%
to obtain estimates for corresponding $b \to u$ transitions.  Finally, the
recently measured branching fractions
%
\begin{eqnarray}
  {\cal B}(B^0 \to \pi^+\pi^-) & = &  (4.3\pm 1.6) \ten{-6},, \nonumber \\
  {\cal B}(B^0 \to K^+\pi^-)   & = & (17.2\pm 2.8) \ten{-6}\,,
\end{eqnarray}
%
are used to estimate $\Lambda_b \to p K^-$ from $\Lambda_b \to p \pi^-$:
%
\begin{equation}
  g_{\pi K} = \frac{17.2}{4.3} = 4.
\end{equation}
%
The resulting branching ratio estimates for the different
$\Lambda_b$ decay modes are summarized in Table~\ref{tab:lb-branchings}.


%...............................................................................
\subsection{Detector Simulation}
\index{CDF!detector simulation}

\begin{table}
\begin{center}
\begin{tabular}{ l  c  c  c  c  c  c }
\hline %----------------------------------------------------------------------
\hline %----------------------------------------------------------------------
$\rm B_s$ decay & $\rm \epsilon_{L1} $ &
$ \rm \epsilon_{L2Bs}$ & $\rm \epsilon_{L2Bd} $ & $\rm \epsilon_{L2Tot}$
& $ \rm \epsilon_{fid} $ & $\rm N_{RunII} $ \\
\hline %----------------------------------------------------------------------
$B_s \to D_s \pi$       & 0.025 & 0.0045 & 0.0029 & 0.0050 & 0.0027 & 36900 \\
$B_s \to D_s \pi\pi\pi$ & 0.018 & 0.0032 & 0.0018 & 0.0033 & 0.0011 & 38300 \\
\hline %----------------------------------------------------------------------
$B_s \to D_s D_s$       & 0.019 & 0.0035 & 0.0015 & 0.0037 & 0.0014 &  2500 \\
$B_s \to D_s^* D_s$     & 0.016 & 0.0031 & 0.0011 & 0.0032 & 0.0013 &  5700 \\
$B_s \to D_s^* D_s^*$   & 0.014 & 0.0030 & 0.0011 & 0.0031 & 0.0012 &  5200 \\
\hline %----------------------------------------------------------------------
\hline %----------------------------------------------------------------------
\end{tabular}
\end{center}
\caption{
  Event yields for hadronic $B_s$ decays relevant for $CP$ violation and
  $\Delta\Gamma_s$ measurements. Only the feasible $D_s$ decays to $\phi \pi$,
  $K^* K$ and $\pi\pi\pi$ are considered.}
\label{tab:hadrtrig-bs}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{ l  c  c  c  c  c  r }
\hline %------------------------------------------------------------------------
\hline %------------------------------------------------------------------------
$\Lambda_b$ (sub)decay & $\rm \epsilon_{L1} $   &
$\rm \epsilon_{L2Bs}$  & $\rm \epsilon_{L2Bd} $ &
$\rm \epsilon_{L2Tot}$ & $ \rm \epsilon_{fid} $ &
$\rm N_{RunII} $ \\
\hline %------------------------------------------------------------------------
$\Lambda_b \to \Lambda_c^+ \pi^- (\Lambda_c^+ \to p K^-\pi^+)$
                         & 0.026 & 0.0040 & 0.0029  & 0.0045 & 0.0031 &  2400 \\
$\Lambda_b \to \Lambda_c^+ \pi^-\pi^+\pi^- (\Lambda_c^+ \to p K^-\pi^+)$
                         & 0.017 & 0.0024 & 0.0014  & 0.0026 & 0.0012 &  3400 \\
$\Lambda_b \to p D^0 \pi^- (D^0 \to K^-\pi^+)$
                         & 0.029 & 0.0043 & 0.0036  & 0.0048 & 0.0032 &  6100 \\
$\Lambda_b \to p D^0 \pi^- (D^0 \to K^-\pi^+\pi^-\pi^+)$
                         & 0.020 & 0.0028 & 0.0018  & 0.0030 & 0.0012 &  4300 \\
$\Lambda_b \to p \pi^-$  & 0.056 & 0.0054 & 0.011   & 0.011  & 0.011  &  1300 \\
$\Lambda_b \to p \pi^-\pi^+\pi^-$
                         & 0.030 & 0.0041 & 0.0032  & 0.0046 & 0.0030 &  1300 \\
$\Lambda_b \to p K^-$    & 0.056 & 0.0053 & 0.011   & 0.011  & 0.011  &  5400 \\
\hline %------------------------------------------------------------------------
\hline %------------------------------------------------------------------------
\end{tabular}
\end{center}
\caption{
  Event yields for most sizeable hadronic $\Lambda_b$ decays.}
\label{tab:hadrtrig-lb}
\end{table}

\paragraph{Hadronic Trigger Only}
%================================
\index{CDF!hadronic trigger}
The Level-1 track trigger is based on a set of kinematic cuts originally
developed for two-body decays of neutral $B$ mesons. The Level-1 triggering
algorithm is therefore based on pairs of XFT trigger tracks. To reduce the background of
inelastic collisions relative to $B$ hadron production, only track pairs in
which both tracks have transverse momentum $p_T$ greater than a specified value
are considered. Because of the time that would be spent on combinatorics, an
event with more than six such tracks passes Level-1 automatically.
For real $B^0$ and $B^0_s$ decays of interest, the two highest $p_T$ tracks are correlated
in angle and generally have opposite charge; consequently, the Level-1 requirements
are chosen as follows:
%highest-$p_T$ tracks often have opposite charge and are correlated in angle, and
%thus, an event is triggered upon at Level-1 if it has a pair of XFT tracks which
%satisfies the following requirements:
%
\begin{list}{$\bullet$}{\setlength{\topsep}{3pt}\setlength{\itemsep}{0pt}
                        \setlength{\parsep}{0pt}}%
\item two tracks having opposite charge
\item individual track $p_T > 2.0~\GeV/c$
\item $p_T,1 + p_T,2 > 5.5~\GeV/c$
\item $\delta \phi   < 135 \deg $
\end{list}

The Level-2 trigger is based on tracking information provided by the
SVT~\cite{Ref:TDR,Ref:PAC}. One application for the hadronic $b$ trigger
is the decay $B^0 \to \pi^+\pi^-$, where the two pions give oppositely-charged
tracks with high transverse momenta and large impact parameters.  The trigger is
also used to select other multibody hadronic $b$ decays, but due to the
different kinematics of these decays, the Level-2 selection criteria are
modified to optimize the efficiencies ~\cite{Ref:PAC}.  An event
passes the Level-2 track trigger if it satisfies either option A) or option
B)\\[10pt]
%
\phantom{HAL}
\begin{minipage}[t]{0.46\textwidth}
A) Multi hadronic $B$ trigger
\begin{list}{$\bullet$}{\setlength{\topsep}{3pt}\setlength{\itemsep}{0pt}
                        \setlength{\parsep}{0pt}}%
\item$120\um <  | d_0 | <  1 \mm$
\item$2\deg < \delta\phi <  90\deg$
\item$ p_T \cdot X_v > 0$
\end{list}
\end{minipage} \hfill
\begin{minipage}[t]{0.46\textwidth}
B) Hadronic pair trigger
\begin{list}{$\bullet$}{\setlength{\topsep}{3pt}\setlength{\itemsep}{0pt}
                        \setlength{\parsep}{0pt}}%
\item $100\um <  | d_0 | <  1 \mm$
\item $20\deg < \delta\phi < 135\deg$
\item $p_T \cdot X_v > 0$
\item $d_{0,B} < 140 \um$
\end{list}
\end{minipage}
%
$ $\\[15pt]
For an event to be useful in offline analysis after it passes the trigger, all the
$b$ hadron decay products have to be reconstructible in the detector. The requirement for a $b$
hadron to be considered reconstructible in this simulation is that all its
stable, charged daughter particles are within $|\eta| < 1$ and have transverse
momenta greater than $400~{\MeV}/c$.
These requirements are probably conservative in two ways: first, track
reconstruction in Run~II will be possible over a larger $\eta$ range;
stand-alone silicon tracking will probably be possible up to $|\eta| <
2$. Second, the reconstruction efficiency for the COT will be similar to that for the CTC
during Run~I. The efficiency for track reconstruction
in the CTC extended down to $p_T \simeq 200~\MeV/c$ and rose over the
transverse momentum range $200~\MeV/c < p_T < 400~\MeV/c$, reaching about
93\% for tracks with $p_T > 400~\MeV/c$.

Applying the trigger to various $B_s$ and $\Lambda_b$ decays, we estimate the event yields at
the two trigger levels. A summary of these estimates is given in
Tables~\ref{tab:hadrtrig-bs} and \ref{tab:hadrtrig-lb} for $B_s$ and
$\Lambda_b$, respectively.

\paragraph{Hadronic Trigger Combined with Lepton}
%================================================
Apart from the purely hadronic trigger there is the possibility of using the
hadronic trigger in conjunction with the lepton triggers. Therefore, a single
lepton requirement is combined with the requirement of a displaced track in the
SVT. This trigger option is studied for the semi-inclusive $B_s \to \nu \ell D_s
X$ sample.

The additional requirement of a displaced track allows a lower threshold for
the lepton momentum, while keeping the trigger rate at a reasonable level. The trigger
cross section for an 8~{\GeV} inclusive electron trigger would need to be prescaled
in Run~II.  However, it is possible to lower the cross section for a 4~{\GeV}
electron trigger below 100~nb by adding the displaced track found by SVT with
$p_T>2~\GeV/c$ and $d_0>120~\mu$m.

Since Run~I data are considered to be most reliable for predictions, the signal
yield for Run~II CDF is obtained by scaling the Run~I analysis results with the
ratio of the acceptances between Run~I and Run~II. The acceptance ratio between
Run~I and Run~II is obtained using a Monte Carlo sample of semileptonic $B_s$
decays containing a $D_s$.
The SVT tracking efficiency is assumed to be $\sim 75\%$ per track, or
$56\%$ for 2-tracks.

The $\ell + D_s$ sample composition is assumed to be,
%
\begin{list}{$\bullet$}{\setlength{\topsep}{3pt}\setlength{\itemsep}{0pt}
                        \setlength{\parsep}{0pt}}%
\item $e D_s$ : $\mu D_s$ = 50\% : 50\%
\item $B_s \to \ell \nu D_s$ : $\ell \nu D_s^*$ : $\ell \nu D_s^{**}$ =
  2 : 5 : 0; ~~~~ $D_s^{**}$ usually decays to $D^{0,\pm}$
\end{list}
%
The $E_T$ and SVT $d_0$ resolutions are taken to be of $14\%/\sqrt{E_T}$ and
$35~\mu$m, respectively.
All tracks ($\ell$, $K$, and $\pi$) are required to have $p_T > 400~\MeV/c$, and
to be in the fiducial volume of the tracking detector. The silicon vertex
detector coverage is $|z|<30$~cm in Run~I and $|z|<45$~cm in Run~II. The
standard analysis Run~I cuts are applied to the final state particles, namely $p_T(K) >
1.2~\GeV/c$, $p_T(\pi) > 0.8~\GeV/c$, and $3~\GeV/c^2 < M(\ell D_s) <
5~\GeV/c^2$.

The event yields for different lepton $p_T$ values are summarized in
Table~\ref{tab:hadrlepttrig-bs}, which shows signal yields per 2~{\ifb}.
Choosing a value of $3~\GeV/c$ lepton $p_T$, roughly 40k semileptonic $B_s$
decay are obtained in 2~{\ifb} of integrated luminosity.

\begin{table}
\begin{center}
\begin{tabular}{ l  c  r }
\hline\hline
Trigger             & Run~II/Run~I & $N_\mathrm{Run~II}$ \\
\hline
$8~\GeV \ell$       & 1.0          & 14000 \\
$2~\GeV \ell$ + SVT & 5.9          & 64000 \\
$3~\GeV \ell$ + SVT & 4.0          & 43000 \\
$4~\GeV \ell$ + SVT & 2.7          & 30000 \\
$5~\GeV \ell$ + SVT & 1.9          & 21000 \\
\hline\hline
\end{tabular}
\end{center}
\caption{
  Event yields corresponding to 2~{\ifb} for semileptonic $B_s$ decays ($B_s \to
  \nu \ell D_s X$) for different lepton $p_T$ trigger thresholds.}
\label{tab:hadrlepttrig-bs}
\end{table}


\paragraph{Rate Estimates -- Hadronic Trigger}
%=============================================
Because the trigger rates depend on the way in which the Tevatron is operated in
Run~II, different XFT trigger cuts were considered for three different running
scenarios.  Scenario A corresponds to a luminosity of less than $1\times
10^{32}\;{\rm cm^{-2}s^{-1}}$ with collisions every 396~ns, while scenarios B
and C correspond to luminosities of $1-2\ten{32}~{\rm cm^{-2}s^{-1}}$ with
collisions every 132~ns and 396~ns, respectively.  The cuts considered for each
scenario are listed in Table~\ref{tab:hadrtrig-rate} along with the total cross
section.  These expectations were derived using tracks recorded in Run~I with
additional hit occupancy close to the beam axis generated using the
MBR~\cite{Ref:MBR} Monte Carlo program.
%
\begin{table}
\begin{center}
\begin{tabular}{ c c c c }
\hline\hline
            & Scenario A & Scenario B & Scenario C \\
\hline
${\cal L}$  &    $<1\times 10^{32}\;{\rm cm^{-1}s^{-1}}$ &
              $1$-$2\times 10^{32}\;{\rm cm^{-1}s^{-1}}$ &
              $1$-$2\times 10^{32}\;{\rm cm^{-1}s^{-1}}$ \\
BX interval           &          396 ns &         132 ns   & 396 ns \\
\hline
$p_T^{(1)},p_T^{(2)}$ &     $>2~\GeV/c$ & $>2.25~\GeV/c$   & $>2.5~\GeV/c$\\
$p_T^{(1)}+p_T^{(2)}$ &   $>5.5~\GeV/c$ & $>6~\GeV/c$      & $>6.5~\GeV/c$ \\
$\delta \phi$         &    $<135^\circ$ & $<135^\circ$     & $<135^\circ$\\
\hline
cross section         & $252\pm18~\mu$b & $152\pm14~\mu $b & $163\pm16~\mu$b \\
%trigger rate          &                 &                  & \\
\hline\hline
\end{tabular}
\end{center}
\caption{
  Level-1 XFT trigger cuts and cross sections for the three Tevatron operating
  scenarios considered.}
\label{tab:hadrtrig-rate}
\end{table}
%
The trigger cuts provide trigger rates which are compatible with the total
Level-1 bandwidth of approximately 50~kHz.

At Level-2, the impact parameter information associated with the tracks is
available, and the cuts described above are used to select $b$ hadron decays.
The requirements are that the impact parameters of both tracks satisfy $120~{\rm
  \mu m} < |d| < 1 \;{\rm mm}$, that their point of intersection occurs with a
positive decay length, and that their opening angle is further restricted to
$\delta \phi < 90^\circ$.  The trigger cross sections are reduced to
approximately 489, 386 and 283~nb for scenarios A, B and C, respectively, and
produce Level-2 trigger rates between 38~Hz and 67~Hz. This is well within the
available Level-2 bandwidth of 300~Hz.

For efficiency estimates in the sections below, scenario A has been chosen.
When implementing trigger option B the numbers of expected events do not
vary significantly compared with the uncertainties quoted above. The
trigger scenario C is not likely to be implemented, asssuming that the Tevatron will be
upgraded to 132~ns bunch spacing.

%===============================================================================
\boldmath
\section{Projections for $\Delta m$}
\unboldmath

%-------------------------------------------------------------------------------
\boldmath
\subsection[$B_s$ mixing measurement at CDF]
{$B_s$ mixing measurement at CDF
$\!$\authorfootnote{Authors: M.~Jones, Ch.~Paus, M.~Tanaka.}}
\unboldmath
\index{CDF!$B_s$ mixing}

The probability that a $B_s$ meson decays at proper time $t$ in the same state,
or has mixed to the $\overline{B}_s$ state is given by
%
\begin{eqnarray}
  P_{\rm unmix}(t) & = & \frac{1}{2}\left(1 + \cos\Delta m_s t\right)\,, \nonumber \\
  P_{\rm mix}(t)   & = & \frac{1}{2}\left(1 - \cos\Delta m_s t\right)\,,
\end{eqnarray}
%
where the mixing frequency, $\Delta m_s$, is the mass difference between the
heavy and light $CP$ eigenstates.

The canonical $B_s$ mixing analysis, in which oscillations are observed and the
mixing frequency, $\Delta m_s$, is measured, proceeds as follows. The $B_s$
meson flavor at the time of its decay is determined by reconstructing a flavor
specific final state. The proper time, $t=m_{B^0_s}L/p c$, at which the decay
occurred is determined by measuring the decay length, $L$, and the $B_s$
momentum, $p$.  Finally, the production flavor must be tagged in order to
classify the event as being mixed or unmixed at the time of its decay.
Oscillations manifest in a time dependence of, for example, the mixed asymmetry:
%
\begin{equation}
  A_{\rm mix}(t) = \frac{N_{\rm mixed}(t)-N_{\rm unmixed}(t)}
                        {N_{\rm mixed}(t)+N_{\rm unmixed}(t)}.
\end{equation}

In practice, the production flavor will be correctly tagged with a probability
${\cal P}_{\rm tag}$ which is significantly smaller than unity. The functional
form of the mixed asymmetry follows
%
\begin{equation}\label{eqn-amix}
  A_{\rm mix}(t) = -D \cos \Delta m_s t
\end{equation}
%
with the dilution, $D$, related to $P_{\rm tag}$ by $D=2P_{\rm tag}-1$.  The
mixing frequency is determined for example by fitting the measured asymmetry to
a function of this form.

So far, $B_s$ oscillations have not been observed experimentally, and the lower limit
on $x_s$ is above 15. This means that $B_s$ mesons oscillate much more rapidly
than $B^0$ mesons. The rapidity of the $B_s$ oscillation implies a
significant difference in the experimental requirements for the $B^0$ and $B_s$ analyses. The
limiting factor in $B^0$ mixing analyses is solely the effective tagging
efficiency, which is equivalent to the effective statistics. In $B_s$ mixing
measurements the resolution of the proper time becomes another very critical
issue. To determine the proper time, not only the positions of primary and
secondary vertices have to be measured precisely, but also the measurement of the
$B_s$ momentum is crucial. Therefore, it is desirable to have fully exclusive
final states such as $B_s \to D_s \pi (D_s \to \phi \pi, \phi \to KK)$.
Semileptonic $B_s$ decays have the intrinsic disadvantage that the neutrino
momentum is undetected.

In Run~I CDF reconstructed 220 and 125 $B_s$ semileptonic decay events
with fully reconstructed $D_s \to \phi \pi$ and $D_s \to K^* K$ channels,
respectively, in low $p_T$ ($>8~\GeV/c$) inclusive lepton ($e$ and $\mu$)
trigger samples~\cite{Ref:SemiBsLifetime}.  An additional 600 semileptonic $B_s$
decays were used, reconstructed in the $D_s \to \phi X$ + track channel. Those
events were part of the dilepton ($\mu\mu$ and $e\mu$) trigger samples, where
the second lepton was used for the $B$ flavor tag.  The best limit on $x_s$ was
given by the dilepton trigger dataset~\cite{Ref:SemiBsMixing}.

In Run~II much more statistics will be available using the lepton trigger, the
lepton trigger plus one secondary vertex track and the all hadronic trigger.
From the event yield estimates in Tables~\ref{tab:hadrlepttrig-bs} and
\ref{tab:hadrtrig-bs}, we expect 40k events in the lepton plus displaced track and 75k
events in the all hadronic trigger.

In the following four sections the measurements of $B_s$ mixing using
semileptonic or hadronic decays are discussed. Since the flavor tagging and
the sensitivity estimates are very similar for the
semileptonic and hadronic $B_s$ decay samples, they will be discussed first.

%...............................................................................
\boldmath
\subsubsection{Projections for Sensitivity to $x_s$}
\unboldmath

The mixing frequency
\index{B mixing@$B_s$ mixing!mixing frequency}
can be determined by calculating, for example, a maximum
likelihood function derived from the measured and expected asymmetries and
minimizing this function with respect to $\Delta m_s$.  The significance of an
observation of mixing is quantified in terms of the depth of this minimum
compared with the second deepest minimum or some asymptotic value at large
$\Delta m_s$.  To a good approximation, the {\it average} significance
\index{B mixing@$B_s$ mixing!average significance}
is given
as
%
\begin{equation}\label{eqn-sig}
  {\rm Sig}(\Delta m_s) = \sqrt{\frac{N\epsilon D^2}{2}}
                          e^{-(\Delta m_s \sigma_t)^2/2}\sqrt{\frac{S}{S+B}}
\end{equation}
%
where $N=S$ is the number of reconstructed $B_s$ signal events, $S/B$ is the
signal-to-background ratio, $\epsilon$ is the efficiency for applying the flavor
tag with associated dilution $D$, and $\sigma_t$ is the average resolution with
which the proper time is measured. This definition is essentially the same as
what would be used to define $n \sigma$ confidence intervals for a Gaussian
probability density function.

Given estimates for these parameters, the limit of sensitivity is defined as the
maximal value of $\Delta m_s$ for which the significance is above a specified
value. The studies described here use the canonical 5 standard deviations to
define an unambiguous observation of mixing. In the following sections the
estimates for $N$, $\epsilon D^2$, $\sigma_t$ and $S/B$ are described.

%...............................................................................
\subsubsection{Flavor Tagging Efficiency}
\index{CDF!flavor tagging}
\index{flavor tagging!CDF}

In Run~II, a Time-of-Flight detector will provide CDF with the ability to
distinguish kaons from pions at the $2\sigma$ level below a momentum of about
$1.6~\GeV/c$.  This allows two new flavor tags to be implemented which rely on
the charge of kaons identified in the event to tag the production flavor of the
$B_s$. As a summary, in Table~\ref{tab:cdf-taggers} the tagging efficiency for $B_s \to D_s
\pi$
\index{decay!$B_s \rightarrow D_s^- \pi^+$}
expected for Run~II is compared to equivalent numbers obtained in Run~I.
We compare the standard figure of merit for each tagger, namely $\epsilon D^2$. The two
additional kaon taggers are briefly explained below.

\begin{table}
\begin{center}
\begin{tabular}{ l  r  r  }
\hline\hline
Method     & Run~I -- $\epsilon D^2$ & Run~II -- $\epsilon D^2$ \\
\hline
SLT        &                  1.7\% &                     1.7\% \\
JQT        &                  3.0\% &                     3.0\% \\
SST(kaon)  &                  1.0\% &                     4.2\% \\
OSK        &                    --- &                     2.4\% \\
\hline
Total      &                  5.7\% &                    11.3\% \\
\hline\hline
\end{tabular}
\end{center}
\caption[Comparison of the various flavor taggers in terms of the 
$\epsilon D^2$ parameter between Run~I and expectations for Run~II.]
{Comparison of the various flavor taggers in terms of the $\epsilon D^2$
  parameter between Run~I and expectations for Run~II. The most significant
  differences are the kaon taggers based on the new Time-of-Flight detector.}
\label{tab:cdf-taggers}
\end{table}

\paragraph{Opposite Side Kaon Tag}
\index{CDF!flavor tagging!opposite side kaon}
%=================================
Due to the $b\to c\to s$ weak decays, $B$-mesons containing a $b$ quark will be
more likely to contain a $K^-$ in the final state than a $K^+$.  As for all the
other opposite side taggers the determination of the quark flavor on the
opposite side determines the flavor on the vertexing side,
since $b\overline b$ quarks are produced in pairs.

For the opposite side kaon tagging, kaon candidates are selected that are well separated from
the reconstructed $B_s$ decay. Kaon candidates coming
from a $b$ hadron decay are separated from prompt kaons by requiring a large
impact parameter. These requirements are implemented by imposing an isolation
cut of
%
\begin{equation}
  \Delta R_{\eta \phi}=\sqrt{\Delta \eta^2+\Delta \phi^2}>1
\end{equation}
%
and a cut on the combined transverse and longitudinal impact parameters of
%
\begin{equation}
  \chi^2_{d_0z_0} = \frac{d_0^2}{\sigma^2_{d_0}}+
                    \frac{z^2_0}{\sigma^2_{z_0}} > 9.
\end{equation}
%
Tagging the production flavor of the $B_s$ using the charge of the kaon selected
in this way gives a contribution to the tagging efficiency of
%
\begin{equation}
  \epsilon D^2 = (2.4\pm 0.2)\%.
\end{equation}

\paragraph{Same Side Kaon Tag}
\index{CDF!flavor tagging!same side kaon}
%=============================
Same side kaon tagging in $B_s$ decays is the equivalent of same side pion
tagging in $B^0$ decays. In the hadronization process, when a $B_s$ meson is
produced, an $s\bar{s}$ pair must be popped from the vacuum during
fragmentation. The remaining $s$ or $\bar{s}$ quark is likely to join with
a $\bar{u}$ or $u$ quark to form a charged kaon. The charge of the kaon thus
depends on the flavor of the $B_s$ meson at production.

To estimate the $\epsilon D^2$ of the same side kaon tagging algorithm the same
side pion tagging algorithm is extended with particle identification
using Time-of-Flight information. It should be noted that $\epsilon D^2$ for this
algorithm is strongly dependent on the momentum spectrum of the $B_s$ meson.
Therefore, a rough simulation of the Level-1 and Level-2 triggers are applied to
the Monte Carlo sample. The $B_s~p_T$ spectrum of this event sample peaks at
around $10~\GeV/c$, and $\epsilon D^2$ is estimated to be
%
\begin{equation}
  \epsilon D^2 = (4.2\pm 0.3)\%.
\end{equation}

%-------------------------------------------------------------------------------
\boldmath
\subsubsection{$B_s$ Mixing with Semileptonic Decays}
\unboldmath
\index{B mixing@$B_s$ mixing!semileptonic}

As discussed in the introduction the key issue for semileptonic $B_s$
decays is the resolution of the proper time measurement. Including the estimates for
flavor-tagged event
yields, the $x_s$ sensitivity determination is straightforward.

\paragraph{Proper Time Resolution}
\index{CDF!proper time resolution!semileptonic}
%=================================
The proper decay time in semileptonic decays is derived as follows
%
\begin{equation}
  ct = \frac{L_T(B_s) M(B_s)}{p_T(B_s)} =
           \frac{L_T(B_s) M(B_s)}{p_T(\ell D_s)} \cdot K,, \qquad
  K  = \frac{p_T(\ell D_s)}{p_T(B_s)} \,,
\end{equation}
%
where the transverse decay length, $L_T(B_s)$, and transverse momentum, $p_T(\ell
D_s)$, are measured from data. The $K$ factor which is used as an average
correction for the the incomplete $B_s$ reconstruction is obtained from Monte
Carlo samples, and $M(B_s)$ is the $B_s$ mass~\cite{Ref:PDG}. The proper time
resolution is given as
%
\begin{equation}
  \sigma_t = \sigma_{t_0} \oplus t \cdot \frac{\sigma K}{K},
\end{equation}
%
where the constant term ($\sigma_{t_0} \sim 60$~fs) is due to the beam spot and the vertex
detector resolution, and the $K$ factor resolution ($\sigma K/K \sim 14\%$) is
due to the momentum spectrum of the undetected particles, namely
the neutrino in the $B_s$ decay or the photon/$\pi^0$ in the subsequent $D_s^*$
decay.

\begin{figure}
\begin{center}
  \mbox{\epsfig{file=ch8/cdf_cor-kfactor.eps,width=0.6\textwidth}}
  \caption{
    $K$ factor as a function of the $M_{\ell D_s}$ (top), and $K$ factor
    distribution after the $M_{\ell D_s}$ correction (bottom).}
  \label{fig:kfac-mld}
\end{center}
\end{figure}

The $K$ factor
\index{B mixing@$B_s$ mixing!K factor}
depends strongly on the lepton
+ $D_s$ invariant mass, $M(\ell D_s)$;
this dependence is shown in Figure~\ref{fig:kfac-mld} for the $D_s$ and $D_s^*$ channels.
Since the invariant lepton +
$D_s$ mass is measured, this dependence is corrected for on an event-by-event
basis to improve the $K$ factor resolution.

For the $B_s \to \ell D_s X$, $X = \nu + x$ channel, the following energy and
\index{decay!$B_s \rightarrow D_s^- \ell^+ \nu$}
momentum conservation rules are given
%
\begin{eqnarray}
  E_{B_S} & = & E_{\mu D_{S}} + E_{X}, \nonumber\\
  p_X     & = & |\vec{p}_{X}|^2= |\vec{p}_{B_S}-\vec{p}_{\mu D_S}|^2
          = p_{B_S}^2 + p_{\mu D_S}^2 - 2p_{B_S} p_{\mu D_S} \cos{\Theta},
\end{eqnarray}
%
where $\Theta$ is the angle in the laboratory frame between the $B_s$ and
$\ell+D_s$ directions, which are obtained using the 3D vertex information of the
Run~II SVX. By assuming $M_X=0$ the quadratic equation can be solved exactly.
Notice that the equation gives generally two solutions, $K^- < K^+$. The equation
sometimes has nevertheless no physical solutions due to the finite detector
resolution. Furthermore, the equation does not give the correct answer for the $B_s
\to \ell\nu D_s^*$ channel because of the missing photon,
$M_{X}=M_{\nu\gamma}>0$.

The $K$ factor distributions after all the trigger and offline selection cuts
are shown in Figure~\ref{fig:kfac-mld-3d}.  A vertex resolution of
$\sigma_{t_0}$ = 60~fs and $\sigma_{L_z}$ = 50~$\mu$m has been assumed.
The $D_s$ and the $D_s^*$ channels are displayed in the upper and lower plots,
respectively.  The probability for the quadratic equation to have a real
solution is approximately 50 percent. However, the correction still improves the
$K$ factor resolution if there is a physical solution.

\begin{figure}
\begin{center}
  \mbox{\epsfig{file=ch8/cdf_3dvc_ds_dsst_wr.eps,width=0.6\textwidth}}
  \caption{
    $K$ factor distributions after 3D vertexing correction with vertex detector
    resolutions of $\sigma_{t_0}$ = 60~fs and $\sigma_{L_z}$ = 50~$\mu$m.}
  \label{fig:kfac-mld-3d}
\end{center}
\end{figure}

Since the 3D vertexing correction strongly depends on the $\Theta$ resolution,
it improves for longer $B_s$ decay lengths.
In Figure~\ref{fig:time-res} the $K$ factor and time resolutions are shown as
functions of the proper decay time for the $D_s$ channel after the $M_{\ell D_s}$
correction and 3D vertexing correction.

To perform the 3D vertexing correction, it is assumed that the correct solution,
$K^-$ or $K^+$, is known, and that the $M(\ell D_s)$ corrected $K$ factor is
used if there are no physical solutions. The $K$ factor resolution is
significantly improved for the longer decay time events. Both channels $D_s$
and $D_s^*$ show similar results.  Unfortunately, the improved $K$ factor resolution
is not sufficient to greatly improve the sensitivity to $x_s$; practically
all of the sensitivity comes from the very short decay length events.

\begin{figure}
\begin{center}
  \mbox{\epsfig{file=ch8/cdf_3dvc_ds_pt_t.eps,width=0.6\textwidth}}
  \caption{
    $K$ factor and time resolution for $B_s \to \ell\nu D_s$ decays.}
  \label{fig:time-res}
\end{center}
\end{figure}

\paragraph{Backgrounds}
%======================
In Run~I the signal to background ratio in $B_s$ reconstruction in the
semileptonic channels was typically 1:1. Since the kinematics of the Run~II
event sample will be somewhat different, a conservative signal to background
ratio of 1:2 is assumed in the following.

\paragraph{Projected Sensitivity}
%================================
The parameters which influence the projected $B_s$ mixing sensitivity,
calculated using equation~\ref{eqn-sig}, are summarized as follows:
%
\begin{eqnarray*}
  N(B_s)       : & &    43k~\mathrm{see~Table~\ref{tab:hadrlepttrig-bs}}\\
  \epsilon D^2 : & & 11.3\%~\mathrm{see~Table~\ref{tab:cdf-taggers}}\\
  \sigma_t     : & &        \mathrm{see~Figure~\ref{fig:time-res}}\\
  S/B          : & &    1:2~\mathrm{as~explained~above}
\end{eqnarray*}

\begin{figure}
\begin{center}
  \mbox{\epsfig{file=ch8/cdf_bs-reach-sltt.eps,width=0.47\textwidth}}\hfill
  \mbox{\epsfig{file=ch8/cdf_bs-reach-sllt.eps,width=0.47\textwidth}}
  \caption[Sensitivity for measuring $x_s$ of the $B_s$ semileptonic decays for
the lepton plus displaced track trigger (left) and the two track trigger
(right).]{
    Sensitivity for measuring $x_s$ using $B_s$ semileptonic decays for the
    hadronic two track trigger (left) and the
    lepton plus displaced track trigger (right).
   The dashed lines show the significance after $M_{\ell D_s}$ correction,
    the solid lines after further applying the 3D vertex correction.}
  \label{fig:bs-semilep-sens}
\end{center}
\end{figure}

The analysis described above for the $3~\GeV/c$ lepton plus displaced track
triggers can easily be extended to the hadronic two-track triggers
defined in Table~\ref{tab:hadrtrig-rate}, where the trigger path
is satisfied by a semileptonic decay; given the large ($\sim 20\%$)
semileptonic combined branching ratio, the hadronic trigger turns out to be
highly efficient for semileptonic decay modes.
The significances for measuring $x_s$ for the two-track trigger (left)
and the lepton plus displaced track (right) are shown in
Figure~\ref{fig:bs-semilep-sens}. The dashed lines show the significance after
the $M_{\ell D_s}$ correction, and the solid lines after
the additional 3D vertexing correction. The $x_s$ reach of the semileptonic decay
sample is estimated to be about 30 for an observation with five standard
deviations. This is significantly less than that for the fully reconstructed hadronic channels,
which are
discussed below, but it does provide an independent trigger path.

%...............................................................................
\boldmath
\subsubsection{$B_s$ Mixing with Hadronic Decays}
\unboldmath
\index{B mixing@$B_s$ mixing!hadronic}

The fully hadronic event sample is particularly important for $B_s$ mixing
analyses since the fully reconstructed decays $B_s \to D_s^- \pi^+$ and $B_s \to
D_s^- \pi^+ \pi^- \pi^+$ have excellent proper time resolution, much smaller
than the expected period of oscillation.

\paragraph{Proper Time Resolution}
\index{CDF!proper time resolution!hadronic}
%=================================
The proper time of a $B_s$ decay is calculated from the measured decay length,
and the reconstructed $B_s$ momentum.  The uncertainty on the proper time is
then given by
%
\begin{equation}
  \sigma_t = t\, \sqrt{\left(\frac{\sigma_L}{L}\right)^2 +
                    \left(\frac{\sigma_P}{P}\right)^2}.
\end{equation}
%
To resolve the rapid oscillations of the $B_s$ it is generally required that
this resolution not be significantly larger than the period of oscillation.  For
partially reconstructed semileptonic $B_s$ decays, the uncertainty in the
momentum is the limiting factor in the mixing analyses.  However, for the fully
reconstructed $B_s$ decays obtained using the XFT+SVT triggers, the momentum
uncertainty will be less than 0.4\%. This does not contribute significantly to
the overall proper time resolution and has been ignored in the projections
described below.

\paragraph{Backgrounds}
%======================
To date, no hadron collider experiment has operated with a displaced track based
trigger. Hence, the level of backgrounds to be expected in the $B_s$ sample is
uncertain. Data recorded by CDF in Run~I based on the single lepton triggers are
used to study the purity of the $B_s$ signal after imposing the XFT and SVT
trigger cuts on the opposite side $b$ hadron decays.

It has been observed that even a modest decay length cut suppresses the light
flavor contribution significantly. Therefore, the main concern is that the
signal is not overwhelmed by background from events containing real $b$- and
$c$-quarks.

Because of the small branching ratios of the $B_s \to D_s^+\pi^-,
D_s^+\pi^+\pi^-\pi^-$ decays, few, if any, such decays are expected to be
present in the Run~I data after imposing the XFT and SVT trigger cuts.  A
similar set of cuts with higher efficiency was used to search for the hadronic
$B^0$ and $B^+$ decays in the $D\pi$ final states. As a result of those studies
it is concluded that a signal to background ratio of 1:1 should be achievable.
To see the dependence of the significance on this parameter, this ratio is
varied between 1:2 and 2:1 in the following projections.

\paragraph{Projected Sensitivity}
%================================
The parameters which influence the projected $B_s$ mixing sensitivity,
calculated using equation~\ref{eqn-sig}, are summarized as follows:
%
\begin{eqnarray*}
  N(B_s)       & = &     75k~\mathrm{see~Table~\ref{tab:hadrlepttrig-bs}}\\
  \epsilon D^2 & = &  11.3\%~\mathrm{see~Table~\ref{tab:cdf-taggers}}    \\
  \sigma_t     & = &   0.045~\mathrm{ps}                                 \\
  S/B          & = & 1:2-2:1~\mathrm{as~explained~above}
\end{eqnarray*}
%
The results are again presented in terms of the dimensionless mixing parameter
$x_s=\Delta m_s \tau_{B_s}$ where the $B_s$ lifetime of
$1.54~\mathrm{ps}$~\cite{Ref:PDG99} is used.  In addition to the analytic
expression for the sensitivity, Equation~\ref{eqn-sig}, an alternative analysis
has been performed using a series of simulated
Monte Carlo experiments. The lifetime distributions of mixed and
unmixed decays are generated and the mixed asymmetry is fitted to
Equation~\ref{eqn-amix}. An example of the mixed asymmetry distribution and the
negative log-likelihood curve is shown in Figure~\ref{fig:example}. The negative
log-likelihood curve is shown as a function of $x_s$, obtained from one of these
Monte Carlo experiments.
%
\begin{figure}
  \mbox{\epsfig{file=ch8/cdf_bsosci.eps,     width=0.47\textwidth}}
  \mbox{\epsfig{file=ch8/cdf_example-sig.eps,width=0.47\textwidth}}
  \caption[Example of a single toy Monte Carlo experiment:
the mixed asymmetry distribution, and negative log-likelihood
from the fit as a function of $x_s$]{
    Example of a single toy Monte Carlo experiment:
the mixed asymmetry distribution (left) and negative log-likelihood
from the fit as a function of $x_s$ (right)}
\label{fig:example}
\end{figure}
%
The comparison of the analytic expression with the averages of many Monte Carlo
simulations indicates that the analytic approximation is very good.

The average significance for
\index{B mixing@$B_s$ mixing!average significance in CDF}
$B_s$ oscillation measurements is shown in
Figure~\ref{fig:sig}. Various event yields and signal-to-background scenarios
are considered.  As stated above, the default event yield is 75k
events and the signal-to-background fraction is 1:1.

\begin{figure}
\begin{center}
  \mbox{\epsfig{file=ch8/cdf_bs-reach-nevt.eps,width=0.47\textwidth}}\hfill
  \mbox{\epsfig{file=ch8/cdf_bs-reach-stob.eps,width=0.47\textwidth}}
  \caption[Average significance of mixing measurements expected as a function of
$x_s$ for various event yields and signal-to-background ratios.]{
    Average significance of mixing measurements expected as a function of the
    mixing parameter $x_s$ for various event yields (left) and
    signal-to-background ratios (right). The default is 75k events at a
    signal-to-background ratio of 1:1. The shaded area is excluded by the
    combined world lower limit on $x_s$.}
  \label{fig:sig}
\end{center}
\end{figure}

From the analytic expression, equation~\ref{eqn-sig}, the following 5 standard deviation sensitivity limits
are derived:
%
\begin{eqnarray*}
  \mbox{Maximum: 75k events}\; x_s &=& \left\{\begin{array}{lcl}
                                                74 & & \mbox{for $S/B=2:1$}\\
                                                69 & & \mbox{for $S/B=1:2$}
                                              \end{array}\right. \\
  \mbox{Maximum: S:B = 1:1}\;  x_s &=& \left\{\begin{array}{lcl}
                                                73 & & \mbox{for 125k events}\\
                                                59 & & \mbox{for  25k events}
                                              \end{array}\right. \\
\end{eqnarray*}
%
The Monte Carlo samples give very similar results as indicated above. It
is concluded that even in the worst case the reach is $x_s \sim 60$.

Fits to various experimental results which assume the Standard Model indicate
that $22.0<x_s<30.8$ at the 95\% confidence level~\cite{Ref:CKM-MATRIX}.  If
mixing occurs near the frequency expected in the Standard Model, it should be
easily observable by CDF in Run~II.  Figure~\ref{fig:lumi-error} (left) shows the
luminosity required to achieve an observation with an average significance of 5
standard deviations as a function of $x_s$.  The figure indicates that if mixing
occurs within the context of the Standard Model, then it should be observed
with a small fraction of 2 fb$^{-1}$, with CDF in a fully operational state.

\begin{figure}
\begin{center}
  \mbox{\epsfig{file=ch8/cdf_bs-reach-lumi.eps,width=0.47\textwidth}}\hfill
  \mbox{\epsfig{file=ch8/cdf_bs-reach-erro.eps,width=0.47\textwidth}}
  \caption[Luminosity required to achieve a 5 standard deviation observation of
mixing (left) and the statistical uncertainty as a function of $x_s$
(right).]{
    Luminosity required to achieve a 5 standard deviation observation of mixing
    (left) and the statistical uncertainty as a function of the mixing
    parameter, $x_s$ (right). The curves on the right are calculated using
    Equation~\ref{eq:error}.}
  \label{fig:lumi-error}
\end{center}
\end{figure}

While the significance of an observation of mixing is determined from the depth
of the minimum in the negative log-likelihood curve, the uncertainty on the
measured value of $x_s$ is determined by how sharp this minimum is.  In the case
of rapid oscillations, many periods will be reconstructed over a few $B_s$
lifetimes and the minimum is expected to be very sharp leading to a small
uncertainty.

The average uncertainty is described approximately by the analytic expression
%
\begin{equation}\label{eq:error}
  \frac{1}{\sigma_{x_s}} = \sqrt{N \epsilon D^2} e^{-(x_s \sigma_t/\tau)^2/2}
                           \sqrt{\frac{S}{S+B}}.
\end{equation}
%
The expected uncertainty from the analytic formula versus the mixing parameter,
$x_s$, is shown in Figure~\ref{fig:lumi-error} (right(). This formula is confirmed using
a series of Monte Carlo samples.


%This approximation is quite good, as can be seen from figure~\ref{fig:errxs} in
%which the calculated uncertainty is compared with the width of the distribution
%of fitted values from {\it ensembles} of Monte Carlo experiments.

%-------------------------------------------------------------------------------
\boldmath
\subsection[$B_s$ mixing measurement at \dzero]
{$B_s$ mixing measurement at \dzero
$\!$\authorfootnote{Authors: N.~Cason, R.~Jesik, and N.~Xuan.}}
\unboldmath
\index{\dzero!$B_s$ mixing}

The expected luminosity of the Tevatron, $2\times10^{32}$ cm$^{-2}$s$^{-1}$, in
Run~II will lead to a huge rate for $b\,\bar{b}$ production, $\sim 10^{11}$
events/year. These enormous statistics combined with the upgraded detector will
allow us to search for $B_{s}$ mixing.  The resulting measurement of $\Delta
m_s$, when used to determine the ratio $\Delta m_d$/$\Delta m_s$ using the
well-measured value for $\Delta m_d$, gives a theoretically clean measurement of
$|V_{td}|^2$/$|V_{ts}|^2$. This puts a precise constraint on the CKM parameters
$\rho$ and $\eta$.

For $B^0_s$ mesons, existing data exclude small values of the mixing parameter
$x_s = \Delta m_s / \Gamma_s$, requiring $x_s>19.0$ at the 95\%
CL~\cite{Ref:PDG}.  Consequently the mass difference $\Delta m_s$ is much larger
than $\Delta m_d$, and the $B^0_s-\overline{B^0_s}$ oscillation frequency will
therefore be much higher than that for the $B^0_d$. Excellent decay length and
momentum resolutions are thus essential in order to observe the rapid
oscillations as a function of proper time.

Various decay modes of $B^0_s$ mesons are under investigation by the \dzero
collaboration. Among them are:
%
\begin{eqnarray}
B^0_s &\rightarrow& D^-_s(K^-K^+\pi^-)\pi^+,\quad\quad\quad\quad\quad\quad 
  (\BR=1.1\times10^{-4}),
  \nonumber\\ 
B^0_s &\rightarrow& D^-_s(K^-K^+\pi^-)3\pi,\,\quad\quad\quad\quad\quad\quad 
  (\BR=2.8\times10^{-4}),
  \nonumber\\
B^0_s &\rightarrow& J/\psi(\mu^+\mu^-,e^+e^-) K^*(K^{\pm}\pi^\mp),\,\quad 
  (\BR=5.1\times10^{-6}),
  \nonumber\\ 
B^0_s &\rightarrow& D^-_s(K^-K^+\pi^-)\ell^+\nu,\,\,\quad\quad\quad\quad\quad 
  (\BR=1.1\times10^{-4}).
\end{eqnarray}
\index{decay!$B_s \rightarrow D_s^- \pi^+$}%
\index{decay!$B_s \rightarrow D_s^- \pi^+ \pi^- \pi^+$}%
\index{decay!$B_s \rightarrow \psi K^* \ov{K}$}%
\index{decay!$B_s \rightarrow D_s^- \ell^+ \nu$}%

Taking advantage of the good SMT resolution, we can select $B_s$ decays by using
displaced secondary vertices or using tracks with large impact parameters. The
$B_s$ final states will be flavor tagged by the charge of the lepton, the charge
of the reconstructed charm meson or the charge of the kaon as appropriate.

The actual measurement strategy for $x_s$ will depend on the frequency of the
oscillation. For smaller oscillation frequencies, semileptonic $B_s$ decays can
be used. The lepton in the final state provides an easy trigger, giving a large
statistics sample.  If nature cooperates, a measurement in this range will come
very early in Run~II.

For higher oscillation frequencies, the measurement becomes more difficult.
Exclusive decays must be used in order to achieve the necessary momentum (and
therefore proper time) resolution. Decays which \dzero has focused on include:
$B_s \rightarrow D_s^- \pi^+(\pi^-\pi^+)$, where the $D_s^-$ decays to $\phi
\pi^-$ or $K^{*-}K^0/K^-K^{*0}$; and $B_s\rightarrow J/\psi \overline{K}{}^{*0}$
followed by $J/\psi\rightarrow e^+e^-$ or $\mu^+\mu^-$ and
$\overline{K}{}^{*0}\rightarrow K^-\pi^+$. We can only trigger on $B_s$ decays
into fully hadronic final states when the other $B$ in the event decays
semileptonically. Using single lepton triggers, we expect to be able to collect
about a thousand reconstructed exclusive $B_s$ decays in each mode in the first
two years, allowing us to measure $x_s$ values up to $\sim 20-30$. 
\index{B mixing@$B_s$ mixing!average significance in \dzero}
We are
presently investigating better trigger scenarios, such as lowering the $p_T$
threshold of the lepton and requiring another moderately high $p_T$ track (or
tracks), which would increase our $\Delta m_s$ reach.

A Monte Carlo study of the $B_s\rightarrow J/\psi \overline{K}{}^{*0}$ decay has
been carried out in order to estimate the number of events which will be in a
data sample based on a 2~fb$^{-1}$ exposure. We have analyzed 20,000 events
using the MCFast program.  The events were generated using Pythia and a
simulation of the upgraded D\O\ detector.  Each event had a $B_s \rightarrow
J/\psi \overline{K}{}^{*0}$ decay as well as a generic $\overline{B}$ decay. The
$J/\psi$ decayed to $\mu^+\mu^-$ (83\% of the time) or $\mu^+\mu^- \gamma$ (17\%
of the time). (The radiative decays are not discussed further here.) The
$\overline{K}{}^{*0}$ decayed to $\pi^+ K^-$.

Event reconstruction efficiency was estimated using the geometric acceptance of
the silicon vertex detector and of the fiber tracker. (Tracking inefficiencies
are not yet included.)  We find that 21\% of the events have all four charged
tracks reconstructed.

\begin{figure}
\centerline{\epsfig{file=ch8/d0_massdists.eps,width=0.95\textwidth}}
\caption[Effective mass distributions for the reconstructed (a)~$\mu^+\mu^-$
system; (b)~$\pi^+ K^-$ system; and (c)~$\mu^+\mu^-\pi^+ K^-$
system.]{Effective mass distributions for the reconstructed: (a) $\mu^+\mu^-$
system, (b) $\pi^+ K^-$ system, and (c) $\mu^+\mu^-\pi^+ K^-$ system.  These
distributions are prior to using vertex and mass constraints.}
\label{massdists}
\end{figure}

Shown in Fig.~\ref{massdists} are the reconstructed $\mu^+\mu^-$, $\pi^+ K^-$,
and $\mu^+\mu^-\pi^+ K^-$ effective mass distributions for the reconstructed
tracks.  The mass resolution of the $B_s$ improves by more than a factor of two
when vertex and $J/\psi$ mass constraints are imposed.  Effective mass
resolutions are given in Table \ref{resolution}.

\begin{table}
\begin{center}
\begin{tabular}{lrr}
\hline\hline
Quantity        &             Level & $\sigma~(MeV/\rm{c}^2$) \\
$M(B_s)$        &    Reconstruction &                      37 \\
$M(B_s)$        & $J/\psi$ mass fit &                      15 \\
$M(\mu^+\mu^-)$ & Reconstruction    &                      29 \\
\hline\hline
\end{tabular}
\end{center}
\caption{Mass Resolutions \label{resolution}}
\end{table}

\begin{figure}
\begin{center}
\epsfig{file=ch8/d0_resolutions.eps,width=0.95\textwidth}
\caption[Reconstructed $B_s$ decay length and proper time
distributions.]{Distributions of: (a) the reconstructed $B_s$ decay length
minus the generated $B_s$ decay length; (b) the reconstructed $B_s$ proper time
minus the generated $B_s$ proper time; and (c) the reconstructed $B_s$ decay
length divided by its error.}
    \label{resolutionfig}
\end{center}
\end{figure}


Resolutions of vertex position, decay length, and proper time were estimated
using the nominal silicon vertex detector and fiber tracker resolutions.  Shown
in Fig.~\ref{resolutionfig} are distributions of the fitted minus measured decay
length of the $B_s$, the fitted minus measured proper decay time of the $B_s$,
and the ratio of the decay length to the error in the decay length ($L/\sigma_L
$). We summarize the resolutions in vertex positions, decay length and proper
time in Table~\ref{resolution2}.

\begin{table}
\begin{center}
\begin{tabular}{lr}
\hline\hline
Quantity &  $\sigma$  \\
\hline
Production vertex (x,y, and z) ($\mu m$) & 34, 34,  80 \\
Decay vertex (x,y, and z) ($\mu m$)      & 50, 50, 140 \\
$B_s$ decay length ($\mu m$)             &         140 \\
Proper time ($ps$)                       &        0.40 \\
\hline\hline
\end{tabular}
\end{center}
\caption{Resolutions \label{resolution2}}
\end{table}

In order to get a realistic estimate of the number of events which would be
available for analysis, additional cuts were placed on the sample.  The muons
were required to have $p_T> 1.5~ GeV/\rm{c}$ and to have $|\eta|< 2$. A total of
24\% of the reconstructed events satisfied these cuts.  Combined with the
reconstruction efficiency of 21\%, we are left with a sample of 5.0\% of the
generated events for further analysis.

For most purposes, additional cuts will be required to obtain a sample of events
with a good signal-to-noise ratio.  To estimate the sample size after such cuts,
we impose a cut on the variable $L/\sigma_L$.  Of the 5.0\% of the events
satisfying all the previous cuts, 83\% have $L/\sigma_L > 2.0$, 73\% have
$L/\sigma_L > 3.0$, and 63\% have $L/\sigma_L > 4.0$. The required cut value
will not be known until the data is in hand, but we use the $L/\sigma_L > 3.0$
cut for further estimates.  Hence the overall combined efficiency which we use
below is (.050)*0.73=0.036.

In order to do mixing studies, it is necessary to tag the flavor of the $B_s$
(or $\overline{B_s}$).  This can be tagged if we identify the sign of the
charged kaon in the $K^*$.  Although D\O\ does not have particle identification
in the traditional sense, it is possible in this event sample to determine
whether the positive or negative particle from the $K^*$ decay is the charged
kaon by calculating the effective mass under the two assumptions ($K^+\pi^-$ and
$K^-\pi^+$) and taking as correct that combination which gives an effective mass
closest to the nominal $K^*$ mass (0.890 $GeV/\rm{c}^2$).  We find that we are
correct using this assignment 67\% of the time. Flavor tagging of the other $b$
quark will have the efficiency and dilution summarized previously in
Table~\ref{flavtag}.

Using estimates of the $b\overline{b}$ production cross section (158~$\mu$b);
the fraction of this cross section producing $B_s$ (0.167); the $B_s\rightarrow
J/\psi \overline{K}{}^{*0}$ branching ratio (5.1 x 10$^{-6}$); the
$J/\psi\rightarrow\mu^+\mu^-$ branching ratio (0.06); the luminosity (2~{\ifb});
and the overall combined efficiency from above (.036), we obtain a signal of
1000 events. We would expect a similar sample will be obtained using the
$J/\psi\rightarrow e^+ e^-$ mode.

%-------------------------------------------------------------------------------
\boldmath
\subsection[Measurement of $B_s$ Mixing in BTeV]
{Measurement of $B_s$ Mixing in BTeV
$\!$\authorfootnote{Author: R.~Kutschke.}}
\unboldmath
\index{BTeV!$B_s$ mixing}

In this section, the $x_s$ reach of BTeV will be demonstrated using $B_s\to
D_s^-\pi^+$ and $B_s\to J/\psi \ov{K}{}^{*0}$.  This study was carried out in
several steps, the first step being a simulation of the BTeV detector response
to signal events.  The output of this step was treated as real data and passed
through a physics analysis program to determine the yield, the time resolution
and the signal-to-background ratio in each mode.  This information was then
passed to a separate program which computed the $x_s$ reach; this program is
discussed in section~\ref{sec_btev_xsreach}.  A separate background study was
performed.

%...............................................................................
\subsubsection{Yields, Resolutions and Signal-to-Background Ratios}

The mode for which BTeV has the most sensitivity to $x_s$ is $B_s\to D_s^-
\pi^+$, 
\index{decay!$B_s \rightarrow D_s^- \pi^+$}
where the $D_s^-$ decays either by $D_s^-\to \phi\pi^-$, $\phi\to
K^+K^-$, or by $D_s^-\to K^{*0}K^-$, $K^{*0}\to K^+ \pi^-$.  Both of these
$D_s^-$ modes have narrow intermediate states and characteristic angular
distributions, both of which can be used to improve the signal-to-background
ratio.

For this study, Monte Carlo events were generated using Pythia and QQ and the
detector response was simulated using BTeVGeant.  The simulated events were
analyzed as real data.  For the $D_s \to \phi \pi$ decay mode the following cuts
were used:
\begin{itemize}
\item All tracks were required to have at least 3 hits in the silicon pixel
  detector.
\item Each of the tracks in the $B_s$ candidate were required to have an impact
  parameter with respect to the primary vertex of $> 3\sigma$.
\item To reduce the background due to tracks that really come from other
  interactions it was required that all 4 tracks have an impact parameter with
  respect to the primary vertex of less than 0.2~cm.
\item At least one of the kaons from the $\phi$ decay was required to be
  strongly identified as a kaon by the RICH detector.  The second kaon was only
  required to be loosely identified.  No particle ID requirements were placed on
  the pion candidates.
\item The $\phi$ and $D_s$ candidates were required to be within
  $\pm{2.5\sigma}$ of their nominal masses.
\item It was required that the distance between the primary vertex and $D_s$
  decay vertex be $L<8.0$ cm and the $D_s$ decay vertex have a decay length
  significance of $L/\sigma_{L}(D_s)>10.0$.
\item It was required that the $B_s$ have decay length significance of
  $L/\sigma_{L}(B_s)>4.0$.
\item The $B_s$ candidate was required to point back to the primary vertex: the
  transverse momentum of the $B_s$ with respect to its line of flight from the
  primary vertex was required to be less than 1.0 GeV/c and the impact parameter
  of the $B_s$ with respect to the primary vertex was required to be less than
  $3\sigma$.
\end{itemize} 
 
The combined geometric acceptance and reconstruction efficiency was found to be
2.7\%.  Of the events that passed these analysis cuts, 74\%\ passed the level 1
trigger.  For the $D_s \to K^* K$ mode we used the same cuts except that both
kaons from the $D_s$ decay were required to be identified in the RICH.  There
was also a broader cut on the intermediate $K^*$ mass.  The combined
reconstruction efficiency and geometric acceptance for the $D_s \to K^* K$ mode
was found to be 2.3\%, and the level 1 trigger efficiency for the events passing
the analysis cuts was 74\%.  For both modes the resolution on the mass of the
$B$ was found to be 18~MeV$/c^2$ and the mean resolution on the proper decay
time was found to be 43~fs.  The nominal acceptance of the BTeV level~2 trigger
for the accepted events is 90\%\ of the events which remain after the level 1
trigger.  The nominal flavor tagging power of BTeV was estimated in chapter~5 to
be $\epsilon D^2=0.1$ which arises from $\epsilon=0.70$ and $D=0.37$.

It is believed that the dominant source of backgrounds will be events of the
form \hbox{$X_b\to D_s^- X$}, where $X_b$ may be any $b$ flavored hadron.  The
background combinations arise when a true $D_s^-$ combination is combined with
some other track in the event.  An MCFast based study of 1~million $B\to D_s^-
X$ events was performed using an older version of the detector geometry, the one
used for the BTeV Preliminary Technical Design Report
(PTDR)~\cite{Ref:BTEV-PTDR}.  Comparisons between BTeVGeant and MCFast, and
comparisons between the old and new detector geometries, show that these
background studies remain valid.  When the 1~million $B\to D_s^- X$ events were
passed through MCFast and analyzed as real data, 8~entries remained in a mass
window 6~times larger than the mass window used to select signal $B_s$
candidates.  From this it is estimated that the signal-to-background ratio in
this channel is 8.4:1.  This study was performed without the proper treatment of
multiple interactions in one beam crossing.  To account for this, the
signal-to-background ratio used in the estimate of the $x_s$ reach is 3:1.

The background from direct charm production has not yet been investigated. While
direct charm production has a cross-section about 10~times higher than that for
production of charm via $B$ decay, it is triggered much less efficiently.
Moreover the the requirement of two, distinct detached vertices greatly reduces
the background from direct charm.  In the end it is expected that the background
from $B\to D_s^- X$ will dominate.

Table~\ref{9:tab:bstodspiyield} gives a summary of the preceding results and
discusses a list of all assumptions which went into the computation of the
yield. The value for ${\cal B}(\bar{b} \to B_s)$ is obtained from
Reference~\cite{Ref:BD-PROD}.  In one year it is expected that 72,000 events
will trigger, survive all analysis cuts and have their birth flavor tagged.

\begin{table}[tb]
\begin{center}
\begin{tabular}{llr}
\hline\hline
 Quantity & Value & 
\multicolumn{1}{c}{Yield} \\
          &       & 
\multicolumn{1}{c}{(Events/year)} \\
\hline
Luminosity:          & $2\times10^{32}\ {\rm cm^{-2} s^{-1}}$ & \\
One Year:            & $10^7\ {\rm s}$ & \\
$\sigma_{b\,\bar{b}}$:                    & $100\ \mu{\rm b}$ &  \\
${\cal B}(B_s \to D_s^-\pi^+)$:             & $3.0\times 10^{-3}$ & \\
${\cal B}(D_s^- \to \phi\pi^-)$:            & $0.030$ & \\
${\cal B}(D_s^- \to K^{*0} K^-)$:     & $0.036$ &\\
${\cal B}(\phi \to K^+ K^-)$:           & $0.49$ & \\
${\cal B}(K^{*0} \to K^+\pi^-)$:  & $0.67$ & \\
${\cal B}(\bar{b}\to B_s)$  & 0.13     &  6,210,000 \\
$\epsilon({\rm Geometry\ +\ cuts}:\ \phi\pi^-)$   & 0.027 &  \\
$\epsilon({\rm Geometry\ +\ cuts}:\ K^{*0}K^- )$  & 0.023 &  \\
$\epsilon({\rm Trigger})$ Level 1 & 0.74      &  \\
$\epsilon({\rm Trigger})$ Level 2 & 0.90      &  \\
$\epsilon({\rm Tag})$     & 0.70      & 72,000  \\
Tagging Dilution          & 0.37      &                \\
$S/B$                     & 3:1       & \\
$\sigma$(Proper Decay time) & 43 fs &  \\
\hline\hline
\end{tabular}
\end{center}
\caption[Projected yield for $B_s\to D_s^-\pi^+$ in one year of BTeV
running.]{Projected yield for $B_s\to D_s^-\pi^+$ in one year of BTeV running.
The numbers in the third column give the expected yield when all of the factors
down to and including that line have been considered.  The branching fraction
${\cal B}(B_s \to D_s^- \pi^+)$ was estimated to be the same as ${\cal
B}(B_d\to D^- \pi^+)$.}
\label{9:tab:bstodspiyield}
\end{table}

Another mode with good $x_s$ sensitivity is $B_s\to J/\psi \ov{K}{}^{*0}$,
\index{decay!$B_s \rightarrow \psi K{}^{*0}$}
$J/\psi\to\mu^+\mu^-$, $\ov{K}{}^{*0}\to K^-\pi^+$.  Although this mode is
Cabibbo suppressed, other factors are in its favor: the final state consists of
a single detached vertex and the state is triggerable with several independent
strategies, including impact parameter triggers, secondary vertex triggers and
dimuon triggers~\cite{Ref:MCBRIDE-STONE}.  While this mode does not have the
$x_s$ reach of $D_s^-\pi^+$ it does cover much of the expected range and it
provides a powerful check with partly independent systematics.

For reasons of time limitations, the simulation of the $J/\psi\ov{K}{}^{*0}$ mode
used MCFast, not BTeVGeant.  The analysis of this mode proceeded as follows.  To
be considered as part of a signal candidate, a track was required to have at
least 20 total hits and at least 4 pixel hits.  The only further requirement
placed on $\pi^{\pm}$ candidates was that they have a momentum greater than
\hbox{0.5~GeV/$c$}.  In order to be considered a muon candidate, a track was
required to have a momentum \hbox{$p>5$~GeV$/c$}, to penetrate the hadron filter
and to leave hits in the most downstream muon chambers.  Kaon candidates were
required to satisfy a simplified model of the RICH system: the track was
required to have a momentum in the range \hbox{$3<p<70\, $~GeV/$c$} and was
required to have hits in the tracking station downstream of the RICH mirror.
True kaons which satisfied this criteria were identified as kaons with an
efficiency of 90\%; other hadrons which satisfied this criteria were
(mis)identified as kaons 3\%\ of the time.

A $\mu^+\mu^-K^-\pi^+$ combination was accepted as a $B_s $ candidate if the
confidence level of fitting all four tracks to a single vertex was greater than
0.005.  It was also required that the resonant substructure requirements be
satisfied.  Combinations were considered for further analysis provided the decay
length of the $B_s$ candidate, $L$, satisfied \hbox{$L/\sigma_L>10$} and the
impact parameter of the $B_s$ candidate with the primary vertex, $d$, satisfied
$d<3\sigma_{d}$.  Each of the four $B_s$ granddaughters were required to have an
impact parameter with the primary vertex, $d$, of $d>2\sigma_d$.  Candidates
with poor time resolution were rejected by demanding $\sigma_{t}\le 0.09$~ps.
Also the mass of the $J/\psi$ was constrained to its PDG value.  The above
procedure found that the efficiency for the 4 tracks to be within the fiducial
volume of the tracking system was $14.2\pm0.3\,$\%\ and the efficiency for the
remaining candidates to pass the analysis cuts was $0.29\pm0.01$.  The
resolution on the mass of the $B_s$ was found to be $8.6\pm 0.3$~MeV$/c^2$ and
the mean resolution on the proper decay time was found to to be 36~fs.

The BTeV Level 1 trigger simulation was run on the $J/\psi\ov{K}{}^{*0}$ sample
and, of the candidates which passed all analysis cuts, $68\pm 2\,$\%\ also
passed the trigger; the error is statistical only.  However, this mode can also
be triggered by the dimuon trigger.  Section~8.3, of the BTeV
proposal~\cite{Ref:BTEV-PROPOSAL}, which describes the algorithms and
performance of the muon trigger, estimates a trigger efficiency of 50\%\ for
this decay mode.  There is, as yet, no calculation of the total Level~1 trigger
efficiency which takes into account the correlations between the two triggers.
For this proposal it will be estimated that the combined Level~1 trigger
efficiency is 85\%.  As for the $D_s\pi$ final state, the Level~2 trigger is
expected to have an efficiency of about 90\%.

By far the dominant background is expected to come from decays of the form
$X_b\to J/\psi X$, $J/\psi\to \mu^+\mu^-$, where $X_b$ is any $b$
flavored hadron.  An MCFast based simulation of 500,000 such decays was
performed and the signal-to-background level was estimated to be about 2:1.
Some sources of background that one might, at first, think to be important turn
out not to be a problem.  First, the more copious $B_s\to J/\psi\,\phi$ final
state is not a significant source of background because of the excellent
particle ID provided by the RICH system.  Second, the mass resolution is
sufficient to separate the decay $B_d\to J/\psi\ov{K}{}^{*0}$.

Finally, the expected yield can be increased by at least 50\%\ by using the
decay mode \hbox{$J/\psi\to e^+ e^-$}.  This mode will have an efficiency for
secondary vertex triggers which is comparable to that for $J/\psi\to \mu^+
\mu^-$ but the acceptance of the ECAL is smaller than that of the muon
detectors. The smaller acceptance of the ECAL is somewhat offset by also using
the RICH for electron identification.

The information reported here is summarized in Table~\ref{9:tab:psiksyield} and
is used in the mini-Monte Carlo described in the next section.  The estimate for
${\cal B}(B_s \to J/\psi \ov{K}{}^{*0})$ is obtained from
Reference~\cite{Ref:MCBRIDE-STONE} and that for ${\cal B}(\bar{b} \to B_s)$ is
obtained from Reference~\cite{Ref:BD-PROD}.

\begin{table}[tb]
\begin{center}
\begin{tabular}{llr}
\hline\hline
 Quantity & Value & 
\multicolumn{1}{c}{Yield} \\
          &       & 
\multicolumn{1}{c}{(Events/year)} \\
\hline
Luminosity:          & $2\times10^{32}\ {\rm cm^{-2} s^{-1}}$ & \\
One Year:            & $10^7\ {\rm s}$ & \\
$\sigma_{b\,\bar{b}}$: & $100\ \mu {\rm b}$ &  \\
${\cal B}(B_s \to  J/\psi \ov{K}{}^{*0})$: & $8.5\times 10^{-5}$ & \\
${\cal B}(J/\psi \to \mu^+\mu^-)$:              & 0.061 & \\
${\cal B}(\ov{K}{}^{*0}\to K^-\pi^+)$:          & 0.667 & \\
${\cal B}(\bar{b}\to B_s)$                      & 0.13      & 180000 \\
$\epsilon({\rm Geometric})$                     & 0.142     & \\
$\epsilon({\rm Analysis\ cuts})$                & 0.26      & 6600 \\
$\epsilon({\rm Trigger})$ Level 1 Tracking only & 0.60      & \\
$\epsilon({\rm Trigger})$ Level 1 Total         & 0.85      &  \\
$\epsilon({\rm Trigger})$ Level 2               & 0.90      & 5100 \\
$\epsilon({\rm Tag})$                           & 0.70      & 3600   \\
Include $J/\psi\to e^+e^-$                      & 1.5       & 5300  \\
Tagging Dilution                                & 0.37      &   \\
$S/B$                                           & 2:1       & \\
$\sigma$(Proper Decay time)                     & 36 fs     &  \\
\hline\hline
\end{tabular}
\end{center}
\caption[Projected yield for $B_s\to J/\psi\ov{K}{}^{*0}$ in one year of BTeV
running.]{Projected yield for $B_s\to J/\psi\ov{K}{}^{*0}$ in one year of BTeV
running. The numbers in the third column give the expected yield when all of
the factors down to and including that line have been considered. The trigger
efficiency is quoted as a fraction of those events which pass the analysis
cuts.}
\label{9:tab:psiksyield}
\end{table}

%...............................................................................
\boldmath
\subsubsection{Computation of the $x_s$ Reach}
\unboldmath
\label{sec_btev_xsreach}
\index{B mixing@$B_s$ mixing!average significance in BTeV}

The final step in the study was to use a mini-Monte Carlo to study the $x_s$
reach of BTeV.  This mini-Monte Carlo generates two lifetime distributions, one
for mixed events and one for unmixed events, smears the distributions and then
extracts a measured value of $x_s$ from a simultaneous fit of the two
distributions.  The time smearing is a Gaussian of fixed width, using the mean
time resolutions determined above.  The model includes the effects of
mistagging, background under the signal, and the minimum time cut which is
implied by the $L/\sigma_L$ cut.  It is assumed that the lifetime distribution
of the background is an exponential with the same mean lifetime as that of the
$B_s$.

Figures~\ref{9:fig:dspitime}~a) and b) show the proper time distributions which
result from one run of the mini-Monte Carlo for a generated value of $x_s=40$.
The simulation is for the decay mode $B_s\to D_s^-\pi^+$ for one month of BTeV
running.  Part a) shows the proper time distribution for unmixed decays while
part b) shows the distribution for mixed decays.  Part c) of the figure shows,
as a function of $x_s$, the value of the unbinned negative log likelihood
function computed from the simulated events.  A clear minimum near the generated
value of $x_s$ is observed and the likelihood function determines the fitted
value to be $x_s=39.96\pm0.08$.  A step of 0.5 in the negative log likelihood
function determines the 1~$\sigma$ error bounds and a line is drawn across the
figure at the level of the 5~$\sigma$ error bound.

This figure nicely illustrates the distinction between two quantities which are
often confused, the significance of the result and the error on $x_s$.  The
significance of the signal is determined by how far the depth of the global
minimum falls below that of the next most significant minimum.  The error on
$x_s$ is determined by the curvature of the likelihood function at the global
minimum.  While these quantities are clearly related, they are distinct; in
particular, the significance of the signal is not the relative error on $x_s$.

\begin{figure}[tb]
\centerline{\epsfxsize=4.5in\epsfbox{ch8/btev_dspitime.eps}}
\caption[Mini Monte Carlo proper lifetime plots of a) unmixed and b)  mixed
decays for a generated value of $x_s=40$.]{Mini Monte Carlo proper lifetime
plots of a) unmixed and b)  mixed decays for a generated value of $x_s=40$. 
The plots simulate the results of the $B_s\to D_s^- \pi^+$ channel after one
month of running.  The oscillations are prominent.  Part c) shows the negative
log likelihood function which was obtained from the entries in parts a) and
b).  A prominent minimum is seen at the generated value of $x_s$.  The dashed
line marks the level above the minimum which corresponds to 5~$\sigma$
significance.}
\label{9:fig:dspitime}
\end{figure}

The error returned by the fit was checked in two ways. First, an ensemble of
mini-Monte Carlo experiments was performed and the errors were found to
correctly describe the dispersion of the measured values about the generated
ones.  Second, the errors returned by the fit were found to be approximately
equal to the Cramer-Rao minimum variance bound.

The mini-Monte Carlo was also used to study the level of statistics below which
the experiment is unable to measure $x_s$.  As the number of events in a trial
is reduced, the negative log likelihood function becomes more and more ragged
and the secondary minima become more pronounced.  Eventually there are secondary
minima which reach depths within 12.5 units of negative log likelihood
(~5~$\sigma$~) of the global minimum.  When this happens in a sufficiently large
fraction of the trials, one must conclude that only a lower limit on $x_s$ can
be established.  In the region of the parameter space which was explored, the
absolute error on $x_s$ was approximately 0.1 when this limit was reached.  This
was independent of the generated value of $x_s$; that is, the discovery
measurement of $x_s$ will have errors of something like $\pm0.1$, even if $x_s$
is large, say~$40$.

It is awkward to map out the $x_s$ reach of the apparatus by running a large
ensemble of mini-Monte Carlo jobs; instead the following automated procedure was
used.  Following ideas from McDonald~\cite{Ref:KTMCDONALD}, the sum over events
in the likelihood function was replaced with an integral over the parent
distribution.  Because the parent distribution does not have any statistical
fluctuations, the fluctuations in the likelihood function are removed, leaving
only the core information.  An example of such a likelihood function is shown in
Fig.~\ref{9:fig:dspilikelihood}.
%
\begin{figure}[tb]
\centerline{\epsfxsize=10cm\epsfbox{ch8/btev_dspilikelihood.eps}}
\caption[The same likelihood function as in part c) of the previous figure but
obtained using the integral method described in the text.]{The same likelihood
function as in part c) of the previous figure but obtained using the integral
method described in the text.  The overall shape is the same but the
statistical fluctuations have been removed. There is also an overall level
shift which is related to the goodness of fit in the previous figure.}
\label{9:fig:dspilikelihood}
\end{figure}

A likelihood function computed in this way has the property that it scales
linearly with the number of events being simulated.  This can be stated formally
as follows.  Let $x_0$ denote the generated value of $x_s$ and let ${\cal
  L}(x;x_0,N)$ denote the value of the likelihood function, evaluated at $x$,
for a sample which has a true value of $x_0$ and which contains N events.  Then,
%
\begin{equation}
  {\cal L}(x;x_0,N) = N\, {\cal L}(x;x_0,1)\,.
\end{equation}
%
Now, one can define the significance of the minimum, $n$, as,
%
\begin{equation}
  n^2 = 2.0\, N\, \big[{\cal L}(\infty;x_0,1)-{\cal L}(x_0;x_0,1)\big]\,.
  \label{9:eq:runtime}
\end{equation}
%
For practical purposes $\infty$ was chosen to be 160.  If one did not have to
worry about the missing statistical fluctuations it would be normal to define a
significant signal as $5\,\sigma$, or $n^2=25$.  Instead, sufficient
significance was defined as $n^2=31.25$, by adding a somewhat arbitrary safety
margin; this allows for the usual $5\,\sigma$ plus a downwards fluctuation of up
to $2.5\,\sigma$ anywhere else in the plot.  Equation~\ref{9:eq:runtime} was
solved for $N$, which was then converted into the running time required to
collect $N$ events.  This procedure was repeated for many different values of
$x_0$ to obtain Fig.~\ref{9:fig:xsreach}.  The solid line shows, for the
$D_s^-\pi^+$ mode, the number of years needed to obtain a measurement with a
significance of $5\,\sigma$ plus the safety margin.  The safety margin reduces
the $x_s$ reach at 3 years by only 3 or 4 units of $x_s$.  For small values of
$x_s$, the effect of the safety margin is not visible.  The dashed line shows
the same information but for the $J/\psi \ov{K}{}^{*0}$ mode; for this mode the
effect of the safety margin is similarly small.

\begin{figure}[tb]
\centerline{\epsfxsize=11cm\epsfbox{ch8/btev_xs_reach_run2.eps}}
\caption[The $x_s$ reach of the BTeV detector.  The curves indicate the number
of years of running required for a measurement with a statistical
significance of $5\,\sigma$]{The $x_s$ reach of the BTeV detector.  The curves
indicate the number of years of running which are required to make a
measurement of $x_s$ with a statistical significance of $5\,\sigma$; a safety
margin, discussed in the text, has been included in the definition of
$5\,\sigma$. The curves are for the two different decay modes indicated on the
figure.}
\label{9:fig:xsreach}
\end{figure}

Inspection of Fig.~\ref{9:fig:xsreach} shows that, using the $D_s^-\pi^+$ mode,
BTeV is capable of observing all $x_s$ less than 75 in one year of running,
which is equivalent to an integrated luminosity of 2~fb$^{-1}$.
\index{B mixing@$B_s$ mixing!average significance in BTeV}

%-------------------------------------------------------------------------------
\subsection{Summary of Projections for Mixing}
\index{summary!$B_s$ mixing projections}

The Standard Model expectation for $B_s$ mixing is $22 < x_s <
31$~\cite{Ref:CKM-MATRIX}. All three experiments CDF, D{\O} and BTeV have shown
that they will be able to reconstruct a substantial amount of $B_s$ decays which
will allow for mixing studies.

With 2~fb$^{-1}$ CDF and BTeV safely cover the range for mixing as predicted by
the Standard Model. The CDF sensitivity for a 5 standard deviation observation
reaches from $x_s$ values of 59 to 74 depending on the event yields and the
signal-to-background ratios. BTeV's sensitivity comfortably covers $x_s$ values
of 75 with even some conservative safety margins included. Due to the large
event yields CDF will be able to observe $B_s$ mixing within the first few month
of data taking provided that the displaced track trigger and the silicon
detector work as advertised.

Once the oscillations are observed the statistical uncertainty on $x_s$ will be
small and in conjunction with an accurate $B^0$ mixing measurement it will
constitute a stringent constraint on the unitarity triangle.

%===============================================================================
\section{Projections for $\Delta\Gamma$}

%-------------------------------------------------------------------------------
\boldmath
\subsection[$B_s$ Lifetime Difference in CDF]
{$B_s$ Lifetime Difference in CDF
$\!$\authorfootnote{Authors: Ch.~Paus, J.~Tseng}}
\unboldmath
\index{width difference $\dg$!measurement of $\dg_s$!at CDF}

The promise of large $B_s$ samples in Run~II puts in reach the measurement of
the width difference between the two weak eigenstates of this meson. With its
analysis of semileptonic decays in Run~I, CDF has already published a limit of
$\Delta\Gamma_s/\Gamma_s < 0.85$ at 95\% confidence level~\cite{Ref:tauBs-lDX}.
This limit was established by fitting the lifetime distribution of the $\ell
D_s$ events with two exponentials.

In Run~II, however, it becomes possible to measure the lifetime of samples in
which the weak eigenstates are separated: for instance, 2~fb$^{-1}$ of data
yield approximately 4000 $B_s \to J/\psi\,\phi$ events, which is expected to be
dominated by the shorter-lived eigenstate, $B_s^H$, and for which an angular
analysis is used to separate the two components~\cite{Ref:polBd-Bs}.  Further it
is expected to yield roughly 75,000 of flavor-specific $B_s \to D_s \pi$ and
$B_s \to D_s \pi\pi\pi$%
\index{decay!$B_s \rightarrow D_s^- \pi^+$}%
\index{decay!$B_s \rightarrow D_s^- \pi^+ \pi^+ \pi^-$}%
decays, which are well-defined mixtures of $B_s^H$ and
$B_s^L$.

It is straightforward to show that computing the difference between two
lifetimes has more statistical power than fitting for two exponentials or
fitting for $\Delta\Gamma_s$ in $B_s$ mixing in flavor-specific samples. The
leading term in taking the difference between two measured lifetimes is
$\Delta\Gamma_s$. On the other hand, the lifetime distribution in a
flavor-specific sample is
%
\begin{equation}
  f(t) = \Gamma_H \left[e^{-\Gamma_Ht} + e^{-\Gamma_Lt}\right] 
       = \Gamma_H\, e^{-\overline{\Gamma}t}\left[e^{-\Delta\Gamma t/2}
                           + e^{+\Delta\Gamma t/2}\right]
       = \Gamma_H\, e^{-\overline{\Gamma}t}\, \bigg[1
                           + \left(\frac{\Delta\Gamma t}{2}\right)^2
                                   + \dots \bigg]\,,
\end{equation}
%
where $\Delta\Gamma$ enters as a second-order effect.

The $B_s$ decay to $D^+_s D^-_s$ is another valuable source of information to
determine the lifetime difference of the $B_s$ meson. First of all $D^+_s D^-_s$
is a pure $CP$ even eigenstate and thus its lifetime is a clean measurement of
the $CP$ even lifetime. In addition its branching fraction directly measures
$\Delta\Gamma/\Gamma$ under certain theoretical
assumptions~\cite{Ref:Bs-NewPhys}. The complication in this particular decay
mode comes from the associated production of $D^{*\pm}_s$.

%...............................................................................
\subsubsection{Lifetime Difference Measurements}

\paragraph{\boldmath$B_s \to J/\psi\,\phi$\unboldmath}
%===================================================
The decay mode $B_s \to J/\psi\,\phi$%
\index{decay!$B_s \rightarrow \psi \phi$}%
~has been analyzed at CDF in two Run~I analyses, examining its
lifetime~\cite{Ref:JpsiBsLifetime} and angular distributions~\cite{Ref:polBd-Bs}
separately. The basic strategy for Run~II is to combine these two analyses into
a maximum likelihood fit of the proper decay time, and {\it transversity angle}
of each candidate. In addition to account properly for the background the
invariant mass distribution will be simultaneously fitted.

The transversity angle,%
\index{transversity angle}%
$\theta_T$, is defined by the angle between the $\mu^+$ and the $z$ axis in the
rest frame of the $J/\psi$ decay, where the $z$ axis is orthogonal to the plane
defined by the $\phi$ and the $K^+$ directions.  This angle allows to
distinguish $CP$ even and $CP$ odd components: the probability density function
for the $CP$ even component is $\frac{3}{8}(1+\cos^2\theta_T)$ and for the $CP$
odd component is $\frac{3}{4}\sin^2\theta_T$.  The amplitude of the $CP$ even
component sums the squares of the unpolarized and linearly polarized state
amplitudes, $|A_0|^2+|A_{||}|^2$, of the $\phi$ in the $J/\psi$ rest frame, and
the $CP$ odd component the square of the transversely polarized state amplitude,
$|A_\perp|^2$, of the same.  The analysis depends upon the weak eigenstates
being also $CP$ eigenstates which is a good approximation for the $B_s \to
J/\psi\,\phi$ decays.

A toy Monte Carlo study was performed to estimate the uncertainty in
$\Delta\Gamma_s$ with 4000 $B_s \to J/\psi\,\phi$ decays.  The background shapes
in mass and proper decay time and the relative fraction of the signal to the
background were assumed to be identical to that in the Run~I lifetime analysis.
The mass resolution was assumed to be the same as in Run~I, and all the lifetime
distributions were convoluted with the 18~$\mathrm{\mu m}$ resolution projected
for Run~II. The background was assumed to have a flat transversity angle
distribution.

The error on $\Delta\Gamma_s/\Gamma_s$ depends on the $CP$ admixture of the
final state. The decay $B_s \to J/\psi\,\phi$ is dominated by $CP$ even
eigenstates. Therefore the smaller the admixture of $CP$ odd component the
larger the sensitivity. Assuming the $CP$ composition as measured in
Run~I~\cite{Ref:polBd-Bs} corresponding to a $CP$ even fraction of $0.77\pm0.19$
the expected uncertainty on $\Delta\Gamma_s/\Gamma_s$ is 0.05.  This uncertainty
varies between 0.08 and 0.035 for $CP$ even fractions of 0.5 and 1.0,
respectively.

\paragraph{\boldmath$B_s\to D_s^{(*)+}D_s^{(*)-}$\unboldmath}
%============================================================
The decay modes $B_s\to D_s^{(*)+}D_s^{(*)-}$%
\index{decay!$B_s \rightarrow D_s^+ D_s^-$}%
\index{decay!$B_s \rightarrow D_s^{*+} D_s^-$}%
\index{decay!$B_s \rightarrow D_s^+ D_s^{*-}$}%
\index{decay!$B_s \rightarrow D_s^{*+} D_s^{*-}$}%
are also promising for lifetime difference studies, though with smaller sample
sizes. These decays are expected to be present among the two-track trigger data.
The decay $B_s \to D_s^+D_s^-$, in particular, is purely $CP$ even, and requires
no angular analysis.  Its companion decays, involving $D_s^*$ decays, are
expected in the heavy quark limit, and in the absence of $CP$ violation, to be
sensitive to $CP$-even $B_s$ states as well~\cite{Ref:ALEKSAN-DSSTAR}.  While
these decays are attractive in that they significantly increase the sample size
over that of $B_s \to D_s^+ D_s^-$ alone, their identification is a challenge,
since the missing photons from the decay $D_s^{*+}\to D_s^+\gamma$ and
$D_s^{*+}\to D_s^+\pi^0$ considerably broaden the $D^+_s D^-_s$ invariant mass
distribution. On the other hand, the missing mass introduces only about 3\% to
the proper lifetime resolution.

In a full GEANT based simulation and reconstruction of $B_s \to D^{(*)+}_s
D^{(*)-}_s$ in the CDF detector it is found that the three different cases are
separated quite cleanly by using the invariant mass spectrum of the charged
decay products. Shifts in the invariant $B_s$ mass are due to the neutral
particles which are not reconstructed. PYTHIA has been used to generate
$b\,\bar{b}$ quarks and fragment them to $b$ hadrons. The $B_s$ mesons are decayed
using the CLEOMC program according to the branching ratios given in
Table~\ref{tab:bs-branchings}.

In Figure~\ref{fig:cdf-bsdsds} the invariant mass spectra of the three
different cases are depicted. The spectra are essentially free of combinatoric
background since the reconstructions of the resonances at each step allow
stringent cuts. For this picture the $D_s$ is always decayed into $K^{*0}K$
which has more combinatorics than the $\phi\pi$ decay mode due to the large
width of the~$K^{*0}$.

\begin{figure}[t]
\begin{center}
  \mbox{\epsfig{file=ch8/cdf_bsdsds.eps,width=0.55\textwidth}}
\caption[$D^+_s,D^-_s$ invariant mass spectra for the three decay modes $B_s
\to D_s D_s$ (solid), $B_s \to D^{(*)}_s D_s$ (large dashes) and $B_s \to
D^{(*)}_s D^{(*)}_s$ (small dashes).]{$D^+_s,D^-_s$ invariant mass spectra for
the three decay modes $B_s \to D_s D_s$ (solid), $B_s \to D^{(*)}_s D_s$ (large
dashes) and $B_s \to D^{(*)}_s D^{(*)}_s$ (small dashes). The $D_s$ is always
decayed into $K^{*0}K$. The shaded background is the sum of the three decay
modes.}
  \label{fig:cdf-bsdsds}
\end{center}
\end{figure}

To estimate the error on the $\Delta\Gamma_s/\Gamma_s$ from this channel
several assumptions have to be made since this mode has not been reconstructed
in Run~I.

In the most conservative estimate only $B_s \to D_s D_s$ is used which is a
clean $CP$ even state. GEANT based Monte Carlo studies indicate that a signal to
background fractions of 1:1 - 1:2 are achievable. This is similar to signal to
background fraction achieved for the decay mode $B_s \to \nu \ell D_s$, measured
in Run~I. The expected error on the lifetime is 0.044~ps is obtained using an
event sample of 2.5k events and a signal to background fraction of 1:1.5. This
converts into an error on $\Delta\Gamma_s/\Gamma_s$ of 0.06.

Assuming that also the other two decay modes involving $D^*_s$ mesons are clean
$CP$ even modes a total of 13k events are available. The estimated error on
$\Delta\Gamma_s/\Gamma_s$ is then reduced to 0.025.

%
% tau(Bs) = 1.36 +- 0.09 (stat) ps -- from 600 Bs -> nu l Ds events
%
% sigma(tau(Bs))   = 0.09 ps sqrt(600/2500) = 0.044 ps
% sigma(DG(Bs)/G)  = 2 * G_world sigma(tau(Bs))
%                  = 2 * 0.670 1/ps 0.044 ps
%                  = 0.059
%
% sigma(tau(Bs))   = 0.09 ps sqrt(600/13000) = 0.019 ps
% sigma(DG(Bs)/G)  = 2 * G_world sigma(tau(Bs))
%                  = 2 * 0.670 1/ps 0.019 ps
%                  = 0.025
%

%...............................................................................
\subsubsection{Related Branching Fractions}

The decay modes $B_s \to D_s^{(*)+}D_s^{(*)-}$ are also interesting because it
is expected that they are the largest contribution to the actual difference
between the heavy and light widths. Indeed, the other decay modes are estimated
to contribute less than 0.01 to the projected $\sim 0.15$ value of
$\Delta\Gamma_s/\Gamma_s$~\cite{Ref:ALEKSAN-DGAMMA}. The branching fraction to
this final state is in the small velocity (SV) limit related to $\Delta\Gamma_s$
by
%
\begin{equation}\label{eq:bs-branching}
  {\cal B}(B_s^H \to D_s^{(*)+} D_s^{(*)-}) =
        \frac{\Delta\Gamma_s}{ \overline{\Gamma_s}
   (1 + \frac{\Delta\Gamma_s}{2\overline{\Gamma_s}})}\,.
\end{equation}
%
This method has been exploited by ALEPH, using $\phi\phi$ correlations, to
obtain a value of $\Delta\Gamma_s/\Gamma_s =
0.25^{+0.21}_{-0.14}$~\cite{Ref:ALEPH-PHIPHI}. However the small velocity
assumption also referred to a Shifman-Voloshin limit may be very approximative.

The following estimates are made assuming the validity of this limit. Further
when measuring branching fractions many systematic effects have to be
considered. For example the tracking efficiency for the kinematics of the
particular decays has to be determined carefully. Since it is difficult if not
impossible to predict those effects only the statistical uncertainties are
discussed below.

The statistical error on the branching fraction for 13k events (see
Table~\ref{tab:hadrtrig-bs}) with a signal to background ratio of 1:1 is 0.012.
Following Equation~\ref{eq:bs-branching} the turns out to be also the statistic
uncertainty on $\Delta\Gamma_s/\Gamma_s$ since the branching ratio is very
small. This uncertainty deteriorates to 0.015 when making the signal to
background ratio 1:2.

%...............................................................................
\subsubsection{Combined CDF Projection}

With the analysis possibilities discussed thus far, the lifetime difference
method conservatively yields a statistical uncertainty of 0.04 on
$\Delta\Gamma_s/\Gamma_s$, utilizing both $J/\psi\,\phi$ and $D_s^+ D_s^-$ decays
and just using the lifetime measurements.%
\index{width difference $\dg$!measurement of $\dg_s$!at CDF}%

If one assumes that the decay modes involving $D_s^*$ are also mostly $CP$ even
the sample for the lifetime measurement is extended and the branching ratios can
be used in the SV limit. This decreases the statistical uncertainty on
$\Delta\Gamma_s/\Gamma_s$ to 0.01.

These numbers refer to the projected Run~II luminosity of 2~$\mathrm{fb}^{-1}$
and bear all the caveats mentioned in the text.

%-------------------------------------------------------------------------------
\boldmath
\subsection[Estimate of Sensitivity on $\Delta\Gamma$ in BTeV]
{Estimate of Sensitivity on $\Delta\Gamma$ in BTeV
$\!$\authorfootnote{Author: H.W.K.~Cheung.}}
\unboldmath
\index{width difference $\dg$!measurement of $\dg_s$!at BTeV}
%%\index{BTeV!width difference $\dg_s$}

Since $\Delta\Gamma_{B_s}$is expected to be much larger than
$\Delta\Gamma_{B^0}$, only projections for measurements of $\Delta\Gamma_{B_s}$
have been studied at BTeV for this report.  The $B_s^0$ decay modes studied
include $CP$-even, $CP$-mixed and
flavor specific decay modes and are listed in Table~\ref{tb_btevdgmodes}.  The
total decay rate for the flavor specific decay $B_s^0\rightarrow D_s^-\pi^+$ 
\index{decay!$B_s \rightarrow D_s^- \pi^+$}
is given by the average of the $CP$-even and $CP$-odd rates.  The decays
$B_s^0\rightarrow J/\psi\eta$, $J/\psi\eta^{\prime}$ 
\index{decay!$B_s \rightarrow \psi \eta $}
\index{decay!$B_s \rightarrow \psi \eta^\prime$}
should be $CP$-even while the
decay to $J/\psi\,\phi$ 
\index{decay!$B_s \rightarrow \psi \phi$}
is predominantly $CP$-even.

\begin{table}[tbh]
\begin{center}
\begin{tabular}{cccc}
\hline\hline
{\boldmath $B_s^0$ \bf{Decay Mode}} & \bf{$CP$ Mode} & 
\bf{Branching Ratio} & \bf{BR Used}\\ \hline
$J/\psi\,\phi$     & Mostly $CP$-even & $(9.3\pm 3.3)\times 10^{-4}$ &
$8.9\times 10^{-4}$\\
$J/\psi\eta$ & $CP$-even & $<0.0038$  &
$3.3\times 10^{-4}$\\
$J/\psi\eta^{\prime}$ & $CP$-even & $<0.0038$  &
$6.7\times 10^{-4}$\\
$D_s^-\pi^+$     & Flavor specific  & $<0.13$  &
$3.0\times 10^{-3}$\\
\hline\hline
\end{tabular}
\end{center}
\caption{The $B_s^0$ decay modes studied for $\Delta\Gamma_{B_s}$
sensitivity studies at BTeV.}
\label{tb_btevdgmodes}
\end{table}

CDF has measured the $CP$-odd fraction of total rate for $J/\psi\,\phi$ to be
$0.229\pm0.188(stat)\pm 0.038(syst)$\cite{Ref:BTEV-CDFJPSIPHI}.  The all charged
mode decay $B_s^0\rightarrow K^+K^-$ has a large enough expected branching
fraction ($\sim 1\times 10^{-5}$) and reconstruction efficiency to get high
statistics. However although the $K^+K^-$ final state is $CP$-even, the decay can
proceed via both a $CP$ conserving Penguin contribution as well as a $CP$ violating
Tree level contribution. Unless the Penguin contribution is completely dominant
or the Penguin and Tree level contributions can be exactly calculated it would
be difficult to use this mode without significant theoretical errors.

%...............................................................................
\subsubsection{Signal yields and backgrounds}

Estimation of the signal yields and signal/background ratios were determined
using a MCFAST simulation for the $B_s^0$ decay mode to $J/\psi\,\phi$, while all
the other modes were simulated using a Geant simulation of BTeV.  Although it is
easy to simulate the signal to determine the reconstruction efficiency, the
background simulation can be more troublesome. The signal sample always has an
average of two embedded min-bias events. Since it takes too much time to
generate enough background one has to determine which backgrounds are dominant
for a particular decay.

The decay $B_s^0\rightarrow J/\psi\,\phi$ was studied through the decay channels
$J/\psi\rightarrow\mu^+\mu^-$ and $\phi\rightarrow K^+K^-$. Studies show that
since the $J/\psi\rightarrow\mu^+\mu^-$ is expected to be so clean, the dominant
backgrounds come from $b\,\bar{b}\rightarrow J/\psi X$.  For this study only
backgrounds from $b\,\bar{b}\rightarrow J/\psi\,\phi X$ have been studied.  Figure
\ref{btev_psimass1}(a) shows the $\mu^+\mu^-$ invariant mass for a
$b\,\bar{b}\rightarrow J/\psi\,\phi X$ background sample compared to an
appropriately normalized signal sample of $B_s^0\rightarrow J/\psi\,\phi$. The two
muons were required to form a vertex with a confidence level of greater than
1\%.  Figure~\ref{btev_psimass1}(b) shows a similar comparison for the $K^+K^-$
invariant mass, again with a vertex requirement of $CL>1$\%.

\begin{figure}[t]
\centerline{
\epsfxsize 7cm \epsffile{ch8/btev_psimass1.eps}
\epsfxsize 7cm \epsffile{ch8/btev_phimass1.eps}}
\caption[The $\mu^+\mu^-$ invariant mass (a) and the $K^+K^-$ invariant mass
(b) for a $b\,\bar{b}\rightarrow J/\psi\,\phi X$ background samples.]{(a) The
$\mu^+\mu^-$ invariant mass and (b) the $K^+K^-$ invariant mass for a
$b\,\bar{b}\rightarrow J/\psi\,\phi X$ background sample (open histogram) and for
an appropriately normalized $B_s^0\rightarrow J/\psi\,\phi$ signal sample (filled
histogram).}
\label{btev_psimass1}
\end{figure}

Figure~\ref{btev_psiphim1}(a) shows the $\mu^+\mu^-K^+K^-$ invariant mass for
the $b\,\bar{b}\rightarrow J/\psi\,\phi X$ background sample without requiring that
the $\mu^+\mu^-$ and $K^+K^-$ masses are consistent with the $J/\psi$ and $\phi$
masses respectively. The four tracks are required to form a single vertex with a
confidence level greater than 1\%. This is compared in the plot to an
appropriately normalized signal sample.  Figure~\ref{btev_psiphim1}(b) shows the
$J/\psi\,\phi$ invariant mass plot with vertexing and requirements on the
$\mu^+\mu^-$ and $K^+K^-$ masses. The $\mu^+\mu^-$ and $K^+K^-$ masses are
required to be within $\pm 2\sigma$ of the true $J/\psi$ and $\phi$ masses
respectively. Both signal and backgrounds are included in the plot and compared
to the signal only sample.  The signal/background ratio is seen to increase when
one applies a primary-to-secondary vertex detachment requirement of
$L/\sigma_L>15$ in Figure~\ref{btev_psiphim1}(c). Where $L$ is the 3-dimensional
distance between the primary vertex and the $B_s^0$ decay vertex and $\sigma_L$
is the error on $L$ calculated for each candidate $B_s^0$ decay.  For
$L/\sigma_L>15$ the reconstruction efficiency is 6.0\%, and the
signal/background ratio is 47/1 with an error of 32\%\ for the backgrounds
considered.

\begin{figure}[ht]
\centerline{
\epsfxsize 4.67cm \epsffile{ch8/btev_psiphim1.eps}
\epsfxsize 4.67cm \epsffile{ch8/btev_psiphim2.eps} 
\epsfxsize 4.67cm \epsffile{ch8/btev_psiphim5.eps}}
\caption[The $\mu^+\mu^-K^+K^-$ invariant mass for $b\,\bar{b}\rightarrow
J/\psi\,\phi X$ background and $B_s^0\rightarrow J/\psi\,\phi$ signal
samples]{(a) The $\mu^+\mu^-K^+K^-$ invariant mass for a 
$b\,\bar{b}\rightarrow J/\psi\,\phi X$ background sample (open histogram) and 
for a correctly normalized $B_s^0\rightarrow J/\psi\,\phi$ signal sample 
(filled histogram). (b) $J/\psi\,\phi$ invariant mass for the background plus
signal sample (open histogram) compared to just the signal sample (filled
histogram). (c) $J/\psi\,\phi$ mass for $L/\sigma_L>15$.}
\label{btev_psiphim1}
\end{figure}

Table~\ref{tb_btevpsiphi1}\ shows projections for the $B_s^0\rightarrow
J/\psi\,\phi$ signal yields for 2~fb$^{-1}$.  A total $b\,\bar{b}$ production
cross-section of 100~$\mu$b is assumed and we take the fraction of
$B_s^0/\overline{B_s^0}$ per $b\,\bar{b}$ from Pythia as 1 in 4.3. The branching
fraction of $B_s^0\rightarrow J/\psi\,\phi$ is taken to be equal to
$BR(B_d^0\rightarrow J/\psi K^0)=8.9\times 10^{-4}$.  A total of $\sim$41400
signal events is expected for 2~fb$^{-1}$.  The expected error on the lifetime
for 2~fb$^{-1}$ was determined with a toy Monte Carlo generating 1000
experiments with 41400 signal events and S/B=47. The background lifetime
distribution was simulated with a short and a long lifetime component as seen in
the background studies and is typical of backgrounds seen in fixed target
experiments. In the toy MC the short and long components were set to be 0.33 and
1.33~ps respectively.  A binned likelihood fit was used to extract the measured
lifetime for each of the 1000 experiments, where the lifetime distribution from
sidebands is used as a measure of the lifetime distribution in the signal
$B_s^0$ mass region. The method is described in
Reference~\cite{Ref:BTEV-E687LIFETIME}.  The expected error is taken to be the
{\it r.m.s.} of the 1000 measured lifetimes and is 0.50\%.

\begin{table}[bht]
\begin{center}
\begin{tabular}{lccc}
\hline\hline
\bf{Quantity}  & \multicolumn{3}{c}{\bf{Value}} \\ \hline
Number of $b\,\bar{b}$ & \multicolumn{3}{c}{$2\times 10^{11}$} \\
Number of $B_s^0/\overline{B_s^0}$ & \multicolumn{3}{c}{$4.7\times 10^{10}$} \\
\hline
           &  $B_s^0\rightarrow J/\psi\,\phi$ & 
              \multicolumn{2}{c}{$B_s^0\rightarrow D_s^-\pi^+$} \\
           & $J/\psi\rightarrow\mu^+\mu^-$ & $D_s^-\rightarrow\phi\pi^-$ &
             $D_s^-\rightarrow K^{\ast 0}K^-$ \\ 
           & $\phi\rightarrow K^+K^-$ & $\phi\rightarrow K^+K^-$ &
             $K^{\ast 0}\rightarrow K^+\pi^-$ \\ \hline
\#\ of Events  & $1.2\times 10^6$  & $2.5\times 10^6$ & $3.1\times 10^6$\\
Reconstruction efficiency (\%) & 6.0 & 2.7  & 2.3 \\
S/B                            & 47:1  & \multicolumn{2}{c}{3:1} \\
L1 Trigger efficiency (\%)     & 70 & \multicolumn{2}{c}{74} \\
L2 Trigger efficiency (\%)     & 90  & \multicolumn{2}{c}{90} \\
\#\ of reconstructed decays    & 41400 & \multicolumn{2}{c}{91700} \\
\hline\hline
\end{tabular}
\end{center}
\caption{Projections for yields of $B_s^0$ decays for 2~fb$^{-1}$ assuming
a total $b\,\bar{b}$ production cross-section of 100~$\mu$b.}
\label{tb_btevpsiphi1}
\end{table}

Although the $J/\psi\,\phi$ signal sample can be obtained through the
$J/\psi\rightarrow \mu^+\mu^-$ trigger, the effect of the Level 1 vertex trigger
\index{width difference $\dg$!measurement of $\dg_s$!effect of vertex trigger}
on this mode was studied to determine the effect of the L1 trigger on the
lifetime analysis. Figure~\ref{btev_ftpsiphi1}(a) shows the proper time
($t=L/\beta\gamma c$) distribution for reconstructed $B_s^0\rightarrow
J/\psi\,\phi$ signal events for a $L/\sigma_L>15$ requirement. The loss of short
lifetime decays is due to the detachment requirement. One can obtain an
exponential distribution if we use the reduced proper time,
$t^{\prime}=t-N\sigma_L/\beta\gamma c$, for a $L/\sigma_L>N$ requirement. This
starts the clock at the minimum required decay time for each decay candidate and
works because the lifetime follows an exponential distribution irrespective of
the when the clock is started. Figure~\ref{btev_ftpsiphi1}(b) shows the reduced
proper time distribution and an exponential fit gives a lifetime of $1.536\pm
0.014$~ps, compared to the generated lifetime of 1.551~ps.

\begin{figure}[t]
\centerline{\epsfxsize 0.95\textwidth \epsffile{ch8/btev_ltpsiphi1.eps}}
\caption{(a) Proper time distribution for reconstructed 
$B_s^0\rightarrow J/\psi\,\phi$; (b) Reduced proper time distribution for
the same decays, the line is an exponential fit.}
\label{btev_ftpsiphi1}
\end{figure}

Figure~\ref{btev_ftpsiphi2}(a) shows the reduced proper time after applying the
Level 1 vertex trigger. Short lifetime decays are again lost because the impact
parameter requirements of the Level 1 trigger effectively gives a larger minimum
required decay time than $N\sigma_L/\beta\gamma c$.  The lifetime acceptance
function is just the observed reduced proper distribution divided by a pure
exponential with the generated lifetime and is given in Figure
\ref{btev_ftpsiphi2}(b).  In order to extract the correct lifetime from the
observed reduced proper time distribution one needs the correct lifetime
acceptance function. This is obtained from Monte Carlo and can be checked by
using decays modes like $J/\psi\,\phi$ that can be obtained with the L1 vertex
trigger and separately through the L1 $J/\psi\rightarrow \mu^+\mu^-$ dimuon
trigger which has no vertexing selection criteria. The L1 trigger lifetime
acceptance correction obtained from MC can also be checked by taking samples of
prescaled triggers that do not have the L1 trigger requirement.

\begin{figure}[bhtp]
\centerline{\epsfxsize 0.95\textwidth \epsffile{ch8/btev_ltpsiphi2.eps}}
\caption[(a) Reduced proper time distribution for reconstructed 
$B_s^0\rightarrow J/\psi\,\phi$ decays that pass the Level 1 vertex trigger;
(b) Lifetime correction function.]{(a) Reduced proper time distribution for
reconstructed  $B_s^0\rightarrow J/\psi\,\phi$ decays that pass the Level 1
vertex trigger; (b) Lifetime correction function, obtained by dividing the
distribution in (a) by a pure exponential distribution with the generated
$B_s^0$ lifetime.}
\label{btev_ftpsiphi2}
\end{figure}

Note that since the L1 trigger can increase the effective minimum required decay
time cut, if one puts on a large impact parameter selection requirement on the
$B$ decay daughters, it may be possible to redefine the reduced proper time to
take this into account and thereby reduce the lifetime acceptance corrections
with some lost of statistics. This still needs to be studied and the details of
the L1 trigger may change since one may be able to redesign it to reduced the
lifetime acceptance correction.

When an acceptance correction like that given in Figure~\ref{btev_ftpsiphi2}(b)
is simulated in the toy Monte Carlo, the expected error on the measured lifetime
increases to 0.58\%, from the previous value of 0.50\%. For the decay mode
$B_s^0\rightarrow D_s^-\pi^+$ where the signal to background is smaller the
effect is similar, the error increases to 0.44\%\ when adding the acceptance
correction effects compared to 0.39\%.

\begin{table}[bht]
\begin{center}
\begin{tabular}{lccc} \hline\hline
\bf{Quantity}  & \multicolumn{3}{c}{\bf{Value}} \\ \hline
Number of $b\,\bar{b}$ & \multicolumn{3}{c}{$2\times 10^{11}$} \\
Number of $B_s^0/\overline{B_s^0}$ & \multicolumn{3}{c}{$4.7\times 10^{10}$} \\
\hline
           &  $B_s^0\rightarrow J/\psi\eta$ & 
              \multicolumn{2}{c}{$B_s^0\rightarrow J/\psi\eta^{\prime}$} \\
           & \multicolumn{3}{c}{$J/\psi\rightarrow\mu^+\mu^-$} \\
           & $\eta\rightarrow \gamma\gamma$ & 
             $\eta^{\prime}\rightarrow \rho^0\gamma$ &
             $\eta^{\prime}\rightarrow\pi^+\pi^-\eta$ \\ \hline
\#\ of Events  & $3.5\times 10^5$  & $5.4\times 10^5$ & $3.2\times 10^5$\\
Reconstruction efficiency (\%) & 0.71 & 1.2  & 0.60\\
S/B                            & 15:1  & \multicolumn{2}{c}{30:1}  \\
L1 Trigger efficiency (\%)     & 75 & \multicolumn{2}{c}{85} \\
L2 Trigger efficiency (\%)     & 90  & \multicolumn{2}{c}{90} \\
\#\ of reconstructed decays    & 1700 & \multicolumn{2}{c}{6400} \\
\hline\hline
\end{tabular}
\end{center}
\caption{Projections for yields of $B_s^0$ decays for 2~fb$^{-1}$ assuming
a total $b\,\bar{b}$ production cross-section of 100~$\mu$b.}
\label{tb_btevpsieta1}
\end{table}

\begin{table}[hbt]
\begin{center}
\begin{tabular}{cccc} \hline\hline
\bf{Decay Mode}      & \multicolumn{3}{c}{\bf{Error on Lifetime (\%)}}\\ \hline
                       & 2~fb$^{-1}$ & 10~fb$^{-1}$ & 20~fb$^{-1}$ \\ \hline
$J/\psi\,\phi$           & 0.50  & 0.23 & 0.16 \\
$J/\psi\eta$           & 2.49  & 1.19 & 0.80 \\
$J/\psi\eta^{\prime}$  & 1.36  & 0.55 & 0.39 \\
$D_s^-\pi^+$           & 0.44  & 0.20 & 0.14 \\
\hline\hline
\end{tabular}
\end{center}
\caption{Projections for statistical errors on lifetimes measured in
different modes for 2, 10 and 20~fb$^{-1}$.}
\label{tb_btevall1}
\end{table}

Tables~\ref{tb_btevpsieta1}\ and \ref{tb_btevpsiphi1}\ show the expected signal
yields and signal/background ratios for the decays modes $B_s^0\rightarrow
J/\psi\eta$, $B_s^0\rightarrow J/\psi\eta^{\prime}$ and $B_s^0\rightarrow
D_s^-\pi^+$.  
\index{decay!$B_s \rightarrow D_s^- \pi^+$}%
\index{decay!$B_s \rightarrow \psi \eta $}%
\index{decay!$B_s \rightarrow \psi \eta^\prime$}%
Backgrounds from $b\,\bar{b}\rightarrow J/\psi X$ were studied for
the decay modes $B_s^0\rightarrow J/\psi\eta^{(\prime)}$, while backgrounds from
$b\,\bar{b}\rightarrow D_s^+X$ were included in the $B_s^0\rightarrow D_s^-\pi^+$
analysis.  Details of the analyses of these modes can be found in
Reference~\cite{Ref:BTEV-PROPOSAL}. Expected errors on the lifetimes were
determined using toy Monte Carlo simulations as before, where acceptance
corrections were also simulated for the $D_s^-\pi^+$ mode.
Table~\ref{tb_btevall1} gives the expected errors on the lifetimes for all
modes.

Note that although only backgrounds from $b\,\bar{b}\rightarrow J/\psi\,\phi X$ were
included in the study of $B_s^0\rightarrow J/\psi\,\phi$ to obtain the value of
S/B=47/1, the effect of lower values of S/B were studied. If the S/B is
decreased to 10/1 which is the expected level for $B_d^0\rightarrow J/\psi
K_s^0$~\cite{Ref:BTEV-PROPOSAL}, the expected error on the measured lifetime
only increases from 0.50\%\ to 0.51\%\ for 2~fb$^{-1}$. (For a much lower
S/B=3/1 the expected error would be 0.58\%.)


\subsubsection{Results for $\Delta\Gamma/\Gamma_{B_s}$ Sensitivity}
\index{width difference $\dg$!measurement of $\dg_s$!sensitivity}
\index{width difference $\dg$!$\Delta\Gamma_{CP}$}

With just two lifetime measurements, like $\tau_{CP+}$ and 
$\tau_{CP-}$, which are defined as $\tau_{CP+}=1/\Gamma(B_s^{even})$
and $\tau_{CP-}=1/\Gamma(B_s^{odd})$, one can determine the error
on $\Delta\Gamma_{CP}/\Gamma$ from the errors on the two lifetimes:
\begin{equation}
\sigma_{\Delta\Gamma_{CP}\over\Gamma} = 4\, {\tau_{CP+}\tau_{CP-}\over
  (\tau_{CP+}+\tau_{CP-})^2}\,
\sqrt{\left({\sigma_{\tau_{CP+}}\over\tau_{CP+}}\right)^2+
  \left({\sigma_{\tau_{CP-}}\over\tau_{CP-}}\right)^2}\,,
\end{equation}
where $\Delta\Gamma_{CP}=\Gamma(B_s^{even})-\Gamma(B_s^{odd})$ and
$\Gamma=(\Gamma(B_s^{even})+\Gamma(B_s^{odd}))/2$.  In the case that one
measures $\tau_{CP+}$ and $\tau_{FS}$ where
$\tau_{FS}=2/(\Gamma(B_s^{even})+\Gamma(B_s^{odd}))$, the error on
$\Delta\Gamma_{CP}/\Gamma$ is
\begin{equation}
\sigma_{\Delta\Gamma_{CP}\over\Gamma}
=2\,{\tau_{FS}\over\tau_{CP+}}\,
\sqrt{\left({\sigma_{\tau_{CP+}}\over\tau_{CP+}}\right)^2+
\left({\sigma_{\tau_{FS}}\over\tau_{FS}}\right)^2}.
\end{equation}

It can be seen that for small $\Delta\Gamma_{CP}/\Gamma$, the error is about 2
times larger when one measures with similar errors $\tau_{CP+}$ and
$\tau_{FS}$ compared to measuring $\tau_{CP+}$ and $\tau_{CP-}$.  Using
$\tau_{CP+}$ from the $J/\psi\eta^{(\prime)}$ decay modes and $\tau_{FS}$
from $D_s^-\pi^+$, we project an error on $\Delta\Gamma_{CP}/\Gamma$ of $0.027$
for $\Delta\Gamma_{CP}/\Gamma=0.15$ for 2~fb$^{-1}$.

Including the lifetime measurement for the decay mode to $J/\psi\pi$ is more
complicated as this is not an equal mixture of $CP$-even and $CP$-odd rates.
Although a combined lifetime and angular analysis should be done to determine
the fraction of $CP$-odd decay in this mode, and the appropriate error on
$\tau_{CP+}$, it was not done for this study.  A simpler determination is made
assuming that the fraction of $CP$-odd has already been determined within some
error in a separate analysis or in some other experiment. If the lifetime for
this decay mode is defined by $\tau_X=1/\Gamma_X$ where
$\Gamma_X=(1-f)\Gamma(B_s^{even})+f\Gamma(B_s^{odd})$, then
\begin{eqnarray}
{\Delta\Gamma_{CP}\over\Gamma} &=& {2(\tau_{FS}-\tau_X)\over (1-2f)\tau_X}\,,
  \\
\sigma_{\Delta\Gamma_{CP}\over\Gamma} &=& {2\tau_{FS}\over (1-2f)\tau_X}\,
\sqrt{\left({\sigma_{\tau_X}\over\tau_X}\right)^2+
  \left({\sigma_{\tau_{FS}}\over\tau_{FS}}\right)^2+
  \left({\tau_{FS}-\tau_X\over\tau_{FS}}\right)^2 {4f^2\over (1-2f)^2}
  \left({\sigma_f\over f}\right)^2}\,. \nonumber
\end{eqnarray}

Setting the value of $f$ to the central valued measured by CDF and assuming the
total error can be improved in Run 2 by a factor of $\sqrt{20}$, then for
$f=0.229\pm 0.043$ and using the $J/\psi\,\phi$ and $D_s^-\pi^+$ modes only, the
projected error on $\Delta\Gamma_{CP}/\Gamma$ is $0.035$ for
$\Delta\Gamma_{CP}/\Gamma=0.15$ and 2~fb$^{-1}$.  Although reducing the error on
$f$ has only a small effect on the projected error on
$\Delta\Gamma_{CP}/\Gamma$, the actual value of $f$ has a huge effect, since as
$f$ approaches 0.5 we lose all sensitivity to $\Delta\Gamma_{CP}$ when only
comparing to the $D_s^-\pi^+$ mode.  It can be seen that even with relatively
low statistics, the $J/\psi\eta^{(\prime)}$ decay modes are just as sensitive to
$\Delta\Gamma_{CP}$.

Table~\ref{tb_btevdgresults}\ shows the projected errors on
$\Delta\Gamma_{CP}/\Gamma$ for different integrated luminosities for the two
different combinations of modes used. The error on $f$ is assumed to reduced by
$\sqrt{R}$ where $R$ is the ratio of integrated luminosities. The total
projected error when all modes are used is also shown. To determine the expected
error when all modes are used, a likelihood fit is used where $f$ is constrained
by a Gaussian probability likelihood term to be within the 0.043 error of the
central value of 0.229. Note that the projected errors have a weak dependence on
the value of $\Delta\Gamma_{CP}/\Gamma$ used but a strong dependence on the
value of $f$ used. It should also be noted that we are assuming that any
systematic errors are insignificant compared to the statistical errors, so that
these projected statistical errors are taken as the total error on
$\Delta\Gamma_{CP}/\Gamma$. For the range of lifetime errors we are considering
this assumption is reasonable since the charm lifetimes can be measured to this
level in fixed target experiments with only small systematics. However the
lifetime acceptance correction in BTeV may be somewhat larger for the hadronic
modes.

\begin{table}[th]
\begin{center}
\begin{tabular}{cccc} \hline\hline
\bf{Decay Modes Used}      & 
       \multicolumn{3}{c}{\bf{Error on $\Delta\Gamma_{CP}/\Gamma$}}\\ \hline
                       & 2~fb$^{-1}$ & 10~fb$^{-1}$ & 20~fb$^{-1}$ \\ \hline
$J/\psi\eta^{(\prime)}$, $D_s^-\pi^+$ & 0.0273 & 0.0135& 0.0081\\
$J/\psi\,\phi$, $D_s^-\pi^+$             & 0.0349 & 0.0158& 0.0082\\
$J/\psi\eta^{(\prime)}$, $J/\psi\,\phi$, $D_s^-\pi^+$ 
                                      & 0.0216 & 0.0095& 0.0067\\ \hline
with $\Delta\Gamma_{CP}/\Gamma=0.03$  & 0.0198 & 0.0088& 0.0062\\
with $f=0.13$                         & 0.0171 & 0.0077& 0.0054\\
with $f=0.33$                         & 0.0258 & 0.0112& 0.0078\\
\hline\hline
\end{tabular}
\end{center}
\caption[Projections for statistical errors on $\Delta\Gamma_{CP}/\Gamma$ for
combining lifetimes from different modes and for using all modes for 2, 10 and
20~fb$^{-1}$.]{Projections for statistical errors on $\Delta\Gamma_{CP}/\Gamma$
for combining lifetimes from different modes and for using all modes for 2, 10
and 20~fb$^{-1}$. The values $\Delta\Gamma_{CP}/\Gamma=0.15$ and $f=0.229$ are
used for the main results and the results for other values of
$\Delta\Gamma_{CP}/\Gamma$ and $f$ are also shown for comparison.}
\label{tb_btevdgresults}
\end{table}

The statistical error can be improved by including also the $J/\psi\rightarrow
e^+e^-$ decay mode for the $J/\psi$ reconstruction. The increase in statistics
is less than by a factor of 2 since the BTeV ECAL acceptance is smaller than the
muon detector and there is no dedicated $J/\psi\rightarrow e^+e^-$ trigger.

It should also be noted that additional measurements of the $CP$-even rate and
especially of the $CP$-odd rate, even with low statistics, can have a very
significant effect on the $\Delta\Gamma_{CP}/\Gamma$ sensitivity. Unfortunately
the $CP$-odd modes look difficult experimentally. For example, two $CP$-odd modes
are $B_s^0\rightarrow J/\psi f_0(980)$ and $B_s^0\rightarrow \chi_{c0}\phi$. 
\index{decay!$B_s \rightarrow \psi f_0$}
\index{decay!$B_s \rightarrow \chi_{c0} \phi $}
The
$f_0(980)$ is relatively broad compared to the $\phi^0$ and decays to $\pi\pi$
or $KK$ and thus will have large backgrounds. The $\chi_{c0}$ has small
branching fractions, and the dominant decays are to non-resonant states with
pions and kaons and thus will also be background challenged. However it will
still be worthwhile looking for these $CP$-odd states.

%-------------------------------------------------------------------------------
\boldmath
\subsection{Summary of Projections for $\Delta\Gamma$}
\unboldmath
\index{width difference $\dg$!measurement of $\dg_s$!summary}
\index{summary!width difference $\dg_s$ projections}

The most sensitive direct measurement of $\Delta\Gamma_{B_s}$ will be from
measuring the lifetime differences between decays to $CP$-specific final states,
({\em i.e.} to $CP$-even, $CP$-odd or $CP$-mixed modes) and flavor specific decays.

The decay modes to flavor specific final states like $D_s\pi$ will be the
most precisely measured.  Other decays with larger branching fractions that can
be reconstructed with high efficiency and good signal-to-background will be to
$CP$-mixed final states, like $J/\psi\,\phi$ and $D_s^{(\ast) +}D_s^{(\ast) -}$
involving at least one $D_s^{(\ast)}$. These decays proceed through an unknown
admixture of $CP$-even and $CP$-odd amplitudes that must be determined
experimentally via an angular analysis.  The error on $\Delta\Gamma_s/\Gamma_s$
obtained using these modes will be very sensitive to the actual fractions of
$CP$-even and $CP$-odd, where the sensitivity is poorest for equal mixtures of
$CP$-even and $CP$-odd.

The decays to purely $CP$-even or purely $CP$-odd final states are difficult
experimentally, either because the backgrounds are larger and/or the branching
fractions are small ({\em e.g.} for $D_s^+D_s^-$, $J/\psi K_s$ and $J/\psi
f_0(980)$), or they contain difficult to reconstruct neutrals in the final state
({\em e.g.} like in $J/\psi\eta^{(\prime)}$). However it is important to try to
use these decay modes as even with small samples of events they improve the
error on the $\Delta\Gamma_s/\Gamma_s$ measurement significantly.

With 2~fb$^{-1}$ CDF should be able to determine $\Delta\Gamma_s/\Gamma_s$ with
a statistical error of 0.04 through lifetime measurements, improving to as good
as 0.025 if the decay to $D_s^{\ast +}D_s^{\ast -}$ proceeds through a 100\%\ 
$CP$-even amplitude. CDF can reach a statistical error of 0.01 on a {\em
model dependent} determination of $\Delta\Gamma_s/\Gamma_s$ using just
branching ratio measurements.

With a vertex (hadronic) trigger at Level-1 and excellent particle
identification and neutral reconstruction, the BTeV experiment should be able to
measure the lifetimes of the purely $CP$-even or purely $CP$-odd final states well
enough to give a model independent determination of $\Delta\Gamma_s/\Gamma_s$
with a statistical error of smaller than 0.02 with 2~fb$^{-1}$ of data. This
error is further decreased as low as 0.01 if the decay to $D_s^{\ast +}D_s^{\ast
  -}$ proceeds through a 100\%\ $CP$-even amplitude. With 20~fb$^{-1}$ of data a
statistical error of 0.005 on $\Delta\Gamma_s/\Gamma_s$ should be obtainable.
Systematic uncertainties are expected to be under control to a similar level.

%===============================================================================
\section{Projections for Lifetimes}

%-------------------------------------------------------------------------------
\boldmath
\subsection[$b$ Hadron Lifetimes at CDF]
{$b$ Hadron Lifetimes at CDF
$\!$\authorfootnote{Authors: Ch.~Paus, J.~Tseng.}}
\unboldmath
\index{lifetime!at CDF}

Lifetime measurements have formed an important part of the CDF research program
in $b$ physics since the 1992 introduction of the silicon vertex detector with
its precision tracking capabilities. This part of the research program has been
very successful, producing some of the most precise lifetime measurements with
semi-inclusive data samples, but also the most precise measurements with
exclusive channels. It is expected that the combination of the Tevatron's large
$\overline{b}b$ cross section and precise tracking capabilities will continue to
reap benefits in Run~II, pushing the comparison between experiment and
theoretical calculation to more stringent levels.

A lifetime measurement consists of reconstructing the decay point of a $b$
hadron by tracing back its long-lived charged descendants.  For instance, the
decay $\overline{B}^0\to e^-\overline{\nu}_eD^+X$ 
\index{decay!$B_d \rightarrow D^- \ell^+ \nu$}
is reconstructed by
intersecting the $e^-$ track with the trajectory of the $D^+$ meson; this
trajectory in turn is reconstructed from its own daughters, such as in the decay
$D^+\to K^-\pi^+\pi^-$.  The distance between the primary interaction
vertex and the $b$ decay vertex gives the flight distance of the $b$ hadron. In
Run~I, this flight distance was measured most precisely in the plane transverse
to the beam and the proper time of decay, $ct$, calculated with the formula
%
\begin{equation}
  ct = \frac{L_{xy}\,m}{p_T} \,,
\end{equation}
%
where $L_{xy}$ is the transverse flight distance, $m$ is the mass of the $b$
hadron, and $p_T$ is the momentum of the $b$ hadron projected in the with
respect to the beam direction transverse plane. The transverse momentum is
calculated by combining the measured momenta of the charged daughter particles.
Unlike the flight distance, this combination requires either the identification
and reconstruction of all the daughter particles, as in the analyses of an
exclusive hadronic channel, or a correction factor for the particles that are
not reconstructed, as in the analyses of much larger semi-inclusive samples. A
third possibility, applicable in some special circumstances, allows a constraint
to be applied to the momentum of a single particle of known mass; this technique
was not applied in Run~I but may become more important in Run~II with the advent
of three-dimensional silicon tracking.

As opposed to the electron--positron machines at the $\Upsilon(4S)$ where only
$B^+$ and $B^0_d$ mesons are produced, Tevatron produces the full spectrum of
$b$ hadrons. Due to this uniqueness in the following particular emphasize is put
on lifetime measurements of $B^0_s$, $B^0_c$ and $\Lambda_b$.

%...............................................................................
\subsubsection{Run~I Results at CDF}

The Run~I CDF lifetime results are summarized in
Figure~\ref{fig:cdf-blifetimes}.  They represent a combination of several
analyses of different types but have one thing in common: all data samples have
been obtained by using the trigger on at least one high-momentum lepton
candidate.

\begin{figure}[t]
\begin{center}
  \mbox{\epsfig{file=ch8/cdf_blifetimes.eps,width=0.6\textwidth}}
  \caption{Summary of lifetime measurements from CDF during Run~I.}
  \label{fig:cdf-blifetimes}
\end{center}
\end{figure}

Three different types of analyses are performed. There are the exclusive, the
semi-exclusive and finally the inclusive analyses. While in the exclusive
analyses one or more distinct $b$ hadron decay channels are fully reconstructed,
in the semi-inclusive analyses some neutral particle cannot be reconstructed,
most typically the neutrino from the semileptonic $b$ hadron decay. In the
inclusive analyses the secondary vertex indicates the presence of a $b$ hadron
but no attempt is made to reconstruct the mass explicitly.

The features of the different analysis types are complementary. Exclusive
analyses usually have small data samples and thus a large statistical
uncertainty but the systematic uncertainty is small. Inclusive analyses have
usually very large data samples and thus small statistical uncertainties but the
systematic errors are large.

The exclusive modes measured also used the $J/\psi$ trigger sample: in Run~I,
these analyses yield $436\pm 27$ decays of the type $B^0 \to \psi(1S,2S)
K^{(*)0}$, $824\pm 36$ of the type $B^+ \to \psi(1S,2S) K^{(*)+}$, and $58 \pm
12$ of the mode $B_s \to J/\psi\,\phi$.  \index{decay!$B_s \rightarrow \psi \phi$}
As will be detailed in the following sections, the $B_s$ lifetime thus measured
is expected to mostly reflect the lifetime of the shorter-lived weak eigenstate
of the $B_s$.  A small sample of $\Lambda_b \to J/\psi\Lambda^0$ decays was also
reconstructed but was too small for lifetime analysis in light of the large $B^0
\to J/\psi K^0_S$ backgrounds. The systematic errors are due to background and
resolution modeling and detector alignment.

For the semi-exclusive modes the $B^0$, $B^+$, $B_s$, and $\Lambda_b$ hadrons
are measured using their semileptonic decays. The charm daughters $D^{(*)}$,
$D_s$, and $\Lambda_c^+$ are reconstructed in data. Since these decays contain
neutrinos as well as possible unreconstructed intermediate states such as the
$D^{**}$, they are subject to systematic uncertainties due to production and
decay modeling, as well as uncertainties in background and resolution and
detector alignment.

An inclusive $b \to J/\psi X$ lifetime is also measured, where the
$J/\psi \to \mu^+\mu^-$ is registered on a dilepton trigger, as well as
the lifetime of the $B_c$ meson through its decays to $J/\psi\ell X$.

The lifetime measurements have also yielded measurements of the lifetime ratio
between charged and neutral $B$ mesons. The measurement of lifetime ratios 
\index{lifetime!ratios}
is
particularly interesting since experimental uncertainties and theoretical
uncertainties cancel out and thus allow more sensitive tests of some aspects of
the theory of heavy quarks. Combining the exclusive and the semileptonic decay
modes yields a ratio of $1.09\pm 0.05$ in agreement with the world average
$1.07\pm 0.02$ and both well within the range of the theoretical prediction
$1.0-1.1$.

The lifetime ratio between $B_s$ and $B^0$ has not yet been measured with
sufficient precision to test the prediction which is within $\approx 1$ percent
of unity.  The experimental value of the lifetime ratio of the $\Lambda_b$ to
the $B^0$ is of similar precision, but lies well below the current theoretical
prediction range of $0.9 - 1.0$. More data is needed to clarify whether there is
a discrepancy.  From the theoretical point of view baryons are more difficult to
calculate and thus the experimental data will hopefully shed some light on this
area.

%...............................................................................
\subsubsection{Run~II Projections for CDF}

To derive the expected lifetime uncertainties for the Run~II data samples it is
assumed that the Run~I measurements are statistically limited. This is certainly
true for the exclusive decay modes. For the other decay channels it is less
clear although experience shows that most systematic errors can be improved when
more statistics is available.  Therefore in the following only the exclusive
measurements are used to estimate the uncertainties on the $b$ hadron lifetime
measurements achievable with the Run~II data samples. This is a conservative
procedure and it is likely that CDF will do better. The increase in statistics
for the exclusive decays involving $J/\psi\to\mu^+\mu^-$ with respect to Run~I
is obtained applying simple scale factors as summarized in
section~\ref{sec:cdf-sin2beta}.

\paragraph{Leptonic Triggers}
\index{lifetime!at CDF!leptonic triggers}
%----------------------------
Considering the lifetime measurement capabilities of CDF in Run~II, it is
useful as a first step to make a direct extrapolation from the Run~I lifetime
analyses of exclusive channels. These are shown in Table~\ref{fig:cdf-ltproj},
assuming 2~{\ifb} of $J/\psi$ dimuon triggers with increased muon coverage,
lower muon trigger thresholds, and increased silicon tracking cover. The
projected uncertainties are only statistical, and, given Run~I experience, are
likely to be comparable to the levels of systematic uncertainty.  Other
improvements, not accounted for in the table, include the possibility of a
di-electron trigger, which adds approximately 50\% more data, and the effect of
smaller-radius silicon as well as three-dimensional microstrip tracking, which
improves $S/N$ and hence the lifetime measurements.

\begin{table}
\begin{center}
\begin{tabular}{  l  c c c  } \hline\hline
            &       Run 1 &       Run 2 & projected          \\
Species     & sample size & sample size & $c\tau$ error [ps] \\
\hline
$B^-$       &         824 &       40000 &               0.01 \\
$B^0$       &         436 &       20000 &               0.01 \\
$B_s$       &          58 &        3000 &               0.03 \\
$\Lambda_b$ &          38 &        2000 &               0.04 \\
\hline\hline
\end{tabular}
\end{center}
\caption{Run~II projections for the Run~I exclusive lifetime measurements at
CDF corresponding to 2~{\ifb}.}
\label{fig:cdf-ltproj}
\end{table}

% Details for the extrapolation (these have not been used but are also fine)
% ==========================================================================
%   B-:    824 x 1.5 x 2 x (2000/110) = 44945 events
%          (0.07ps) x sqrt(824/44945) = 0.009477
%
%   B0:    436 x 1.5 x 2 x (2000/110) = 23782
%          (0.09ps) x sqrt(436/23782) = 0.012185
%
%   BS:    see Wester (4000) or
%   ---------------------------
%   BS:    58 x 1.5 x 2 x (2000/110)  = 3164
%          (0.21ps) x sqrt(58/3164)   = 0.028
%
%   LB:     38 x 1.5 x 2 x (2000/110) = 2073 events
%          (0.264ps x sqrt(38/2073)   = 0.035743
%
% Barry and Manfred did more details but essentially it is the same as quoted
% here.

It is evident from the table that already the projections of only the exclusive
decay modes as measured by CDF in Run~I improves on the current world-averaged
$B^-/B^0$ lifetime ratio and combined with other experiments, including the $B$
factories, this ratio will be very precisely known.

In the case of the $B_s$, however, since the $J/\psi\,\phi$ final state is mostly
sensitive to one weak eigenstate, it is useful primarily in measuring
$\Delta\Gamma_s$ rather than the average $\Gamma_s$ which is to be compared with
$\Gamma_d$.  The $\Lambda_b \to J/\psi\Lambda^0$ 
\index{decay!$\Lambda_b \rightarrow J\psi\Lambda$}
lifetime is expected to
be known within 0.04~ps --- significantly better than the current world average
--- but depends upon being able to more effectively distinguish the signal from
$B^0 \to J/\psi K^0_S$ decays, which are topologically very similar.

Comparable projections for $B_c$ and $\Xi_b$ exclusive lifetime analyses cannot
be made, since their exclusive decays were not observed in Run~I data.

\paragraph{Hadronic Triggers}
\index{lifetime!at CDF!hadronic triggers}
%----------------------------
Beyond the $J/\psi \to \ell^+\ell^-$ trigger samples, there are expected
to be large data samples that will be made available by the hadronic
displaced-track trigger.  Although those samples will also improve on the $B^-$
and $B^0$ lifetimes most interestingly they will improve on the $B_s$ and
$\Lambda_b$ lifetime measurements. Since the hadronic trigger is a new hardware
device all predictions are less certain than the predictions for decay modes
originating from the leptonic triggers.

Most branching fractions for the decay modes used below are not measured and
have to be estimated. This is particularly difficult for the $\Lambda_b$ decay
modes.  Errors of 50 percent for the $B_s$ decays and 100 percent for the
$\Lambda_b$ decays are assumed.

Another principle difficulty in these data samples lies in the understanding of
the effect of the trigger on the lifetime distributions. The displaced vertex
trigger prefers large lifetime events and introduces a bias in the proper time
distribution. In the following it is assumed that it is possible to model the
trigger bias and use all of the statistical power of the projected yields.

% assume 75000 Bs, S/N(Bs): 1.0
%
%    sqrt(75,000 * 2)/75,000 * 1.36 = 0.007
%
For the $B_s$ decay modes $D_s \pi$ and $D_s \pi\pi\pi$ 
\index{decay!$B_s \rightarrow D_s^- \pi^+$}
\index{decay!$B_s \rightarrow D_s^- \pi^+ \pi^+ \pi^-$}
there are approximately
75k events projected. This includes the $D_s$ decay modes to $\phi \pi$ and
$K^*K$ only, as indicated in Table~\ref{tab:hadrtrig-bs}. Further it is expected
that the signal to background ratios is one, which is rather conservative. This
results in an uncertainty of the $B_s$ lifetime of approximately 0.007~ps.

For the lifetime ratio of $B_s$ and $B^0$ this corresponds to an error of
roughly half a percent and is thus in the same order as the theoretical
prediction for the deviation from unity.

% assume 24400 Lb, S/N(Lb): 1.0
%
%    sqrt(24,400 * 2)/24,400 * 1.229 = 0.011
%
For the $\Lambda_b$ decay modes $\Lambda_c \pi (\Lambda_c\to pK\pi)$, $pD^0\pi
(D^0 \to K \pi$ and $D^0 \to K \pi\pi\pi)$, 
\index{decay!$\Lambda_b \rightarrow \Lambda_c^+\pi^-$}
\index{decay!$\Lambda_b \rightarrow pD^0\pi^-$}
$p\pi$ and $pK$ there are
approximately 24.4k events projected. Again assuming a signal to background
ratio of one, a statistical uncertainty of 0.01~ps is obtained. This is more
precise than the expectation from the $J/\psi\Lambda^0$ decay mode. A stringent
test will be available for the theoretical predictions of the lifetime ratio of
$\Lambda_b$ to $B^0$.

More inclusive strategies, using triggerable combinations of leptons and
displaced tracks, will also significantly increase the precision of the lifetime
ratios of the rarer $b$ hadrons, such as the $B_c$ and so far by CDF not
observed baryons as $\Xi_b$ might be accessible. These strategies have yet to be
investigated in detail.

%-------------------------------------------------------------------------------
\subsection[Lifetime measurements at \dzero]
{Lifetime measurements at \dzero
$\!$\authorfootnote{Author: W.~Taylor.}}

\index{lifetime!at \dzero}

A rich spectroscopy and lifetime measurement program is planned for both beauty
mesons and baryons. We will study any species that has significant decay modes
that result in at least one lepton.  One of the highlights of this program is a
measurement of the $\Lambda_b$ lifetime in the exclusive decay mode $\Lambda_b
\rightarrow J/\psi + \Lambda$. 
This is particularly interesting since the
measured ratio $\tau(\Lambda_b) / \tau(B_d) = 0.78 \pm 0.04$ \cite{Ref:LAMBLIFE}
is significantly different from the naive spectator model prediction of unity.
Current theoretical understanding of non-spectator processes such as final-state
quark interference and $W$ boson exchange cannot account for such a large
deviation.  Another important measurement which we will make is a measurement of
the $B_s$ lifetime using modes such as $B_s\rightarrow J/\psi \phi$.

Measurements of the $\Lambda_b^0$ baryon lifetime have long been hampered by a
dearth of statistics.  For this reason, the lifetime was measured in the
semileptonic decay mode, where the branching fraction is orders of magnitude
higher than for any fully reconstructed
mode~\cite{Ref:ALEPH,Ref:CDFI,Ref:DELPHI,Ref:OPAL}.  The disadvantage of the
semileptonic mode is that the neutrino information is lost.  To obtain the true
decay length in the absence of neutrino information, the ratio $p_T(\Lambda_c
\ell)/p_T(\Lambda_b^0)$ (called the $K$ factor) was obtained from a Monte Carlo
calculation and used in the lifetime fit.  The limited knowledge of the value of
the $K$ factor represented a modest contribution to the uncertainty of this
measurement.

In Run~II at D\O\, we can expect to collect 2~fb$^{-1}$ of $p\overline{p}$
collisions or twenty times the statistics of Run~1. This provides the
opportunity to probe the $\Lambda_b^0$ lifetime in a fully reconstructed mode:
$\Lambda_b^0 \rightarrow J/\psi \Lambda^0$, 
\index{decay!$\Lambda_b \rightarrow J\psi\Lambda$}
where the $J/\psi$ meson decays into
two leptons (muon or electron) and the $\Lambda^0$ baryon decays into a proton
and a pion.  Using a fully reconstructed channel avoids the introduction of the
$K$ factor and any uncertainty associated with it.

To study this mode, we generated 75~000 $\Lambda_b^0$ baryon events using
Pythia, simulated the D\O\ Run~II detector using MCFast, and forced the decay
mode $\Lambda_b^0 \rightarrow J/\psi \Lambda^0$, followed by $J/\psi \rightarrow
\mu^+\mu^-$ and $\Lambda^0\rightarrow p \pi^-$ with QQ, the CLEO Monte Carlo
program.  We use this sample to predict the trigger and reconstruction
acceptances and to estimate the lifetime resolution.  For the purpose of this
study, we assume that electrons from $J/\psi$ decays will be reconstructed with
a similar efficiency to that of the muons, an assumption which is expected to be
approximately true.

The D\O\ $J/\psi$ trigger%
\index{\dzero!$J/\psi$ trigger}%
identifies events resembling $J/\psi$ decays.  The presence of two muon tracks,
each with $|\eta|\leq2.0$ and $p_T \geq 1.5$~GeV/$c$, is sufficient for passing
the $J/\psi$ trigger criteria.  The trigger acceptance is found to be 16.8\% for
the muons in these generated Monte Carlo events.  We introduce by hand an
additional acceptance cut of 55\% for the holes in the muon detection system not
modeled in MCFast.

Track quality cuts are applied to all the tracks in the events passing the
trigger.  The muon tracks from the $J/\psi$ decay are required to have at least
four stereo hits in the tracking system (silicon detector and central fiber
tracker combined) and at least eight hits (stereo and axial combined) in the
silicon detector.  For tracks in the central region ($|\eta|\leq1.7$), at least
fourteen hits are required in the central fiber tracker.  The transverse
momentum is required to be above 400~MeV/$c$.

The $\Lambda^0$ baryon daughters are required to have at least three stereo hits
in the tracking system (silicon detector and central fiber tracker combined) and
a minimum of eight hits in the central fiber tracker.  The transverse momentum
is required to be above 400~MeV/$c$. The long lifetime of the $\Lambda^0$ baryon
prevents the use of strict silicon hit requirements, as the $\Lambda^0$ baryon
often decays outside of the fiducial volume of the silicon detector.

The muons are constrained to come from a common vertex in three dimensions as
are the $\Lambda^0$ daughters.  Finally, the location of the $\Lambda_b^0$ decay
vertex is obtained by extrapolating the $\Lambda^0$ baryon momentum direction in
three dimensions back to the $J/\psi$ decay vertex.  The resulting vertex
defines the decay length of the $\Lambda_b^0$ baryon.

The acceptance for these selection criteria is 9.6\%.  We introduce by hand an
additional acceptance cut of (0.95)$^4$=0.81 for the tracking efficiency not
modeled in MCFast.  The final acceptance times efficiency for the trigger and
reconstruction criteria is 0.72\%.  We therefore predict about 15~000 events to
be reconstructed in 2~fb$^{-1}$. Table~\ref{tab:yield} shows the values used to
obtain this prediction for the yield.

\begin{table}
\centerline{
\begin{tabular}{lr} \hline\hline
  \multicolumn{1}{c}{Multiplier} & \multicolumn{1}{c}{Events} \\
  \hline
  &\\[-1em]
  ${\cal L} = 10^{32}$ cm$^{-2}$s$^{-1}$ =  $10^{36}$ m$^{-2}$s$^{-1}$ & \\
  $t=2\times10^7$ s                                                    & \\
  $\sigma_{b\overline{b}}$ = 158 $\mu$b $\times 10^{-28}$ m$^2$/b      & \\
  $b,\overline{b}$ = 2                               & $6.32\times10^{11}$ \\
  ${\cal B}(b\rightarrow \Lambda_b^0)=0.090$         & $5.7\times10^{10}$ \\
  ${\cal B}(\Lambda_b^0\to J/\psi\Lambda^0)=4.7\times10^{-4}$
                                                     & $2.7\times10^{7}$ \\
 ${\cal B}(J/\psi\rightarrow \mu\mu,ee)=2\times0.06$ & $3.2\times10^6$ \\
  ${\cal B}(\Lambda^0 \rightarrow p \pi)=0.639$      & $2.0\times10^6$ \\
  Detector acceptance = 0.55                         & $1.1\times10^6$ \\
  $\epsilon$(trigger) = 0.168                        & $1.9\times10^5$ \\
  $\epsilon$(track efficiency) = 0.95$^4$            & $1.5\times10^5$ \\
  $\epsilon$(reconstruction) = 0.096                 & 14~850 \\
  \hline\hline
\end{tabular}}
\vspace*{6pt}
\caption{Event yield determination after 1 year or 2~fb$^{-1}$.}
\label{tab:yield}
\end{table}

The mass distributions for the $J/\psi$ and $\Lambda^0$ particles are shown in
Figures~\ref{8fig:jpsimass} and \ref{fig:lambdamass}, respectively.  With the
application of constraints on both the $J/\psi$ mass and the $\Lambda^0$ mass,
we predict the $\Lambda_b^0$ mass resolution to be 16~MeV/$c^2$, as indicated in
Figure~\ref{fig:lambdabmass}.  The lifetime resolution is found to be 0.11~ps
(Figure~\ref{fig:tausig}).

%-------------------------------------------------------------------------------
\subsection{Summary of Projections for Lifetimes}
\index{summary!lifetime projections}
\index{lifetime!experimental lifetime}

The exclusive $b$ hadron event samples involving $J/\psi\to\mu^+\mu^-$ alone
will enable CDF and D{\O} during Run~II to push the precision of lifetime
measurements to values close to 1~fs. Further inclusion of other hadronic decay
modes will decrease the statistical error further below 1~fs.

In particular the lifetime measurements of $B_s$ and $\Lambda_b$ which cannot be
measured by the $B$ factories will achieve a statistical precision of about
0.007~ps and 0.01~ps, respectively. Those measurements determine the ratio of
$\tau(B_s)/\tau(B^0)$ to roughly half a percent which allows a first test of the
theoretical predictions.

Lifetimes of other interesting $b$ hadrons like the $B_c$ will be improved
significantly, and there is a good chance to observe $\Xi_b$ for the first time
and measure its lifetime.

\begin{figure}[t]
\centerline{\psfig{figure=ch8/d0_jpsi-mass.eps,width=6.5cm}}
\caption{\label{8fig:jpsimass} $J/\psi$ mass distribution.}
\end{figure}

\begin{figure}
\centerline{\psfig{figure=ch8/d0_lambda-mass.eps,width=6.5cm}}
\caption{\label{fig:lambdamass}$\Lambda^0$ mass distribution.}
\end{figure}

\clearpage

\begin{figure}[thb]
\centerline{\psfig{figure=ch8/d0_lambdab-mass.eps,width=6.5cm}}
\caption{\label{fig:lambdabmass}$\Lambda_b^0$ mass distribution.}
\end{figure}

\begin{figure}[thb]
\centerline{\psfig{figure=ch8/d0_tausig.eps,width=6.5cm}}
\caption{\label{fig:tausig}$\tau(\Lambda_b^0)_{meas}-\tau(\Lambda_b^0)_{gen}$.}
\end{figure}

%===============================================================================

%===============================================================================
\section{Conclusions}

The Tevatron provides a unique testing ground for mixing and lifetime studies.
For these measurements it is superior to the $B$-factories, because the large
boosts of the produced hadrons and the higher statistics allow to study the
decay distributions more precisely.  Moreover, studies of $B_s$ mesons and
$b$-flavored baryons are not possible at current $B$-factories running at the
$\Upsilon(4S)$ resonance. At Run~II \bbms\ will be discovered and the mass
difference $\dm_s$ will be determined very precisely.  This measurement is of
key importance for the phenomenology of the unitarity triangle.  Our knowledge
of the lifetime pattern of $b$-flavored hadrons will significantly improve, the
yet undetected width difference $\dg_s$ between the $B_s$ mass eigenstates is
within reach of Run~II and the $\Lambda_b$ lifetime puzzle will be addressed.


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%%%%%%% end JoAnne

%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CDF Intro %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\bibitem{Ref:PETERSON}

  C.~Peterson {\it et al.}, Phys.~Rev. {\bf D27} (1983) 105.

\bibitem{Ref:CLEOMC}

  P.~Avery, K.~Read and G.~Trahern, Cornell Internal Note CSN-22, 1985
  (unpublished).

\bibitem{Ref:CDF-XS-BD}
  
  T.~A.~Keaffaber, J.~D.~Lewis, M.~W.~Bailey, D.~Bortoletto, S.~Tkaczyk,
  A.~F.~Garginkel, {\it Measurement of the B Meson Differential Cross Section
    Using the Exclusive Decay $B \rightarrow J/\psi K$ }, CDF Note 4911 (to be
  updated, 2000).

\bibitem{Ref:CDF-FSFUFD}
  
  %S.~Gadomski, P.~K.~Sinervo, W.~Taylor, W.~Trischuk, {\it A Combined
  %  Measurement of the B Meson Fragmentation Fraction Ratio ${f_s/(f_u + f_d)}$
  %  Using Run~I Inclusive Electron and Dimuon Data}, CDF Note 5007, (2000).

  F.~Abe {\it et al.}, The CDF Collaboration,
  % Measurement of b Quark Fragmentation Fractions in the Production of Strange
  % and Light B Mesons in p anti-p Collisions at $\sqrt{s} = 1.8~\TeV$
  Phys.~Rev.~{\bf D60} (1999) 92005.

\bibitem{Ref:PDG}
  
  C. Caso, {\it et al.} [Particle Data Group], Eur.~Phys.~Jor. {\bf C15} (2000)
  1.

\bibitem{Ref:TDR}

  CDF Run~II Technical Design Report.

\bibitem{Ref:PAC}
  
  The CDF Collaboration, Update to Proposal P-909: {\it Physics Performance of
    the CDF II Detector with An Inner Silicon Layer and A Time of Flight
    Detector}, Submitted to the Fermilab Director and PAC (1999)

\bibitem{Ref:MBR}

  S. Belforte {\it et al.}, {\it A Complete Minimum Bias Event Generator}, CDF
  Note 4219.

\bibitem{Ref:PDG99}
  
  C. Caso, {\it et al.} [Particle Data Group], Eur.~Phys.~Jor. {\bf C3} (1998)
  1.

\bibitem{Ref:CKM-MATRIX}

  F.~Parodi, P.~Roudeau and A.~Stocchi,
  %Constraints on the Parameters of the V(CKM) Matrix by End 1998
  Nuovo~Cim.~{\bf A112} (1999) 833.

\bibitem{PDG} Particle Data Group, Eur. Phys. J. {\bf C 15} (2000) 1.

%%%%%%%%%%%%%%%%%%%%%%%%%%%% CDF Mixing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{Ref:SemiBsLifetime}

  F.~Abe {\it et al.}  [CDF Collaboration],
  % Measurement of the $B^0_s$ Meson Lifetime using Semileptonic Decays
  Phys.~Rev.~{\bf D59} (1999) 032004.

\bibitem{Ref:SemiBsMixing}

  F.~Abe {\it et al.}  [CDF Collaboration],
  % A search for B/s0 anti-B/s0 oscillations using the semileptonic
  % decay $B_s^0 \to \phi \ell X \nu$
  Phys.~Rev.~Lett.~{\bf 82} (1999) 3576.

%%%%%%%%%%%%%%%%%%%%%%%%%%%% D0 Mixing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%% BTeV Mixing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{Ref:BTEV-PTDR}
  
  BTeV Preliminary Technical Design Report.

\bibitem{Ref:BD-PROD}
  
  F.~Abe {\it et. al.} [The CDF Collaboation], Phy.~Rev.~{\bf D54}, 6596
  (1996).

\bibitem{Ref:MCBRIDE-STONE}
  
  P.~McBride and S.~Stone, Nucl. Instrum. Meth. {\bf A368}, 38 (1995).

\bibitem{Ref:BTEV-PROPOSAL}
  
  BTeV Collaboration, {\it Proposal for an Experiment to Measure Mixing, $CP$
    Violation and Rare Decays in Charm and Beauty Particle Decays at the
    Fermilab Collider -- BTeV}, May 2000.

\bibitem{Ref:KTMCDONALD}
  
K.T.~McDonald, {\it Maximum Likelihood Analysis of $CP$ Violating
Asymmetries},\\
  PRINCETON-HEP-92-04, Sep 1992. 12pp, unpublished.

%%%%%%%%%%%%%%%%%%%%%%%%%%%% CDF Delta Gamma %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{Ref:tauBs-lDX}

  F.~Abe {\it et al.}  [CDF Collaboration],
  % Measurement of the $B^0_s$ Meson Lifetime using Semileptonic Decays
  Phys.~Rev.~{\bf D59} (1999) 032004.

\bibitem{Ref:tauBs-JPsiPhi}

  F.~Abe {\it et al.}  [CDF Collaboration],
  % Measurement of the Lifetime of the $B^0_s$ Meson using the Exclusive Decay
  % Mode $B^0_s \to J/\psi\,\phi$
  Phys.~Rev.~Lett.~{\bf 77} (1996) 1945.

\bibitem{Ref:polBd-Bs}

  F.~Abe {\it et al.}  [CDF Collaboration],
  % Measurement of the Polarization in the Decays of the $B^0 \to J/\psi
  % K^{*0}$ and $B^0_s \to J/\psi\,\phi$
  Phys.~Rev.~Lett.~{\bf 75} (1995) 3068, Phys.~Rev.~Lett.~{\bf 85} (2000) 4668.

\bibitem{Ref:Bs-NewPhys}

  I.~Dunietz, R.~Fleischer, U.~Nierste,
  {\it In Pursuit of New Physics with $B_s$ Decays},
  CERN-TH/2000-333 .

\bibitem{cdf:bslife}

  F.~Abe {\it et al.}, {\it Phys. Rev. Lett.} 77(1996)1945.

\bibitem{Ref:ALEKSAN-DSSTAR}

  R.~Aleksan, A.~Le~Yaouanc, L.~Oliver, O.~P\`ene, J.-C.~Raynal, {\it Phys.
    Lett.} {\bf B317} (1993) 173.

\bibitem{Ref:ALEKSAN-DGAMMA}

  R.~Aleksan, A.~Le~Yaouanc, L.~Oliver, O.~P\`ene, J.-C.~Raynal, {\it Phys.
    Lett.} {\bf B316} (1993) 567.

\bibitem{Ref:ALEPH-PHIPHI}

  The ALEPH Collaboration, CERN-EP/2000-036, to be submitted to {\it Phys.
    Lett.} {\bf B}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BTeV Delta Gamma %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{Ref:BTEV-CDFJPSIPHI}
  
  T. Affolder {\it et al.}, CDF Collaboration, Phys. Rev. Lett. {\bf 85} (2000)
  4668.

\bibitem{Ref:BTEV-E687LIFETIME}
  
  P.~L.~Frabetti {\it et al.}, E687 Collaboration, Phys. Rev. Lett. {\bf 70}
  (1993) 1755.

%%%%%%%%%%%%%%%%%%%%%%%%%%%% CDF Lifetimes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{Ref:JpsiBsLifetime}

  F.~Abe {\it et al.}, {\it Phys. Rev. Lett.} 77(1996)1945.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D0 Lifetimes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{Ref:LAMBLIFE}

  LEP B Lifetime Working Group, 
  http://home.cern.ch/$\sim$claires/lepblife.html.

\bibitem{Ref:ALEPH}
  
  ALEPH Collaboration, R.~Barate {\em et al.}, {\em Measurement of the $b$
    Baryon Lifetime and Branching Fractions in $Z$ Decays}, Eur. Phys. J.~{\bf
    C2}, (1998) 197.

\bibitem{Ref:CDFI}
  
  CDF Collaboration, F.~Abe {\em et al.}, {\em Measurement of the $\Lambda_b^0$
    Lifetime Using $\Lambda_b^0 \rightarrow \Lambda_c^+ \ell^- \overline{\nu}$},
  Phys. Rev.  Lett.~{\bf 77}, (1996) 1439.

\bibitem{Ref:DELPHI}
  
  DELPHI Collaboration, P.~Abreu {\em et al.}, {\em Determination of the Average
    Lifetime of $b$ Baryons}, Z.  Phys.~{\bf C71}, (1996) 199.

\bibitem{Ref:OPAL}
  
  OPAL Collaboration, K.~Ackerstaff {\em et al.}, {\em Measurements of the
    $B_s^0$ and $\Lambda_b^0$ Lifetimes}, Phys.  Lett.~{\bf B426}, (1998) 161.

\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter[Production, Fragmentation and Spectroscopy]
{Production, Fragmentation and Spectroscopy}

\def\et{$E_T$} 
\def\ptrel{$P_T^{rel}$}
\def\bbar{$\overline{b}$}
\def\bbbar{$b\overline{b}$}
\def\into{\rightarrow} 


\authors{
G.~Bodwin, 
E.~Braaten, 
K.~Byrum, 
R.K.~Ellis, 
G.~Feild, 
S.~Fleming, 
W.~Hao,
B.~Harris, 
V.V.~Kiselev, 
E.~Laenen, 
J.~Lee, 
A.~Leibovich,
A.~Likhoded, 
A.~Maciel, 
S.~Menary,
P.~Nason, 
E.~Norrbin, 
C.~Oleari,
V.~Papadimitriou
G.~Ridolfi, 
K.~Sumorok, 
W.~Trischuk,
R.~Van Kooten
}
\label{ProductionFragmentationandSpectroscopy}

\input \chninehome/preamble.tex
\input \chninehome/sec1/prod
\input \chninehome/sec2/quarkonium
\input \chninehome/sec3/bc
\input \chninehome/sec4/dhb
\input \chninehome/sec5/frag
\clearpage
\input \chninehome/ref


\begin{center}
\sl\vspace*{-26pt}
${}^{1}$Argonne National Laboratory, Argonne, Illinois \\
${}^{2}$Brandeis Universtiy, Waltham, Massachusetts \\
${}^{3}$Brookhaven National Laboratory, Upton, New York \\
${}^{4}$University of California, Riverside, California \\
${}^{5}$University of California, San Diego, California \\
${}^{6}$Carnegie Mellon University, Pittsburgh, Pennsylvania \\
${}^{7}$CERN, Geneva, Switzerland \\
${}^{8}$Deutsches Elektronen-Synchrotron (DESY), Hamburg, Germany \\
${}^{9}$Fermi National Accelerator Laboratory, Batavia, Illinois \\
${}^{10}$Glasgow University, Glasgow, United Kingdom \\
${}^{11}$Harvard University, Cambridge, Massachusetts \\
${}^{12}$University of Illinois, Urbana-Champaign, Illinois \\
${}^{13}$Imperial College, London, United Kingdom \\
${}^{14}$Indiana University, Bloomington, Indiana \\
${}^{15}$INFN, Genoa, Italy \\
${}^{16}$Institute for Theoretical and Experimental Physics, Moscow, Russia \\
${}^{17}$Institute for High Energy Physics, Protvino, Russia \\
${}^{18}$Iowa State University, Ames, Iowa \\
${}^{19}$The Johns Hopkins University, Baltimore, Maryland \\
${}^{20}$University of Kentucky, Lexington, Kentucky \\
${}^{21}$Lawrence Berkeley National Laboratory, Berkeley, California \\
${}^{22}$Lund University, Lund, Sweden \\
${}^{23}$Massachusetts Institute of Technology, Cambridge, Massachusetts \\
${}^{24}$INFN \& Universit\`a degli Studi di Milano, Milan, Italy \\
${}^{25}$University of Minnesota, Minneapolis, Minnesota \\
${}^{26}$Robert Morris College, Moon Township, Pennsylvania \\
${}^{27}$NIKHEF, Amsterdam, The Netherlands \\
${}^{28}$Northern Illinois University, DeKalb, Illinois \\
${}^{29}$University of Notre Dame, Notre Dame, Indiana \\
${}^{30}$The Ohio State University, Columbus, Ohio \\
${}^{31}$Oxford University, Oxford, United Kingdom \\
${}^{32}$University of Pennsylvania, Philadelphia, Pennsylvania \\
${}^{33}$INFN \& Universit\`a e Scuola Normale Superiore di Pisa, Pisa, Italy \\
${}^{34}$Max-Planck-Institut f\"ur Physik, Munich, Germany \\
${}^{35}$Stanford Linear Accelerator Center, Stanford, California \\
${}^{36}$State University of New York, Stony Brook, New York \\
${}^{37}$Syracuse University, Syracuse, New York \\
${}^{38}$Technion--Israel Institute of Technology, Haifa, Israel \\
${}^{39}$Texas Tech University, Lubbock, Texas \\
${}^{40}$University of Toronto, Toronto, Canada \\
${}^{41}$University of Tsukuba, Tsukuba, Japan  \\
${}^{42}$Vanderbilt University, Nashville, Tennessee \\
${}^{43}$Weizmann Institute of Science, Rehovot, Israel \\
${}^{44}$University of Wisconsin, Madison, Wisconsin \\
${}^{45}$Yale University, New Haven, Connecticut \\
${}^{46}$York University, Toronto, Canada
\end{center}
\begin{titlepage}

\vspace*{-1.6cm}
\centerline{\raisebox{11.2pt}{\epsfxsize 1cm\epsffile{fnal.ps}} \hfill 
  \vbox{\hbox{FERMILAB-Pub-01/197}\hbox\hbox{December, 2001}}}

\begin{center}

\vspace*{1cm}
{\bf\Huge \boldmath $B$ Physics at the Tevatron: \\[12pt]
  Run II and Beyond}
\vspace*{1.75cm}

\includegraphics*[scale=0.8]{logo.ps}

\vspace*{1.25cm}

{\bf Authors} \vspace*{-2mm}

\addtolength{\hsize}{-.4cm}
\parbox{\hsize}{\input{authors}}
\addtolength{\hsize}{.4cm}

\vspace*{-1.25cm}
\end{center}

\end{titlepage}


\thispagestyle{empty}
\input{institutions}

\input{preface_zl}

%%\input{preface_ask}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter*{Workshop Structure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The workshop formed four semi-autonomous working groups.  Each was led by two
theory conveners and a convener from each experiment. 

\begin{itemize}

\item
{\bf CP Violation} \\ 
Yossi Nir, Helen Quinn, Manfred Paulini (CDF), 
Rick Jesik (D\O), Tomasz Skwarnicki (BTeV) 

\item
{\bf Rare and Semileptonic Decays} \\ 
Aida El-Khadra, Mike Luke, Jonathan Lewis (CDF), 
Andrzej Zieminski (D\O), Ron Poling (BTeV) 

\item
{\bf Mixing and Lifetimes} \\ 
Ulrich Nierste, Mikhail Voloshin, Christoph Paus (CDF), 
Neal Cason (D\O), Harry Cheung (BTeV) 

\item
{\bf Production, Fragmentation, Spectroscopy} \\  
Eric Braaten, Keith Ellis, Eric Laenen, 
William Trischuk (CDF), Rick Van Kooten (D\O), Scott Menary (BTeV) 

\end{itemize} 
Chapters \mbox{6 -- 9} of this report present the work done in these groups.

The programs of the general meetings, including transparencies of all talks,
working group information, and other documentation for this workshop can be
found on the WorldWideWeb at \verb"http://www-theory.lbl.gov/Brun2/".

The workshop was organized by
Rick Jesik, 
Andreas Kronfeld,
Rob Kutschke,
Zoltan Ligeti,
Manfred Paulini, and
Barry Wicklund.


\OMIT{
In addition, Chapter 1 provides a theoretical introduction, Chapters 2 -- 5
give introductions to the detectors and simulation tools, and Chapter 10
highlights the main results.  Chapter 1 was convened by U.~Nierste, Z.~Ligeti,
and A.~Kronfeld; Chapter 2 was convened by R.~Kutschke, R.~Jesik, and
M.~Paulini; Chapter 3 was convened by M.~Paulini and B.~Wicklund; Chapter 4 was
convened by R.~Jesik; and Chapter 5 was convened by R.~Kutschke.
}
%%  This file contains pieces from the BaBar PhysBook's symbols.tex
%%  and def.tex, together with some of Uli's and Zoltan's def's.
%%

\renewcommand{\d}{{\rm d}}
\def\OMIT#1{{}}

% General Equation Shortcuts
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}} 
\newcommand{\beqa}{\begin{eqnarray}}
\newcommand{\eeqa}{\end{eqnarray}}
\newcommand{\bey}{\begin{eqnarray}}
\newcommand{\eey}{\end{eqnarray}}
\newcommand{\nn}{\nonumber \\}
\newcommand{\no}{\nonumber }

%% CKM
\def\Vud{V_{ud}} \def\Vus{V_{us}} \def\Vub{V_{ub}}
\def\Vcd{V_{cd}} \def\Vcs{V_{cs}} \def\Vcb{V_{cb}}
\def\Vtd{V_{td}} \def\Vts{V_{ts}} \def\Vtb{V_{tb}}

% General
\def\ifmath#1{\relax\ifmmode#1\else$#1$\fi}
\def\etal{{\it et~al.,}}
\def\etc{{\it etc.}}
\def\ie{{\it i.e.}}
\def\eg{{\it e.g.}}
\def\vs{{\it vs.}}

\def\D0{D\O}

%%%%%%%%%%%%%% Stuff for math environment:
\newcommand{\lt}{\left}
\newcommand{\rt}{\right}
\newcommand{\ov}{\overline}

\renewcommand{\Im}{{\rm Im}\,}
\renewcommand{\Re}{{\rm Re}\,}
\newcommand{\imag}{\Im}
\newcommand{\real}{\Re}
\newcommand{\diag}{{\rm diag}\,}

\def\BR{{\rm BR}}
\def\epsK{\varepsilon_K}
\def\epsp{\varepsilon^\prime_K}
%\def\gsim{{~\raise.15em\hbox{$>$}\kern-.85em
%          \lower.35em\hbox{$\sim$}~}}
%\def\lsim{{~\raise.15em\hbox{$<$}\kern-.85em
%          \lower.35em\hbox{$\sim$}~}}
\def\gsim{\gtrsim}
\def\lsim{\lesssim}

\def\lesssim{\mathrel{\mathpalette\vereq<}}
\def\gtrsim{\mathrel{\mathpalette\vereq>}}
\def\vereq#1#2{\lower4pt\vbox{\baselineskip1pt\lineskip1pt
    \ialign{\\$#1\hfill##\hfil\\$\crcr#2\crcr\sim\crcr}}}


%%%%%% A unitmatrix, taken from Manfred and Revtex:
\def\openone{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}%

% Units
\newcommand{\gev}{\,\mbox{GeV}}
\newcommand{\mev}{\,\mbox{MeV}}
\newcommand{\keV}{\ensuremath{\mathrm{ke\kern -0.1em V}}}
\newcommand{\eV}{\ensuremath{\mathrm{e\kern -0.1em V}}}
\newcommand{\MeV}{\ensuremath{\mathrm{Me\kern -0.1em V}}}
\newcommand{\GeV}{\ensuremath{\mathrm{Ge\kern -0.1em V}}}
\newcommand{\TeV}{\ensuremath{\mathrm{Te\kern -0.1em V}}}


\def\degrees{\mbox{$^{\circ}$}}
\def\vek#1{\mbox{\protect\boldmath $#1$}}

%%%%%%%%%%%%% Stuff for tables/arrays:
\newcommand{\ds}{\displaystyle}


%%%%%%%%%%%%% Physics stuff:
\renewcommand{\C}{\ensuremath{C}}
\newcommand{\p}{\ensuremath{P}}
\newcommand{\CP}{\ensuremath{CP}}
\newcommand{\T}{\ensuremath{T}}
\newcommand{\CPT}{\ensuremath{C\!PT}}

\newcommand{\Bbar}{\,\overline{\!B}}
\newcommand{\Dbar}{\,\overline{\!D}}
\newcommand{\Kbar}{\,\overline{\!K}}
\def\B0bar{\Bbar{}^0}

\newcommand{\adg}[1]{\ensuremath{{\cal A}^{#1}_{\rm \Delta\Gamma}}}
%%\newcommand{\adg}{\ensuremath{{\cal A}_{\rm \Delta\Gamma}}}
\newcommand{\brunt}[1]{ \ensuremath{\BR [ { #1} ]}}
\newcommand{\gunt}[1]{\ensuremath{ \Gamma \lt[#1,t\rt] }}
%%\newcommand{\guntf}{\ensuremath{\Gamma (\Bun (t)\rightarrow f )}}
\newcommand{\guntf}{\ensuremath{\Gamma  [f,t] }}
\newcommand{\guntfb}{\ensuremath{\Gamma  [\ov{f},t] }}
\newcommand{\guntfcpp}{\ensuremath{\Gamma [ f_{\rm CP+} ,t] }}
\newcommand{\bbd}{$B_d$--$\Bbar{}_d\,$}

\def\lqcd{\Lambda_{\rm QCD}}
\newcommand{\as}{\ensuremath{\alpha_s}}
\newcommand{\e}{\epsilon}
\newcommand{\bra}[1]{\ensuremath{\langle #1 |}}
\newcommand{\ket}[1]{\ensuremath{| #1 \rangle }}
\newcommand{\braket}[2]{\ensuremath{\langle #1 | #2 \rangle }}
%%%%%% Asymmetric errors, example: \epm{2}{3} gives +2 stacked on -3  
\newcommand{\epm}[2]{
 \raisebox{-0.5ex}{\shortstack[l]{$\scriptstyle+#1$\\$\scriptstyle-#2$}}}


%%%%%%%%%%%%% Standard referencing
\newcommand{\eq}[1]{(\ref{#1})}
\newcommand{\fig}[1]{Fig.~\ref{#1}}
\newcommand{\tab}[1]{Table~\ref{#1}}

%%%%%%%%%%%%% Journals:
\newcommand{\prl}{Phys.\ Rev. Lett.}
\newcommand{\prd}{Phys.\ Rev.~D}
\newcommand{\prep}{Phys.\ Rep.}
\newcommand{\plb}{Phys.\ Lett.~B}
\newcommand{\npb}{Nucl.\ Phys.~B}
\newcommand{\rmp}{Rev.\ Mod.\ Phys.}
\newcommand{\zpc}{Z.~Phys.~C}
\newcommand{\ncim}{Nuo.~Cim.}
\newcommand{\nim}{Nucl.\ Instr.\ and Methods}
\newcommand{\ijmp}{Int.\ Jour.\ Mod.\ Phys.}

\newcommand{\PRL}[3]{{\em Phys. Rev. Lett.} {\bf #1},\ #2 (#3)}
\newcommand{\PRD}[3]{{\em Phys. Rev. } {\bf #1},\ #2 (#3)}
\newcommand{\PLB}[3]{{\em Phys. Lett.} {\bf #1},\ #2 (#3)}
\newcommand{\NPB}[3]{{\em Nucl. Phys.} {\bf #1},\ #2 (#3)}
\newcommand{\RMP}[3]{{\em Rev. Mod. Phys.} {\bf #1},\ #2 (#3)}
\newcommand{\ZPC}[3]{{\em Z. Phys.} {\bf #1},\ #2 (#3)}
\newcommand{\EPJ}[3]{{\em Eur. Phys. J.} {\bf #1},\ #2 (#3)}
\newcommand{\NPproc}[3]{{\em Nucl. Phys. B} (Proc. Suppl.){\bf #1},\ #2 (#3)}


%%%%%%%%%%%%%%%%%%%%%%%%%%  DEF.TEX  %%%%%%%%%%%%%%%%%%%%%%%%

\def\bsg{\ifmmode B\to X_s\gamma\else $B\to X_s\gamma$\fi}

\def\bxnn{\mbox{$b \to X\,\nu\,\bar\nu$}}
\def\bsnn{\mbox{$b \to X_s\,\nu\,\bar\nu$}}
\def\bdnn{\mbox{$b \to X_d\,\nu\,\bar\nu$}}
\def\bqnn{\mbox{$b \to X_q\,\nu\,\bar\nu$}}

\def\bcln{\mbox{$b \to X_c\,l \,\bar\nu$}}
\def\bqln{\mbox{$b \to X_q\,l \,\bar\nu$}}
\def\bctn{\mbox{$b \to X_c\,\tau\,\bar\nu$}}
\def\bqtn{\mbox{$b \to X_q\,\tau\,\bar\nu$}}

\def\Btn{\mbox{$B^- \to \tau\,\bar\nu$}}
\def\Bmn{\mbox{$B^- \to \mu\, \bar\nu$}}
\def\Ben{\mbox{$B^- \to   e\, \bar\nu$}}
\def\Bln{\mbox{$B^- \to \,l\, \bar\nu $}}

\def\Btt{\mbox{$B \to \,\tau^+\,\tau^-$}}
\def\Bstt{\mbox{$B_s \to \,\tau^+\,\tau^-$}}
\def\Bqtt{\mbox{$B_q \to \,\tau^+\,\tau^-$}}
\def\Bdtt{\mbox{$B_d \to \,\tau^+\,\tau^-$}}

\def\Bdem{\mbox{$B_d \to \,e^\pm\,\mu^\mp$}}
\def\Bdet{\mbox{$B_d \to \,e^\pm\,\tau^\mp$}}
\def\Bdmt{\mbox{$B_d \to \,\mu^\pm\,\tau^\mp$}}

\def\Bmm{\mbox{$B \to \,\mu^+\,\mu^-$}}
\def\Bsmm{\mbox{$B_s \to \,\mu^+\,\mu^-$}}
\def\Bdmm{\mbox{$B_d \to \,\mu^+\,\mu^-$}}
\def\Bqmm{\mbox{$B_q \to \,\mu^+\,\mu^-$}}

\def\Bee{\mbox{$B \to \,e^+\,e^-$}}
\def\Bsee{\mbox{$B_s \to \,e^+\,e^-$}}
\def\Bdee{\mbox{$B_d \to \,e^+\,e^-$}}
\def\Bqee{\mbox{$B_q \to \,e^+\,e^-$}}

\def\Bll{\mbox{$B \to  \,l^+\,l^-$}}
\def\Bsll{\mbox{$B_s \to  \,l^+\,l^-$}}
\def\Bdll{\mbox{$B_d \to  \,l^+\,l^-$}}
\def\Bqll{\mbox{$B_q \to  \,l^+\,l^-$}}

\def\bxtt{\mbox{$b \to X\,\tau^+\,\tau^-$}}
\def\bstt{\mbox{$b \to X_s\,\tau^+\,\tau^-$}}
\def\bdtt{\mbox{$b \to X_d\,\tau^+\,\tau^-$}}
\def\bqtt{\mbox{$b \to X_q\,\tau^+\,\tau^-$}}

\def\bxmm{\mbox{$b \to X\,\mu^+\,\mu^-$}}
\def\bsmm{\mbox{$b \to X_s\,\mu^+\,\mu^-$}}
\def\bdmm{\mbox{$b \to X_d\,\mu^+\,\mu^-$}}
\def\bqmm{\mbox{$b \to X_q\,\mu^+\,\mu^-$}}

\def\bxll{\mbox{$b \to X\, l^+\, l^-$}}
\def\bsll{\mbox{$b \to X_s\,l^+\,l^-$}}
\def\bdll{\mbox{$b \to X_d\,l^+\,l^-$}}
\def\bqll{\mbox{$b \to X_q\,l^+\,l^-$}}

%%%%%%%%%%B-mixing 
\newcommand{\bbm}{$B^0 - \Bbar{}^0$ mixing}
\newcommand{\bbmd}{$B_d^0\!-\!\Bbar{}_d^0\,$ mixing}
\newcommand{\bbms}{$B_s^0\!-\!\Bbar{}_s^0\,$ mixing}
\newcommand{\kkm}{$K^0\!-\!\ov{K}{}^0\,$ mixing}
\newcommand{\ddm}{$D^0\!-\!\ov{D}{}^0\,$ mixing}
\newcommand{\dbt}{\ensuremath{|\Delta B| \! = \!2}}
\newcommand{\dbo}{\ensuremath{|\Delta B| \! = \!1}}
\newcommand{\dst}{\ensuremath{|\Delta S|  = 2}}
\newcommand{\dso}{\ensuremath{|\Delta S|  = 1}}
\newcommand{\dft}{\ensuremath{|\Delta F|  = 2}}
\newcommand{\dfo}{\ensuremath{|\Delta F|  = 1}}
\newcommand{\hbo}{\ensuremath{H^\mathrm{|\Delta B| = 1}}}
\newcommand{\dm}{\ensuremath{\Delta m}}
\newcommand{\dms}{\ensuremath{\Delta m_{B_s}}}
\newcommand{\dmd}{\ensuremath{\Delta m_{B_d}}}
\newcommand{\dg}{\ensuremath{\Delta \Gamma}}
\newcommand{\dgs}{\ensuremath{\Delta \Gamma_{B_s}}}
\newcommand{\dgd}{\ensuremath{\Delta \Gamma_{B_d}}}
\newcommand{\gtf}{\ensuremath{\Gamma (B^0(t) \rightarrow f )}}
\newcommand{\gbtf}{\ensuremath{\Gamma (\Bbar{}^0(t) \rightarrow f )}}
\newcommand{\gtfb}{\ensuremath{\Gamma (B^0(t) \rightarrow \ov{f} )}}
\newcommand{\gbtfb}{\ensuremath{\Gamma (\Bbar{}^0(t) \rightarrow \ov{f} )}}
%%\newcommand{\Bun}{\ensuremath{\stackrel{\scriptscriptstyle \lt( -\rt)}{B}}}
%%\newcommand{\Bdun}{\ensuremath{\stackrel{\scriptscriptstyle \lt( -\rt)}{B}_d}}
%%\newcommand{\Bsun}{\ensuremath{\stackrel{\scriptscriptstyle \lt( -\rt)}{B}_s}}
\newcommand{\Bun}{\raisebox{7.5pt}{$\scriptscriptstyle(\hspace*{6.5pt})$}
  \hspace*{-9.7pt}\!\Bbar}
\newcommand{\Bdun}{\raisebox{7.5pt}{$\scriptscriptstyle(\hspace*{6.5pt})$}
  \hspace*{-9.7pt}\!\Bbar_{d}}
\newcommand{\Bsun}{\raisebox{7.7pt}{$\scriptscriptstyle(\hspace*{6.5pt})$}
  \hspace*{-9.7pt}\!\Bbar_{s}}
\newcommand{\ega}{a}
\begin{theindex}

  \item $\ega$
    \subitem definition, 20
  \item $A_{CP}^{\rm dir}$ and $A_{CP}^{\rm mix}$ definitions, 24
  \item $A_{\Delta \Gamma}$
    \subitem and untagged decay, 359
    \subitem definition, 23
  \item $D$(tagging dilution), 83
  \item $\epsilon D^2$ (tagging power), 83
  \item $\eta$ (psuedo-rapidity)
    \subitem $\beta$ vs $\eta$, 78
    \subitem coverage of detectors, 76
    \subitem definition of, 75
  \item $\lambda_f$
    \subitem $\imag \lambda_f\neq 0$, 27
    \subitem $\vert\lambda_f\vert\neq 1$, 26
    \subitem definition, 21
  \item $\vert\Delta B\vert=2$ transition, 18
  \item $e^+e^-$ machines
    \subitem comparison with, 94
  \item $q/p$, 20
  \item $y$ (rapidity)
    \subitem definition of, 75

  \indexspace

  \item absorptive part, 27, 57
  \item anomalous fermion couplings, 273
  \item anomalous triple gauge boson couplings, 272
  \item anti-unitary transformation, 16, 17

  \indexspace

  \item $B \to X_s\ell^+\ell^-$, 246
    \subitem $Q\bar{Q}$ background, D\O, 282
    \subitem charmonium resonances, 249
    \subitem convergence of OPE, 250
    \subitem decay amplitude, 246
    \subitem dimuon mass spectrum, D\O, 283
    \subitem event rates, D\O, 283
    \subitem forward-backward asymmetry, 248
    \subitem intended approach, BTeV, 300
    \subitem SM prediction, 246
  \item $B \to \ell^+ \ell^-$, 268
    \subitem branching ratio, 269
    \subitem event rates, CDF, 294
    \subitem event rates, D\O, 285
    \subitem SM prediction, 270
  \item $B\to D^{(*)}\ell\bar\nu$, 304
    \subitem $Q^2$ reconstruction, BTeV, 319
    \subitem $Q^2$ resolution, BTeV, 319
    \subitem background rates, BTeV, 317
    \subitem event rates, BTeV, 318
  \item $B\to K\ell^+\ell^-$, 254
    \subitem $CP$ asymmetries, 265
    \subitem analysis cuts, BTeV, 299
    \subitem backgrounds, BTeV, 300
    \subitem event rates, BTeV, 300
    \subitem form factors, 251
    \subitem in large energy limit, 257
  \item $B\to K^*\ell^+\ell^-$, 254
    \subitem $CP$ asymmetries, 265
    \subitem analysis cuts, BTeV, 296
    \subitem analysis cuts, D\O, 280
    \subitem angular distribution, 262
    \subitem backgrounds, BTeV, 295, 297
    \subitem charmonium resonances, 260
    \subitem event rates, BTeV, 297
    \subitem event rates, CDF, 288
    \subitem event rates, D\O, 281
    \subitem form factors, defined, 252
    \subitem forward-backward asymmetry, 263
    \subitem forward-backward asymmetry, BTeV, 297
    \subitem forward-backward asymmetry, CDF, 289, 290
    \subitem forward-backward asymmetry, D\O, 282
    \subitem forward-backward asymmetry, zero-point extraction, CDF, 
		290
    \subitem in large energy limit, 257
    \subitem phase space, 278
    \subitem SM prediction, 260
    \subitem trigger strategy, CDF, 288
  \item $B\to K^*\gamma$, 266
    \subitem event rates, CDF, 287
    \subitem form factors, defined, 252
    \subitem HQET prediction, 256
    \subitem measured branching fractions, 266
    \subitem trigger, CDF, 286
  \item $B\to K_2^*(1430)\gamma$, 266
  \item $B\to X_d\gamma$, 246
  \item $B\to X_s\gamma$, 244
    \subitem decay amplitude, 244
    \subitem measured branching fraction, 245
    \subitem SM prediction, 245
  \item $B\to \pi(\rho)\ell\bar\nu$, 306
  \item $B\to\rho\gamma$, 267
    \subitem long distance contribution, 267
  \item $B$ decay
    \subitem nonleptonic, 44
    \subitem semileptonic, 41, 335, 379
  \item $B$ decays
    \subitem $B^0 \to D^- X$, 384
    \subitem $B^0 \to D^- \ell^+ \nu$, 384, 426
    \subitem $B^0 \to D^- \pi^+ \pi^+ \pi^-$, 384
    \subitem $B^0 \to D^- \pi^+$, 26, 384
    \subitem $B^0 \to D^0 K^-$, 191, 196, 210, 214, 233
    \subitem $B^0 \to D^\pm \pi^\mp$, 29
    \subitem $B^0 \to D^{(*)+} D^{(*)-}$, 384, 385
    \subitem $B^0 \to D^{(*)} \ell \nu$, 60
    \subitem $B^0 \to D^{*-} X$, 384
    \subitem $B^0 \to D^{*-} \pi^+ \pi^+ \pi^- \pi^0$, 384
    \subitem $B^0 \to D^{*-} \pi^+ \pi^+ \pi^-$, 384
    \subitem $B^0 \to D^{*-} \pi^+$, 384
    \subitem $B^0 \to D^{*\pm} \pi^\mp$, 29
    \subitem $B^0 \to D_s^{(*)+} D_s^{(*)-} K_S$, 385
    \subitem $B^0 \to K \pi$, 151, 153, 171, 182, 233
    \subitem $B^0 \to KK$, 171
    \subitem $B^0 \to X \ell^+ \nu$, 22, 26, 379, 384
    \subitem $B^0 \to \ov{D}{}^0 \rho^0$, 384, 385
    \subitem $B^0 \to \ov{D}{}^{*0} \pi^+ \pi^-$, 385
    \subitem $B^0 \to \ov{D}{}^{*0} \rho^0$, 384, 385
    \subitem $B^0 \to \ov{D}{}^{0} \pi^+ \pi^-$, 385
    \subitem $B^0 \to \pi \ell \nu$, 60
    \subitem $B^0 \to \pi \pi$, 29, 150, 152, 168, 170, 171, 179, 181, 
		187, 188, 232, 379
    \subitem $B^0 \to \psi K^+ \pi^-$, 384
    \subitem $B^0 \to \psi K_S \rho^0$, 385
    \subitem $B^0 \to \psi K_S$, 28, 59, 150, 157, 158, 160, 164, 168, 
		232, 370, 372, 379, 489
    \subitem $B^0 \to \psi \phi K^*$, 385
    \subitem $B^0 \to \psi \phi K_S$, 385
    \subitem $B^0 \to \psi \rho^0$, 385
    \subitem $B^0 \to \rho \pi$, 29, 152, 215, 217, 233
  \item $B$ meson
    \subitem $CP$ transformation, 15
    \subitem decay constant $f_B$, 15
    \subitem mass difference \dm, 19, 20, 331, 352
      \subsubitem $\dm_d$, 352, 379
      \subsubitem $\dm_s$, 353, 382
      \subsubitem new physics, 379, 382
      \subsubitem SM prediction for $\dm_s$, 354
    \subitem width difference $\dg$, 19, 20, 354
      \subsubitem $\dg_s$ and branching ratios, 360, 362, 368
      \subsubitem $\dg_s$ and lifetimes, 362
      \subsubitem measurement of $\dg_d$, 370
      \subsubitem measurement of $\dg_s$, 358
      \subsubitem new physics, 363
      \subsubitem sign convention, 19
      \subsubitem SM prediction for $\dg_d$, 358
      \subsubitem SM prediction for $\dg_s$, 357
  \item $B$ mixing, 17--22, 352
    \subitem $B^0$ mixing, 352, 379
    \subitem $B_s^0$ mixing, 331, 353, 382
    \subitem new physics, 18
  \item $B_s^0$ mixing, 138
    \subitem average significance, 394
    \subitem average significance in BTeV, 410, 413
    \subitem average significance in CDF, 401
    \subitem average significance in D\O, 404
    \subitem hadronic, 399
    \subitem K factor, 397
    \subitem mixing frequency, 394
    \subitem semileptonic, 396
  \item $b$-jet
    \subitem $p_t$ spectrum, 447
    \subitem production, 445, 447
  \item $b$-quark production
    \subitem  impact on experiments, 69
    \subitem Coulomb singularities, 443
    \subitem high energy behavior, 443
    \subitem Run~II, 451
    \subitem Run~II reach, 445
    \subitem Sudakov logarithms, 443
    \subitem theory, 441
    \subitem threshold resummation, 443
    \subitem via gluino decay, 445
  \item bag constants, 342
    \subitem $B_{B_q}$, 356
    \subitem $B_{B_q}^S$, 356
  \item baryons
    \subitem doubly-heavy, 499
    \subitem triply-heavy, 499
  \item $B_c$ decays
    \subitem $B_c \to B_s^0 \pi^+$, 497
    \subitem $B_c \to B_s^{(*)}\ell^{+}\nu$, 485
    \subitem $B_c \to B_s^{(*)}\pi^{+}$, 486
    \subitem $B_c \to B_s^{(*)}\rho^{+}$, 486
    \subitem $B_c \to \eta_c \ell^+\nu$, 485
    \subitem $B_c \to \eta_c \pi^+$, 486
    \subitem $B_c \to \eta_c \rho^+$, 486
    \subitem $B_c \to \psi \ell^+\nu$, 485, 488, 498
    \subitem $B_c \to \psi \pi^+$, 486, 488, 490, 492, 495
    \subitem $B_c \to \psi \rho^+$, 486
  \item $B_c$ meson, 476
    \subitem decay, 481
      \subsubitem leptonic, 485
      \subsubitem nonleptonic, 486
      \subsubitem semileptonic, 485
    \subitem lifetime, 482
    \subitem mass, 477
    \subitem production, 478
      \subsubitem cross section, 480
    \subitem reconstruction, 488
    \subitem triggering, 488
  \item $\bar{b}c$-system
    \subitem spectroscopy, 477
  \item beam drag, 520, 524
  \item bottomonium
    \subitem decay
      \subsubitem $\eta_b(1S) \to \psi \psi$, 457
    \subitem $\eta_b(1S)$, 456
    \subitem polarization
      \subsubitem CDF, 473
    \subitem production, 454
  \item $B_s^0$ decays
    \subitem $B_s^0 \ra \psi \eta^{(\prime)}$, 221, 225, 230, 233
    \subitem $B_s^0 \to D^{(*)+} D^{(*)-}$, 385
    \subitem $B_s^0 \to D_s^- K^+$, 147, 190, 193, 203, 214, 233
    \subitem $B_s^0 \to D_s^- \ell^+ \nu$, 384, 397, 403
    \subitem $B_s^0 \to D_s^- \pi^+ \pi^+ \pi^- \pi^0$, 384
    \subitem $B_s^0 \to D_s^- \pi^+ \pi^+ \pi^-$, 370, 384, 387, 403, 
		414, 429
    \subitem $B_s^0 \to D_s^- \pi^+$, 22, 26, 194, 359, 364, 370, 384, 
		387, 394, 403, 406, 414, 418, 422, 429
    \subitem $B_s^0 \to D_s^\pm K^\mp$, 29
    \subitem $B_s^0 \to D_s^{(*)+} D_s^{(*)-}$, 24, 155, 355, 361, 364, 
		367, 368, 372, 384, 385, 387, 416
      \subsubitem $CP$-odd fraction, 368, 372
    \subitem $B_s^0 \to D_s^{*-} \pi^+ \pi^+ \pi^- \pi^0$, 384
    \subitem $B_s^0 \to D_s^{*-} \pi^+ \pi^+ \pi^-$, 370, 384
    \subitem $B_s^0 \to D_s^{*-} \pi^+$, 370, 384
    \subitem $B_s^0 \to K K$, 182, 188, 361, 362
    \subitem $B_s^0 \to K \pi$, 233
    \subitem $B_s^0 \to X \ell^+ \nu$, 22, 364, 370, 384
    \subitem $B_s^0 \to \chi_{c0} \phi$, 385, 425
    \subitem $B_s^0 \to \ov{D}{}^{(*)0} K_S$, 384, 385
    \subitem $B_s^0 \to \pi \pi$, 171
    \subitem $B_s^0 \to \psi K^- \pi^+$, 384
    \subitem $B_s^0 \to \psi K^{(*)} \ov{K}{}^{(*)}$, 385, 403
    \subitem $B_s^0 \to \psi K_S \pi^0$, 385
    \subitem $B_s^0 \to \psi K_S$, 145, 385
    \subitem $B_s^0 \to \psi K{}^{*0}$, 408
    \subitem $B_s^0 \to \psi \eta^{(\prime)}$, 385, 418, 422
    \subitem $B_s^0 \to \psi \phi \phi$, 385
    \subitem $B_s^0 \to \psi \phi$, 145, 155, 158, 367, 368, 372, 385, 
		415, 418, 427
    \subitem $B_s^0 \to \psi f_0$, 385, 425
  \item BTeV
    \subitem $B^0 \to D^0 K^-$ prospects, 210
    \subitem $B^0 \to \pi \pi$ prospects, 181
    \subitem $B^0 \to \rho \pi$ prospects, 217, 220
    \subitem $B_s^0 \ra \psi \eta^{(\prime)}$ prospects, 225
    \subitem $B_s^0 \to D_s^- K^+$ prospects, 203
    \subitem $B_s^0$ mixing, 406
    \subitem $\sin 2\beta$ prospects, 164, 167
    \subitem $\sin 2\beta_s$ prospects, 225, 230
    \subitem $\sin\gamma$ prospects, 203, 208
    \subitem detector, 121
      \subsubitem charged particle identification, 126
      \subsubitem data acquisition system, 128
      \subsubitem detached vertex trigger, 125
      \subsubitem electromagnetic calorimeter, 126
      \subsubitem forward tracking system, 127
      \subsubitem forward vs central, 121
      \subsubitem key features, 122
      \subsubitem muon detection, 128
      \subsubitem pixel vertex detector, 123
      \subsubitem RICH, 126
      \subsubitem sketch, 122
    \subitem flavor tagging, 131
      \subsubitem  away side combined, 132
      \subsubitem  away side vertex charge, 132
      \subsubitem  same and away combined, 133
      \subsubitem away side kaon, 132
      \subsubitem away side lepton, 131
      \subsubitem same side, 133
    \subitem presumed $b\bar{b}$ cross-section, 121
    \subitem presumed luminosity, 121
    \subitem schedule, 133
    \subitem software
      \subsubitem analysis tools, 128
      \subsubitem calorimeter clustering, 130
      \subsubitem RICH simulation, 130
      \subsubitem simulation tools, 128
      \subsubitem tracking, 129
      \subsubitem trigger simulation, 130

  \indexspace

  \item Cabibbo-Kobayashi-Maskawa matrix
    \subitem $V_{cb}$, 304
    \subitem $V_{td}$, 57, 58, 352
    \subitem $V_{ts}$, 57, 58, 353
    \subitem $V_{ub}$, 379
  \item CDF
    \subitem $B^0 \to D^0 K^-$ prospects, 196
    \subitem $B^0 \to \pi \pi$ prospects, 170
    \subitem $B_s^0 \ra \psi \eta^{(\prime)}$ prospects, 221
    \subitem $B_s^0 \to D_s^- K^+$ prospects, 193
    \subitem $B_s^0$ mixing, 393
    \subitem $\Upsilon$ polarization, 473
    \subitem $\psi$ polarization, 472
    \subitem $\sin 2\beta$ prospects, 158
    \subitem $\sin 2\beta_s$ prospects, 221
    \subitem $\sin\gamma$ prospects, 170, 193
    \subitem $B_c$, 491
    \subitem Bgenerator, 386
    \subitem calorimetry, 102
    \subitem detector improvements, 99, 386
    \subitem detector simulation, 109, 389
    \subitem dimuon trigger, 159
    \subitem flavor tagging, 159, 394
      \subsubitem opposite side kaon, 395
      \subsubitem same side kaon, 395
    \subitem hadronic $B$ decays, 202
    \subitem hadronic trigger, 170, 202, 389
    \subitem muon systems, 102
    \subitem proper time resolution
      \subsubitem hadronic, 399
      \subsubitem semileptonic, 396
  \item charge conjugation, 14, 16
  \item charmonium
    \subitem decay
      \subsubitem $\chi_{cJ}(1P) \to \psi + \gamma$, 455
      \subsubitem $\eta_c \to \phi \phi$, 456
      \subsubitem $\eta_c(2S) \to \psi + \gamma$, 455
      \subsubitem $h_c \to \psi \pi^0$, 455
    \subitem $\eta_c(1S)$, 455
    \subitem $\eta_c(2S)$, 455
    \subitem $h_c(1P)$, 455
    \subitem polarization
      \subsubitem CDF, 472
    \subitem production, 454
  \item chromomagnetic energy of a heavy quark, 336
  \item CKM angle
    \subitem $\gamma$, 189, 191, 214
  \item color evaporation model, 458
  \item color octet model, 459
  \item color singlet model, 458, 479
  \item $CP$ asymmetry
    \subitem $\sin2\beta$, 28
    \subitem in $B_s^0$ decay, 144
    \subitem in charged $B$ decay, 23
    \subitem in flavor-specific decay, 26, 371, 381
      \subsubitem SM prediction, 372
    \subitem interference type, 24
    \subitem mixing-induced, 24
    \subitem semileptonic, 26, 156, 371
    \subitem time-dependent, 23, 28, 148, 155
    \subitem time-integrated, 24, 155
  \item $CP$ conjugate state, 21
  \item $CP$ transformation
    \subitem $B$ meson, 15, 360, 361
    \subitem Higgs boson, 16
    \subitem quark currents, 15
    \subitem vector boson, 16
  \item $CP$ violation
    \subitem clean modes, 28, 29
    \subitem complex couplings, 16
    \subitem direct, 24, 26, 29
    \subitem in $\Delta F=1$ transitions, 29, 56
    \subitem in decay, 24, 26, 141, 148, 174
      \subsubitem kaon, 56
    \subitem in mixing, 26, 140, 174
      \subsubitem kaon, 55
    \subitem in non-$CP$ eigenstates, 29
    \subitem indirect, 29
    \subitem interference type, 24, 27, 141, 148
      \subsubitem kaon, 55, 56, 59
    \subitem kaon physics, 53
      \subsubitem $\e_K$, 30, 55
      \subsubitem $\e_K$ constraint on $\ov{\rho},\ov{\eta}$, 58, 59
      \subsubitem $\e_K^\prime$, 30, 55
      \subsubitem semileptonic asymmetry, 53
    \subitem Kobayashi-Maskawa mechanism, 7, 17
    \subitem Standard Model, 16, 137
    \subitem superweak, 30
    \subitem \T\ violation, 17
    \subitem tagged decay, 141
    \subitem untagged decay, 144
  \item $CPT$ theorem, 17
  \item $CPT$ transformation, 17
  \item $CPT$ violation
    \subitem string theory, 17
  \item \ensuremath {C\mskip -\thinmuskip PT}\ violation, 19
  \item cross-sections
    \subitem D\O\ measurement, 77
    \subitem table of, 70

  \indexspace

  \item $D$ meson
    \subitem mass difference $\dm_D$, 377
      \subsubitem new physics, 382
    \subitem width difference $\dg_D$, 377
  \item $D$ mixing, 18, 377, 382
  \item D\O
    \subitem $B^0 \to \pi \pi$ prospects, 179
    \subitem $B_s^0$ mixing, 402
    \subitem $\psi$ trigger, 430
    \subitem $\sin 2\beta$ prospects, 160, 162
    \subitem $B_c$, 488
    \subitem flavor tagging, 164
  \item D\O\ calorimetry, 113
  \item D\O\ central fiber tracker, 111
  \item D\O\ dielectron trigger, 118
  \item D\O\ muon detector, 113
  \item D\O\ muon trigger
    \subitem  Level 1
      \subsubitem  $\psi$ event rates, 115
      \subsubitem  $Q{\bar Q}$ dimuon rates, 115
      \subsubitem  description, 114
      \subsubitem  efficiency for dimuon events, 115
      \subsubitem  notation, 114
      \subsubitem  trigger rates, 115
    \subitem Level 2 and 3
      \subsubitem  description, 116
  \item D\O\ Silicon Tracker Trigger, 116
    \subitem  efficiency, 118
    \subitem  impact parameter resolution, 116
    \subitem  prompt $\psi$ separation, 118
  \item D\O\ silicon vertex detector, 111
  \item D\O\ trigger system, 113
  \item decay amplitude, 21, 139
  \item decay matrix $\Gamma$, 18
    \subitem eigenvalues, 20
    \subitem eigenvectors, 20
  \item detector
    \subitem  desired properties, 87
    \subitem central vs forward, 88
  \item dilepton mass spectrum
    \subitem $Q\bar{Q}$ production, 277
    \subitem  sequential decays, 278
    \subitem trigger rates, D\O, 279
  \item discrete ambiguity of $CP$ phase, 372
  \item dispersive part, 57
  \item doubly-heavy baryon decays
    \subitem $\Omega^{++}_{ccc}(ccc)$, 508
    \subitem $\Omega^{+}_{cc}(ccs)$, 508
    \subitem $\Omega^{-}_{bb}(bbs)$, 508
    \subitem $\Omega^{0}_{bc}(bcs)$, 508
    \subitem $\Xi^{++}_{cc}(ccu)$, 508
    \subitem $\Xi^{+}_{bcc}(bcc)$, 508
    \subitem $\Xi^{+}_{bc}(bcu)$, 508
    \subitem $\Xi^{+}_{cc}(ccd)$, 508
    \subitem $\Xi^{-}_{bb}(bbd)$, 508
    \subitem $\Xi^{0}_{bbc}(bbc)$, 508
    \subitem $\Xi^{0}_{bb}(bbu)$, 508
    \subitem $\Xi^{0}_{bc}(bcd)$, 508
  \item doubly-heavy baryons, 499
    \subitem decays, 503
    \subitem lifetimes, 503
    \subitem observability, 505
    \subitem production, 500
    \subitem spectrum, 499

  \indexspace

  \item effective Hamiltonian, 242
    \subitem  Wilson coefficients defined, 242
    \subitem for $B \to \ell^+ \ell^-$, 268
    \subitem renormalization group, 242, 244, 269
    \subitem Wilson coefficients in SM, 242, 269
  \item EM Calorimeter
    \subitem BTeV, 126
    \subitem overview of, 92
  \item $\eta_b(1S)$, 456
  \item $\eta_c(1S)$, 455
  \item $\eta_c(2S)$, 455
  \item event
    \subitem  cartoon of a typical, 72

  \indexspace

  \item FCNC transitions, 241
  \item flavor tagging
    \subitem BTeV, 131
    \subitem CDF, 394
    \subitem description of
      \subsubitem away side, 83
      \subsubitem overall strategy, 86
      \subsubitem same side, 84
    \subitem dilution ($D$), 83
    \subitem motivation for, 82
    \subitem power ($\epsilon D^2$), 83
  \item flavor-specific, 22
    \subitem decay modes, 352
  \item fourth generation, 384
  \item fragmentation, 508
    \subitem color string model, 521
    \subitem experimental impact, 528
    \subitem heavy quark
      \subsubitem $e^+ e^-$ collisions, 511
      \subsubitem $p\bar{p}$ collisions, 514
      \subsubitem ambiguities, 525
    \subitem non-perturbative, 517
    \subitem perturbative, 511
  \item fragmentation function, 508
    \subitem heavy quarks, 510
    \subitem in HQET, 518
    \subitem Peterson, 512, 517

  \indexspace

  \item gluino production, 445

  \indexspace

  \item hadronic parameter
    \subitem $B_{B_q}$, 356
    \subitem $B_{B_q}^S$, 356
    \subitem $\widehat{B}_B$, 37, 353
      \subsubitem relation to $B_{B_q}$, 356
    \subitem $\widehat{B}_K$, 57
    \subitem $\xi$, 354
    \subitem $f_{B^0}$, 353
    \subitem $f_{B_s}$, 353
    \subitem and $SU(3)_F$ symmetry, 354
    \subitem in lattice QCD, 50
  \item Hamiltonian
    \subitem $\vert\Delta B\vert=1$, 32
    \subitem $\vert\Delta B\vert=2$, 35
  \item $h_c(1P)$
    \subitem $h_c \to \psi \pi^0$, 455
  \item heavy baryons
    \subitem $SU(3)$ antitriplet, 343
    \subitem average decay rate, 347
    \subitem matrix elements, 344
  \item heavy quark expansion, 40--45
  \item heavy quark symmetry, 37
  \item Higgs sector, 275
    \subitem strong dynamics, 275
  \item HQET (heavy quark effective theory), 37--40
    \subitem $\lambda_1$ and $\lambda_2$, 39, 43
    \subitem constraints on semileptonic form factors, 303
    \subitem flavor symmetry relations, 255
    \subitem for lattice QCD, 51
    \subitem in inclusive decays, 43
    \subitem perturbative corrections, 303
    \subitem spin symmetry relations, 255

  \indexspace

  \item inclusive weak decay rates
    \subitem effective Lagrangian, 334
    \subitem kinematical integrals, 337
    \subitem operator product expansion, 333
    \subitem Pauli interference, 335
    \subitem semileptonic decay of charm, 341
    \subitem spectator quark effects, 339
    \subitem splitting for baryons, 346
    \subitem theoretical predictions summary, 351
    \subitem weak scattering, 335
  \item interactions
    \subitem mean rates of, 80
    \subitem Poisson distribution of, 81

  \indexspace

  \item $K$ decays
    \subitem $K \to \pi \pi$, 54
    \subitem $K^+ \to \pi^+ \nu\bar\nu$, 58
    \subitem $K_L \to \ell^{\pm}\nu\,\pi^{\mp}$, 53
    \subitem $K_L \to \pi^0 \nu\bar\nu$, 59
  \item $K$ mixing, 17, 53
  \item kaon
    \subitem $q/p$, 53
    \subitem isospin, 54
    \subitem mass difference $\dm_K$, 53
    \subitem mass eigenstates, 53
    \subitem rare decay, 58
    \subitem width difference $\dg_K$, 53
  \item kinetic energy of a heavy quark, 336

  \indexspace

  \item $\Lambda_b\to\Lambda\gamma$
    \subitem event rate, CDF, 287
    \subitem trigger strategy, CDF, 287
  \item $\Lambda_b\to\Lambda_c\ell\bar\nu_{\ell}$, 304
    \subitem $Q^2$ efficiency, CDF, 312
    \subitem $Q^2$ reconstruction, BTeV, 319
    \subitem $Q^2$ reconstruction, CDF, 312
    \subitem $Q^2$ resolution, BTeV, 319
    \subitem $Q^2$ resolution, CDF, 312
    \subitem analysis cuts, CDF, 312
    \subitem background from excited hadrons, 305
    \subitem backgrounds, BTeV, 317, 320
    \subitem backgrounds, CDF, 315
    \subitem event rates, BTeV, 318
    \subitem event rates, CDF, 312
    \subitem trigger, CDF, 309
  \item $\Lambda_b$ decays
    \subitem $\Lambda_b \to \Lambda_c^+\pi^-$, 389, 429
    \subitem $\Lambda_b \to \Lambda_c^+\pi^-\pi^+\pi^-$, 389
    \subitem $\Lambda_b \to \psi\Lambda$, 388, 429, 430
    \subitem $\Lambda_b \to pD^0\pi^-$, 389, 429
  \item large energy limit, 257
    \subitem corrections, 258
    \subitem zero in forward-backward asymmetry, 263
  \item lattice QCD, 253
    \subitem $B\to D^{(*)}\ell\bar\nu$, 304
    \subitem $B\to \pi(\rho)\ell\bar\nu$, 306
    \subitem lattice spacing effects
      \subsubitem heavy quarks, 50--52
      \subsubitem light quarks and gluons, 50
    \subitem lattice spacing errors, 253
    \subitem momentum dependent errors, 254
    \subitem predictions for $B\to K^*\gamma$, 266
    \subitem quenched approximation, 49
    \subitem semileptonic decays, 304
    \subitem sources of uncertainty, 253
    \subitem unquenched $f_B$, 49
  \item lifetime
    \subitem $B^+$, 331
    \subitem $B^0$, 331
    \subitem $B_s^0$, 331
    \subitem $\Lambda_b$, 331
    \subitem $b$-flavored hadron, 331
    \subitem at CDF, 426
      \subsubitem hadronic triggers, 429
      \subsubitem leptonic triggers, 428
    \subitem at D\O, 430
    \subitem experimental lifetime, 432
    \subitem ratios, 428
  \item light-cone sum rules, 259, 266
  \item luminosity, 121
  \item Lund string model, 521

  \indexspace

  \item mass difference \dm, 331, 352
    \subitem $\dm_d$, 352, 379
      \subsubitem new physics, 379
    \subitem $\dm_s$, 138, 353, 382
      \subsubitem new physics, 382
      \subsubitem SM prediction, 354
  \item mass eigenstates, 18, 138
  \item mass matrix $M$, 18
    \subitem eigenvalues, 20
    \subitem eigenvectors, 20
  \item mixing asymmetry, 22
  \item mixing parameter
    \subitem $x$, 24, 138
    \subitem $y$, 24, 138
  \item muon detection
    \subitem BTeV, 128
    \subitem overview of, 92

  \indexspace

  \item new physics
    \subitem fourth generation, 384
    \subitem in $B_s^0$ mixing, 154
    \subitem left-right symmetry, 381
    \subitem strong dynamics, 381
    \subitem supersymmetry, 381
    \subitem two-Higgs doublet model, 384
      \subsubitem flavor-changing Higgs, 381, 384
  \item nonleptonic weak interaction
    \subitem QCD renormalization, 337
  \item NRQCD (nonrelativistic QCD), 457, 479

  \indexspace

  \item OPE (operator product expansion), 31, 32, 482, 483
  \item operator
    \subitem $O_\pm$, 33
    \subitem $O_{1-2}$, 33
    \subitem $O_{3-6}$, 33
    \subitem $O_{7-10}^{\rm ew}$, 34
    \subitem $O_{7}$, 34
    \subitem $O_{8}$, 33
    \subitem $O_{9-11}$, 34
    \subitem $Q$, 352
    \subitem chromomagnetic, 33
    \subitem current-current, 33
    \subitem electroweak penguin, 34
    \subitem magnetic, 34
    \subitem QCD penguin, 33
    \subitem semileptonic, 34

  \indexspace

  \item parity, 14, 16
  \item particle identification
    \subitem BTeV RICH, 126
    \subitem overview, 90
  \item Pauli interference, 483
  \item penguins
    \subitem in $B$ decay, 148
    \subitem in $B^0\to \psi K_S$, 150
    \subitem in $B^0\to K\pi$, 153
    \subitem in $B^0\to\pi\pi$, 150
    \subitem pollution, 148, 151, 168
  \item phase
    \subitem in decay matrix, 20
      \subsubitem kaon, 57
    \subitem in mass matrix, $\phi_M$, 20, 21
      \subsubitem kaon, 56
    \subitem of $CP$ violation in mixing, $\phi$, 20, 357
      \subsubitem kaon, 54, 55
      \subsubitem SM prediction for $\phi_d$ and $\phi_s$, 357
  \item phase convention, 25
    \subitem CKM matrix, 15
  \item phase transformation
    \subitem quark field, 15
  \item pixel detector, 123
  \item potential models, 482
  \item production
    \subitem $b$-quark, 441
  \item proper decay time, 86
  \item psuedo-rapidity ($\eta$)
    \subitem $\beta$ vs $\eta$, 78
    \subitem coverage of detectors, 76
    \subitem definition of, 75

  \indexspace

  \item QCD improved parton model, 442
  \item quark models, 258
  \item quark-hadron duality, 243
  \item quarkonium
    \subitem polarization, 468
      \subsubitem CDF, 472, 473
    \subitem production, 454, 457
    \subitem spectroscopy, 454

  \indexspace

  \item rapidity ($y$)
    \subitem definition of, 75
  \item renormalization group, 33, 34
  \item resummation, 444

  \indexspace

  \item Schr\"odinger equation, 18, 19
  \item semileptonic decays
    \subitem lepton-displaced track trigger, CDF, 310
    \subitem muon identification, BTeV, 317
    \subitem simulation method, CDF, 308
    \subitem trigger strategy, CDF, 307
  \item short distance cross section, 442, 479
  \item spectator model, 483
  \item strong phase shifts
    \subitem from lattice QCD, 52
    \subitem from perturbative QCD, 13
  \item $SU(3)$, 256
    \subitem breaking, 256
    \subitem relations, 152, 178
  \item summary
    \subitem $B_s^0$ mixing projections, 413
    \subitem lifetime projections, 432
    \subitem width difference $\dg_s$ projections, 425
  \item supersymmetry, 262, 381

  \indexspace

  \item Tevatron parameters, 70
  \item time evolution
    \subitem decay rate, 21
      \subsubitem $D$ meson, 378
    \subitem untagged $B$, 23
  \item time reversal, 14, 16
  \item tracking system
    \subitem BTeV, 127
    \subitem required properties, 90
  \item transversity angle, 415
  \item triggering
    \subitem $B_c$ meson, 488
    \subitem BTeV, 125
    \subitem detached vertex, 125
    \subitem overview of, 91
  \item twist expansion, 243
  \item two-Higgs doublet model, 381, 384
    \subitem flavor-changing Higgs, 381, 384

  \indexspace

  \item unitarity triangle
    \subitem and \bbmd, 59, 331, 352
    \subitem and \bbms, 59, 331
    \subitem and $V_{ub}$, 59, 379
    \subitem angle $\alpha$, 379
    \subitem angle $\beta$, 355, 379
    \subitem angle $\beta_s$, 355
    \subitem angle $\gamma$, 355
    \subitem kaon physics, 58, 59
  \item untagged $B$
    \subitem branching ratio, 23
    \subitem two-exponential decay, 359, 365, 368, 369
  \item $\Upsilon$
    \subitem polarization
      \subsubitem CDF, 473
    \subitem production, 454

  \indexspace

  \item vertex detector
    \subitem  BTeV pixel, 123
    \subitem  desired properties, 89

  \indexspace

  \item width difference $\dg$, 354
    \subitem $B_s^0$, 45, 138, 155
    \subitem $\Delta\Gamma_{CP}$, 422
    \subitem measurement of $\dg_d$, 370
    \subitem measurement of $\dg_s$, 358
      \subsubitem at BTeV, 418
      \subsubitem at CDF, 414, 418
      \subsubitem branching ratios, 360, 368
      \subsubitem effect of vertex trigger, 421
      \subsubitem lifetimes, 362
      \subsubitem sensitivity, 422
      \subsubitem summary, 425
    \subitem new physics, 363
    \subitem SM prediction for $\dg_d$, 358
    \subitem SM prediction for $\dg_s$, 357
  \item Wigner-Weisskopf approximation, 19
  \item Wilson coefficient, 32, 34, 352
    \subitem $\vert\Delta B\vert=2$, 37
    \subitem table, 36
  \item Wolfenstein parameters
    \subitem for $V_{cd}V_{cb}^*$, 355
    \subitem for $V_{cs}V_{cb}^*$, 355
    \subitem for $V_{td}$, 353
    \subitem for $V_{td}V_{tb}^*$, 355
    \subitem for $V_{ts}$, 353
    \subitem for $V_{ts}V_{tb}^*$, 355
    \subitem for $V_{ud}V_{ub}^*$, 355
    \subitem for $V_{us}V_{ub}^*$, 355

\end{theindex}
\documentclass[openany,11pt]{book}

%%%% \twosided=0 puts pagenumbers on the outside for 1up duplex printing
%%%% \twosided=1 puts pagenumbers on the outside for 2up printing
\newcount\twosided \twosided=0

\usepackage{Bbook}
\usepackage{axodraw}  %% needed for ch9
%%\usepackage{side}

%%%% uncomment next line to shift text up
\setlength{\topmargin}{-.5cm}
%%%% uncomment next line to have labels on margin as in draft mode of harvmac
%%\usepackage[notref,notcite]{showkeys}
%%\usepackage{showidx}
%%%% uncomment next line to have "draft" printed on postscript
%%\usepackage[light,first,bottomafter]{draftcopy}
%%\usepackage[light]{draftcopy}

\input{newcommands}

%%%% comment this out for the arxives submission
%%\makeindex

\begin{document}

\frontmatter
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% --- Introductory stuff
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\include{intro}
%
\tableofcontents
%
\listoffigures
\addcontentsline{toc}{chapter}{List of Figures}
\listoftables
\addcontentsline{toc}{chapter}{List of Tables}

\clearpage{\pagestyle{empty}\cleardoublepage}

\mainmatter
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% --- Real Chapters begin
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\include{ch1}
\clearpage{\pagestyle{empty}\cleardoublepage}
\include{ch2}
\clearpage{\pagestyle{empty}\cleardoublepage}
\include{ch3}
\clearpage{\pagestyle{empty}\cleardoublepage}
\include{ch4}
\clearpage{\pagestyle{empty}\cleardoublepage}
\include{ch5}
\clearpage{\pagestyle{empty}\cleardoublepage}
\include{ch6}
\clearpage{\pagestyle{empty}\cleardoublepage}
\def\chsevenfigs{ch7/wg2figs/}
\include{ch7}
\clearpage{\pagestyle{empty}\cleardoublepage}
\include{ch8}
\clearpage{\pagestyle{empty}\cleardoublepage}
\def\chninehome{ch9/}
\include{ch9}
\clearpage{\pagestyle{empty}\cleardoublepage}
\include{ch10}
\clearpage{\pagestyle{empty}\cleardoublepage}

\backmatter
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% --- Index and stuff
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%%  must run `makeindex draft' from the command line
\addcontentsline{toc}{chapter}{Index}
\printindex
 
\end{document}

