%Paper: 
%From: PHRA0MG@TECHNION.TECHNION.AC.IL
%Date: Thu, 24 Nov 94 16:01:51 IST


\magnification 1200
\vsize=21.0cm
\hsize=14.1cm
%\nopagenumbers
\baselineskip=6truemm

\def\ubar{\overline{u}}
\def\cbar{\overline{c}}
\def\tbar{\overline{t}}
\def\dbar{\overline{d}}
\def\sbar{\overline{s}}
\def\bbar{\overline{b}}
\def\fbar{\overline{f}}
\def\pbar{\overline{p}}
\def\Abar{\overline{A}}
\def\Kbar{\overline{K}^0}
\def\Bbar{\overline{B}^0}
\def\Pbar{\overline{P}^0}
\def\tepsilon{\tilde{\epsilon}}
\def\bd{B^0}
\def\bs{B_s^0}
\def\bdb{\overline{B}^0}
\def\bsb{\overline{B}_s^0}
\def\bu{B_u^+}
\def\bub{B_u^-}
\def\Dbar{\overline{D}^0}
\def\nubar{\overline{\nu}}
\def\Gammabar{\overline{\Gamma}}
\def\to{\rightarrow}

\line{\hfil TECHNION-PH-94-64}
\line{\hfil November, 1994}
\null\vskip 1.5truecm

\centerline{\bf TESTING THE STANDARD MODEL OF CP VIOLATION }
\centerline{\bf IN THE $B$ SYSTEM}
\vskip 10mm
\centerline{\it Michael Gronau}
\centerline{\it Department of Physics,
Technion -- Israel Institute of Technology}
\centerline{\it 32000 Haifa, Israel}

\vskip 2.0cm

\magnification 1200
\baselineskip=6truemm
\noindent
\centerline{\bf Abstract}
\vskip 3mm
\noindent
We review the subject of CP violation in the $B$ meson system within the
Standard Model of CP violation, which is based on a
complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix.
Three kinds of CP
nonconservation are studied:  In
$B^0-\Bbar$ mixing, in mixed neutral $B$
decays to states which are common decay products of $B^0$ and $\Bbar$ and in
direct decays of charged $B$ mesons.
Measurements of certain CP asymmetries are shown to determine the three angles
of the CKM unitarity triangle. $\beta$ is measured cleanly in $B^0\to\psi K_S$.
Uncertainties due to penguin amplitudes in the
determination of $\alpha$ using the asymmetry in $B^0\to \pi^+\pi^-$ (or
$B^0\to
\rho^+\pi^-$) can be eliminated (or in the worst case only
estimated) by relating the rate
of this process to decay rates involving other two light pseudoscalars.
The angle
$\gamma$ can be measured in charged $B$ decays, provided that color suppression
in $B\to DK$ (or $B\to DK_i(1400)$)
is not too strong and/or that certain rescattering effects in $B$
decays to two light pseudoscalars are not too large.

\vskip 2.5cm
\centerline{\it Invited talk presented at the International Workshop on $B$
Physics}
\centerline{\it Physics Beyond the Standard Model at the $B$ Factory}
\centerline {\it October 26 - 28, Nagoya, Japan}

\vfill \eject

\baselineskip=6truemm
\noindent
{\bf 1. INTRODUCTION}
\vskip 3mm

The Standard Model [1] provides a suitable framework for understanding the CP
violation observed in the neutral $K$ meson mixing [2]. The single source of CP
violation in the theory is a phase in the Cabibbo-Kobayashi-Maskawa
(CKM) matrix [3]. So far the only
observation of CP violation has been the CP impurity in the
neutral $K$ meson system [4]. The measured value of $\epsilon_K$ can be
accommodated in the CKM theory, consistent with a sizable, however badly
determined, CKM phase. The prediction of direct CP violation in
strangeness-changing processes, such as the value of $\epsilon'/\epsilon$ in
$K\to \pi\pi$, involves large theoretical uncertainties [5]. A future
measuremen
of a nonzero value of $\epsilon'/\epsilon$  would demonstrate CP violation
outside $K^0-\Kbar$ mixing, however this would not be a powerful test of the
Standard Model. On the other hand, the $B$ meson system provides a wide variety
of independent CP asymmetry measurements, which can be related cleanly to
phases of corresponding CKM matrix elements. The determination of these phases,
which are fundamental parameters of the Standard Model, just as the
electron mass, is a crucial step towards a precision test of the
CKM mechanism of CP violation.

In this theoretical review we discuss predictions of CP violation in the $B$
meson system within the Standard Model. We begin in Section 2 by introducing
the
CKM matrix. We summarize the available information on the magnitude of its
elements and on their CP violating phases. CP nonconservation in neutral $B$
mixing is described in Section 3.
Section 4 deals with CP violation which occurs when mixed neutral
$B$ mesons decay to states which are common decay products of $B^0$ and
$\Bbar$.
Direct CP violation in charged $B$ decays is the subject of Section 5. Our
focus
is mainly on CP asymmetries which can be related to fundamental CKM phases in a
manner which involves no, or very small, theoretical uncertainties. In
Section 6 we describes two different methods of neutral $B$
flavor-tagging, which is needed for CP asymmetry measurements in neutral $B$
decays. Section 7 summarizes with a few concluding remarks.

\vskip 0.5truecm
\noindent
{\bf 2. CP VIOLATION IN THE STANDARD MODEL: THE CKM MATRIX}
\vskip 3mm

In the standard model the $SU(3)_C\times
SU(2)_L\times U(1)_Y$  gauge group is spontaneously broken by the vacuum
expectation value of a single scalar Higgs doublet. CP violation occurs
in the interactions of the three families of left-handed quarks with the
charged
gauge boson:
$$
-\cal L =
\left(
\matrix{\ubar&\cbar&\tbar\cr}\right)
\left(\matrix{m_u&~&~\cr ~&m_c&~\cr ~&~&m_t\cr}\right)
\left(\matrix{u\cr c\cr t\cr}\right)
+ \left(
\matrix{\dbar&\sbar&\bbar\cr}\right)
\left(\matrix{m_d&~&~\cr ~&m_s&~\cr ~&~&m_b\cr}\right)
\left(\matrix{d\cr s\cr b\cr}\right)
$$
$$
+{g\over\sqrt{2}}
\left(
\matrix{\ubar&\cbar&\tbar\cr}\right)_L \gamma^{\mu} V
\left(\matrix{d\cr s\cr b\cr}\right)_L W_{\mu}^+ +...
\eqno(1)
$$
CP violation requires a complex Cabibbo-Kobayashi-Maskawa
[3] (CKM) mixing matrix
$V$. The quark mass terms exhibit a symmetry under phase redefinitions
of the six quark fields. This freedom leaves a single phase in $V$. The unitary
matrix $V$, which can be defined in terms of this phase ($\gamma$) and three
Euler-like mixing angles, is approximated for most practical purposes by the
following form:
$$
V\approx\left(
\matrix{1&\vert V_{us}\vert&\vert V_{ub}\vert e^{-i\gamma}\cr
-\vert V_{us}\vert&1&\vert V_{cb}\vert\cr
\vert V_{us}V_{cb}\vert-\vert V_{ub}\vert e^{i\gamma}&-\vert
V_{cb}\vert&1\cr}\right)~.\eqno(2)
$$

The measured values of the three mixing angles ($\sin\theta_{12}\equiv\vert
V_{us}\vert,~\sin\theta_{23}\equiv\vert V_{cb}\vert,~
\sin\theta_{13}$

\noindent
$\equiv\vert V_{ub}\vert$) have a hierarchical pattern in
generation space [6],
$$
\vert V_{us}\vert=0.220\pm 0.002~(\lambda)~,
$$
$$
\vert V_{cb}\vert=0.038\pm
0.005~({\cal O}(\lambda^2))~,
$$
$$
\vert V_{ub}\vert=0.0035\pm 0.0015~({\cal
O}(\lambda^3))~,\eqno(3)
$$
often characterized by powers of a parameter $\lambda\equiv\sin\theta_c=0.22$
[7]. This structure was used with unitarity to obtain the approximate
expressions of  the three $t$ quark couplings in $V$. It is amusing to note
that
the yet unmeasured value of $\vert V_{tb}\vert$ obtained from unitarity is the
most accurately known parameter of the mixing matrix.

Unitarity of $V$ can be
represented geometrically in terms of triangles, such as the one depicted in
Fig. 1 representing the relation
$$
V_{ud}V^*_{ub}+V_{cd}V^*_{cb}+V_{td}V^*_{tb}=0~.\eqno(4)
$$

\vskip 5.5truecm
\centerline{\rm Figure 1: The CKM unitarity triangle}
\vskip 0.5truecm

\noindent
The three angles of the unitarity triangle, $\alpha,~\beta$ and $\gamma$
(which appears as a phase in (2)), are rather badly known at present. Current
constarints, which depend on uncertainties in $K$- and $B$-meson hadronic
parameters, can be approximately summarized by the following ranges [8]:
$$
10^0\leq\alpha\leq 150^0,~~~5^0\leq\beta\leq 45^0,~~~20^0\leq\gamma\leq 165^0~.
\eqno(5)
$$
As we will show, certain CP asymmetries in $B$ decays are directly related to
these angles in a manner which is free of hadronic uncertainties, and can
provide a more precise determination for some of these fundamental parameters.

One can draw similar unitarity triangles describing the orthogonality of
other pairs of columns or rows of the CKM matrix. Knowledge of the angles of
all
these triangles, which can be related to CP asymmetries, suffice to determine
the entire matrix [9]. All such triangles have equal areas, however they
involve one side which is much shorter than the other two sides, and
consequently one of their angles is very tiny. This is in contrast to the
angles
$\alpha, \beta$ and $\gamma$ which are naturally large, since  all the three
sides of the unitarity triangle of Fig. 1 are of comparable (${\cal
O}(\lambda^3)$) magnitude. Thus, for instance, the neutral $K$ meson triangle,
built of elements $V_{qd}V^*_{qs}~(q=u,c,t)$, has two long sides (length
$\lambda$) and one extremely short side (length ${\cal O}(\lambda^5)$). This
explains why CP asymmetries in K decays, which are related to the tiny angle of
this triangle (${\cal O}(\lambda^4)$), are of order $10^{-3}$.

\vskip 0.5truecm
\noindent
{\bf 3. CP VIOLATION IN $B^0-\Bbar$ MIXING}
\vskip 3mm

The flavor states $B^0$ and $\Bbar$ mix through the weak interactions to form
the "Light" and "Heavy" mass-eigenstates $B_L$ and $B_H$:
$$
|B_L\rangle = p|B^0\rangle  + q|\Bbar\rangle~,
$$
$$
|B_H\rangle = p|B^0\rangle  - q|\Bbar\rangle~.
\eqno(6)
$$
These states have masses $m_{L,H}$ and widths $\Gamma_{L,H}$.
The Hamiltonian eigenvalue equation (using CPT)
$$
\left(
\matrix{M-{i\over 2}\Gamma&M_{12}-{i\over 2}\Gamma_{12}\cr
M^*_{12}-{i\over 2}\Gamma^*_{12}&M-{i\over 2}\Gamma\cr}\right)
\left(\matrix{p\cr \pm q\cr}\right)=(m_{L,H}-{i\over 2}\Gamma_{L,H})
\left(\matrix{p\cr \pm q\cr}\right)\eqno(7)
$$
has the following solution for the mixing parameter:
$$
{q\over p}= \sqrt{{M^*_{12}-{i\over 2}\Gamma^*_{12}\over
M_{12}-{i\over 2}\Gamma_{12}}}
=-{2(M^*_{12}-{i\over 2}\Gamma^*_{12})\over \Delta m-{i\over 2}\Delta \Gamma}
{}~,\eqno(8)
$$
where $\Delta m\equiv m_H-m_L, \Delta\Gamma\equiv \Gamma_H-\Gamma_L$.
$M_{12}$ and $\Gamma_{12}$ describe respectively transitions from $B^0$ to
$\Bbar$ via virtual states and contributions from decay channels which are
common to $B^0$ and $\Bbar$.

The CP impurity parameter $\tilde{\epsilon}$, related to $q/p$ by
$q/p\equiv(1-\tilde{\epsilon})/ (1+\tilde{\epsilon})$,
gives the mass-eigenstates in terms of states with well-defined CP
$$
|B_L\rangle={1\over\sqrt{1+\vert\tepsilon\vert^2}}(|B^0_1\rangle + \tepsilon
|B^0_2\rangle)~,
$$
$$
|B_H\rangle={1\over\sqrt{1+\vert\tepsilon\vert^2}}(|B^0_2\rangle + \tepsilon
|B^0_1\rangle)~,\eqno(9)
$$
$$
|B_1\rangle={1\over\sqrt{2}}(|B^0\rangle + |\Bbar\rangle)~,
$$
$$
|B_2\rangle={1\over\sqrt{2}}(|B^0\rangle - |\Bbar\rangle)~.\eqno(10)
$$

$q/p$ has a phase freedom under redefinition of the phases of the
flavor states $B^0,~\Bbar$: $|B^0\rangle\to e^{i\xi} |B^0\rangle,~
|\Bbar\rangle\to e^{-i\xi} |\Bbar\rangle~\Rightarrow ~(q/p)\to
e^{2i\xi}(q/p)$. Thus the phase of $q/p$ can be rotated away and $|q/p|=1$
means CP conservation in $B^0-\Bbar$ mixing. The deviation of $|q/p|$ from
one measures CP violation in the mixing:
$$
1-|{q\over p}|\approx 2{\rm Re}\tepsilon~,\eqno(11)
$$
where ${\rm Re}\tepsilon$ is phase-convention independent. For convenience,
we will use the quark phase convention in which the CKM matrix (2) is written.

In the $B$ system one has $|\Gamma_{12}|\ll |M_{12}|$. $\Gamma_{12}$ is
given by the absorptive part of the box diagram, Fig. 2(a), arising from decay
channels which are common to $B^0$ and $\Bbar$. On the other hand, $M_{12}$ is
the dispersive part of the diagram, Fig. 2(b), governed by the $t$ quark mass.
Crudely speaking $|\Gamma_{12}/M_{12}|\sim m^2_b/m^2_t$. Thus CP violation
in  $B^0-\Bbar$ mixing is expected to be very small in the Standard
Model [10], $2{\rm Re}\epsilon\approx 1-|q/p|\sim {\cal O}(10^{-3})$. This
estimate, which is about the level of violation measured in the neutral $K$
meso
system [2], involves hadronic uncertainties and cannot provide a useful
quantitative test of the Standard Model. A much larger value of
${\rm Re}\tepsilon$ would be evidence against the CKM mechanism.

\vskip 5.5truecm
\centerline{\rm Figure 2: Box diagrams of $\Gamma_{12}$ (a) and $M_{12}$ (b)}
\vskip 0.5truecm

CP violation in $B^0-\Bbar$ mixing is expected to show up as a charge asymmetry
in semileptonic decays to "wrong charge" leptons, namely leptons to which only
a
mixed neutral $B$ can decay:
$$
A_{SL}={\Gamma(\Bbar(t)\to\ell^+\nu X)-\Gamma(B^0(t)\to\ell^-\overline{\nu}X)
\over \Gamma(\Bbar (t)\to\ell^+\nu X)+\Gamma(B^0(t)\to\ell^-\overline{\nu}
X)}~.
\eqno(12)
$$
$B^0(t)~(\Bbar(t))$ is a time-evolving state, which is created as a
$B^0~(\Bbar)
state at $t=0$. The asymmetry can be easily shown to be time-independent:
$$
A_{SL}={1-|q/p|^4\over 1+|q/p|^4}\approx 4{\rm Re}\tepsilon~.\eqno(13)
$$
There exists already an experimental upper limit from CLEO
[11], $|{\rm Re}\tepsilon|<45\times 10^{-3}$ ($90\%$ c.l.), which is about two
orders of magnitude above the Standard Model prediction. It will be extremely
difficult to observe an asymmetry at the level predicted by the model. However,
further efforts to improve this limit, for the semileptonic decays which have a
large branching ratio, is definitely worthwhile

Let us note in passing that while CP violation in the $B^0-\Bbar$ {\it mixing}
i
expected to be as small as about the one observed in $K^0-\Kbar$
mixing, the  asymmetries expected in neutral $B$ {\it decays} are much larger
than those of $K$ decays. Thus, when discussing neutral $B$ decay asymmetries
in
the following section we will take $|q/p|=1$ which is a very good
approximation.
In this approximation the mixing parameter is a pure phase
$$
{q\over p}\approx \sqrt{M^*_{12}\over M_{12}}\equiv e^{-2i\phi_M}=\cases
{e^{-2i\beta}~&for $\bd$~,\cr 1~&for $\bs$~.\cr}
\eqno(14)
$$
We will also assume $\Gamma_L=\Gamma_H$, which is a good
approximation, in particular for $\bd$ where it is expected to hold with an
accuracy better than $1\%$

\vskip 5mm
\noindent
{\bf 4. CP VIOLATION IN DECAYS OF MIXED $B^0-\Bbar$}
\vskip 3mm

\noindent
{\bf 4.1 Time-dependent asymmetries in the general case}
\vskip 3mm

Consider the time-evolution of a state which is identified at time $t=0$ as a
$B^0$:
$$
t=0:~~~~|B^0\rangle ={e^{-i\phi_M}\over\sqrt{2}}(|B_L\rangle+|B_H\rangle)~.
\eqno(15)
$$
The time-evolutions of the states $B_{L,H}$ are given simply by their masses
and
by their equal decay width $\Gamma$: $|B_{L,H}(t=0)\rangle \to
|B_{L,H}(t)\rangle= \exp[-i(m_{L,H}-{i\over 2}\Gamma)t]|B_{L,H}(t=0)\rangle$.
Thus, the $B^0$ oscillates into a mixture of $B^0$ and $\Bbar$:
$$
t:~~~~~~|B^0(t)\rangle=e^{-i\overline{m}t}e^{-{\Gamma\over 2}t}[\cos({\Delta m
t\over 2})|B^0\rangle+ie^{-2i\phi_M}\sin({\Delta mt\over 2})|\Bbar\rangle]~,
\eqno(16)
$$
where $\overline{m}\equiv(m_H+m_L)/2,~\Delta m\equiv m_H-m_L$.
Now, assume that both $B^0$ and $\Bbar$ can decay to a common state $f$, with
amplitudes $A$ and $\Abar$, respectively. The time-dependent decay rate to $f$
of an initial $B^0$ and the corresponding rate for an initial $\Bbar$ are
$$
\Gamma(B^0(t)\to f)=e^{-\Gamma t}|A|^2[\cos^2({\Delta mt\over 2})+|\Abar/A|^2
\sin^2({\Delta mt\over 2})-{\rm Im}(e^{-2i\phi_M}\Abar/A)\sin(\Delta mt)]~,
$$
$$
\Gamma(\Bbar (t)\to f)=e^{-\Gamma t}|A|^2[|\Abar/A|^2\cos^2({\Delta mt\over 2})
+\sin^2({\Delta mt\over 2})+{\rm Im}(e^{-2i\phi_M}\Abar/A)\sin(\Delta mt)]~.
\eqno(17)
$$
In the special case that $f$ is an eigenstate of CP,
$CP|f\rangle=\pm|f\rangle$, CP violation is manifest when
$\Gamma(t)\equiv\Gamma(B^0(t)\to f)\ne \Gamma(\Bbar (t)\to
f)\equiv\overline{\Gamma}(t)$. The CP asymmetry is given
by [12]:
$$
Asym.(t)\equiv {\Gamma(t)-\Gammabar(t)\over \Gamma(t)+\Gammabar(t)}=
{(1-|\Abar/A|^2)\cos(\Delta mt)-2{\rm
Im}(e^{-2i\phi_M}\Abar/A)\sin(\Delta mt)\over 1+|\Abar/A|^2}~.
\eqno(18)
$$
The two terms in the numerator represent different sources of CP violation.
The first term
follows from CP violation in the direct decay of a neutral $B$
meson, whereas the second term is induced by $B^0-\Bbar$ mixing.

\vskip 5mm
\noindent
{\bf 4.2 Decays to CP eigenstates dominated by a single CKM phase}
\vskip 3mm

Let us first consider the case of no direct CP violation, $|\Abar|=|A|$, in
whic
a single weak amplitude (or rather a single weak phase)
dominates the decay [13]. This is the case of a
maximal interference term in Eq.(17). Denoting the weak and strong phases by
$\phi_f$ and $\delta$, respectively, we have $A=|A|\exp(i\phi_f)\exp(i\delta),~
\Abar=\pm|\Abar|\exp(-i\phi_f)\exp(i\delta)$, and the asymmetry is given
simply by
$$
Asym.(t)=\pm\sin2(\phi_M+\phi_f)\sin(\Delta mt)~.
\eqno(19)
$$
The sign is given by $CP(f)$. The time-integrated asymmetry is
$$
Asym.= \pm\big({\Delta m/\Gamma\over 1+(\Delta m/\Gamma)^2}\big )\sin2(\phi_M+
\phi_f)~.
\eqno(20)
$$
That is, in this case {\it the CP asymmetry measures a CKM phase with no
hadronic uncertainty}. The integrated asymmetry in $\bd$ decays may be as large
as $(\Delta m/\Gamma)/[1+(\Delta m/\Gamma)^2]=0.47$.

The best example is the well-known and much studied case [14] of $\bd\to \psi
K_S$, for which a branching ratio of about $5\times 10^{-4}$ has already been
measured [15]. In this case
$\phi_M=\beta,~\phi_f={\rm Arg}(V^*_{cb}V_{cs})=0,~CP(\psi K_S)=-1$. Another
case is $\bd\to\pi^+\pi^-$, for which a combined branching ratio
$BR(\bd\to\pi^+\pi^-~{\rm and}~K^+\pi^-)=(2.3\pm 0.8)\times 10^{-5}$ has been
measured [16], with a likely solution in which the two modes have about equal
branching ratios. In this case $\phi_f={\rm
Arg}(V^*_{ub}V_{ud})= \gamma,~CP(\pi^+\pi^-)=1$. Consequently one has in these
two cases    $$ Asym.(\bd\to\psi K_S;t)=-\sin2\beta\sin(\Delta mt)~,
$$
$$
Asym.(\bd\to\pi^+\pi^-;t)=-\sin2\alpha\sin(\Delta
mt)~.\eqno(21)
$$
In the case of decay to two pions the asymmetry obtains, however, corrections
from a second (penguin) CKM phase. This problem will be discussed below.

\vskip 5mm
\noindent
{\bf 4.3 Decays to non-CP eigenstates}
\vskip 3mm
\noindent

Angles of the unitarity triangle can also be determined from neutral B decays
to states $f$ which are not eigenstates of CP [17]. This is feasible when
both a $B^0$ and a $\Bbar$ can decay to a final state which appears in only one
partial wave, provided that a single CKM phase dominates each of the
corresponding decay amplitudes.

The time-dependent rates for states which are $B^0$ or $\Bbar$ at $t=0$ and
decay at time $t$ to a state $f$ or its charge-conjugate $\fbar$ are given
by [18]:
$$
\Gamma_f(t)=e^{-\Gamma t}[|A|^2\cos^2({\Delta mt\over 2})+|\Abar|^2\sin^2
({\Delta mt\over 2})+|A\Abar|\sin(\Delta\delta+\Delta\phi_f+2\phi_M)\sin(\Delta
mt)]~,
$$
$$\Gammabar_f(t)=e^{-\Gamma t}[|\Abar|^2\cos^2({\Delta mt\over
2})+|A|^2\sin^2 ({\Delta mt\over
2})-|A\Abar|\sin(\Delta\delta+\Delta\phi_f+2\phi_M)\sin(\Delta mt)]~,
$$
$$
\Gamma_{\fbar}(t)=e^{-\Gamma t}[|\Abar|^2\cos^2({\Delta mt\over
2})+|A|^2\sin^2 ({\Delta mt\over
2})-|A\Abar|\sin(\Delta\delta-\Delta\phi_f-2\phi_M)\sin(\Delta mt)]~,
$$
$$
\Gammabar_{\fbar}(t)=e^{-\Gamma t}[|A|^2\cos^2({\Delta mt\over
2})+|\Abar|^2\sin
({\Delta mt\over 2})+|A\Abar|\sin(\Delta\delta-\Delta\phi_f-2\phi_M)\sin(\Delta
mt)]~.\eqno(22)
$$
Here $\Delta\delta,~(\Delta\phi_f)$ is the difference between the strong
(weak) phases of $A$ and $\Abar$. The four rates depend on four
unknown quantities, $|A|,~|\Abar|,~\sin(\Delta\delta+\Delta\phi_f+2\phi_M),~
\sin(\Delta\delta-\Delta\phi_f-2\phi_M)$. Measurement of the rates allows a
determination of the weak CKM phase $\Delta\phi_f+2\phi_M$ apart from a
two-fold ambiguity [17].

There are two interesting examples to which this method can be applied. In the
first case, $\bd\to\rho^+\pi^-$, one must neglect a second contribution of a
penguin amplitude, a problem which will be addressed in the following
subsection. Assuming for a moment that tree diagrams, shown in Figs. 3(a),
3(b),
dominate $A$ and $\Abar$, one can measure in this manner the angle $\alpha$,
since in this case $\Delta\phi_f+2\phi_M=2(\gamma+\beta)=2(\pi-\alpha)$. A
second case, which may be used to measure $\gamma$, is $\bs\to D^+_s K^-$,
in which only one amplitude contributes to $A$ and another amplitude - to
$\Abar$.
\vfil\eject

{}~~
\vskip 5.0truecm
\centerline{\rm Figure 3: Diagrams of $\bd\to\rho^+\pi^-$ (a) and
$\bdb\to\rho^+\pi^-$ (b)}
\vskip 0.5truecm

\vskip 5mm
\noindent
{\bf 4.4 Corrections from penguin amplitudes}
\vskip 3mm

A crucial question is, of course, how good is the assumption of a single
dominant CKM phase, which is needed for a clean determination of an angle
of the  unitarity triangle. One may try to answer this question experimentally
by looking for an extra $\cos(\Delta mt)$ term in the time-dependent asymmetry
o
Eq.(18) which describes CP violation in the direct decay of $B^0$. There is,
however, the danger that this term will be unobservably small, just because
the final state interaction phase difference happens to be small. The effect of
second amplitude on the coefficient of $\sin(\Delta mt)$, which is proportional
to the cosine of this phase-difference [12], may still be large. This will be
demonstrated below for $\bd\to\pi^+\pi^-$.

In a wide variety of decay processes there exists
a second amplitude due to ``penguin" diagrams [19] in addition to the usual
``tree" diagram. In general, the new contribution becomes more disturbing when
the process involves a stronger CKM-suppression.
The penguin-to-tree ratio of amplitudes is proportinal to the ratio of the
corresponding CKM factors and to a QCD factor $(\alpha_s(m^2_b)/12\pi){\rm ln}
(m^2_t/m^2_b)$. This ratio may be estimated for a given process. A few examples
of final states in $\bd$ decays,
with different levels of CKM suppression, are [12]:
$$
{{\rm Penguin}\over {\rm Tree}}= \cases
{10^{-3}~& $\psi K_S$~,\cr 0.05~& $D^+D^-~(D^{*+}D^-)$~,\cr 0.20~&
$\pi^+\pi^-~(\rho^+\pi^-)$~,\cr {\cal O}(1)~& $K_S\pi^0$~.} \eqno(23)
$$
These numbers represent quite crude estimates, since there exists no reliable
method to calculate hadronic matrix elements of penguin operators. One way to
obtain information about these matrix elements would be to measure pure penguin
processes, such as $\bd\to\phi K_S$. Another way will be mentioned when
discussing charged $B$ decays.

We see from Eqs.(23) that the decay $\bd\to \psi K_S$ remains a pure case,
with less than $1\%$ corrections, also
in the presence of penguin contributions. On the other hand, penguin effects on
the CP asymmetry of $\bd\to\pi^+\pi^-$ may be substantial. This is demonstrated
in Fig. 4, taken from Ref.20, which shows the coefficient of the $\sin(\Delta
mt)$ term in the asymmetry as function of the angle $\alpha$ for a zero final
state interaction phase difference. The range of values comes from taking the
ratio (Penguin/Tree) to be anywhere between 0.04 and 0.20. For a ratio of
0.20, an asymmetry as large as 0.40 can possibly be measured even when
$\sin(2\alpha)=0$.

\vskip 5.5truecm
\centerline{\rm Figure 4: Asymmetry in $\bd\to\pi^+\pi^-$ as function of
$\alpha$}

\vskip 5mm
\noindent
{\bf 4.5 Removing penguin corrections in $\bd\to\pi^+\pi^-$}
\vskip 3mm

It is possible to disentangle the penguin contribution in $\bd\to\pi^+\pi^-$
fro
the tree-dominating asymmetry by measuring also the rates of
$B^+\to\pi^+\pi^0$ and $\bd\to\pi^0\pi^0$. The method [21] is based on the
observation that the two weak operators contributing to the three
isospin-related processes have different isospin properties just as in $K\to
2\pi$. Whereas the tree operator is a mixture of $\Delta I=1/2$ and $\Delta
I=3/2$, the penguin operator is pure $\Delta I=1/2$. Denoting the  physical
amplitudes of $B\to \pi^+\pi^-, \pi^0\pi^0, \pi^+\pi^0$  by the charges of the
two corresponding pions, one finds from an isospin decomposition
$$
{1\over\sqrt{2}}A^{+-}=A_2-A_0~,~~~A^{00}=2A_2+A_0~,~~~A^{+0}=3A_2~,
\eqno(24)
$$
where $A_0$ and $A_2$ are the amplitudes for a $\bd$ or a $B^+$ to decay into a
$\pi\pi$ state with $I=0$ and $I=2$, respectively. This yields the complex
triangle relation
$$
{1\over\sqrt{2}} A^{+-} + A^{00} = A^{+0}~.\eqno(25)
$$
There is a similar triangle relation for the charge-conjugated processes:
$$
{1\over\sqrt{2}}{\overline{A}}^{+-}
+ {\overline{A}}^{00} = {\overline{A}}^{-0}~.\eqno(26)
$$
Here, ${\overline{A}}^{+-}$, ${\overline{A}}^{00}$, and
${\overline{A}}^{-0}$ are the amplitudes for the processes
$\bdb\to\pi^+\pi^-$, $\bdb\to\pi^0\pi^0$, and $B^-\to\pi^-\pi^0$,
respectively. The ${\overline{A}}$ amplitudes are obtained from the $A$
amplitudes by simply changing the sign of the CKM phases (the strong phases
remain the same).

The crucial point in the analysis is that the "tree" contribution to $A_2$
has a well-defined weak phase, which is given by the angle $\gamma$ of the
unitarity  triangle,
$$
A_2=\vert A_2\vert e^{i\delta_2}e^{i\gamma}~,~~~
{\overline{A}}_2=\vert A_2\vert e^{i\delta_2}e^{-i\gamma}~.
\eqno(27)
$$
where $\delta_2$ is the $I=2$ final-state-interaction phase.
We neglect an
electroweak penguin contribution which introduces a second order
correction,
whereas the gluonic penguin is first order. It is convenient to
define $\tilde{A}=\exp(2i\gamma)\overline{A}$ so that $\tilde{A_2}=A_2$ and
$\tilde{A}^{-0}=A^{+0}$. The two complex triangles representing Eqs. (25)(26)
(where $\overline{A}$ is replaced by $\tilde{A}$)  are shown in Fig. 5. They
have a common base (CP is conserved in $B^+\to\pi^+\pi^0$); however the length
of their corresponding sides are different. That is, CP is violated in
$\bd\to\pi\pi$.

\vskip 5.5truecm
\centerline{\rm Figure 5: Isospin triangles of $B\to\pi\pi$}
\vskip 0.5truecm

The six sides of the two triangles are measured by the decay rates of $B^{\pm}$
and by the time-integrated rates of $\bd~(\bdb)$. This determines the two
triangles within a two-fold ambiguity; each triangle may be turned
up-side-down. The coefficients of the
$\sin(\Delta mt)$ term in $\bd\to\pi^+\pi^-$ measures the quantities
$$
{\rm Im}\thinspace(e^{-2i(\beta+\gamma)}{\tilde
{A}^{+-}\over A^{+-}})={\vert\tilde{A}^{+-}\vert\over\vert
A^{+-}\vert}\sin(2\alpha+\theta_{+-}),\eqno(28)
$$
where $\theta_{+-}$ is obtained from Fig. 5. (This angle vanishes in the
absence
of the penguin correction). This determines the angle $\alpha$.

The application of this method in asymmetric $e^+e^-$ $B$-factories
[22] may suffer from the very small branching ratio of
$\bd\to\pi^0\pi^0$ which is expected to be color-suppressed. One
would have to observe the two neutral pions. Of course, if the penguin
amplitude
is small, its effect on the asymmetry of $\bd\to\pi^+\pi^-$ will be small. When
discussing charged $B$ decays we will mention how these decays can tell us
something about the magnitude of the penguin term in $\bd\to\pi\pi$.

A similar isospin analysis
was carried out for other decays in which penguin
amplitudes are involved [23]. In general, the precision of determining a
CKM phase becomes worse when a larger number of amplitudes must be related.
Also a few ambiguities show up in this case. In the case of
$\bd\to\rho\pi$ (and $B^+\to\rho\pi$) five physical decay amplitudes appear.
In this case the ambiguity can be resolved if a full Dalitz plot analysis
can be made for the three pion final states [24].

\vskip 0.5truecm
\noindent
{\bf 5. CP VIOLATION IN CHARGED $B$ DEACYS}
\vskip 3mm

\noindent
{\bf 5.1 A theoretical difficulty}
\vskip 3mm

The simplest manifestations of $CP$ violation are different partial decay
widths for a particle and its antiparticle into corresponding decay modes.
Consider a general decay $B^+\to f$ and its charge-conjugate process
$B^-\to\fbar$. In order that these two proceses have different rates, two
amplitudes ($A_1, A_2$) must contribute, with different CKM phases ($\phi_1 \ne
\phi_2$) and different final state interaction phases ($\delta_1\ne\delta_2$):
$$  A(B^+\to f)~=~\vert A_1\vert  e^{i\phi_1}e^{i\delta_1}~+ ~\vert
A_2\vert e^{i\phi_2}e^{i\delta_2}~,
$$
$$
{}~~~~~\Abar (B^-\to \fbar)~=~\vert A_1\vert  e^{-i\phi_1}e^{i\delta_1}~+
{}~\vert
A_2\vert e^{-i\phi_2}e^{i\delta_2}~,
$$
$$
{}~~~~~~~~~\vert A \vert^2-\vert\Abar\vert^2=2\vert A_1
A_2\vert\sin(\phi_1-\phi_2)\sin(\delta_1-\delta_2)~. \eqno(29)
$$
The theoretical difficulty of relating an asymmetry in charged $B$ decays to a
pure CKM phase follows from having two unknowns in the problem: The
ratio of amplitudes, $\vert A_2/A_1\vert$, and the final state phase
difference,
$\delta_2-\delta_1$. Both quantities involve quite large theorertical
uncertainties.

This is demonstrated in Fig. 6, which describes the two amplitudes $A_1$ and
$A_2$ for $B^+\to K^+\pi^0$, given by the ``penguin"  and ``tree"
diagrams, respectively.

\vskip 5.5truecm
\centerline{\rm Figure 6: Penguin (a) and tree (b) diagrams in $B^+\to
K^+\pi^0$}
\vskip 0.5truecm

\noindent
In this case $\phi_1=\pi,~\phi_2=\gamma$. A few calculations of the
asymmetry in this process exist [25], based on model-dependent estimates of the
tree-to-penguin ratio of amplitudes and of the strong phase difference.
The strong phase includes a phase
due to the absorptive part of the physical $c\cbar$ quark pair in the penguin
diagram, which may be viewed as describing rescattering processes such as
$B\to\overline{D} D_s\to K\pi$. All such model-dependent calculations
involve large theoretical uncertainties.

\vskip 5mm
\noindent
{\bf 5.2 Measuring $\gamma$ in $B^{\pm}\to D^0 K^{\pm}$}
\vskip 3mm

The decays
$B^{\pm}\to D^0_1(D^0_2) K^{\pm}$ and a few other processes of this
type provide a unique case [26], in which one can measure separately the
magnitudes of the two contributing amplitudes, and thereby determine the CKM
phase  $\gamma$. $D^0_1(D^0_2)=(D^0+(-)\Dbar)/\sqrt{2}$ is a CP-even (odd)
state, which is  identified by its CP-even (odd) decay products. For
instance, the states $K_S\pi^0,~K_S\rho^0,~K_S \omega,~K_S \phi$ identify a
$D^0_2$, while $\pi^+\pi^-,~ K^+K^-$ represent a $D^0_1$. The decay amplitudes
of the above two charge-conjugate processes can be written (say for $D^0_1$) in
the form
$$
\eqalign{ \sqrt{2}A(B^+\to D^0_1 K^+)~=~\vert
A_1\vert\exp(i\gamma)\exp(i\delta_1)~+~\vert
A_2\vert\exp(i\delta_2)~~,\cr \sqrt{2}A(B^-\to D^0_1 K^-)~=~\vert
A_1\vert\exp(-i\gamma)\exp(i\delta_1)~+~\vert
A_2\vert\exp(i\delta_2).\cr}\eqno(30)
$$
$A_1$ and $A_2$ are the two weak amplitudes, shown in Fig. 7(b) and 7(a),
respectively. Their CKM
factors $V^*_{ub}V^{~}_{cs}$ and $V^*_{cb}V^{~}_{us}$ are of comparable
magnitudes. Their weak phases are $\gamma$ and zero. Since $A_1$ leads to final
states with isospin $0$ and $1$, whereas $A_2$ can only lead to isospin $1$
states, one generally expects [27] $\delta_1\ne\delta_2$.

\vskip 5.5truecm
\centerline{\rm Figure 7: Diagrams decribing $B^+\to\Dbar K^+$ (a) and
$B^+\to D^0 K^+$ (b)}
\vskip 0.5truecm

As shown in Fig. 7, the two amplitudes on the right-hand-sides
of the first of Eqs. (30) are the amplitudes of $B^+\to D^0 K^+$ and
$B^+\to \Dbar K^+$, respectively. Similarly, the two terms in the second
equation describe the amplitudes of $B^-\to\Dbar  K^-$ and $B^-\to D^0 K^-$,
respectively. The flavor states $D^0$ and $\Dbar$ are
identified by the charge of the decay lepton or kaon. Thus one has:
$$
\eqalign{
\sqrt{2}A(B^+\to D^0_1 K^+)~=~A(B^+\to
D^0 K^+)~+~A(B^+\to \Dbar K^+),\cr \sqrt{2}A(B^-\to D^0_1 K^-)~=~A(B^-\to \Dbar
K^-)~+~A(B^-\to D^0 K^-).\cr}\eqno(31)
$$
These relations can be described by two triangles in the complex plane  as
shown
in Fig. 8.

\vskip 5.5truecm
\centerline{\rm Figure 8: Triangles describing Eqs.(31)}
\vskip 0.5truecm

The two triangles represent the complex $B^+$ and $B^-$ decay amplitudes. Note
that
$$
\eqalign{
A(B^+\to\Dbar K^+)=A(B^-\to D^0 K^-)~~~~~~~~~~,\cr
A(B^+\to D^0 K^+)=\exp(2i\gamma)A(B^-\to \Dbar K^-),\cr
\vert A(B^+\to D^0_1 K^+)\vert\ne\vert A(B^-\to D^0_1
K^-)\vert~~~~~~~~~.\cr}\eqno(32)
$$
This implies that CP is conserved in $B^{\pm}\to D^0(\Dbar) K^{\pm}$ but is
violated in  $B^{\pm}\to D^0_1 K^{\pm}$. In the last of Eqs.(32) we assumed
$\gamma\ne 0$, $\delta_1\ne\delta_2$. The asymmetry in the rates of
$B^{\pm}\to D^0_1 K^{\pm}$ depends on $\gamma$ and $\delta_2-\delta_1$;
clearly
$$
\eqalign{&
\vert A(B^+\to D^0_1 K^+)\vert^2-\vert A(B^- \to D^0_1 K^-)\vert^2 \cr &
=2\vert A(B^+\to\Dbar K^+)\vert\vert A(B^+\to D^0
K^+)\vert\sin(\delta_2-\delta_1)\sin\gamma. } \eqno(33)
$$

The procedure for obtaining $\gamma$ is straightforward. Measurements of the
rates of the above six proccesses, two pairs of which are equal, determine the
lengths of all six sides of the two triangles.
When the two triangles are formed, $2\gamma$ is the angle between
$A(B^+\to D^0 K^+)$ and $A(B^-\to \Dbar K^-)$. This determines the magnitude of
$\gamma$ within a two-fold ambiguity
related to a possible interchange of $\gamma$ and $\delta_1-\delta_2$. This
ambiguity may be resolved by carrying out this analysis for other decay
processes of the type $B^{\pm}\to D^0(\Dbar, D^0_{1(2)})X^{\pm}$, where
$X^{\pm}$ is any other state with the flavor quantum number of a $K^{\pm}$.

The feasibility of observing a CP asymmetry in $B^+\to D^0_{1(2)} K^+$ depends
on the branching ratios of the three related decay processes, and on the values
of the weak and strong phases. One may estimate $BR(B^+\to \Dbar K^+)\approx
2\times 10^{-4}$, using the corresponding measured Cabibbo-allowed branching
ratio  of $B^+\to\Dbar \pi^+$ [15]. The process
$B^+\to D^0 K^+$, in which the two quarks of the $c\sbar$ current
enter two different meson states, is likely to be "color-suppressed".
Color suppression has already been seen in $B\to D \pi$ [15]. If the same
suppression factor applies also to $B^+\to D^0 K^+$, then the branching ratio
of this process is at most at the level of $10^{-5}$. Using a value
of $5\times 10^{-6}$, the feasibility for observing a CP asymmetry in $B^+\to
D^0_{1(2)} K^+$ was studied [28] as function of $\gamma$ and
$\delta_2-\delta_1$, for a (symmetric) $e^+e^-\to\Upsilon(4S)$ $B$-factory with
an integrated luminosity of $20 fb^{-1}$. The discovery region was found
to cover a significant part of the ($\gamma, \delta_2-\delta_1$) plane. For
small final state phase differences the experiment is sensitive mainly to
values
of $\gamma$ around $90^0$. Large values of $\delta_2-\delta_1$ allow a
useful measurement of $\gamma$ in the range $50^0\leq\gamma\leq 130^0$.

Present experiments are reaching the level of being able to observe the first
Cabibbo suppressed decays $B\to DK$. The question of color-suppression in these
decays needs to be studied. It is possible that the final state phase
difference
$\delta_2-\delta_1$ is too small to allow a good measurement of $\gamma$ if
this angle is not around $90^0$. A recent study [29] generalized this method to
quasi-two-body decays  $B\to DK_i\to D K\pi$, where $K_i$ are excited kaon
resonance states with masses around 1400 MeV. The resonance effect
gives give
rise to large final state phases and thus enhances the CP asymmetry.

\vskip 5mm
\noindent
{\bf 5.3 Using SU(3) to determine $\gamma$ from $B^+\to\pi K$ and
$B^+\to\pi\pi$}
\vskip 3mm

Flavor SU(3) symmetry can be used to relate $B$ decays to $\pi\pi,~\pi K$ and
$KK$ states [30]. Recently this idea was applied [31] jointly with the
dynamical
assumption that annihilation-like diagrams are small in these two-body
decays.  This assumption means that certain rescattering effects are small.
That
is, final states which are produced, for instance, by tree decay amplitudes
have
small rescattering amplitudes to states created by quark-antiquark
annihilation. This assumption is motivated by the high $B$ meson mass (compared
to $f_B$). It is supported by the experimental evidence for color-suppression
and for factorization in two body $B$ decays [15], two features which are
expected to be spoiled by large rescattering amplitudes. It was shown that the
assumption of negligible rescattering is equivalent to assuming that certain
final state phases are equal to others [32]. Neglecting such rescattering
effects leads to simple testable predictions, such as $A(B^0\to K^+ K^-)=0$,
and
to useful information about weak and strong phases. Here we wish to demonstrate
this idea through a rather simple case [33].

Consider the decay $B^+\to \pi^0 K^+$ for which the two contributing
amplitudes $A_1$ and $A_2$ are described in Fig. 6. The penguin amplitude
$A_1$ is related by isospin
to the amplitude of $B^+\to \pi^+K^0$, in which the annihilation contribution
is neglected, $A_1=A(\pi^+K^0)/\sqrt{2}$. The tree amplitude $A_2$
is related by SU(3) to the amplitude of $B^+\to\pi^+\pi^0$, which receives no
penguin contribution. Using factorization to introduce SU(3) breaking into this
relation, one has $A_2=(f_K/f_{\pi})|V_{us}/V_{ud}|A(\pi^+\pi^0)$. Thus one
obtains a simple relation between the three $B^+$ decay amplitudes:
$$
A(\pi^0 K^+)={1\over \sqrt{2}}A(\pi^+ K^0) + {f_K\over f_{\pi}}|{V_{us}\over
V_{ud}}|A(\pi^+\pi^0)~.\eqno(34)
$$
A similar relation holds among the corresponding $B^-$ decay amplitudes.
There are phase relations, $A(\pi^- \Kbar)=A(\pi^+ K^0),
A(\pi^-\pi^0)=\exp(-2i\gamma)A(\pi^+\pi^0)$, since the weak
phases of these amplitudes are $\pi$ and $-\gamma$, respectively. The two
relations among the $B^+$ and among the $B^-$ amplitudes are analogous to
Eqs.(31). They can be descrlibed by two triangles very similar to those of
Fig. 8. In the present case the two triangles share a common base given by
$A(\pi^- \Kbar)=A(\pi^+ K^0)$, and the angle between the sides describing
$A(\pi^-\pi^0)$ and $A(\pi^+\pi^0)$ is $2\gamma$. Measurements of the four
rates
into $\pi^0 K^+, \pi^0 K^-, \pi^+ K^0, \pi^+\pi^0$, suffices to determine
$\gamma$. CP violation is demonstrated by $A(\pi^0 K^+)\ne A(\pi^0 K^-)$. About
100 events of this mode, which is expected to have a branching ratio of about
$10^{-5}$, are needed to measure $\gamma$ to a statistical accuracy of $10^0$
[33].

This method is not as clean as the one using $B^{\pm}\to D^0K^{\pm}$ decays,
since it is based on certain dynamical assumptions. Also, it was recently noted
[34] that contributions from electroweak penguin diagrams can spoil the
relation
(34). Information about the effect of
these diagrams, of SU(3) breaking and of annihilation-like diagrams, and the
separate magnitudes of tree and penguin amplitudes, can be obtained from a
systematic study of all the possible $B$ decay modes to two light pseudoscalar
mesons [31]. Such a detailed study may then be used to evaluate the precision
to
which the weak phase can be determined.

\vskip 0.5truecm
\noindent
{\bf 6. FLAVOR-TAGGING OF NEUTRAL $B$ MESONS}
\vskip 3mm
\noindent
{\bf 6.1 Tagging by the associated $B$ decay}
\vskip 3mm

In order to measure CP asymmetries in neutral $B$ decays one must identify the
flavor of the decaying meson at some reference time $t=0$. In a
$e^+e^-\to\Upsilon(4S)$ $B$-factory this is achieved [14] by observing a
lepton, or a cascade charged kaon from $B\to D\to K$, from the decay of the
othe
neutral $B$. Since at any time after production the two neutral $B$ mesons form
{\it coherent}  $C(B^0\Bbar)=-1$ EPR pair, the charge of the lepton serves to
"tag" the opposite flavor of the other $B$ at the time of  semileptonic decay.
Furthermore, the CP asymmetry is odd in the time-difference of the two decays,
and consequently asymmetric storage rings are required for an asymmetry
measurement.

A similar method of determining the flavor of neutral $B$ mesons
in high energy $e^+e^-$ or in hadronic collisions [35] uses as a "tag"
the lepton from a semileptonic decay of an associated $b$-meson or $b$-baryon.
The flavor is misidentified part of the time as a result of $B^0-\Bbar$ mixing.
The probability of misidentification and its effect on diluting the measured CP
asymmetry can be crudely estimated. Since the $B^0$ and $\Bbar$
are usually  produced with many other particles, it is commonly assumed that
the
are in an {\it incoherent} mixture.

\vskip 5mm
\noindent
{\bf 6.2 Tagging by correlated charged pions}
\vskip 3mm

An alternative method of flavor identification [36] uses an expected
correlation
between the decaying neutral $B$ and a charged pion making a low-mass
$B-\pi$ system. There are two arguments for such a correlation.
The first argument is based on the existence of positive-parity ``$B^{**}$"
resonances, with $J^P=0^+,~1^+,~2^+$ and masses below about 5.8 GeV/$c^2$ [37].
Using Heavy Quark Symmetry, this mass value
is obtained from the corresponding observed
``$D^{**}"$ masses (2420, 2460 GeV/$c^2$). The $B^{**}$ resonances decay to
$B\pi$ and/or $B^*\pi$ mesons in $I=1/2$  states. That is, a $\pi^+$ will
accompany a $B^0$ and not a $\Bbar$. A similar method [38] has been used to
tag neutral charmed mesons, where the decays $D^{*+}\to D^0\pi^+,~D^{*-}\to
\overline{D}^0 \pi^-$ are kinematically allowed.
The second argument is that in $b$-quark fragmentation the leading pion carries
information about the flavor of the neutral $B$. A neutral $B$ meson
containing an initially produced $b$ quark is a $\Bbar$ which contains a
$\dbar$ quark. The next charged pion down the fragmentation chain must contain
a $d$, and hence must be a $\pi^-$. Similarly, a $B^0$ will be correlated
with a $\pi^+$.

The efficiency of this method depends on the degree of the correlation, which
can be studied in neutral $B$ decays to states of identified flavor, such as
$D^-\pi^+$ or $\psi K^{*0}$ (with $K^{*0}\to K^+\pi^-$). Usually $B$ mesons are
produced in an isospin-independent manner and one can find this correlation
using charged $B$ mesons as well. The time-dependent CP asymmetry measured with
this tagging method is diluted by the degree of correlation. Aside from using
the asymmetry to determine weak phases, it can also be used to test the
assumption that the produced $B^0$ and $\Bbar$ are incoherent with respect to
one another [39].

\vskip 0.5truecm
\noindent
{\bf 7. SUMMARY}
\vskip 3mm

We have shown how in certain cases CP asymmetries in $B$ decays can determine
CK
phases in manners which are free of hadronic uncertainties. A determination of
the three angles of the unitarity triangle is based theoretically on different
types of asymmetries and is expected to involve different levels of
experimental difficulty. The most promising measurement at an asymmetric
$e^+e^-$ $B$ factory seems to be that of the angle $\beta$ in $\bd\to
\psi K_S$, which is the purest case of CP violation in decays of mixed
$B^0\Bbar$. In the asymmetry of $\bd\to\pi^+\pi^-$ which measures the angle
$\alpha$ one will have to disentangle penguin effects from the
measured asymmetry. This requires measuring decays to neutral pions or, at
least, estimating penguin effects from other processes. A time-independent
determination of $\gamma$ from direct decay CP violation in $B^{\pm} \to
D^0_{1,2} K^{\pm}_{(i)}$
may be feasible also in a symmetric $e^+e^- B$ factory if
$BR(B^+\to D^0 K^+_{(i)})$ is not too strongly color-suppressed.
An alternative
measurement of $\gamma$, by comparing $B^{\pm}\to\pi K$ to $B^{\pm}\to \pi\pi$,
requires a few dynamical assumptions which must be tested to assess the
accuracy
of this method.

Testing the Standard Model of CP violation requires more precise information
than available today about magnitudes and phases of CKM elements. At the very
least, B decay asymmetries will allow direct measurements of some of these
fundamental phases. With an improved knowledge of the magnitudes of CKM
elements, these phase measurements may eventually serve to overconstrain the
CKM
matrix. One would hope to find some inconsistencies which could be clues for
physics beyond the Standard Model.



\vskip 0.5truecm
\noindent
{\bf ACKNOWLEDGEMENTS}
\vskip 3mm

It is a pleasure to thank David Atwood, Gad Eilam, Oscar Hern\'andez, David
London, Alex Nippe, Jonathan Rosner, Amarjit Soni and Daniel Wyler for very
enjoyable collaborations on various topics presented here. This work was
supported in part by the United States - Israel Binational Science Foundation
under Research Grant Agreement 90-00483/3, by the German-Israeli Foundation of
Scientific Research and Development and by the Fund for Promotion of Research
at
the Technion.


%\vskip 0.5truecm
\vfil\eject
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\vskip0.3truecm

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