\documentclass[12pt]{article}
\addtolength{\textwidth}{100pt}
\addtolength{\evensidemargin}{-50pt}
\addtolength{\oddsidemargin}{-50pt}

\usepackage{graphicx}

\begin{document}
\title{The $CP$ Violation Problem}
\author{Ramy Naboulsi}
\maketitle

Tokyo Institute of Technology, Department of Physics, O-Okoyama, Meguro-ku, Tokyo\\\\

\hspace{75 mm}\textbf{Abstract}\footnotesize\\

A review on $CP$ violation in the $B$ and $K$ mesons systems is given, which, we hope may familiarize new Belle members in the basic ingredients of this topic, which is one of the most challenging problems of experimental high energy physics.\\\\\noindent
  
\normalsize
\textbf{I. Introduction}\\

The violation of $CP$ symmetry is one of the most interesting topics of high-energy physics today. Experimentally, it is one of the least tested properties of the Standard Model. $CP$ violation is the violation of the combined conservation laws associated with charge conjugation $C$ and parity $P$ by the weak nuclear force, which is responsible for reactions such as the decay of atomic nuclei. Charge conjugation is a mathematical operation that transforms a particle into an antiparticle, for example, changing the sign of the charge. Charge conjugation implies that every charged particle has an oppositely charged antimatter counterpart, or antiparticle. The antiparticle of an electrically neutral particle may be identical to the particle, as in the case of the neutral pi meson, or it may be distinct, as with the antineutron. Parity, or space inversion, is the reflection in the origin of the space coordinates of a particle or particle system; i.e., the three space dimensions x, y, and z become, respectively, -x, -y, and -z. Stated more concretely, parity conservation means that left and right and up and down are indistinguishable in the sense that an atomic nucleus throws off decay products up as often as down and left as often as right. For years it was assumed that charge conjugation and parity were exact symmetries of elementary processes, namely those involving electromagnetic, strong, and weak interactions. The same was held true for a third operation, time reversal $T$, which corresponds to reversal of motion. Invariance under time implies that whenever a motion is allowed by the laws of physics, the reversed motion is also an allowed one. A series of discoveries from the mid-1950s caused physicists to alter significantly their assumptions about the invariance of $C$, $P$, and $T$.\\
   
To understand whether a given theory can accomodate $CP$ violation,
One needs to know the transformation properties of the fields under
the various discrete symmetries. In particular for a Dirac spinor
\begin{equation}
P\psi (t,x)P = \gamma^0\psi (t,-x),
\end{equation}
\begin{equation}
T\psi (t,x)T = -\gamma^1\gamma^3\psi (-t,x),
\end{equation}
\begin{equation}
C\psi (t,x)C = -i(\bar{\psi}(t,x)\gamma^0\gamma^2)^T.
\end{equation}

The Lagrangian being a Lorentz scalar, can only depend on terms
bilinear in fermion fields (and not on single fermion fields). The
transformation properties of various fermion bilinears under $CP$ are
summerized in the table below. Here the shorthand $(-1)^\mu \equiv 1$ for $\mu = 0$ and $(-1)^\mu \equiv -1$ for $\mu = 1,2,3$ (namely, $(-1)^\mu a^\mu = a_\mu )$ is used.
\begin{equation}
\begin{array}{ccccc}
\mbox{term} & \bar{\psi}_i\psi_j & i\bar{\psi}_i\gamma^5\psi_j & \bar{\psi}_i\gamma^\mu\psi_j & \bar{\psi}_i\gamma^\mu\gamma^5\psi_j\\
\mbox{CP-transformed term} & \bar{\psi}_j\psi_i & -i\bar{\psi}_j\gamma^5\psi_i & -(-1)^\mu\bar{\psi}_j\gamma^\mu\psi_i & -(-1)^\mu\bar{\psi}_j\gamma^\mu\gamma^5\psi_i\end{array}
\end{equation}

Similarly, the transformation properties of scalar (\textbf{H}), pseudoscalar (\textbf{A}) and vector
boson (\textbf{W}) fields, and also of the derivative operator are given by
\begin{equation}
\begin{array}{ccccc}
\mbox{term} & H & A & W^{\pm\mu} & \partial_\mu \\
\mbox{CP-transformed term} & H & -A & -(-1)^\mu W^{\mp\mu} & (-1)^\mu\partial_\mu \end{array}
\end{equation}

Taking into account the Lorentz invariance and hermiticity of the Lagrangian, the above
CP transformation rules imply that each of the combinations of fields and derivatives, 
that appear in the Lagrangian, transforms under $CP$ to its hermitian conjugate. However, 
there are coefficients in front of these expressions which represent either coupling 
constants or particle masses and which do not transform under $CP$. If any of these quantities 
are complex, then the coefficients in front of CP-related terms are complex conjugates of 
each other. In such a case, $CP$ is not necessarily a good symmetry of the Lagrangian. When
the rates of physical processes that depend on these Lagrangian parameters are calculated, there
can be $CP$ violating effects, namely rate differences between pairs of $CP$ conjugate processes. An apparent lack of the conservation of parity in the decay of charged $K$ mesons into two or three $\pi$ mesons prompted the Chinese-born American theoretical physicists Chen Ning Yang and Tsung-Dao Lee to examine the experimental foundation of parity itself. In 1956 they showed that there was no evidence supporting parity invariance in weak interactions.\\

Experiments conducted the next year verified decisively that parity was violated in the weak interaction beta decay. Moreover, they revealed that charge conjugation symmetry also was broken during this decay process. The discovery that the weak interaction conserves neither charge conjugation nor parity separately, however, led to a quantitative theory establishing combined $CP$ as a symmetry of nature. Physicists reasoned that if $CP$ were invariant, time reversal $T$ would have to remain so as well. But further experiments, carried out in 1964, demonstrated that the electrically neutral $K$ meson, which was thought to break down into three $\pi$ mesons, decayed a fraction of the time into only two such particles, thereby violating $CP$ symmetry. In fact the neutral $K$ mesons with their medium-sized masses and their capacity of interaction both weakly and strongly, seem to be specially selected by nature to demonstrate through a few typical phenomena the reality of quantum effects.\\

No completely satisfactory explanation of $CP$ violation has yet been devised. The size of the effect, only about two parts per thousand, has prompted a theory that invokes a new force, called the ``superweak" force, to explain the phenomenon. This force, much weaker than the nuclear weak force, is thought to be observable only in the $K$-meson system or in the neutron's electric dipole moment, which measures the average size and direction of the separation between charged constituents. Another theory, named the Kobayashi-Maskawa model after its inventors, posits certain quantum mechanical effects in the weak force between quarks as the cause of $CP$ violation. The attractive aspect of the superweak model is that it uses only one variable, the size of the force, to explain everything. Furthermore, the model is consistent with all measurements of $CP$ violation and its properties. The Kobayashi-Maskawa model is more complicated, but it does explain $CP$ violation in terms of known forces.\\

$CP$ violation has important theoretical consequences. The violation of $CP$ symmetry, taken as a kind of proof of the $CPT$ theorem, enables physicists to make an absolute distinction between matter and antimatter. The distinction between matter and antimatter may have profound implications for cosmology. One of the unsolved theoretical questions in physics is why the universe is made chiefly of matter. With a series of debatable but plausible assumptions, it can be demonstrated that the observed matter-antimatter ratio may have been produced by the occurrence of $CP$ violation in the first seconds after the ``Big Bang", the violent explosion that is thought to have resulted in the formation of the universe. Direct $CP$ violation has been observed at Fermilab by the KTeV collaboration. An important way of apprehending the basic nature of time and space is to ask ``what if" questions. For example, will a collision between particles be altered if we view the whole thing in a mirror? Or what if we turn all the particles into antiparticles? These propositions, called respectively parity $P$ and charge conjugation $C$ conservation, are upheld by all the forces of nature except the weak nuclear force. And even the weak force usually conserves the compound proposition of $CP$. In only one small corner of physics---the decay of $K$ mesons---has $CP$ violation been observed, although physicists suspect that $CP$ violation must somehow operate on a large scale since it undoubtedly helped bring about the present-day preponderance of matter over antimatter.\\

$K$ mesons (kaons) are unstable and do not exist outside the interiors of neutron stars and particle accelerators, where $K$'s might be born courtesy of the strong nuclear force, but the rest of their short lives are under control of the weak force, which compels a sort of split personality: neither the $K$ nor $\bar{K}$ leads a life of its own. Instead each transforms repeatedly into the other. A more practical way of viewing the matter is to suppose that the $K$ and $\bar{K}$ are each a combination of two other particles, a short lived entity called $K_1$ which usually decays to two pions (giving $K_1$ a $CP$ value of +1) and a longer-lived entity, $K_2$, which decays into three pions (giving $K_2$ $CP$ value of -1). This bit of bookkeeping enshrined the idea then current that $CP$ is conserved.  All of this was overthrown when in 1964 the experiment of Jim Cronin and Val Fitch showed that a small fraction of the time (about one case in every 500, a fraction called epsilon) the $K_2$ turns into a $K_1$, which subsequently decays into two pions. This form of $CP$ violation is said to be indirect since the violation occurs in the way that $K$'s mix with each other and not in the way that $K$'s decay. One theoretical response was to say that this lone $CP$ indiscretion was not the work of the weak force but of some other novel ``superweak" force. Most theorists came to believe, however, that the weak force was responsible and, moreover, that $CP$ violation should manifest itself directly in the decay of $K$2 into two pions. The strength of this direct $CP$ violation, characterized by the parameter epsilon prime, would be far weaker than the indirect version. For twenty years detecting a nonzero value of epsilon prime has been the object of large-scale experiments at Fermilab and for nearly as long at CERN. In each case, beams of $K$'s are sent down long pipes in which the $K$-decay pions could be culled in sensitive detectors. In the quantum theory there are conservation laws corresponding to discrete transformations.\\

The work of Lee and Yang questioned the assumption in 1956[1], and the subsequent experiments by Wu et al. and Garwin et al. in 1957 independently demonstrated the violation of $P$ and $C$ invariance in weak decays in nuclei and of pions and muons[1][2]. This violation can be visualized by the longitudinal polarization of neutrinos emerging from a weak vertex: they are left-handed when they are particles and right-handed when antiparticles. Application of $P$ or $C$ to a neutrino leads to an unphysical state (Figure 1). The combined operation $CP$, however, transforms a left-handed neutrino into right-handed antineutrino, thus connecting two physics states. $CP$ invariance was therefore considered to be replacing the separate $P$ and $C$ invariance of weak interactions.\\ 

In 1964, Christenson, Cronin, Fitch and Turlay discovered the violation of the $CP$ symmetry in $K^0$ meson decays. There are two neutral strange mesons, $K^0$ and $\bar{K}^0$. The existence of these states results in a second property called strangeness oscillations: a pure strangeness eigenstate, say $K^0$, produced at a given time becomes at a later time a mixture of $K^0$ and $\bar{K}^0$. These states are mixed by the weak interaction, which does not conserve strangeness, to produce two states quite similar in their masses (which differ only by $\Delta m = 3.49\times 10^{-6}eV = 5.30\times 10^{9}s^{-1}$) but very dissimilar in their distinctive decay modes and their lifetimes. The two mesons $K^0$ and $\bar{K}^0$ are quite distinct in the presence of strong interactions which conserve strangeness. In other words $K^0$ is as distinct from $\bar{K}^0$ as the neutron is from the antineutron. Both $K^0$ and $\bar{K}^0$ can decay into pions via strangeness-violating weak transitions. Thus the transmutation of $K^0$ into $\bar{K}^0$, or inversely of $\bar{K}^0$ into $K^0$ can proceed through common intermediates states of pions as in $K^0\rightarrow (2\pi ,3\pi )\rightarrow\bar{K}^0$. The quark content of the $K^0$ is $\bar{s}d$, and $\bar{K}^0$ is $s\bar{d}$. With definition of $CP|K^0\rangle = -|\bar{K}^0\rangle$, and $CP|\bar{K}^0\rangle = -|K^0\rangle$ we can create $CP$ eigenstates of neutral $K$ mesons, labeled with 1 and 2 as linear combinations of $|K^0\rangle$ and $|\bar{K}^0\rangle$ as\\
\begin{equation}
|K_1\rangle = \frac{1}{\sqrt{2}}\left(|K^0\rangle - |\bar{K}^0\rangle\right),\hspace{10 mm} |K_2\rangle = \frac{1}{\sqrt{2}}\left(|K^0\rangle + |\bar{K}^0\rangle\right)
\end{equation}
where $CP$ eigenvalues of $|K_1\rangle$ and $|K_2\rangle$ are +1 and -1, respectively, that is $CP|K^0_1\rangle = +|K^1_0\rangle$, \\$CP|K^0_2\rangle = - |K^0_2\rangle$. To determine their decay modes into pions, it suffices to find the corresponding $CP$-parities of the multi pions states. It turns out that neutral two-pion states ($\pi^0+\pi^0$ and $\pi^++\pi^-$) are $CP$-even and neutral three-pion states ($\pi^0+\pi^0+\pi^0$ and $\pi^0+\pi^++\pi^-$) are $CP$-odd. Since $CP$ is assumed to be conserved, the only allowed decay modes are $K^0_1\rightarrow 2\pi$ and $K_2^0\rightarrow 3\pi$. Experimentally, what was observed was that the neutral $K$ mesons decayed in two different hadronic channels at two different time scales. The first type goes through two-pion channels, with lifetime $\tau_S = 8.92\times 10^{-11}$s, which is called $K_S$. The second type, which can decay into three pions with characteristic time $\tau_L = 5.17\times 10^{-8}$s is called $K_L$. Assuming $CP$ conservation, one may identify $K_S$ with the $CP$-even state $K_1^0$, and $K_L$ with the $CP-odd$ state $K_2^0$. This double property, mass generation and distinct lifetimes, is unique to the neutral $K$ mesons. It was also measured that $K_S$ decays into two-pions and $K_L$ decays into three pions. Since $CP$ eigenvalue of two pion state is +1 and that of three pions was -1 the common consideration, at that period was that the $K_S$ corresponds to the $K_1$, and the $K_L$ corresponds to the $K_2$, respectively. The branching fraction was order of $\sim$ $10^{-3}$. Even though it was small, it was an evidence for that  $K_S$ and $K_L$ are not ``real" $CP$ eigenstates and should be rewritten as:
\begin{equation}
|K_S\rangle = \frac{1}{\sqrt{1 + |\epsilon_m|^2}}\left(|K_1\rangle + \epsilon_m|K_2\rangle\right)
\end{equation}
\begin{equation}
|K_L\rangle = \frac{1}{\sqrt{1 + |\epsilon_m|^2}}\left(|K_2\rangle + \epsilon_m|K_1\rangle\right)
\end{equation}\\\\

\setlength{\unitlength}{1mm}
\hspace{50 mm}
\begin{picture}(0,0)
\thicklines
\put(20,-8){$\nu_L$}
\put(10,-10){\vector(1,0){20}}
\thinlines
\put(25,-15){\vector(-1,0){10}}
\put(35,-2){\line(0,-1){20}}
\put(35,0){$P$}
\thicklines
\put(60,-10){\vector(-1,0){20}}
\thinlines
\put(55,-15){\vector(-1,0){10}}
\put(50,-8){$\nu_R$}

\thicklines
\put(20,-38){$\nu_L$}
\put(10,-40){\vector(1,0){20}}
\thinlines
\put(25,-45){\vector(-1,0){10}}
\put(35,-32){\line(0,-1){20}}
\put(35,-30){$C$}
\thicklines
\put(40,-40){\vector(1,0){20}}
\thinlines
\put(55,-45){\vector(-1,0){10}}
\put(50,-38){$\bar{\nu}_L$}

\thicklines
\put(20,-68){$\nu_L$}
\put(10,-70){\vector(1,0){20}}
\thinlines
\put(25,-75){\vector(-1,0){10}}
\put(35,-62){\line(0,-1){20}}
\put(35,-60){$CP$}
\thicklines
\put(60,-70){\vector(-1,0){20}}
\thinlines
\put(55,-75){\vector(-1,0){10}}
\put(50,-68){$\bar{\nu}_R$}
\end{picture}\vspace{85 mm}\\\noindent
Figure 1: Mirror images of left-handed neutrino under P, C, and $CP$ symmetry. Long and short arrows represent momenta and spins of the neutrinos.\\\\\\\noindent


Subsequent observations of $K_L\rightarrow\pi^0\pi^0$ decay[3], and charge asymmetries in $K_L\rightarrow\pi^{\pm}e^{\mp}\nu$ and $K_L\rightarrow\pi^{\pm}\mu^{\mp}\nu$[4] confirmed the $CP$ violation in neutral $K$ meson system.\\

The theoretical explanation within the standard model for the origin of the $CP$ violation was proposed by Kobayashi and Maskawa in 1973[5].\\\\\\\noindent	%Here can be lessened
\textbf{II. $CP$ Violation in Standard Model}\\

In terms of mass eigenstates, a Lagrangian of charged-current weak interaction forms
\begin{equation}
-\mathit{L}_{cc} = -\frac{g}{\sqrt{2}}\bar{u}_L\gamma^\mu Vd_LW^+_\mu + [h.c.]
\end{equation}

In general, $\bar{u}_L$ and $d_L$ are an ``array" of mass eigenstates of quark-fermions and $V$ is a mixing matrix, which is unitary: $V^\dagger V = I$. When irreducible phases exist in the mixing matrix, they remain in the amplitude with different signs between $K^0\rightarrow \bar{K}^0$, and $\bar{K}^0\rightarrow K^0$ and then they enable $CP$ violation to occur. This is a key idea of the KM-mechanism to describe the $CP$ violation within the Standard Model. In this section, first we trace the historical development of weak interaction within the Standard Model. In the history, we see the KM-mechanism is a natural expansion of the theory at the period. In the successive paragraphs, we describe the ``unitarity triangles", which are derived from the unitarity condition of $V$. They are worth being described because the unitarity triangles are related to the $CP$ violation. In the description, we review the approximated parameterization of the matrix elements of $V$ to discuss the shapes of the unitarity triangles. After the discussion, we find more feasible $CP$ violation in the $B$ meson system rather than the $K$ meson system.\\

The first proposal of ``quark" model was proposed by Gell-Mann and Zweig in 1964 including three flavors of quarks, up ($u$), down ($d$), and strange ($s$)[6].Citing a paper[7] written by Cabibbo, Gell-Mann also proposed that weak interaction would couple the $u$-quark to a combined state of $d$ and $s$ as $|u^\prime\rangle = \cos\theta |d\rangle + \sin\theta |s\rangle$. With this definition, all (known) weak interactions could be described by a single coupling constant. In 1970, S. L. Glashow, J. Illiopoulos, and L. Maiani proposed a theory that a fourth flavor of quark, which they labeled ``charm'' ($c$), can explain why $K^0$ meson was not observed to decay to $\mu^+\mu^-$[2]. Figure 2(a) shows a diagram of $K^0$ decay to $\mu^+\mu^-$ whose amplitude is proportional to $\sin\theta_C\cos\theta_C$. To explain the small decay rate of $K^0\rightarrow\mu^+\mu^-$ process, they introduced a coupling of $c$ quark and $\mid s\rangle$ state defined as $\mid s^\prime\rangle = -\sin\theta_C\mid d\rangle + \cos\theta_C\mid s\rangle$ to ``cancel'' the amplitude. The canceling diagram shown in Figure 2(b) has proportional amplitude to $-\cos\theta_C\sin\theta_C$.\\\\

\begin{picture}(0,0)
\put(0,3){(a)}
\thicklines
\put(15,-2){$+\cos\theta_C$}
\put(-3,-5){\textit{d}}
\put(0,-5){\vector(1,0){10}}
\put(10,-5){\line(1,0){10}}
\put(20,-5){$\ldots\ldots\ldots\ldots$}
\put(30,-10){W}
\put(20,-5){\vector(0,-1){10}}
\put(20,-15){\line(0,-1){10}}
\put(20,-25){\vector(-1,0){10}}
\put(10,-25){\line(-1,0){10}}
\put(-3,-25){\textit{s}}
\put(16,-29){$+\sin\theta_C$}
\put(17,-16){$u$}
\put(20,-25){$\ldots\ldots\ldots\ldots$}
\put(30,-23){W}
\put(40,-5){\vector(1,0){10}}
\put(50,-5){\line(1,0){10}}
\put(62,-5){$\mu^-$}
\put(40,-5){\line(0,-1){10}}
\put(40,-25){\vector(0,1){10}}
\put(40,-25){\line(1,0){10}}
\put(60,-25){\vector(-1,0){10}}
\put(62,-25){$\mu^+$}
\put(42,-16){$\nu_\mu$}

\put(90,3){(b)}
\put(105,-2){$-\sin\theta_C$}
\put(87,-5){\textit{d}}
\put(90,-5){\vector(1,0){10}}
\put(100,-5){\line(1,0){10}}
\put(110,-5){$\ldots\ldots\ldots\ldots$}
\put(120,-10){W}
\put(110,-5){\vector(0,-1){10}}
\put(110,-15){\line(0,-1){10}}
\put(110,-25){\vector(-1,0){10}}
\put(100,-25){\line(-1,0){10}}
\put(87,-25){\textit{s}}
\put(106,-29){$+\cos\theta_C$}
\put(107,-16){$c$}
\put(110,-25){$\ldots\ldots\ldots\ldots$}
\put(120,-23){W}
\put(130,-5){\vector(1,0){10}}
\put(140,-5){\line(1,0){10}}
\put(152,-5){$\mu^-$}
\put(130,-5){\line(0,-1){10}}
\put(130,-25){\vector(0,1){10}}
\put(130,-25){\line(1,0){10}}
\put(150,-25){\vector(-1,0){10}}
\put(152,-25){$\mu^+$}
\put(132,-16){$\nu_\mu$}
\end{picture}

\vspace{35 mm}\noindent
Figure 2: Paths for the decay of $K^0\rightarrow\mu^+\mu^-$. The diagram (b) exchanges $c$ quark for $d$ quark is introduced in the GIM-mechanism.\\\\

Now, the combined states of $d$- and $s$-quarks are described by ``mixing matrix'' as 
\begin{equation}
\left(\begin{array}{c}
	d^\prime\\
	s^\prime\end{array}\right) = \left(\begin{array}{cc}
					\cos\theta_C & \sin\theta_C\\
					-\sin\theta_C & \cos\theta_C\end{array}\right) \left(\begin{array}{c}
	d\\
	s\end{array}\right)
\end{equation}
where $\theta_C$ is known as Cabibbo-angle. The angle is measured to be $\sin\theta_C \sim 0.22$.\\

In 1973, M. Kobayashi and T. Maskawa proposed that when at least three quark ``generations" exist, the $CP$ violation could be explained within the Standard Model [8]. It was the extension of a dimension of the quark-mixing matrix from two to three or more. A general $n\times n$ complex matrix, $V = \{V_{ij}\}$, has $2n^2$ free parameters. The $SU(n)$ symmetry of the mixing matrix requires $\sum_j V_{ij}V^\ast_{jk} = \delta_{ik}$, where $\delta_{ik}$ is Kronecker's $\delta$, and it yields $n$ constraints for $i$ = $k$ and $n^2 - n$ constraints for $i\neq k$. Thus, $n\times n$ unitary matrix has $n^2$ free parameters. The phases of the quark fields can be rotated freely. Since the overall phase is irrelevant, $2n - 1$ relative phase can be removed from $V$. Accordingly, $V$ has $(n - 1)^2$ free observables. On the other hand, a general $n\times n$ orthogonal matrix has $\frac{n(n - 1)}{2}$ rotational Euler angle. Thus, $\frac{(n - 1)(n - 2)}{2}$ parameters corresponding to irreducible ``phase" remain in $V$. To summarize, general $n\times n$ matrix representing mixing matrix consists of $\frac{n(n - 1)}{2}$ rotational Euler angles and $\frac{(n - 1)(n - 2)}{2}$ irreducible phases independentley. The least number for $n$ to generate irreducable phase in $V$ is three. The expanded mixing matrix from equation (10) was named Cabibbo-Kobayashi-Maskawa matrix (CKM matrix).\\

Let us start by studying some properties of the CKM matrix related to the $CP$ violation. Define $\Delta_{\gamma k} = V_{\alpha i}V_{\beta j}V_{\alpha j}^\ast V_{\beta i}^\ast$ where $(\alpha ,\beta ,\gamma ) = (1,2,3) = (u,c,t)$ or any other cyclic permutation, and similarly $(i,j,k) = (1,2,3) = (d,s,b)$.
\noindent
These nine complex numbers $\Delta_{\gamma k}$ have exaclty equal imaginary parts. Their common imaginary parts will be denoted as $J \equiv \Im (\Delta_{\gamma k}) = J\sum_{\gamma k}(\epsilon_{\alpha\beta\gamma}\epsilon_{ijk}).$ The term $J$ is a universal number in the sense that it does not depend on how the CKM matrix is to be parametrized. It stores this property with $|V_{\alpha i}|$, $J$ and $|V_{\alpha i}|$ are also invariant to the phase redefinition of the quark fields that define the matrix representation, and independant of the parametrization of the CKM matrix. Since any physical quantity that violates $CP$ symmetry is proportional to $J$, that quantity must equally possess these properties. It can be shown that the nine $\Delta_{\gamma k}$ have equal imaginary parts and there exists just one independant $\Delta_{\gamma k}$, the other eight being expressible in terms of it and of the magnitudes of the CKM matrix elements. The common imaginary part is given by: 
\begin{equation}
J = |c_{12}c_{23}c_{13}^2s_{12}s_{23}s_{13}\sin\delta |        
\end{equation}
or
\begin{equation}
J = A^2\lambda^6\eta
\end{equation}\\\noindent
where $\delta$ is between 0 and $2\pi$, $c_{ij} = \cos\theta_{ij}$, and $s_{ij} = \sin\theta_{ij}$ (all mixing angles are different from 0, $\frac{\pi}{2}$ and $\delta\neq 0,\pi$)\\

It can also be shown that $J$ is given by twice the area of any of the six triangles defined by the following unitarity relations (three of them are useful for understanding the Standard Model predictions for $CP$ violation):
\begin{equation}
V_{ud}V_{us}^\ast + V_{cd}V_{cs}^\ast + V_{td}V_{ts}^\ast = 0
\end{equation}
\begin{equation}
V_{us}V_{ub}^\ast + V_{cs}V_{cb}^\ast + V_{ts}V_{tb}^\ast = 0
\end{equation}
\begin{equation}
V_{ud}V_{ub}^\ast + V_{cd}V_{cb}^\ast + V_{td}V_{tb}^\ast = 0
\end{equation}

Each of these relations requires the sum of three complex quantities to 
vanish and so can be geometrically represented in the complex plane as a
triangle. These are ``the unitary triangles." Note that the term ``Unitary
Triangle" is reserved for the relation (3rd tiangle) only (for reasons to be understood).\\

It is instructive to draw the three triangles, knowing the experimental values (within errors) for
the various $|V_{ij}|$. This is done in figure 3. In the first two triangles one side is much shorter
than the other two and so they almost collapse tp a line. This would give an intuitive understanding of why $CP$ violation is so small in the leading K decays (the first triangle) and in leading $B_S$
decays (the second triangle). Decays related to the short side of these triangles ($K_L\rightarrow\pi\nu\bar{\nu}$)
are rare but could exhibit significant $CP$ violation. The most exiting physics of $CP$ violation lies
in the B system, related to the third triangles. The openness of this triangle predicts a large
CP assymetries in B decays.\\\\\noindent
\begin{figure}
\centering
\includegraphics[width = 150 mm, height = 200 mm]{figure3.ps}
\begin{picture}(0,0)

\put(-130,180){(a)}
\put(-140,170){$V_{td}V^\ast_{ts}$}
\put(-100,175){$V_{ud}V^\ast_{us}$}
\put(-100,162){$V_{cd}V^\ast_{cs}$}

\put(-130,153){(b)}
\put(-100,147){$V_{us}V^\ast_{cs}$}
\put(-140,143){$V_{ub}V^\ast_{cb}$}
\put(-100,132){$V_{ud}V^\ast_{cd}$}

\put(-125,123){(c)}
\put(-135,110){$V_{us}V^\ast_{ub}$}
\put(-105,110){$V_{cs}V^\ast_{cb}$}
\put(-115,95){$V_{ts}V^\ast_{tb}$}

\put(-60,115){(d)}
\put(-70,105){$V_{td}V^\ast_{cd}$}
\put(-40,110){$V_{ts}V^\ast_{cs}$}
\put(-40,95){$V_{tb}V^\ast_{cb}$}

\put(-120,75){(e)}
\put(-135,60){$V_{td}V^\ast_{ud}$}
\put(-100,65){$V_{ts}V^\ast_{us}$}
\put(-115,45){$V_{tb}V^\ast_{ub}$}

\put(-55,80){(f)}
\put(-70,65){$V_{ud}V^\ast_{ub}$}
\put(-40,70){$V_{td}V^\ast_{tb}$}
\put(-40,45){$V_{cd}V^\ast_{cb}$}
\put(-150,30){Figure 3: Six unitary triangles constructed by the unitary conditions from (13) through (15)}
\put(-150,25){In a phase reparametrization of the quark fields that build the CKM matrix, the triangles}
\put(-150,20){change their orientation in the plane, but their shape remains unaffected.} 
\end{picture}
\end{figure}

Equations (13-15) has striking implications for the unitary triangles:

1. All triangles are equal in area.

2. The area of each unitary triangle equals $\frac{|J|}{2}$

3. The sign of J gives the direction of the complex vectors.\\

Figures 3(a) through (f) show the unitary triangles constructed by equations (13) through (15)
where the lenghts and angles correspond to the complex phase that can cause assymmetry of the amplitudes between quark and antiquark transition. Therefore an observation of the $CP$ violation can lead to a measurement of the angles. According to the matrix elements of each side one can find the tiangle in figure 3(a) corresponds to the deacays in the neutral meson K system. Because two sides are of the order of $O(\lambda )$ and one side is of the order of $O(\lambda^5)$ this 
triangle is extremely squashed and thus it is hard to measure the side and the angles exept for the two long sides. This cuases relative smaller $CP$ violation in K meson system. Because of the same reason the triangle in figure 1(b) also has an extremely squashed shape. The angles of the triangles 3(e) and 3(f) are equally large since the length of all three sides are of the order $O(\lambda^3)$. The triangle of figure 3(f) relates to the B meson decay. Tus we can expect relative larger $CP$ violation in the B meson system than in the K system. The triangle of figure 3(e) relates to top quark decays. We can also image a large $CP$ violation in the t-quark system, however, the examination of the t-quark decays is far beyond our current experimental technique.  
\\
\begin{figure}
\centering
\includegraphics[width = 75 mm, height = 75 mm]{figure4.ps}
\begin{picture}(0,0)

\put(-65,13){O}
\put(-65,25){$\frac{V_{ud}V^\ast_{ub}}{V_{cd}V^\ast_{cb}}$}
\put(-55,19){$\phi_3$}

\put(-36,18){$\phi_1$}
\put(-25,12){(0,1)}
\put(-40,33){$\frac{V_{td}V_{tb}^\ast}{V_{cd}V_{cb}^\ast}$}

\put(-50,30){$\phi_2$}
\put(-53,40){$(\bar{\rho},\bar{\eta})$}

\put(-80,-2){Figure 4: Scaled triangle of Figure 3 (f) by $\frac{1}{V_{cd}V_{cb}^\ast}$}
\end{picture}
\end{figure}

The rescaled Unitary Triangle (Figure 4) is derived from (15) by (a) choosing a phase convection such that $(V_{cd}V_{cb}^\ast )$ is real, and (b) dividing the lengths of all sides by $|V_{cd}V_{cb}^\ast |$; (a) aligns one side of the triangle with the real axis, and (b) makes the length of this side 1. The form of the triangle is unchanged. Two verticies of the rescaled Unitary triangle are thus fixed at (0,0) and (1,0). The coordinates of the remaining vertex are denoted $(\rho,\eta )$. It is customary these days to express the CKM-matrix in terms of four Wolfenstein parameters ($\lambda ,A,\rho ,\eta $) with $\lambda = |V_{us}| = 0.22$ playing the role of the expansion parameter and $\eta$ representing the CP-violating phase[9]:
\begin{equation}
V = \left(\begin{array}{ccc}
1 - \frac{\lambda^2}{2} & \lambda & A\lambda^3(\rho i\eta (1 - \frac{\lambda^2}{2}))\\
-\lambda & 1 - \frac{\lambda^2}{2}-i\eta A^2\lambda^4 & A\lambda^2(1 + i\eta\lambda^2)\\
A\lambda^3(1 - \rho - i\eta) & -A\lambda^2 & 1\end{array}\right)
\end{equation}

The parameter $\eta$ represents the complex phase responsible for $CP$ violation. A, $\rho$, and $\eta$ can be extracted from data on B meson decays with the results A = 0.794$\pm$ 0.054 and $\sqrt{\rho^2 + \lambda^2} = 0.363\pm 0.073$. The necesary condition for $CP$ violation is that none of the nine matrix elements $V_{ij}$ is zero. We express all three factors found in it as:
\[\Im (V_{cd}^\ast V_{cs})^2 = -2J\]
\begin{equation}
\Im (V_{td}^\ast V_{ts})^2 = 2A^2\lambda^4(1-\rho )J
\end{equation}
\[\Im (V_{td}^\ast V_{ts}V_{cd}^\ast V_{cs}) = +J\] 
$\lambda$ is small for each element in V, the expansion parameter is usually $\lambda^2$. Hence it is sufficient to keep only the first few terms in this expansion.\\

The definition of $(\lambda ,A,\rho ,\eta )$ is useful because it allows an elegant improvement of the accuracy of the original Wolfenstein parametrization. In particular, up to O$(\lambda^6)$ corrections
\begin{equation}
V_{us} = \lambda , V_{cb} = A\lambda^2 , V_{ub} = A\lambda^3(\rho - i\eta ),
\end{equation}
\begin{equation}
V_{td} = A\lambda^3(1 - \bar{\rho} - i\bar{\eta}),
\end{equation}
\begin{equation}
\Im V_{cd} = -A^2\lambda^5\eta , \Im V_{ts} = -A\lambda^4\eta,
\end{equation}
where
\begin{equation}
\bar{\rho} = \rho (1 -\frac{\lambda^2}{2}), \bar{\eta}(1 - \frac{\lambda^2}{2})
\end{equation}

These are excellent approximations to the exact expressions[10]. Depicting the rescaled Unitary Triangle in the ($\bar{\rho},\bar{\eta}$) plane, the lengths of the two complex sides are:
\begin{equation}
R_b\equiv\sqrt{\bar{\rho}^2 + \bar{\eta}^2} = \frac{1-\frac{\lambda^2}{2}}{\lambda}\left|\frac{V_{ub}}{V_{cb}}\right|, R_t\equiv\sqrt{(1-\bar{\rho})^2 + \bar{\eta}^2} = \frac{1}{\lambda}\left|\frac{V_{td}}{V_{cb}}\right|.
\end{equation}
 
The three angles of the Unitary triangle denoted by $\phi_2 , \phi_1$ , and $\phi_3$ [11]:
\begin{equation}
\phi_2 = arg\left[-\frac{V_{td}V_{tb}^\ast}{V_{ud}V_{ub}^\ast}\right], \phi_1 = arg\left[\frac{-V_{cd}V_{cb}^\ast}{V_{td}V_{tb}^\ast}\right]
\end{equation}
The third angle is then
\begin{equation}
\phi_3 = \left[-\frac{V_{ud}V_{ub}^\ast}{V_{cd}V_{cb}^\ast}\right]\equiv \pi - \phi_2 -\phi_1
\end{equation}

These are physical quantities and, as discussed below, can be measured by $CP$ assymetries in various $\phi_1$ decays. The consitency of the various measurements provide tests of the Standard Model.\\

The angle $\phi_1$ gives, to a good approximation, the Standard Model phase between the neutral B mixing amplitude and its leading decay amplitudes. It is interresting to define the analog phases for $B_S$ meson $\phi_{1S}$, and the K meson, $\phi_{1K}$:
\begin{equation}
\phi_{1S}\equiv arg\left[-\frac{V_{ts}V_{tb}^\ast}{V_{cs}V_{cb}^\ast}\right], \phi_{1K}\equiv arg\left[-\frac{V_{cs}V_{cd}^\ast}{V_{us}V_{ud}^\ast}\right]
\end{equation}

The angles $\phi_{1S}$ and $\phi_{1K}$ can be seen to be small angles of the second and first unitary triangles (14) and (13), respectively.\\\\\noindent
It is straightforward to express the angles of the triangle in terms of $\bar{\rho}$ and $\bar{\eta}$. For example the following two relations are useful:
\begin{equation}
\sin 2\phi_2 = \frac{2\bar{\eta}[\bar{\eta}^2 + \bar{\rho}(\bar{\rho}  1)]}{[\bar{\eta}^2 + (1 - \rho )^2][\bar{\eta}^2 + \bar{\rho}^2]}, \sin 2\phi_1 = \frac{2\bar{\eta}(1-\bar{\rho})}{\bar{\eta}^2 + (1 - \bar{\rho})^2}
\end{equation}

Note that the unitarity is a fundamental property of any field theory. When one speaks of testing the unitarity of the CKM matrix one is not only looking for violations of unitarity, but for violations of the consequences of unitarity in the three generation theory. Such violations would simply imply the presence of other channels, particles not included in the Standard Model theory, contributing in some ways to the decay under study. To call these effects ``unitary violations" is perhaps misleading, but it is a common terminology of the field.\\\\\noindent
\textbf{III. $CP$ Violation in B Decays}\\

In 1980, A. Carter, I. I. Bigi, and A. I. Sanda pointed out that the KM-mechanism indicates the possibility of sizable $CP$ violation in the $B$ meson system [12]. When neutral mesons, namely $B$ meson, mix with their antiparticles, In the $B$ meson system, sizable $CP$ asymmetries are expected in the interference between decays into a same final state with and without the $B^0-\bar{B}^0$ mixing. The $CP$ asymmetry is observed in the difference between the time-dependent decay rates of $B^0$ and $\bar{B}^0$ me sons into a common $CP$ eigenstate. In this section, we first explain the phenomenology of time evolution of neutral $B$ mesons. Then we consider the case that both $B^0$ and $\bar{B}^0$ decay into the same $CP$ eigenstate and the $CP$ violation in these decays. Finally, we relate the $CP$ violation in $B^0\rightarrow J/\psi K_S$, of which final state is the $CP$ eigenstate.\\\\\\\\\noindent
\textbf{1. $B^0 - \bar{B}^0$ Mixing}\\

Once the neutral $B$ mesons are produced in pairs, their semileptonic decays (inclusive or exclusive) provide an execllent method to measure the $B^0-\bar{B}^0$ mixing. $B^0$ and $\bar{B}^0$ can mix through second order weak interactions via diagrams shown in Figure 5 known as ``box diagrams". In this subsection, we see the time-evolution of $B^0$ and $\bar{B}^0$. Due to the mixing, an arbitrary neutral $B$ meson state is written as the admixture of $B^0$ and $\bar{B}^0$
\begin{equation}
|\Phi (t)\rangle = |B^0(t)\rangle + |\bar{B}^0(t)\rangle
\end{equation}
where $CP|B^0\rangle = -|\bar{B}^0\rangle$. The time-dependent Schr$\ddot{o}$dinger equation of $|\Phi (t)\rangle$ is
\begin{equation}
i\frac{\partial|\Phi (t)\rangle}{\partial t} = \textit{H}|\Phi (t)\rangle
\end{equation}
\textit{H} is an Hamiltonian defined as 
\begin{equation}
\textit{H} \equiv \textit{M} - i\Gamma \equiv \left(\begin{array}{cc}
					M_{11} - \frac{i}{2}\Gamma_{11} & M_{12} - \frac{i}{2}\Gamma_{12}\\
M_{21} - \frac{i}{2}\Gamma_{21} & M_{22} - \frac{i}{2}\Gamma_{22}\end{array}\right)
\end{equation}

Using $\textit{H}^\dagger = \textit{H}$, eigenvalues $\mu_{\pm}$, and eigenvectors $\mid B_{\pm}\rangle$ of above equation can be given as
\begin{equation}
\mu_{\pm} = M_0 - \frac{i}{2}\Gamma_0 \pm\sqrt{\left( M_{12} - \frac{i}{2}\Gamma_{12}\right)\left( M_{12}^\ast - \frac{i}{2}\Gamma_{12}^\ast\right)}
\end{equation}
and 
\begin{equation}
|B_{\pm}\rangle = \frac{1}{\sqrt{|p|^2 + |q|^2}}(p|B^0\rangle\pm q|\bar{B}^0\rangle )
\end{equation}\\\\\noindent
\begin{equation}
\left(\frac{q}{p}\right)_B = \sqrt{\frac{M_{12}^\ast - \frac{i}{2}\Gamma_{12}^\ast}{M_{12} - \frac{i}{2}\Gamma_{12}}}
\end{equation}\\


\begin{figure}
\includegraphics[width = 175 mm, height = 40 mm]{figure5.ps}
\begin{picture}(0,0)

\put(0,13){\textit{d}}
\put(0,38){\textit{b}}
\put(45,16){W}
\put(45,40){W}
\put(65,27){\textit{t}}
\put(30,27){\textit{t}}
\put(83,38){\textit{d}}
\put(83,13){\textit{b}}

\put(90,13){\textit{d}}
\put(90,38){\textit{b}}
\put(130,16){\textit{t}}
\put(130,40){\textit{t}}
\put(145,28){W}
\put(115,28){W}
\put(170,38){\textit{d}}
\put(170,13){\textit{b}}
\put(40,0){Figure 5: Two diagrams contributing for $B^0-\bar{B}^0$ mixing.} 
\end{picture}
\end{figure}

We consider $p$ and $q$ are normalized as $|p|^2 + |q|^2$ = 1. Here the lifetimes and masses of $B^0$ and $\bar{B}^0$ are assumed to be equal because of the $CPT$ theorem, i.e. $M_0 \equiv M_{11} = M_{22}$, and $\Gamma_0 \equiv \Gamma_{11} = \Gamma_{22}$. Equation (30) indicates that the general mass eigenstates are deviated from the eigenstates of the weak interaction.\\\\\noindent
We redefine mass eigenvectors to fit to usual convention, 
\begin{equation}
|B_H\rangle = p|B^0\rangle - q|\bar{B}^0\rangle
\end{equation}
\begin{equation}
|B_L\rangle = p|B^0\rangle + q|\bar{B}^0\rangle
\end{equation}
where corresponding eigenvalues are $\mu_H \equiv \mu_-$ and $\mu_L \equiv \mu_+$, respectively. Defining mass and width of $B_{H,L}$ as $M_{H,L}$ and $\Gamma_{H,L}$, the time evolutions of $|B_H\rangle$ and $|B_L\rangle$ are given by 
\begin{equation}
|B_H(t)\rangle = e^{-i\mu_Ht}|B_H\rangle \equiv e^{-iM_Ht}e^{-\frac{\Gamma_Ht}{2}}|B_H\rangle
\end{equation}
\begin{equation}
|B_L(t)\rangle = e^{-i\mu_Lt}|B_L\rangle \equiv e^{-iM_Lt}e^{-\frac{\Gamma_Lt}{2}}|B_L\rangle
\end{equation}

Next, we transform the base from the mass eigenstates in equations (35) and (36) to the weak interaction eigenstates. First, suppose a neutral $B$ meson being $|B^0\rangle$ at time $t = 0$.
The wave function of the particle can be expressed with equations (31), (33), and (34) as $|B^0\rangle = \frac{(|B_H\rangle + |B_L\rangle )}{2p}$ and therefore, the time-evolution of $|B^0\rangle$ is written using equations from (33) through (36) as
\begin{equation}
|B^0(t)\rangle = \frac{1}{2}\left[\left(e^{-i\mu_Ht} + e^{-i\mu_Lt}\right)|B^0\rangle - \frac{q}{p}\left(e^{-i\mu_Ht} - e^{-i\mu_Lt}\right)|\bar{B}^0\rangle\right]
\end{equation}

In the same manner, the time-evolution of $|\bar{B}^0(t)\rangle$ is written as 
\begin{equation}
|\bar{B}^0(t)\rangle = -\frac{1}{2}\left[\frac{p}{q}\left(e^{-i\mu_Ht} + e^{-i\mu_Lt}\right)|B^0\rangle - \left(e^{-i\mu_Ht} - e^{-i\mu_Lt}\right)|\bar{B}^0\rangle\right]
\end{equation}\\\\\\\noindent
\textit{\textbf{2. $B$ Meson Decay into $CP$ Eigenstate}}\\

In this subsection, we see the time evolution of $B$ meson decay into $CP$ eigenstate. We start from the calculation of four amplitudes of generic decay final state for $B^0\rightarrow f$ and its $CP$ conjugation modes: $\bar{B}^0\rightarrow f, B^0\rightarrow \bar{f}$, and $\bar{B}^0\rightarrow \bar{f}$. Then we set $f = \bar{f} = f_{CP}$. After calculating the decay rates, we discuss the difference of them, and finally we see that the difference is induced by the $B^0-\bar{B}^0$ mixing.\\

We define instananeous decay amplitudes of $B^0$ and $\bar{B}^0$ to $f$ and $\bar{f}$ as
\[a \equiv A(B^0\rightarrow f) = \langle f|H_w|B^0\rangle, a^\prime \equiv A(\bar{B}^0\rightarrow f) = \langle f|H_w|B^0\rangle\]
\begin{equation}
b \equiv A(\bar{B}^0\rightarrow f) = \langle \bar{f}|H_w|\bar{B}^0\rangle, b^\prime \equiv A(B^0\rightarrow\bar{f}) = \langle \bar{f}|H_w|B^0\rangle
\end{equation}
where $H_w$ is a weak-decay Hamiltonian.\\

Let $A_{B\rightarrow f}(t)$ be the time-dependent decay amplitude for a pure $B^0$ state at $t = 0$ to decay into a final state $f$ at time $t$. It can be obtained by the replacement of $|B^0\rangle$ and $|\bar{B}^0\rangle$ in equation (37) by $a$ and $a^\prime$, respectively. Similarly, $A_{\bar{B}\rightarrow f}(t)$can be obtained by replacing $|B^0\rangle$ and $|\bar{B}^0\rangle$ in equation (38) by $b^\prime$ and $b$, respectively. The amplitudes can be written as
\begin{equation}
A_{B\rightarrow f}(t) = \frac{a}{2}\left[\left(e^{-i\mu_Ht} + e^{-i\mu_Lt}\right) - \alpha\left(e^{-i\mu_Ht} - e^{-i\mu_Lt}\right)\right]
\end{equation}
\begin{equation}
A_{\bar{B}\rightarrow \bar{f}(t)} = \frac{a}{2}\left[\left(e^{-i\mu_Ht} + e^{-i\mu_Lt}\right) - \beta\left(e^{-i\mu_Ht} - e^{-i\mu_Lt}\right)\right]
\end{equation}
where
\begin{equation}
\alpha \equiv \frac{qa^\prime}{pa},\hspace{10 mm} \beta \equiv \frac{pb^\prime}{qb}
\end{equation}

The amplitudes for the $CP$ conjugation modes, $A_{\bar{B}\rightarrow f}(t)$ and $A_{B\rightarrow\bar{f}}(t)$, are similarly expressed as
\begin{equation}
A_{\bar{B}\rightarrow f}(t) = -\frac{a}{2}\frac{p}{q}\left[\left(e^{-i\mu_Ht} + e^{-i\mu_Lt}\right) - \alpha\left(e^{-i\mu_Ht} - e^{-i\mu_Lt}\right)\right]
\end{equation}
\begin{equation}
A_{B\rightarrow \bar{f}}(t) = -\frac{a}{2}\frac{q}{p}\left[\left(e^{-i\mu_Ht} + e^{-i\mu_Lt}\right) - \alpha\left(e^{-i\mu_Ht} - e^{-i\mu_Lt}\right)\right]
\end{equation}
Here, we introduce two new quantities: mass difference, $\Delta M \equiv M_H - M_L$, and width difference, $\Delta\Gamma \equiv \Gamma_H - \Gamma_L$. Consequently, we obtain
\begin{equation}
\mu_H - \mu_L = \Delta M + i\Delta\Gamma
\end{equation}

We also define the average width of $B_H$ and $B_L$ as
\begin{equation}
\Gamma \equiv \frac{\Gamma_H + \Gamma_L}{2}
\end{equation}

Now we set $f = \bar{f} = f_{CP}$. Squaring the amplitudes of equations from (40) and (41) the corresponding decay distributions are obtained:
\[\Gamma_{B\rightarrow f_{CP}}(t) = \frac{|a|^2}{2}e^{-\Gamma t}[\left(1 + |\lambda_{CP}|^2\right)\cosh\frac{\Delta\Gamma}{2}t + 2\Re (\lambda_{CP})\sinh\frac{\Delta\Gamma}{2}t\]
\begin{equation}
+ (1 - |\lambda_{CP}|^2)\cos\Delta Mt - 2\Im (\lambda_{CP})\sin\Delta Mt]
\end{equation}
\[\Gamma_{\bar{B}\rightarrow f_{CP}}(t) = \frac{|a^\prime|^2}{2}e^{-\Gamma t}[\left(1 + |\lambda_{CP}|^{-2}\right)\cosh\frac{\Delta\Gamma}{2}t + 2\Re (\lambda_{CP}^{-1})\sinh\frac{\Delta\Gamma}{2}t\]
\begin{equation}
+ (1 - |\lambda_{CP}|^{-2})\cos\Delta Mt - 2\Im (\lambda_{CP}^{-1})\sin\Delta Mt]
\end{equation}\\\noindent
We used $a = b^\prime$, $a^\prime = b$, and $\alpha = \beta^{-1}$ derived from equation (32), and we defined 
\begin{equation}
\lambda_{CP} \equiv \alpha
\end{equation}

Squares of equations (43) and (44) yield same result as equations (47) and (48). In the $B^0$ meson system, $\frac{\Delta\Gamma}{\Gamma} ~ O(10^{-3})$ can be explicitly derived from the Standard Model, and therefore, the relation $e^{(\Gamma + \frac{\Delta\Gamma}{2})t} \simeq e^{\Gamma t}$ holds. Consequent approximations of equations (47) and (48) are:
\[\Gamma_{B\rightarrow f_{CP}}(t) \simeq \frac{|a|^2}{2}e^{-\Gamma t}[\left(1 + |\lambda_{CP}|^2\right) + (1 - \lambda_{CP}^2)\cos\Delta Mt\]
\begin{equation}
- 2\Im (\lambda_{CP})\sin\Delta Mt]
\end{equation}
\[\Gamma_{\bar{B}\rightarrow f_{CP}}(t) \simeq \frac{|a^\prime|^2}{2}e^{-\Gamma t}[\left(1 + |\lambda_{CP}|^{-2}\right) + (1 - \lambda_{CP}^{-2})\cos\Delta Mt\]
\begin{equation}
- 2\Im (\lambda_{CP}^{-1})\sin\Delta Mt]
\end{equation}

The $CP$ invariance is violated when the time-dependent decay rate of $B^0\rightarrow f_{CP}$ and its $CP$ conjugation decay are different for any possible $t$:
\begin{equation}
\Gamma_{B\rightarrow f_{CP}}(t) \neq \Gamma_{\bar{B}\rightarrow f_{CP}}(t)
\end{equation}

A time-dependent asymmetry $a_{CP}(t)$ is defined as the normalized decay rate difference:
\[a_{CP}(t) \equiv \frac{\Gamma_{\bar{B}\rightarrow f_{CP}}(t) - \Gamma_{B\rightarrow f_{CP}}(t)}{\Gamma_{\bar{B}\rightarrow f_{CP}}(t) + \Gamma_{B\rightarrow f_{CP}}(t)}\]
\begin{equation}
= \frac{(|\lambda_{CP}|^2 - 1)\cos\Delta Mt + 2\Im(\lambda_{CP})\sin\Delta Mt}{1 + |\lambda|^2} 
\end{equation}\\\\\noindent
\textbf{\textit{3- $CP$ Violation in }}$B^0\rightarrow J/\psi K_S$\\

Among all $B^0\rightarrow$charmonium $+ K_S$ channels, $B^0\rightarrow J/\psi K_S$, where $K_S$ decays into two charged pions, is the most promising decay mode to extract $\phi_1$ experimentally because of relatively large branching fractions with small backgrounds. In addition, as we see later in this subsection, the final state has only negligible theoretical uncertainty in $\phi_1$ measurement. These are the reasons why this mode has earned the name of ``golden mode" (GM). We use $\lambda_{J/\psi K_S}$ instead of $\lambda_{CP}$ to indicate the decay final state being $J/\psi K_S$ explicitly, hereafter.\\\\\noindent 
$\lambda_{J/\psi K_S}$ can be expressed as 
\begin{equation}
\lambda_{J/\psi K_S} = \frac{A(B^0\rightarrow J/\psi K_S)}{A(\bar{B}^0\rightarrow J/\psi K_S)}\frac{q}{p}
\end{equation}\\

\begin{figure}
\includegraphics[width = 180 mm, height = 45 mm]{figure6.ps}
\begin{picture}(0,0)

\put(0,15){\textit{d}}
\put(0,40){\textit{b}}
\put(40,30){$W^-$}

\put(83,14){\textit{d}}
\put(83,40){\textit{c}}
\put(83,35){\textit{c}}
\put(83,20){\textit{s}}

\put(95,12){\textit{d}}
\put(95,22){\textit{b}}
\put(107,17){$W^-$}
\put(115,30){\textit{t}}

\put(172,14){\textit{d}}
\put(172,40){\textit{c}}
\put(172,30){\textit{c}}
\put(172,21){\textit{s}}
\put(5,50){(a)}
\put(100,50){(b)}
\put(0,5){Figure 6: Tree (a) and penguin (b) diagrams for $B^0\rightarrow J/\psi K_S$. The penguin contamination}
\put(0,0){is negligable for $\sin 2\phi_1$ measurement}
\end{picture}
\end{figure}
\noindent In the following discussion, first we compute $\frac{A(B^0\rightarrow J/\psi K_S)}{A(\bar{B}^0\rightarrow J/\psi K_S)}$, then we compute $\frac{q}{p}$.\\

The decay of $B^0\rightarrow J/\psi K_S$ is based on the quark transition $b\rightarrow c\bar{c}s$. Contributing Feynman diagrams to $b\rightarrow c\bar{c}s$ decays are shown in figure 6 where the left diagram is called ``tree diagram" and the right diagram holding one loop is called ``strong (penguin) diagram". In the case of electroweak penguin contribution, the gluons are replaced by a $Z$ or a $\gamma$. The amplitude of the tree diagram is\\
\begin{equation}
\langle J/\psi K^0|H_t|B^0\rangle = V^\ast_{cb}V_{cs}A^t
\end{equation}    \\\noindent
and the amplitude of the penguin diagram is
\begin{equation}\\
\langle J/\psi K_S|H_p|B^0\rangle = V^\ast_{ub}V_{us}A^p_u + V^\ast_{cb}V_{cs}A^p_c + V^\ast_{tb}V_{ts}A^p_t
\end{equation}    \\\noindent
Here, $A^t$ and $A^p_i (i = u,c,t)$ are the amplitudes apart from the explicitly shown CKM factors. Because $\sum_{i=u,c,t}V_{ib}^\ast V_{is}$ is zero due to the unitarity condition, following equation holds:\\
\begin{equation}
\langle J/\psi K_S|H_p|B^0\rangle = V^\ast_{cb}V_{cs}(A^p_c - A_t^p) + V^\ast_{ub}V_{us}(A^p_u - A^p_t)
\end{equation}\\

According to the Wolfenstein parameterization, the first term has the same weak phase as the tree diagram and the second term has different weak phase from the tree diagram.
We can also read from the parameterization that $\frac{(V_{ub}^\ast V_{us})}{(V_{cb}^\ast V_{cs})}$ is order of $O(\lambda^2)$. Thus, the second term is negligible with respect to the first term. Therefore, the penguin diagram possesses the same weak phase as the tree diagram up to very small correction. When there is only one amplitude (or more than but with the same weak phase), $|B^0\rightarrow f_{CP}| = |\bar{B}^0\rightarrow f_{CP}|$ holds. Thus, we can conclude
\begin{equation}
|A(B^0\rightarrow J/\psi K_S)| = |A(\bar{B}^0\rightarrow J/\psi K_S)|
\end{equation}
and
\begin{equation}
\frac{A(B^0\rightarrow J/\psi K_S)}{A(\bar{B}^0\rightarrow J/\psi K_S)} = \frac{(V_{cb}V_{cs}^\ast)}{(V_{cb}^\ast V_{cs})}
\end{equation}\\

Then we calculate $\frac{q}{p}$. The dominating quark in the internal loop of the mixing diagrams is $t$ because of its heaviest mass. Neglecting the contribution from $u$- and $c$-quarks, the $M_{12}$ is computed as[13]:
\begin{equation}
M_{12} \simeq \frac{G^2_Fm^2_W}{4\pi^2}\cdot\eta^{tt}_{QCD}\cdot (V_{td}^\ast V_{tb})^2\cdot\left(\frac{m_t}{m_W}\right)^2\cdot\langle B^0|(\bar{d}_L\gamma_\mu b_L)^2|\bar{B}^0\rangle
\end{equation}
On the other hand, the $\Gamma_{12}$ is computed as:
\begin{equation}
\Gamma_{12} \simeq \frac{3G^2_Fm^2_W}{8\pi}\cdot (V_{td}^\ast V_{tb})^2\cdot\left(\frac{m_b}{m_W}\right)^2\cdot\langle B^0|(\bar{d}_L\gamma_\mu b_L)^2|\bar{B}^0\rangle
\end{equation}
From above equations, we have
\begin{equation}
\left|\frac{\Gamma_{12}}{M_{12}}\right| \simeq \frac{3\pi m^2_b}{2m_t^2} \sim O(10^{-2})
\end{equation}
Using equations (32), (60), and (62), $\frac{q}{p}$ is approximated as
\begin{equation}
\frac{q}{p} \simeq \sqrt{\frac{M_{12}^\ast}{M_{12}}} = \frac{V_{tb}^\ast V_{td}}{V_{tb}V_{td}^\ast} = e^{-2i\phi_1}
\end{equation}

We have to be aware of $K^0 - \bar{K}^0$ mixing because we have $K_S$ in the decay final state.
Similar discussion as equation (33) is available for $K^0 - \bar{K}^0$ mixing:
\begin{equation}
\langle K_S| = p_K^\ast \langle K^0| - q_K^\ast \langle\bar{K}^0|
\end{equation}
Writing the transition of $K^0\rightarrow \bar{K}^0$ explicitly, the amplitudes for $B^0\rightarrow J/\psi K_S$ are expressed as
\begin{equation}
\langle J/\psi K_S|B^0\rangle = \langle K_S|K^0\rangle\langle J/\psi K^0|B^0\rangle = p_K^\ast V_{cb}^\ast V_{cs}A
\end{equation}
\begin{equation}
\langle J/\psi K_S|\bar{B}^0\rangle = p_K^\ast V_{cb}V_{cs}^ast A
\end{equation}
Here, we used the facts of $CP|B^0\rangle = -|\bar{B}^0\rangle$, $CP|K^0\rangle = -|\bar{K}^0\rangle$, $CP|J/\psi\rangle = |J/\psi\rangle$, and $CP|J/\psi K^0\rangle = -|J/\psi\bar{K}^0\rangle$, where the third equation is derived from $J^{CP} = 1^{--}$ for $J/\psi$ and the last equation is from angular momentum of the system being 1. Therefore, 
\[\langle J/\psi K_S|\bar{B}^0\rangle = -q_K^\ast\langle J/\psi \bar{K}^0|(CP)^{-1}(CP)H(CP)^{-1}(CP)|\bar{B}^0\rangle\]
\begin{equation}
	= q^\ast_K\langle J/\psi K^0|H^{CP}|B^0\rangle
\end{equation}
$(\frac{q}{p})_K$ is obtained from the calculation of $K^0-\bar{K}^0$ mixing diagrams.
In this case, using $|V_{cd}^\ast V_{cs}| \gg |V_{td}^\ast V_{ts}|$, we obtain the additional factor to $\lambda_{J/\psi K_S}$ as
\begin{equation}
\left(\frac{q}{p}\right)_K \simeq \frac{V_{cs}V_{cd}^\ast}{V_{cs}^\ast V_{cd}}
\end{equation}
Finally, we obtain
\[\lambda_{J/\psi K_S} = \frac{A(\bar{B}^0\rightarrow J/\psi K_S)}{A(B^0\rightarrow J/\psi K_S)}\cdot\left(\frac{q}{p}\right)\cdot\left(\frac{q}{p}\right)_k\]
\[\simeq -\frac{V_{cb}V_{cs}^\ast}{V_{cb}^\ast V_{cs}}\cdot \frac{V_{tb}^\ast V_{td}}{V_{tb}V_{td}^\ast}\cdot\frac{V_{cs}V_{cd}^\ast}{V_{cs}^\ast V_{cd}}\]
\begin{equation}
= -e^{-2i\phi_1}
\end{equation}
and 
\begin{equation}
\Im (\lambda_{J/\psi K_S}) = \sin 2\phi_1
\end{equation}
The minus sign in front of equation (69) is due to the state of $|J/\psi K_S\rangle$ being $CP$ odd. Because of $|B^0\rightarrow J/\psi K_S| = |\bar{B}^0\rightarrow J/\psi K_S|$  and $|\frac{q}{p}| = 1$, $|\lambda_{J/\psi K_S}| = 1$ holds. As a result, equations (50), (51) and (53) become much simpler:
\begin{equation}
\Gamma_{B\rightarrow f_{CP}}(t) = |a|^2e^{-\Gamma t}(1 - \sin 2\phi_1\cdot\sin\Delta Mt)
\end{equation}
\begin{equation}
\Gamma_{\bar{B}\rightarrow f_{CP}}(t) = |a|^2e^{-\Gamma t}(1 + \sin 2\phi_1\cdot\sin\Delta Mt)
\end{equation}
\begin{equation}
a_{CP}(t) = \sin 2\phi_1\cdot\sin\Delta Mt
\end{equation}\\

Finally, consider neutral B decays into final $CP$ eigenstates  $f_{CP}$[21,22,23]. Such states are accessible in both $B^0$ and $\bar{B}^0$ decays. The quantity of interest here that is independant of phase conventions and physically meaningful $\lambda$ of (49) $\lambda = \eta_{f_{CP}}\frac{q}{p}\frac{\bar{A}_{\bar{f}_{CP}}}{A_{f_{CP}}}$. When $CP$ is conserved $|\frac{q}{p}|=1$ and $|\frac{\bar{A}_{\bar{f}_{CP}}}{A_{f_{CP}}}|=1$ furthermore the relative phase between $(\frac{q}{p})$  and $(\frac{\bar{A}_{\bar{f}_{CP}}}{A_{f_{CP}}})$ vanishes therfore (49) implies
\begin{equation}
\lambda\neq\pm 1 \Rightarrow \mbox{ $CP$ violation }
\end{equation}\vspace{2 mm}

Note that both $CP$ violation in decay (54) and $CP$ violation (59) lead to (63) through $|\lambda |\neq 1 $. However, it is possible that to a good approximation $|\frac{q}{p}|=1$ and $|\frac{\bar{A}}{A}|=1$ yet there is a $CP$ violation
\begin{equation}
|\lambda |=1, \Im\lambda\neq 0
\end{equation}\vspace{2 mm}

This type of $CP$ violation is called $CP$ violation in the interference between decays with and without mixing; sometimes this is abbreviated as ``interference between mixing and decay". This type of $CP$ violation has also been obsserved in the neutral Kaon system.\\

For the neutral B system, $CP$ violation in the interference between decays with and without mixing can be observed by comparing decays into the final $CP$ eigenstates of a time-evolving neutral B state that begins at time zero as $B^0$ to those of the state that begin as a $\bar{B}^0$:
\begin{equation}
a_{f_{CP}} = \frac{\Gamma (B^0_{phys}(t)\rightarrow f_{CP})-\Gamma (\bar{B}^0_{phys}(t)\rightarrow f_{CP})}{\Gamma (B^0_{phys}(t)\rightarrow f_{CP})+\Gamma (\bar{B}^0_{phys}(t)\rightarrow f_{CP})}
\end{equation}
It was shown above (59) that this time dependant symmetry is given by:
\begin{equation}
a_{f_{CP}} = \frac{(1-|\lambda_{f_{CP}}|^2)\cos (\Delta m_Bt) - 2\Im\lambda_{f_{CP}}\sin (\Delta m_Bt)}{1 + |\lambda_{f_{CP}}|^2}
\end{equation}

This asymmetry will be non-vanishing if any of the three types of $CP$ violations are present. However, for decays such that $|\lambda | = 1$ (the 'clean' modes), (77) simplifies considerably:
\begin{equation}
a_{f_{CP}} = -\Im\lambda_{f_{CP}}\sin (\Delta m_Bt)
\end{equation}
One point concerning this type of asymetries is worth clarifying. Consider the decay amplitudes of $B^0$ into two different final $CP$ eigenstates $A_a$ and $A_b$. A non-vanishing difference between $\eta_aA_a$ and $\eta_bA_b$,
\begin{equation}
\eta_aA_a - \eta_bA_b = \frac{q}{p}\left(\frac{\bar{A}_{\bar{a}}}{A_a} - \frac{\bar{A}_{\bar{b}}}{A_b}\right)\neq 0
\end{equation}   
would establish the existance of $CP$ violation in $\Delta b =1$ processes. For this reason, this type of $CP$ violation is also called sometimes ``direct $CP$ violation". Yet unlike the case of $CP$ violation in decay, no non-trivial strong phases are necessary. The richness of possible final $CP$ eigenstates in B decays makes it very likely that various asymetries will exhibit (79). (A measurement of $\phi_1 (K_L\rightarrow\pi\nu\bar{\nu}) \ge 10^{-2}$ can establish the existence [24,25,26] of a similar effect as $\Delta s = 1$ $CP$ violation that does not depend on strong phase shifts.) Either this type of observation or the observation of $CP$ violation in decay would rule out super weak models for $CP$ violation.\\

$CP$ violation in the interference between decays with and without mixing can be cleanly related to Lagrangian parameters when it occurs with no $CP$ violation in decay. In particular for $B_d$ decays that are dominated by a single $CP$ violating phase so that the effect of $CP$ violation in decay is negligible $A_{f_{CP}}$ is cleanly translated into a value for $\Im\lambda$ (see (78)) which in turn is cleanly interpreted in terms of purely electroweak Lagrangian parameters. (As discussed below $\Im\epsilon_K$ which describes $CP$ violation in the interference between decays with and without mixing in the K system is cleanly translated into a value of $\phi_{12}$ the phase between $M_{12}(K)$ and $\Gamma_{12}(K)$. It is difficult however to interpret $\phi_{12}$ cleanly in terms of electroweak Lagrangian parameters.)\\

When there is $CP$ violation in decay at the same time as in the interference between decays with and without mixing, the symmetry (76) depends also on the ratio of the different amplitudes and their relative strong phases, and thus the prediction had hadronic uncertainties. In some cases, however, it is possible to remove any large hadronic uncertainties by measuring several isospin-related rates (see e.g. [27,28,29]) and therby extract a clean measurement of CKM phases.\\

There are so many final states for B decay that have $CP$ self-conjugate particle content but are not $CP$ eigenstates because they contain a mixtures of different angular momenta and hence different parities. In certain angular analyses of the final state can be used to determine the amplitudes for each different $CP$ contribution separately. Such final states can then also be used for clean comparison with theoretical models [30].\\

As mentioned above, in the $B_d$ system the result $\Gamma_{12} \ll M_{12}$ is model independent. Moreover, within the Standard Model and assuming that the box diagram (with a cut) is appropriate to estimate $\Gamma_{12}$, one can actually calculate the two quantities from the quark diagrams of Figure (6). The calculation gives [31]
\begin{equation}
\frac{\Gamma_{12}}{M_{12}} = -\frac{3\pi}{2}\frac{1}{f_2(\frac{m_t^2}{m_W^2})}\frac{m_b^2}{m_t^2}\left( 1+ \frac{8}{3}\frac{m_c^2}{m_b^2}\frac{V_{cb}V_{cd}^\ast}{V_{tb}V_{td}^\ast}\right)
\end{equation}
This confirms the order of magnitude estimate, $|\frac{\Gamma_{12}}{M_{12}}| \le 10^{-2}$. The derivation of $|\frac{q}{p}|$
from unity is proportional to $\Im (\frac{\Gamma_{12}}{M_{12}})$ which is even suppressed by another order of magnitude. Thus to a very good approximation
\begin{equation}
\frac{q}{p} = -\frac{M_{12}^\ast}{|M_{12}|} = \frac{V_{tb}^\ast V_{td}}{V_{tb}V_{td}^\ast}e^{2i\zeta B}
\end{equation}
Note that equation (80) allows an estimate of $CP$ violation in mixing namely
\begin{equation}
1 - \left|\frac{q}{p}\right| = \frac{1}{2}\Im\frac{\Gamma_{12}}{M_{12}} = \frac{4\pi}{f_2(\frac{m_t^2}{m_W^2})}\frac{m_c^2}{m_t^2}\frac{J}{|V_{tb}V_{td}^\ast |^2} \sim 10^{-3}
\end{equation}
The last term is the ratio of the area of the Unitary Triangle to the length of one of its sides squared, so it is O(1).
The only suppresion factor then $(\frac{m_c^2}{m_t^2})$. The uncertainty in the calculation comes from the use of a quark
diagram to describe $\Gamma_{12}$ and could easily be of order 30\% but not three orders of magnitude. (A similar expresion (81) holds for $B_S$, execpt that the last terms is $\frac{J}{|V_{tb}V_{td}^\ast |} \sim 10^{-2}$, as can be seen from the relevent unitarity triangle in figure 3).\\

Most channels have contributions from both tree and three types of penguin diagrams[32]. The latter are classified aaccording to the identity of the quark in the loop, as diagrams with different intermediate quarks may have both different strong phases and different weak phases. On the other hand, the subdivision of tree processes into spectator, exchange, and annihilation diagrams is unimportant in this respect since they all carry the same weak phase. In addition to gluonic penguins there are also electro weak penguin contributions, with a photon or Z boson. In certain cases the latter contribution can be significant becuase it is enhanced by a factor $\frac{M_t^2}{M_Z^2}$ which partially compensates the relative suppression of electro weak versus QCD couplings.\\
\begin{figure}
\includegraphics[width = 150 mm, height = 150 mm]{figure9.ps}
\begin{picture}(0,0)
\put(-120,-10){Figure 7: Quark diagrams contributing to b decays.}
\end{picture}
\end{figure}

\noindent
Figure 7 shows the quark diagrams for tree, penguin and electroweak penguin contributions. While quark diagrams can be easily classified in this way, the description of B decays is not so neatly divided into tree and penguin contributions once long distance physics effects are taken into account. Rescattering processes can change the quark content of the final state and confuse the identification of a contribution. There is no physical distinction between rescattered tree diagrams and long-distance contributions to the cuts of a penguin diagram. While these issues complicate estimates of various rates they can always be avoided in describing the weak-phase structure of B-decay amplitudes. The decay amplitudes for $b\rightarrow q\bar{q}q^\prime$ can always be written as a sum of three terms with definite CKM coefficients:
\begin{equation}
A(q\bar{q}q^\prime ) = V_{tb}V_{tq^\prime}^\ast P^t_{q^\prime} + V_{cb}V_{cq^\prime}(T_{c\bar{c}q^\prime}\delta_{qc} + P^c_{q^\prime}) + V_{ub}V_{uq^\prime}(T_{u\bar{u}q^\prime}\delta_{qu} + P^u_{q^\prime})
\end{equation}

Here P and T denote contributions from tree and penguin diagrams, excluding the CKM factors.
As they stand, the P terms are not well defined because of the divergences of the penguin diagrams.
Only differences of penguin diagrams are finite and well defined. (However, as will be seen,
introduction of a common high momentum cut off in the loop diagrams does not affect the final
answer, since it depends only on differences of penguin amplitudes). This can be seen by using
Eqs. (13) and (14) to eliminate one of the three terms, by writing its CKM coefficient as minus
the sum of the other two.
In the case of $q\bar{q}s$ decays it is convenient to remove the $V_{tb}V_{ts}^\ast$ term. Then
\[A(c\bar{c}s) = V_{cb}V_{cs}^\ast (T_{c\bar{c}s} + P_s^c - P^t_s) + V_{ub}V_{us}^\ast (P_s^u - P_s^t)\]
\begin{equation}
A(u\bar{u}s) = V_{cb}V_{cs}^\ast (P_s^c - P_s^t) + V_{ub}V_{us}^\ast (T_{u\bar{u}s} + P^u_s - P^t_s)
\end{equation}
\[A(s\bar{s}s) = V_{cb}V_{cs}^\ast (P_s^c - P_s^t) + V_{ub}V_{us}^\ast (P^u_s - P^t_s)\]

In these expressions only differences of penguin contributions occur, which makes the cancelation
of the ultraviolet divergences of these diagrams explicit. Further, the second term has a CKM
coefficient that is much smaller than the first. Hence this grouping is useful in classifying the
expected direct $CP$ violations. (Note that terms $b\rightarrow d\bar{d}s$, which have only penguin contributions,
mix strongly with the uus terms and hence cannot be separated from them. Thus P terms in
$A(u\bar{u}s)$ include contributions from both $d\bar{d}s$ and $u\bar{u}s$ diagrams.)
In the case of $q\bar{q}d$ decays the three CKM coefficients are all of similar magnitude. The convention
is then to retain the $V_{tb}V_{td}$ term because, in the Standard Model, the phase difference between this
weak phase and half the mixing weak phase is zero. Thus only one unknown weak phase enters
the calculation of the interference between decays with and without mixing. One can choose to
eliminate whichever of the other terms does not have a tree contribution. In the cases q = s or
d, since neither has a tree contribution either term can be removed. Thus the amplitudes can be
written
\[A(c\bar{c}d) = V_{tb}V_{td}^\ast (P_d^t - P^u_d) + V_{cb}V_{cd}^\ast (T_{c\bar{c}d} + P_d^c - P_d^u)\]
\begin{equation}
A(u\bar{u}d) = V_{tb}V_{td}^\ast (P_d^t - P_d^c) + V_{ub}V_{ud}^\ast (T_{u\bar{u}d} + P^u_d - P^c_d)
\end{equation}
\[A(s\bar{s}d) = V_{tb}V_{td}^\ast (P_d^t - P_d^u) + V_{cb}V_{cd}^\ast (P^c_d - P^u_d)\]

Again only differences of penguin amplitudes occur. Furthermore the difference of penguin terms
that occurs in the second term would vanish if the charm and up quark masses were equal, and thus
is GIM (Glashow-Illiopoulos-Maiani) suppressed. However, particularly for in modes with no tree
contribution, $(s\bar{s}d)$, the interference of the two terms can still give significant direct $CP$ violation,
and thus complicate the simple predictions for the interference of decays with and without mixing
[33] obtained by ignoring this term.\\

The penguin processes all involve the emission of a neutral boson, either a gluon (strong penguins)
or a photon or Z boson (electroweak penguins). Excluding the CKM coefficients, the ratio of
the contribution from the difference between a top and light quark strong penguin diagram to the  contribution from a tree diagram is of order
\begin{equation}
\gamma_{PT} = \frac{P^t - P^{light}}{T_{q\bar{q}q^\prime}} \simeq \frac{\alpha_s}{12\pi}ln(\frac{m_t^2}{m_b^2})
\end{equation} 

This is a factor of O(0.03). However this estimate does not include the effect of hadronic matrix
elements, which are the probability factor to produce a particular final state particle content from
a particular quark content. Since this probability differs for different kinematics, color flow, and
spin structures, it can be different for tree and penguin contributions and may partially compensate
the coupling constant suppression of the penguin term. Electroweak penguin difference terms are
even more suppressed since they have an additional $\frac{\alpha_{em}}{\pi}$ or $\frac{\alpha_w}{\pi}$ compared to tree diagrams, but certain Z-contributions are enhanced by the large top quark mass and so can be non-negligible [34].\\

The most efficient tool to analyze B decays is that of the low-energy effective Hamiltonian. The
meaning and use of this tool is discussed further in the following chapter. Here the conventional
notations used for the B decay Hamiltonian are simply noted. This section is based on Ref. [35],
where a more detailed discussion can be found.
Low-energy effective Hamiltonians are constructed using the operator product expansion (OPE)
which yields transition matrix elements of the structure
\begin{equation}
\langle f|H_{eff}|i\rangle \propto \sum_k\langle f|Q_k(\mu )|i\rangle C_k(\mu )
\end{equation}
where $\mu$ denotes an appropriate renormalization scale. The OPE allows one to separate the ``long-distance" contributions to that decay amplitude from the ``short-distance" parts. Whereas the former pieces are not calculable and are relegated to the nonperturbative hadronic matrix elements $\langle f|Q_k(\mu )|i\rangle$ the latter are described by perturbatively calculable Wilson coefficient function $C_k(\mu )$ In the case of $|\Delta B| = 1$, $\Delta C = \Delta U = 0$ transitions one finds 
\begin{equation}
H_{eff} = H_{eff}(\Delta B = -1) + H_{eff}(\Delta B = -1)^\dagger
\end{equation}
with
\begin{equation}
H_{eff}(\Delta = -1) = \frac{G_F}{\sqrt{2}}\left[\sum_{j=u,c}V_{jq}^\ast V_{jb}\lbrace\sum_{k=1}^2 Q^{jq}_kC_k(\mu ) + \sum_{k=3}^{10}Q_k^qC_k(\mu )\rbrace\right].
\end{equation}
Here $G_F$ denotes the Fermi constant, the renormalization scale $\mu$ is of $O(m_b)$, the flavor label q in \{d,s\} corresponds to $b\rightarrow d$ and $b\rightarrow s$ transitions, respectively, and $Q^{jq}_k$ are four-quark
operators that can be divided into three categories:\\\\\noindent
(i) current-current operators:
\[Q_1^{jq} = (\bar{q}_\alpha j_\beta )_{V-A}(\bar{j}_\beta b_\alpha )_{V-A}\]
\begin{equation}
Q_2^{jq} = (\bar{q}_\alpha j_\alpha )_{V-A}(\bar{j}_\beta b_\beta )_{V-A}
\end{equation}
(ii) QCD penguin operators:
\[Q_3^{q} = (\bar{q}_\alpha b_\alpha )_{V-A}\sum_{q^\prime}(\bar{q}^{\prime}_\beta q^{\prime}_\beta )_{V-A}\]
\[Q_4^{q} = (\bar{q}_\alpha b_\beta )_{V-A}\sum_{q^\prime}(\bar{q}^{\prime}_\beta q^{\prime}_\alpha )_{V-A}\]
\begin{equation}
Q_5^{q} = (\bar{q}_\alpha b_\alpha )_{V-A}\sum_{q^\prime}(\bar{q}^{\prime}_\beta q^{\prime}_\beta )_{V+A}
\end{equation}
\[Q_6^{q} = (\bar{q}_\alpha b_\beta )_{V-A}\sum_{q^\prime}(\bar{q}^{\prime}_\beta q^{\prime}_\alpha )_{V+A}\]
(iii) EW penguin operators:
\[Q_7^{q} = \frac{3}{2}(\bar{q}_\alpha b_\alpha )_{V-A}\sum_{q^\prime}e_{q^\prime}(\bar{q}^{\prime}_\beta q^{\prime}_\beta )_{V+A}\]
\[Q_8^{q} = \frac{3}{2}(\bar{q}_\alpha b_\beta )_{V-A}\sum_{q^\prime}e_{q^\prime}(\bar{q}^{\prime}_\beta q^{\prime}_\alpha )_{V+A}\]
\begin{equation}
Q_9^{q} = \frac{3}{2}(\bar{q}_\alpha b_\alpha )_{V-A}\sum_{q^\prime}e_{q^\prime}(\bar{q}^{\prime}_\beta q^{\prime}_\beta )_{V-A}
\end{equation}
\[Q_{10}^{q} = \frac{3}{2}(\bar{q}_\alpha b_\beta )_{V-A}\sum_{q^\prime}e_{q^\prime}(\bar{q}^\prime_\beta q^{\prime}_\alpha )_{V-A}\]
Here $\alpha$ and $\beta$ denote $SU(3)_c$ color indices, $V\pm A$ refers to the Lorentz structures $\gamma_\mu (1\pm \gamma_5)$, respectively, $q^\prime$ runs over the quark flavors active at the scale $\mu = O (m_b)$, i.e., $q^\prime$ in \{u,d,c,s,b\} and $e_{q^\prime}$ are the corresponding electrical quark charges. The current-current, QCD, and EW penguin
operators are related to the tree, QCD, and EW penguin processes, depicted in Figure 7.\\

In the case of transitions of the type $b\rightarrow q\bar{u}c$ and $b\rightarrow q\bar{c}u$ with q in \{d,s\} , only current-current operators contribute. The structure of the corresponding low-energy effective Hamiltonians is
completely analogous to (89). To obtain it, one replaces both the CKM factors $V_{jq}^\ast V_{jb}$ and the flavor contents of the current-current operators (90) straightforwardly with the appropriate quark flavor structure, and omits the sum over penguin operators.\\

Direct $CP$ violations require two contributions to the decay process which differ in both their strong phases and their weak phases so that $\left|\frac{\bar{A}}{A}\right|\neq 1$. Purely leptonic and semileptonic decays are dominated by a single diagram and thus are unlikely to exhibit any measurable direct $CP$ violation. Nonleptonic decays often have two terms that are comparable in magnitude and hence could have significant direct $CP$ violations. The theoretical calculation of $CP$ asymmetries requires knowledge of strong phase shifts and of absolute values of
various amplitudes. The estimates therefore necessarily have hadronic
uncertainties. In contrast, a clean relationship between measured asymmetries and CKM phases is
obtained when studying $CP$ violation in the interference between decays with and without mixing
for $CP$ eigenstate modes dominated by a single term in the decay amplitude.
B decays can thus be grouped into five classes. Classes 1 and 2 are expected to have relatively
small direct $CP$ violations and hence are particularly interesting for extracting CKM parameters
from interference of decays with and without mixing. In the remaining three classes, direct CP
violations could be significant and the neutral decay asymmetries cannot be cleanly interpreted in
terms of CKM phases.\\

1. Decays dominated by a single term: $b\rightarrow c\bar{c}s$ and $b\rightarrow s\bar{s}s$. The Standard Model cleanly
predicts zero (or very small) direct $CP$ violations because the second term is Cabibbo sup-pressed.
Any observation of large direct CP-violating effects in these cases would be a clue
to beyond Standard Model physics. The modes $B^+\rightarrow\psi K^+$ and $B^+\rightarrow\phi K^+$ are examples
of this class. The corresponding neutral modes have cleanly predicted relationships between
CKM parameters and the measured asymmetry from interference between decays with and
without mixing.\\

2. Decays with a small second term: $b\rightarrow c\bar{c}s$ and $b\rightarrow u\bar{u}d$. The expectation that penguin-only
contributions are suppressed compared to tree contributions suggests that these modes will have small direct $CP$ violation effects, and an approximate prediction for the relationship between measured asymmetries in neutral decays and CKM phases can be made.\\
 
3. Decays with a suppressed tree contribution: $b\rightarrow u\bar{u}s$. The tree amplitude is suppressed by
small mixing angles, $V_{ub}V_{us}^\ast$ . The no-tree term may be comparable or even dominate and give large interference effects. An example is $B\rightarrow\rho K$\\

4. Decays with no tree contribution: $b\rightarrow s\bar{s}d$. Here the interference comes from penguin
contributions with different charge $\frac{2}{3}$ quarks in the loop. An example is $B\rightarrow KK$\\ 

5. Radiative decays: $b\rightarrow s\gamma$. The mechanism here is the same as in case 4 except that the leading contributions come from electromagnetic penguins. An example is $B\rightarrow K^\ast\gamma$\\

Recent CLEO results on $B(B\rightarrow K\pi )$ and $B(B\rightarrow\pi\pi )$ [36] suggest that the matrix element of penguin operators is enhanced compared to that of tree operators. If this enhancement is significant, then some of the decay modes listed in Class 2 might actually fit better to Class 3; that is it becomes more difficult to relate a measured asymmetry to a CKM phase. For example, it is possible that $b\rightarrow u\bar{u}d$ decays have comparable contributions from tree and penguin amplitudes. On the other hand, this would also mean that some modes listed in Class 3 could be dominated by a single penguin term. For such cases an approximate relationship between measured asymmetries in neutral decays and CKM phases can be made.\\\\\noindent
Some more detailed examination  of the effects in a variety of theories beyond the Standard Model is given in Ref.[37].\\

By now the Standard Model and its particle content are so well established that any future theory
will certainly contain them. However extensions that go beyond the Standard Model inevitably
introduce additional fields. Along with them there often come additional coupling constants
and hence the possibility of additional CP-violating phases. Even if no new phases occur there
can be changes in the relationship between various physical quantities and CKM matrix element magnitudes and phases. Effects of physics beyond the Standard Model can manifest themselves
in two ways, as additional contributions to the mixing of $B^0$ and $\bar{B}^0$ states, and/or as additional
contributions to some set of decays.\\

An additional contribution to the mixing would have two effects: a change in the relationship
between $x_d$ and $|V_{td}V_{tb}|$ and a change in the relationship between the
phase of $\frac{q}{p}$ and the phase of $V_{tb}V_{td}^\ast$. However, since all $\lambda_f$ have a common factor $\frac{q}{p}$, it would not change the relative phases between various $\lambda_f$.\\

Additional contributions to the decays can only be unambiguously and model-independently observed
in cases where an amplitude is dominated by a single weak-phase term in the Standard
Model. Then such terms destroy the relationship between the asymmetry and a CKM matrix phase and so lead to inconsistencies. For example, various modes that have the same Standard Model asymmetry may actually give different asymmetries [38]. In cases where two competing terms with different weak phases occur in the Standard Model expression, any additional term,
whatever its phase, can always be absorbed into these two terms, appearing simply as changes in their magnitudes. Since these magnitudes cannot as yet be calculated in a model-independent and
reliable fashion, this makes it quite difficult to identify changes from the Standard Model in these
cases. However by a systematic study of expected patterns and improved theoretical calculations
of matrix elements, one may be able to identify the impact of contributions beyond the Standard
Model in these cases as well.\\

The angle $\phi_1$ can be obtained through measurements of CP-violating asymmetries involving final
states which can be either $CP$ eigenstates or not. If the state of interest is a $CP$ eigenstate, it
was shown how to relate an asymmetry to CKM parameters. If only a single weak-decay
amplitude contributes to $B\rightarrow f_{CP}$ , that is, if penguin contributions are negligible, then $\left|\frac{\bar{A}_{f_{CP}}}{A_{f_{CP}}}\right| = 1$ so that
\begin{equation}
a_{f_{CP}} = -\Im\lambda_{f_{CP}}sin(\Delta m_Bt).
\end{equation}\vspace{2 mm}

In this case $\lambda_{f_{CP}}$ is a pure phase, i.e.,$\Im\lambda$ is one of the angles of the unitarity triangle. In particular, for the decay $B\rightarrow\pi\pi$, $\Im\lambda_{\pi\pi} = \sin 2\phi_1$.
The case where the final state $f$ is not a $CP$ eigenstate is considerably more complicated. There
are four separate amplitudes for $B^0$ and $\bar{B}^0$ to decay to $f$ and $\bar{f}$:
\[A(B^0\rightarrow f)\equiv A_f = |A_f|e^{i\theta_f}\]
\[A(\bar{B}^0\rightarrow f)\equiv \bar{A}_f = |\bar{A}_f|e^{i\bar{\theta}_f}\]
\[A(B^0\rightarrow\bar{f})\equiv A_{\bar{f}} = |A_{\bar{f}}|e^{i\theta_{\bar{f}}}\]
\begin{equation}
A(\bar{B}^0\rightarrow \bar{f})\equiv \bar{A}_{\bar{f}} = |\bar{A}_{\bar{f}}|e^{i\bar{\theta}_{\bar{f}}}
\end{equation}
The rates for the physical, time-evolved $B^0_{phys}(t)$ and $\bar{B}^0_{phys}(t)$ states to decay into $f$ can then be
written:
\[\Gamma (B^0_{phys}(t)\rightarrow f) = e^{-\Gamma t}A^2 \times \lbrace 1 + R\cos (\Delta m_Bt) - D\sin (2\phi_M - \theta_f + \bar{\theta}_f)\sin (\Delta m_Bt)\rbrace\]
\begin{equation}
\Gamma (\bar{B}^0_{phys}(t)\rightarrow f) = e^{-\Gamma t}A^2 \times \lbrace 1 - R\cos (\Delta m_Bt) + D\sin (2\phi_M - \theta_f +\bar{\theta}_f)\sin (\Delta m_Bt)\rbrace
\end{equation}
where $\phi_M$ is the phase of $B^0 - \bar{B}^0$ mixing, and
\begin{equation}
A^2\equiv\frac{1}{2}\left( |A_f|^2 + |\bar{A}_f|^2\right) , R\equiv\frac{|A_f|^2 - |\bar{A}_f|^2}{|A_f|^2 + |\bar{A}_f|^2} , D\equiv 2\frac{|A_f|^2|\bar{A}_f|^2}{|A_f|^2 + |\bar{A}_f|^2}.
\end{equation}
Similarly, the rates for $B^0_{phys}(t)$ and $\bar{B}^0_{phys}(t)$ states to decay into $\bar{f}$ are
\[\Gamma (B^0_{phys}(t)\rightarrow \bar{f}) = e^{-\Gamma t}\bar{A}^2 \times \lbrace 1 - \bar{R}\cos (\Delta m_Bt) - \bar{D}\sin (2\phi_M + \bar{\theta}_{\bar{f}} - \theta_{\bar{f}})\sin (\Delta m_Bt)\rbrace\hspace{5 mm}\mbox{     }\]
\begin{equation}
\Gamma (\bar{B}^0_{phys}(t)\rightarrow \bar{f}) = e^{-\Gamma t}\bar{A}^2 \times \lbrace 1 + \bar{R}\cos (\Delta m_Bt) - \bar{D}\sin (2\phi_M + \bar{\theta}_{\bar{f}} - \theta_{\bar{f}})\sin (\Delta m_Bt)\rbrace
\end{equation}
where
\begin{equation}
\bar{A}^2\equiv\frac{1}{2}\left( |A_{\bar{f}}|^2 + |\bar{A}_{\bar{f}}|^2\right) , R\equiv\frac{|A_{\bar{f}}|^2 - |\bar{A}_{\bar{f}}|^2}{|A_{\bar{f}}|^2 + |\bar{A}_{\bar{f}}|^2} , D\equiv 2\frac{|A_{\bar{f}}|^2|\bar{A}_{\bar{f}}|^2}{|A_{\bar{f}}|^2 + |\bar{A}_{\bar{f}}|^2}.
\end{equation}
CP conservation requires that
\begin{equation}
|A_{\bar{f}}| = |\bar{A}_f|, |\bar{A}_{\bar{f}}| = |A_f|
\end{equation}
\begin{equation}
\sin (2\phi_M - \theta_f + \bar{\theta}_f) = \sin (2\phi_M + \bar{\theta}_{\bar{f}} - \theta_{\bar{f}}).
\end{equation}  

CP violation occurs if any of these equalities is not satisfied.
The above expressions for the decays of $B^0_{phys}(t)$ and $B^0_{phys}(t)$ to $f$ and $\bar{f}$ are completely general.
However, when one assumes that each decay is dominated by a single weak amplitude (i.e., that
penguins and any new physics effects are negligible), the expressions simplify. In this case, the
parameters in the amplitudes for the decays of $B^0$ and $\bar{B}^0$ to $f$ and $\bar{f}$ obey certain equalities which
reflect their $CP$ transformation relationships:
\[|A_{\bar{f}}| = |\bar{A}_f| , |\bar{A}_{\bar{f}}| = |A_f|\]
\begin{equation}
\theta_f = \phi_{D_f} + \delta , \bar{\theta}_{\bar{f}} = -\phi_{D_f} + \delta
\end{equation}
\[\theta_{\bar{f}} = \phi_{D_{\bar{f}}} + \delta^\prime , \bar{\theta}_f = -\phi_{D_{\bar{f}}} + \delta^\prime\] 
In the above, $\phi_{D_f}$ and $\phi_{D_{\bar{f}}}$ represent the weak-CKM phases of the decays of $B^0$ to $f$ and $\bar{f}$ respectively, while $\delta$ and $\delta^\prime$ are the strong phases. With these equalities, the expressions in Eqs. (96) and (98) become very similar: $A^2 = \bar{A}^2$, $R = -\bar{R}$, $D = \bar{D}$. The above equalities give
\[\sin (2\phi_M - \theta_f + \bar{\theta}_f) = \sin (2\phi_M - \phi_{D_f} - \phi_{D_{\bar{f}}} - \Delta\delta )\]
\begin{equation}
\sin (2\phi_M - \bar{\theta}_f + \theta_{\bar{f}}) = \sin (2\phi_M - \phi_{D_f} - \phi_{D_{\bar{f}}} + \Delta\delta ).
\end{equation}
where $\Delta\delta = \delta - \delta^\prime$. The CP-violating weak-CKM phase is given by the quantity $2\Phi \equiv 2\phi_M - \phi_{D_f} - \phi_{D_{\bar{f}}}$. From measurements of the time-dependent decay distributions one can obtain $S\equiv\sin (2\Phi + \Delta\delta )$ and $\bar{S}\equiv\sin(2\Phi - \Delta\delta)$, and from these one can extract $\sin^22\Phi$ up to a two-fold ambiguity:
\begin{equation}
\sin^22\Phi = \frac{1}{2}\left[ 1 + S\bar{S} \pm \sqrt{(1 - S^2)(1 - \bar{S})^2}\right]
\end{equation}
The true value of $\sin^22\Phi$ is given by one of the signs on the right-hand side, while the other gives
$\cos^2\Delta\delta$ However, this discrete ambiguity can in principle be removed by comparison with other
final states which have the same weak-phase $2\Phi$, but different strong phases. Note that, if the
three different time-dependent terms (1, cos, sin) can be isolated with sufficient accuracy, it is not
necessary to measure all four time-dependent rates $B^0_{phys}(t), \bar{B}^0_{phys}(t)\rightarrow f, \bar{f}$. The measurement of one of the rates in Eq. (95) and one of the rates in Eq. (97) is sufficient to obtain the above
phase information.\\

Thus, assuming penguins make negligible contributions, this technique can be used to extract the
CP angle $\phi_1$: one must measure the decays of $B^0_{phys}(t)$ and $\bar{B}^0_{phys}(t)$ to such final states as $\rho^+\pi^-$ or $\alpha_1^+\pi^-$. When penguins are significant these methods yield a quantity, denoted $\alpha_{eff}$, which differs from the true $\alpha$ by an unknown amount, which we will denote as $\frac{\kappa_f}{2}$. This quantity is channel-dependent because it depends on the ratio of tree-dominated to penguin-only contributions. Model-dependent
calculations can be used to estimate this shift in but significant theoretical uncertainty
remains. Eventually, this uncertainty may be reduced by restricting the value of the penguin
terms from other measurements, for example via the SU(3) relationships discussed below. As
models become better tested by a variety of measurements the uncertainties that arise due to their
application may also be reduced, even without the use of such SU(3) relationships. Eventually,
however, one would like to be able to use model-independent methods that take the penguin
contributions into account correctly. These are discussed below.\\

In most cases there is in fact more than one weak-decay amplitude contributing to a decay, which
can always be written as a tree-dominated plus a penguin-only term. In the channels of interest here, the weak-phase difference between these terms is $\phi_1$.\\

For the case where f is a $CP$ eigenstate the amplitudes for $B\rightarrow f_{CP}$ and $\bar{B}\rightarrow f_{CP}$ can then be
written as
\[A_{f_{CP}} = Te^{i\phi_T}e^{i\delta_T} + Pe^{i\phi_P}e^{i\delta_P},\]
\begin{equation}
\bar{A}_{f_{CP}} = Te^{-i\phi_T}e^{i\delta_T} + Pe^{-i\phi_P}e^{i\delta_P}
\end{equation}
where T, $\phi_T$ and $\delta_T$ (P, $\phi_P$ and $\delta_P$) represent the magnitude, the weak phase and the strong phase
of the tree-dominated (penguin-only) amplitude, respectively.\\

Now suppose that penguin contributions are non-negligible and that $\phi_T \neq \phi_P$. In this case it is clear
from Eq. (104) that $\lambda_{f_{CP}}$ depends on a function of tree and penguin parameters, so
that it no longer cleanly measures a single CKM phase. Thus the presence of significant ``penguin
pollution" spoils the clean extraction of the angles of the unitarity triangle from CP-violating
asymmetries. In general, the presence of non-negligible penguin contributions will also lead to
direct $CP$ violation, that is,$|\lambda_{f_{CP}}| = 1$. In the presence of direct $CP$ violation,
the time-dependent $CP$ asymmetry contains a $\cos(\Delta m_Bt)$ term, the coefficient of which can also
be measured. However, this need not be the case. If the strong phases are equal, $\delta_T = \delta_P$ , then
$\lambda_{f_{CP}}$ is a pure phase (i.e. $|\lambda_{f_{CP}}| = 1)$. However, this phase depends on both tree and penguin
parameters, so that there is still a shift in $\phi_1$ due to penguin contributions, even though there is no
direct $CP$ violation.\\

Although the above discussion has been made for the case where the final state is a $CP$ eigenstate,
it applies equally well when the final state is not a $CP$ eigenstate (e.g. $\rho\pi$ , $a_1\pi$, etc.). If penguin
contributions are important in decays to such final states, then the $CP$ asymmetries alone do not cleanly probe the angles of the unitarity triangle.\\

In fact, present experimental information suggests that penguin pollution may well be significant
in $B^0\rightarrow\pi^+\pi^-$. CLEO has observed the decay $B^0\rightarrow\pi^-K^+$: they have $N_{K_\pi} = 21.6^{+6.8}_{-6.0}$ events, which translates into a branching ratio of $1.5\times 10^{-5}$. For $B^0\rightarrow\pi^+\pi^-$ they have $9.9^{+6.0}_{-5.1}$ events, leading to an upper limit of $1.5\times 10^{-5}$ . While one cannot draw rigorous conclusions from these data, one can still make a back-of-the-envelope estimate as follows. The quantity of interest is a $\frac{P}{T}$, where
\begin{equation}
P\equiv A(B^0\rightarrow\pi^+\pi^-)|_{penguin} , T\equiv A(B^0\rightarrow\pi^+\pi^-)|_{tree}
\end{equation}
This ratio can be written
\begin{equation}
\frac{P}{T}=\frac{A(B^0\rightarrow\pi^-K^+)|_{penguin}}{T}\frac{P}{A(B^0\rightarrow\pi^-K^+)|_{penguin}}
\end{equation}
This ratio can be estimated with the help of some assumptions. First, take the central values of
the number of events at face value, so that the branching ratio for $B^0\rightarrow\pi^+\pi^-$ is half that of
$B^0\rightarrow\pi^-K^+$. Second, assume that the observed events for $B^0\rightarrow\pi^- K^+$ and $B^0\rightarrow\pi^+\pi^-$ are due only to the $b\rightarrow s$ penguin and the $b\rightarrow u\bar{u}d$ tree amplitudes, respectively. This implies that
\begin{equation}
\frac{A(B^0\rightarrow\pi^-K^+)|_{penguin}}{T} = \sqrt{2}
\end{equation}
The second term in Eq. (106) is the ratio of $b\rightarrow d$ and $b\rightarrow s$ penguins. This can be written
\begin{equation}
\frac{P}{A(B^0\rightarrow\pi^-K^+)|_{penguin}} = \left|\frac{V_{td}}{V_{ts}}\right|\times \mbox{ an SU(3)-breaking factor}
\end{equation}
The size of the SU(3)-breaking effects is not known. However, as a crude guess, take this factor to be roughly $\frac{f_\pi}{f_K} \sim \frac{1}{1.2}$. The ratio of CKM matrix elements is constrained to be in the range 
\begin{equation}
0.15\le\left|\frac{V_{td}}{V_{ts}}\right|\le 0.34
\end{equation}
Putting all the factors together, gives
\begin{equation}
0.18\le\frac{P}{T}\le0.4
\end{equation}
These numbers should not be taken literally, since they neglect both theoretical and experimental
uncertainties other than the range of $\left|\frac{V_{td}}{V_{ts}}\right|$. However they show that the CLEO data suggests that penguins are likely to be significant in $B^0\rightarrow \pi^+\pi^-$ and, by extension, in $B^0\rightarrow\rho\pi$ and $a_1\pi$.\\

The tool to separate the tree and penguin contributions is isospin analysis. Isospin amplitudes $I_{\Delta I,I_f}$ If can be labeled by t$\Delta_I$ value of the b-quark decay and by the If of the final state, which includes the spectator quark. The key observation is that a gluon is pure I = 0, so that the dominant gluonic $b\rightarrow d$ penguins are pure $\Delta I = \frac{1}{2}$. On the other hand, the tree-level $b\rightarrow u\bar{u}d$ decays have both $\Delta_I = \frac{3}{2}$ and $\Delta I = \frac{1}{2}$ components. Thus, if the $\Delta I = \frac{3}{2}$ piece can be isolated, then the tree contribution, which contains the weak phase to be measured, is thereby isolated. Inclusion of the spectator quark then gives final isospin of 0 or 1 for the gluonic penguin contributions, but 0, 1 or 2 for the tree contributions. (Similar arguments apply to $b\rightarrow s$ penguins and $b\rightarrow u\bar{u}s$ tree amplitudes.)\\\\
\begin{figure}
\includegraphics[width = 175 mm, height = 120 mm]{figure8.ps}
\begin{picture}(0,0)
\put(0,150){\textbf{Table 1} Isopin decompostion for $B\rightarrow\pi\pi$, $B\rightarrow\pi K$ and $B\rightarrow\rho\pi$ in terms of the isopin}
\put(0,145){amplitudes $A_{\Delta I,I_f}$ where $\Delta I$ and $I_f$ are the isopin change of the transition and the final-state}
\put(0,140){isopin, respectively. The CP-even part of $B\rightarrow\rho\rho$ decays follow the same pattern as $B\rightarrow\pi\pi$,}
\put(0,135){$B\rightarrow\pi K^\ast$ and $B\rightarrow\rho K$ are analogous to $B\rightarrow\pi K$; $B\rightarrow a_1\pi$ is similar to $B\rightarrow\rho\pi$}       
\end{picture}
\end{figure}
\noindent
Isospin analysis can be used for a variety of final states : $\pi\pi , \rho\pi , a_1\pi , \rho\rho , K\pi , K^\ast\pi , K\rho$ etc. Isospin analysis for some final-state particle pairs will be discussed separately below. Table 1
lists the isospin amplitudes for all relevant channels for these states. Note that, in all cases, there is at least one isospin amplitude which can be reached only via tree diagrams: $A_{\frac{3}{2},2}$  for $B\rightarrow\pi\pi$, $A_{1,\frac{3}{2}}$ for $B\rightarrow\pi K$, $A_{\frac{3}{2},1}$  for $B\rightarrow\rho\pi$. Isolation of such isospin amplitudes allows
the removal of penguin pollution. (Note: this statement is only true to the extent that electroweak penguins can be neglected The effects of such contributions in the context of the various final states are discussed below.) Note also that the decay $B\rightarrow DD$ is not included in the list. This is because both tree and penguin diagrams correspond to $\Delta I = \frac{1}{2}$ transitions. Thus, an isospin analysis cannot be used to remove penguin pollution in this case.\\\\\noindent
\textbf{IV- Measurement of $\sin 2\phi_1$ at B-Factory}\\

In the B-factory, $B$ mesons are produced from $b\bar{b}$ resonance state of $\Upsilon (4S)$, because the $\Upsilon (4S)$ is the lowest bound state that can decay into two $B$ mesons. In this system, we observe the $CP$ violation in proper-time difference distribution. In the following subsections, first we describe the proper-time difference of two $B$ meson decays produced from $\Upsilon (4S)$. Then we describe the overview to observe the $CP$ violation at the B-factory. Experimental constraints and recent measurement of $\sin 2\phi_1$ are also described.\\\\\noindent
\textit{\textbf{1 Proper-Time Difference}}\\

$\Upsilon (4S)$ decays into a coherent $B^0-\bar{B}^0$ state with a $C$ odd configuration. Subsequently oscillations take place preserving the $C$ odd configuration: Bose statistics tells us that if one of the mesons is a $B^0$ at some time, the other one cannot be $B^0$ at that time, since the state must be odd under exchange of two mesons. Let us consider the decay rate of such pair of $B$ mesons. We label each of the $B$ mesons with its momentum: $\vec{k}$ or $-\vec{k}$. Assuming $\Delta\Gamma = 0$, a state with $B_{\vec{k}}$ at time $t = t_{\vec{k}}$ and $B_{-\vec{k}}$ at time $t = t_{-\vec{k}}$ can be expressed as
\[|B_{\vec{k}}B_{-\vec{k}}(t_{\vec{k}},t_{-\vec{k}})\rangle = \frac{1}{\sqrt{2}}e^{-\frac{\Gamma}{2}(t_{\vec{k}}+t_{-\vec{k}}\rangle)}\]
\[\times [i\sin\frac{\Delta M(t_{-\vec{k}}+t_{\vec{k}})}{2}\left(\frac{p}{q}|B^0_{\vec{k}}B^0_{-\vec{k}}\rangle - \frac{q}{p}|\bar{B}^0_{\vec{k}}\bar{B}^0_{-\vec{k}}\rangle\right)\]
\begin{equation}
+\cos\frac{\Delta M(t_{-\vec{k}}+t_{\vec{k}})}{2}\left(|B^0_{\vec{k}}\bar{B}^0_{-\vec{k}}\rangle - |\bar{B}^0_{\vec{k}}B^0_{-\vec{k}}\rangle\right)]
\end{equation}
where $B^0_{\vec{k}}(B^0_{-\vec{k}})$ denotes $B_{\vec{k}}(B_{-\vec{k}})$ is $B^0$ and $\bar{B}^0_{\vec{k}}(\bar{B}^0_{-\vec{k}})$ denotes $B_{\vec{k}}(B_{-\vec{k}})$ is $\bar{B}^0$. This equation is obtained from the time evolution of 
\begin{equation}
\frac{1}{\sqrt{2}}(|B^0_{\vec{k}}(t)\rangle|\bar{B}^0_{-\vec{k}}(t)\rangle - |\bar{B}^0_{\vec{k}}(t)\rangle|B^0_{-\vec{k}}(t)\rangle )
\end{equation}
where the state is chosen to be anti-symmetric due to the $C$ odd state of the system. The decay rates are computed for the case in which one of the particles decays in a ``flavor-specific" way, while the other one decays in a "flavor-nonspecific" way, e.g.:
\[\bar{B}^0\rightarrow l^- + X \not\leftarrow B^0\]
\[\bar{B}^0\not\rightarrow l^+ + \bar{X} \leftarrow B^0\]
\[\bar{B}^0\rightarrow f_{CP} \leftarrow B^0\]
Using the definitions of $\langle l^+X|B^0\rangle = \langle l^-\bar{X}|\bar{B}^0\rangle \equiv A_{SL}, \langle f_{CP}|B^0\rangle \equiv a$, and $\langle f_{CP}|\bar{B}^0\rangle \equiv a^\prime = \lambda_{CP}\cdot\frac{ap}{q}$, the decay rates can be calculated as 
\[\left|\langle (l^-X)_{\vec{k}},(f_{CP})_{-\vec{k}}|B_{\vec{k}}B_{-\vec{k}}(t_{\vec{k}},t_{-\vec{k}})\rangle\right|^2 = \frac{1}{4}e^{-\Gamma (t_{\vec{k}}+t_{-\vec{k}})}|A_{SL}|^2|a|^2\]
\begin{equation}
\times [(1 + |\lambda_{CP}|^2) + (1-|\lambda_{CP}|^2)\cos\Delta M(t_{-\vec{k}}-t_{\vec{k}}) - 2\Im (\lambda_{CP}\sin\Delta M(t_{-\vec{k}} - t_{\vec{k}})]
\end{equation}
and 
\[\left|\langle (l^+\bar{X})_{\vec{k}},(f_{CP})_{-\vec{k}}|B_{\vec{k}}B_{-\vec{k}}(t_{\vec{k}},t_{-\vec{k}})\rangle\right|^2 = \frac{1}{4}e^{-\Gamma (t_{\vec{k}}+t_{-\vec{k}})}|A_{SL}|^2|a|^2\left|\frac{q}{p}\right|^2\]
\begin{equation}
\times [(1 + |\lambda_{CP}|^2) - (1-|\lambda_{CP}|^2)\cos\Delta M(t_{-\vec{k}}-t_{\vec{k}}) + 2\Im (\lambda_{CP}\sin\Delta M(t_{-\vec{k}} - t_{\vec{k}})]
\end{equation}
approximations of $|\lambda_{CP}| \simeq 1$, and $|\frac{q}{p} \simeq 1|$ for $B^0\rightarrow J/\psi K_S$ simplify equations (113) and (114) into:
\begin{equation}
G_{l\pm}(t_{-\vec{k}},t_{\vec{k}}) \propto e^{-\Gamma (t_{\vec{k}}+t_{-\vec{k}})}[1\pm\sin 2\phi_1\sin\Delta M(t_{-\vec{k}}-t_{\vec{k}})]
\end{equation}
where $G_{l\pm} \equiv |\langle(l^\pm X)_{\vec{k}},(f_{CP})_{-\vec{k}}|B_{\vec{k}}B_{-\vec{k}}(t_{\vec{k}},t_{-\vec{k}})\rangle|^2$ and $\sin 2\phi_1 = \Im (\lambda_{CP})$.\\

Let $t_{-\vec{k}}$ as the time when one of two $B$ meson decays into $f_{CP}$ state, and $t_{\vec{k}}$ as the time when another $B$ meson decays in flavor-specific way, such as semileptonic decay. We label the $B$ meson decaying into $CP$ eigenstate as $B_{CP}$ and the remaining $B$ meson as $B_{tag}$, hereafter. A duration measurement from $B_{tag}$ decay to $B_{CP}$ decay, defined as $\Delta t \equiv t_{-\vec{k}}-t_{\vec{k}}$, provides $\sin 2\phi_1$ instead of the measurements of $t_{-\vec{k}}$ and $t_{\vec{k}}$, as follows. Because we care about neither individual decay times of $t_{-\vec{k}}$ nor $t_{\vec{k}}$, we have to integrate equation (115) with respect to $t_{-\vec{k}}$ and $t_{\vec{k}}$ under the constraint of $\Delta t = t_{-\vec{k}} - t_{\vec{k}}$. Using $t_{-\vec{k}} \ge 0$ and $t_{\vec{k}} \ge 0$, we obtain
\begin{equation}
\int\int dt_{-\vec{k}}dt_{\vec{k}}G_{l\pm}(t_{-\vec{k}},t_{\vec{k}})\delta (t_{-\vec{k}}-t_{\vec{k}}-\Delta t) \propto e^{-\Gamma |\Delta t|}(1\pm\sin 2\phi_1\sin\Delta M\Delta t)
\end{equation} 
It is worth mentioning what happens if we swap $\vec{k}$ and $-\vec{k}$ in equations (113) and (114). In this case, ``flavor-specific" mode is associated to $B_{-\vec{k}}$, and ``flavor-nonspecific" mode is associated to $B_{\vec{k}}$, and therefore the definition of $\Delta t$ is flipped as $\Delta \equiv t_{\vec{k}} - t_{-\vec{k}}$. Thus, we have exactly same functions as (113) and (114).\\\\\noindent
To summarize, in the $C$ odd system, if we measure the proper-time difference and if we identify the flavor of $B_{tag}$, we can extract $\sin 2\phi_1$ from the $\Delta t$ distributions.\\

Calculating a normalization factor for equation (116), we obtain a $\Delta t$ distribution function as
\begin{equation}
f_{l\pm}(\Delta t;\sin 2\phi_1) = \frac{1}{2\tau_{B^0}}e^{-\frac{|\Delta t|}{\tau^{B^0}}}(1\pm\sin 2\phi_1\sin\Delta M\Delta t)
\end{equation}
where $\tau_{B^0} \equiv \frac{1}{\Gamma}$ is a lifetime of $B^0$ meson. In general, $l^+$ denotes that flavor-specific $B$ meson was $B^0$ and $l^-$ denotes that flavor-specific $B$ meson was $\bar{B}^0$. Inputs of $\sin 2\phi_1$, $\tau_{B^0}$, and $\delta M$ are 0.60, 1.548 ps, and 0.472 ps$^{-1}$, respectively.\\\\\noindent
\textit{\textbf{2. Belle Experiment}}\\

In this subsection, we give introduction of the experimental procedure to determine $\sin 2\phi_1$ in the Belle experiment, which is the B-factory experiment at KEK.\\

In the KEK B-factory, a $B\bar{B}$ meson pair is produced by a decay of $\Upsilon (4S)$, where $\Upsilon (4S)$ is a $b\bar{b}$ resonance state possessing a minimum mass to decay $B^0-\bar{B}^0$ pair. One of the $B$ meson pair is fully reconstructed to identify its decay final state. We reconstruct $B$ meson with $J/\psi$ and $K_S$ mesons. $J/\psi$ and $K_S$ are 
constructed via $J/\psi\rightarrow l^+l^- (l = e,u)$ and $K_S\rightarrow\pi^+\pi^-$ decay, respectively. Efficient determination of the charged particles and good lepton identification capability is essential.\\

It is necessary to determine flavor of the associated $B$ meson. This process is called ``flavor tagging". In addition to the semileptonic decays, the presence of the following particles can be used to tag the flavor of the $B$ mesons: secondary lepton in $b\rightarrow c\rightarrow l^+$ decays, fast pions, which reflects the charge of virtual $W$ in $b\rightarrow c + W^-$, slow pions coming from $D^{\ast\pm}$ whose charge reflects a charge of $c$, and kaons and $W$ from cascade decays of $b\rightarrow c\rightarrow s$. When the probability to incorrectly assign the flavor of the associated $B$ meson is $w$, (called wrong tag fraction), the observed $\Delta t$ distribution becomes
\[f_{l\pm}(\Delta t;\sin 2\phi_1) = \frac{1}{2\tau^{B^0}}e^{-\frac{|\Delta t|}{\tau_{B^0}}}[1\pm (1-w)\cdot\sin 2\phi_1\sin\Delta M\Delta t]\]
\[+\frac{1}{2\tau_{B^0}}e^{-\frac{|\Delta t|}{\tau_{B^0}}}[1\mp w\cdot\sin 2\phi_1\sin\Delta M\Delta t]\]
\begin{equation}
= \frac{1}{2\tau_{B^0}}e^{-\frac{|\Delta t|}{\tau_{B^0}}}[1\pm (1-2w)\cdot\sin 2\phi_1\sin\Delta M\Delta t]
\end{equation} 
Good particle identification, in particular lepton and kaon identification, is required to minimize $w$.\\\\\noindent
The $\Delta t$ is obtaind from the distance of two decay vertices $B_{tag}$ and $B_{CP}(\Delta z)$ as
\begin{equation}
\Delta t = \frac{\Delta z}{c(\beta\phi_1 )_B}
\end{equation}\\\noindent 
where $(\beta\phi_1 )_B$ is a motion of $B$ meson. Because the mass of $\Upsilon (4S)$ is close to a sum of two $B^0$ masses, two $B$ mesons are produced almost at rest in the $\Upsilon (4S)$ rest frame.

\begin{figure}
\includegraphics[width = 200mm, height = 60 mm]{figure7.ps}
\begin{picture}(0,0)

\put(10,40){electron}
\put(10,35){(8Gev)}
\put(40,40){$\Upsilon (4S)$}
\put(40,35){resonance}
\put(50,30){positron}
\put(50,25){(3.5Gev)}
\put(70,38){$B_{CP}$}
\put(70,25){$B_{tag}$}

\put(135,60){$\mu^+$}
\put(143,53){$\mu^-$}
\footnotesize
\put(128,54){J/$\psi$}
\put(128,43){$K_S$}
\normalsize
\put(165,47){$\pi^+$}
\put(165,37){$\pi^-$}

\put(87,23){$B^0$}
\put(87,41){$\bar{B}^0$}
\footnotesize
\put(77,52){$\Delta t = 0$}
\put(100,15){$\Delta z$}
\normalsize
\put(113,29){$\nu_\mu$}
\footnotesize
\put(123,23){$\pi^-$}
\normalsize
\put(123,28){$\bar{D}^0$}
\put(144,26){$\pi^-$}
\footnotesize
\put(136,20){$K^+$}
\put(132,18){$\mu^+$}
\normalsize
\put(10,0){Figure 8: Schematic drawing of the vertex reconstruction of two B decay vertices.}
\put(10,-5){In the illustration $B_{tag}$ and $B_{CP}$ is assumed $B^0$ at the decay time $B_{tag}$ $(\Delta t = 0)$}
\end{picture}
\end{figure}\vspace{10 mm}

Thus, we can approximately assume that $(\beta\phi_1)_B$ is common for both $B$ mesons. To make the $\Delta z$ length sizable, we boost $B$ meson by an asymmetric energy of the $e^+e^-$ collider, 3.5 GeV for $e^+$ and 8.0 GeV for $e^-$. The produced $\Upsilon (4S)$ possesses $(\beta\phi_1)_\Upsilon = 0.425$. Because $B$ meson pair is produced at the rest frame of the $\Upsilon (4S)$, $B$ mesons also possess $(\beta\phi_1)_B = 0.425$. Since $B$ meson lifetime is 1.5 - 1.7 ps, the produced $B$ mesons run about $200\mu$m before they decay, which are sizable length by the detector. The decay vertex of $B_{CP}$ is reconstructed by leptons from $J/\psi$ decay and the decay vertex of $B_{tag}$ is reconstructed by all remaining tracks after the $J/\psi K_S$ reconstruction. Figure 8 shows the schematic drawing of the vertex reconstruction of two $B$ decay vertices. As equation (73) states, accurate measurement of $\Delta z$ is crucial issue because the integration of equation (73) with $\Delta t$ vanishes $\sin 2\phi_1$. For the precise measurement of the $B$ decay vertex, the Belle is equipped with the silicon vertex detector.\\

The wrong tagging probability varies event by event according to the $B_{tag}$ decay products. The detector response also varies event by event by multiple scattering and energy loss of the tracks, and/or resolution of each hit on the detector, and so on. To take into account the event-by-event effect, we determine $\sin 2\phi_1$ from the asymmetric $\Delta t$ distribution by the unbinned-maximum-likelihood method.\\\\\noindent
\textit{\textbf{3. Constraints on $\sin 2\phi_1$ and Unitarity Triangle}}\\

The review of current experimental constraits is given in this subsection. Those measurements define the preferable area for $\phi_1$ by specifying the apex of $(\tilde{\rho},\tilde{\eta})$ in figure 4(b).\\

The entries in the first two rows of the CKM matrix are accessible in so-called direct (tree-level) processes, i.e. in weak decays of hadrons containing the corresponding quarks. $|V_{ud}|$ and $|V_{us}|$ are known to an accuracy of better than 1\%, $|V_{cb}|$ is known to 5\% and $|V_{cd}|$ and $|V_{us}|$ are known to about 10-20\%. Hence, the two Wolfenstein parameters $\lambda$ and $A$ are rather well determined experimentally:
\[\lambda = |V_{us}| = 0.2205 \pm 0.0018 ,\hspace{10 mm} A = \left|\frac{V_{cb}}{V_{us}^2}\right| = 0.80 \pm 0.04\]
On the other hand, $|V_{ub}|$ has an uncertainty of $\sim$ 30\%, and the same is true for $|V_{td}|$, which is obtained from $B^0-\bar{B}^0$ mixing. This implies rather significant uncertainty in $\rho$ and $\eta$. A more precise determination of these parameters will be done by the B-factory experiments.\\

To determine the shape of the triangle, one can aim for measurements of the two sides and three angles. So far, experimental information is available only on the sides of the triangle. Using the Wolfenstein parameterization and equation (18), the two sides of the unitarity triangle are 
expressed as
\begin{equation}
R_b \equiv |(0,0) \leftrightarrow (\tilde{\rho},\tilde{\eta})| = \sqrt{\tilde{\rho}^2 + \tilde{\eta}^2} = \frac{1}{\lambda}(1 - \frac{\lambda^2}{2})\frac{|V_{ub}|}{|V_{cb}|}
\end{equation}
\begin{equation}
R_t \equiv |(1,0) \leftrightarrow (\tilde{\rho},\tilde{\eta})| = \sqrt{(1 - \tilde{\rho}^2) + \tilde{\eta}^2} \simeq \frac{1}{A\lambda^3}|V_{td}V_{tb}^\ast|
\end{equation}
$|V_{td}V_{tb}^\ast |$ is accessible through $B^0-\bar{B}^0$ mixing by the measurement of mass difference, $\Delta M$. the theoretical prediction by the Standard Model is [13]
\begin{equation}
|V_{td}V_{tb}^\ast |= \frac{6\pi^2\Delta M}{G^2_Fm^2_Wn^{tt}_{QCD}B_{B_d}f^2_{B_d}m_{B_d}S(\frac{m_t}{m_W})}
\end{equation}
where $n^{tt}_{QCD}$ is known as QCD correction factor measured to be $n^{tt}_{QCD} = 0.55\pm 0.01$ [14], and $S(\frac{m_t}{m_W})$ is a function of the top quark mass. The product, $B_{B_d}f^2_{B_d}$, parameterizes the hadronic matrix element of a local four-quark operator between $B_d$ meson states. Another way to improve the determination of $R_t$ is through a measurement of $B^0_S-\bar{B}^0_S$ mixing, 
\begin{equation}
R_t^2 = \frac{f^2_{B_S}B_{B_S}}{f^2_{B_s}B_{B_S}}\cdot\frac{m_{B_S}}{m_{B_d}}\cdot\frac{\Delta M_{B_S}}{\Delta M}\cdot\frac{1-\lambda^2(1-2\tilde{\rho})}{\lambda^2}
\end{equation}

The advantage of this way to the one from $\Delta M$ alone is that th elimination of dependence of $m_t$ and the ratio of $\frac{f_{B_S}}{f_{B_d}}$ can be more precisely determined than each decay constant itself. Presently only a lower limit on $\Delta M_{B_S}$ is obtained, and thus it gives upper limit of $R_t$.\\\\\noindent
Another constraint is given by $K^0-\bar{K}^0$ mixing parameter $\epsilon_K$. the constraint arising in the $\tilde{\rho}-\tilde{\eta}$ plane forms hyperbola, depending on a hadronic parameter $B_K$.\\

The preferred region for $\phi_1$ is 
\[0.47<\sin 2\phi_1<0.89\]
at 95\% confidence level[15], or $\sin 2\phi_1 = 0.70 \pm 0.07$ at 68\% confidence level[16].\\\\\noindent
\textbf{4}. \textit{\textbf{Previous Measurements of }}$\mathbf{ \sin 2\phi_1}$\\

The first direct measurement of $\sin 2\phi_1$ was presented by the OPAL collaboration in 1998. They selected 24 candidates of $B^0\rightarrow J/\psi K_S$ decay with a purity of $\sim$ 60\% from $4.4\times 10^6$ hadronic $Z^0$ decays. They obtained [17] 
\[\sin 2\phi_1(OPAL) = 3.2^{+1.8}_{-2.0}(stat)\pm 0.5(syst).\]
The ALEPH collaboration recorded 23 candidates of $B^0\rightarrow J/\psi K_S$ from $4.2 \times 10^6$ hadronic $Z^0$ decays with estimated purity of 71\%. The result was [18]
\[\sin 2\phi_1(ALEPH) = 0.84^{+0.82}_{-1.04}(stat)\pm 0.16(syst).\]
The CDF collaboration also reported $\sin 2\phi_1$ value. In $110 pb^{-1}$ of proton antiproton collison at $\sqrt{s}=1.8Tev$, they accumulated $395\pm31$ events of $B^0\rightarrow J/\psi K_S$ candidates, with a signal-to-noise ratio $\frac{S}{N}$ of 0.7. Their conclusion was [19] 
\[\sin 2\phi_1(CDF) = 0.79^{+0.41}_{-0.44}(stat+syst)\]
Recently the CDF collaboration updated their result wih improvement of analysis thechnique [20]. The result is: $\sin 2\phi_1(CDF) = 0.91\pm 0.32(stat)+\pm (syst)$, but this is still preliminary.\\\\\\\noindent
\textbf{Conclusion}\\

The violation of $CP$ symmetry enables physicists to make an absolute distinction between matter and antimatter. The distinction between matter and antimatter may have profound implications for cosmology. One of the unsolved theoretical questions in physics is why the universe is made chiefly of matter. With a series of debatable but plausible assumptions, it can be demonstrated that the observed matter-antimatter ratio may have been produced by the occurrence of $CP$ violation in the first seconds after the ``Big Bang", the violent explosion that is thought to have resulted in the formation of the universe. In the Standard Model it turns out that there is only one way to introduce a parameter that gives $CP$ violation. The observed $CP$ violation in K decay then leads to predicted relationships between a number of $CP$ violating effects in B meson decays. A primary purpose of the B Factory at KEKB is to look for these effects and study whether or not they fit the pattern predicted by the Standard Model. KEK physicits predicts the observation of $CP$ violation in B-meson system.  Most physicists hope that they do not because this will then give some clues to the nature of physics beyond the Standard Model.\\\\\noindent
\textbf{References:}\\\\\noindent
[1] C.S.Wu, E. Ambler, R.W. Hayward, D.D. Hoppes and R.P. Hudson, $Phys. Rev.$ 105 1413 (1957).\\\\\noindent
[2] L. Wolfenstein, $Phys. Rev. Lett.$ 13, 362 (1964).\\\\\noindent
[3] S. Bennet et al., $Phys. Rev. Lett.$ 19 993 (1967).\\\\\noindent
[4] M. Gell-Mann, $Phys. Lett.$ 8 214 (1964).\\\\\noindent
[5] M. Kobayashi and T. Maskawa, $Prog. Theor. Phys.$ 49, 652 (1973).\\\\\noindent
[6] N. Cabibbo, $Phys. Rev. Lett.$ 10 531 (1963).\\\\\noindent
[7] S.L. Glashow, J. Illipoulos and L. Maiani, $Phys. Rev.$ D2 1585 (1970).\\\\\noindent
[8] L. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, $Phys. Rev. Lett.$ 13 138 (1964)\\\\\noindent
[9] L. Wolfenstein in $Phys. Rev. Lett.$ 51, 1945(1983).\\\\\noindent  
[10] A.J. Buras, M.E. Lautenbacher and G. Ostermaier, $Phys. Rev.$ D 50, 3433 (1994).\\\\\noindent
[11] C.O. Dib et al.. $Phys. Rev.$ D 41, 1522(1990).\\\\\noindent
[12] T. Inami and C.S. Lim, $Prog. Theor. Phys.$ 65 297 (1981) [Erratum-ibid. 65 1772 (1981)];
F.J. Gilman and M.B. Wise, $Phys. Rev.$ D27,1128 (1983);
A.J. Buras, M. Jamin and P.H. Weisz, $Nucl. Phys.$ B 347 491 (1990).\\\\\noindent
[13] J.S. Hagelin, $Nucl. Phys.$ B 193 123 (1981).\\\\\noindent
[14] CKM Fitter:
ckmfitter.in2p3.fr
A. Hocker, H. Lacker, S. Laplace, and F. Le Diberder, $Eur. Phys.$ J. C 21 225 (2001).\\\\\noindent
[15] M. Ciuchini et al., JHEP 0107 013 (2001) .\\\\\noindent
[16] H. Hirano et al., $Nucl. Instrum. Meth.$ A 455 259 (2000).\\\\\noindent
[17] R. Barate et al., [ALEPH Collaboration], $Phys. Lett.$ B 492 259 (2000).\\\\\noindent
[18] T. Affodler et al., [CDF Collaboration], $Phys. Rev.$ D 61, 072005 (2000).\\\\\noindent
[19] V. Papadimitriou, The Fifth  KEK Topical Conference, (2001).\\\\\noindent
[20] G. Alimonti et al., [BELLE Collaboration], $Nucl. Instrum. Meth.$ A 453 71 (2001).\\\\\noindent
[21] A.B. Carter and A.I. Sanda, $Phys. Rev. Lett.$ 45,952 (1980); $Phys. Rev.$ D23, 1567 (1981).\\\\\noindent
[22] I.I. Bigig and A.I. Sanda, $Nucl. Phys.$ B193,85 (1981); $Nucl. Phys.$ B281, 41 (1987).\\\\\noindent   
[23] I. Duneitz and J. Rosner, $Phys. Rev.$ D34, 1404 (1986).\\\\\noindent
[24] L.S. Littenberg $Phys. Rev.$ D39, 3322 (1989).\\\\\noindent	
[25] G. Buchalla and A.J. Buras, $Nucl. Phys.$ B400, 225 (1993);
     A.J. Buras $Phys. Lett.$ B333, 476 (1994).\\\\\noindent	 
[26] Y. Grossman and Y. Nir, $Phys. Lett.$ B398, 163 (1997).\\\\\noindent 
[27] M. Gronan and D. London, $Phys. Rev. Lett.$ 65, 3381 (1990).\\\\\noindent
[28] Y. Nir and H.R. Quinn, $Phys. Rev. Lett.$ 67, 541 (1991).\\\\\noindent
[29] H.J. Lipkin et al.. $Phys. Rev.$ D44, 1454 (1991).\\\\\noindent
[30] I. Duneitz et al.. $Phys. Rev.$ D43, 2193 (1991).\\\\\noindent
[31] I.I. Bigi et al.. in $CP$ $violation$, ed. C. Jarlskog, Wolrd Science Singapore (1992).\\\\\noindent
[32] M. Bander, S. Silverman and A. Soni $Phys. Rev. Lett.$ 43, 242(1979).\\\\\noindent  
[33] R. Fleischer, Z. Phys. C341, 205 (1995);
     A.J. Buras R. Fleisher, $Phys. Rev. Lett.$ B336, 471 (1994).\\\\\noindent   	 
[34] R. Fleisher Z. Phys. C62, 81 (1994);
     N.G. Deshpande and X.-G. He, $Phys. Rev. Lett.$ 80, 3456 (1998).\\\\\noindent  
[35] R. Fleisher, Int. I. Mod. Phys. A12, 2459 (1997).\\\\\noindent  
[36] R. Godang et al.. (CLEO Collaboration), $Phys. Rev. Lett.$ 78, 3257 (1997).\\\\\noindent
[37] C.O. Dib, D.London and Y. Nir, Int. J. Mod. Phys. A6, 1253 (1991);
     M.Gronau and D.London, $Phys. Rev.$ D55, 2845 (1997);
     Y. Grossman, Y. Nir and R. Rattazzi, (1997)  to appear in Heavy Flavours II, eds. A.J. Buras and M. Lindner, Advanced Series on Direction in High Energy Physics, World. Scientific Publishing, Singapore.\\\\\noindent  
[38] Y. Grossman and M. Worah, $Phys. Rev. Lett.$ B395, 241 (1997).
\end{document} 

