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\documentclass[preprint,preprintnumbers]{revtex4}
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\begin{document}

\preprint{}

\title{CP Violation}% Force line breaks with \\

\author{J.C. Yoon}
\email{jcyoon@u.washington.edu} \affiliation{University of
Washington}

\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
CP violation theory is based on the nonorthogonality of physical
particles. In order to avoid this, CP violation with orthogonal
physical particles is introduced and the charge asymmetry of
$\overline{K}^{0}$ is suggested as a possible experimental test.
\end{abstract}

\pacs{}% PACS, the Physics and Astronomy
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\maketitle
\section{Introduction.}
CP violation theory is based on the effective Hamiltonian from the
Weisskopf-Wigner approximation \cite{WW}. The time evolution of
the neutral meson(denoted by $P^0$, $\overline{P}^{0}$) can be
described by
\begin{equation}
i{d \over {d\tau}} \Psi
  =  (M -  {i \over 2} \Gamma)\Psi
\end{equation}
where two hermitian matrices M and $\Gamma$ in the basis of $P^0$,
$\overline{P}^{0}$ are
\begin{equation}
M = \left( \begin{array}{c c} M_{11} & M_{12} \\ M^{*}_{12} &
M_{22}
\end{array} \right) ~~~\mathrm{and}~~~
\Gamma = \left( \begin{array}{c c} \Gamma_{11} & \Gamma_{12} \\
\Gamma^{*}_{12} & \Gamma_{22}
\end{array} \right)
\end{equation}
The matrix $M -  {i \over 2} \Gamma$ has eigenvalues
\begin{eqnarray}
M_{S,L} - {i \over 2} \Gamma_{S,L} &=& M_{0} - {i \over 2}
\Gamma_{0} \nonumber \\
&\pm& \sqrt{(M_{12}-{i \over 2}\Gamma_{12})(M^{*}_{12}-{i \over
2}\Gamma^{*}_{12})}
\end{eqnarray}
where $M_{11} = M_{22} \equiv M_{0}$ and $\Gamma_{11} =
\Gamma_{22} \equiv \Gamma_{0}$ under the assumption of CPT
invariance. The corresponding eigenstates are given by
\begin{eqnarray}
|P_{S}\rangle & = & [(1 + \epsilon_{P} + \delta_P)|{P^0}\rangle
+(1 - \epsilon_{P} -
\delta_{P})|{\overline{P}^{0}}\rangle]/{\sqrt2}
\quad  \nonumber\\
|{P_{L}} \rangle & = & [(1 + \epsilon_P - \delta_P)|{P^0}\rangle
-(1 - \epsilon_P + \delta_P)|{\overline{P}^{0}}\rangle]/{\sqrt 2}
\quad
\end{eqnarray}
The complex parameter $\epsilon_P$ represents a CP violation with
T violation, while the complex parameter $\delta_P$ represents a
CP violation with CPT violation.

\section{Problem of CP violation.}
If quantum states are not orthogonal, it implies one state can be
observed as the other state at any given time. Therefore, it is
ambiguous how the physical particles evolve in time with their own
eigenvalues while they are not orthogonal. Also, the definition of
off-diagonal elements in the decay matrix is not clear. While the
diagonal elements are defined by the decay rate of $P^0$ or
$\overline{P}^{0}$ into the final states, the initial state of the
decay rate $\Gamma_{12}$ is not well-defined.

This is because we start with the effective Hamiltonian and obtain
the eigenvalues from this equation using the decay rate matrix.
Also, in Weisskopf-Wigner approximation, it is presumably assumed
that the initial quantum states are already eigenstates. However,
the initial quantum states of neutral mesons are not eigenstates
when we apply this approximation. Since we do not have a strong
evidence why Shr$\rm{\ddot{o}}$dinger equation fails, here I will
suggest to start with Shr$\rm{\ddot{o}}$dinger equation and apply
the effective Hamiltonian.

\section{The Bell-Steinberger Equation.}
We consider a coherent mixture of $P_S$ and $P_L$ whose amplitude
at its proper time $t$ is described by the wave function
\begin{eqnarray}
\Psi(t) & = & a_S e^{-i(M_S - {i \over 2}{\Gamma_S}) t} |S \rangle
+ a_L e^{-i(M_L - {i \over 2} {\Gamma_L})t } |L\rangle \nonumber
\end{eqnarray}
where $|a_{S}|^2 + |a_{L}|^2 = 1$. Based on the assumption that
$P_S$ and $P_L$ are not orthogonal \cite{LeeYangOehme}, the
Bell-Steinberger relation is derived from identifying the decay
rate
\begin{eqnarray}
- {d \over d\tau} \langle \Psi |
 \Psi \rangle  =
|a_{S}|^{2}\Gamma_{S}e^{-\Gamma_{S}t} +
|a_{L}|^{2}\Gamma_{L}e^{-\Gamma_{L}t} \nonumber \\
 + a_{S}^{*}a_{L}(i \Delta M +\Gamma_{0})e^{-i \Delta M\tau
-\Gamma_{0}t}\langle S|L\rangle \nonumber
\\
-a_{S}a_{L}^{*} (i \Delta M -\Gamma_{0})e^{+i \Delta M t
-\Gamma_{0} t } \langle L| S\rangle \label{leftBS}
\end{eqnarray}
by the total transition rate
\begin{eqnarray}
\sum |\langle {f}| T |\Psi \rangle |^{2}
 =  |a_{S}|^{2}\sum |\langle f |T |S \rangle|^{2}
e^{-\Gamma_{S}t} \nonumber \\
+ |a_{L}|^{2}\sum |\langle f |T |L \rangle |^{2} e^{-\Gamma_{L} t} \nonumber\\
 + a_{S}^{*}a_{L}\sum \langle f | T | S \rangle ^{*}\langle
f |T | L \rangle e^{-i \Delta M\tau -\Gamma_{0} t} \nonumber \\
 + a_{S}a_{L}^{*} \sum \langle f |T | S \rangle \langle f|T|L
\rangle ^{*}e^{+i \Delta M t -\Gamma_{0} t
 } \label{rightBS}
\end{eqnarray}
However, it is in question whether these equations are the same
since the interactions are not considered in Eq. (\ref{leftBS})
and it is presumably assumed that the final states inherits the
properties of $P^{0}$ and $\overline{P}^{0}$ states in decay
amplitude. If we sum the decay rates over all types of decays,
only $2 \pi$ and $3 \pi$ decays are responsible for the
interference terms since those of the semileptonic decays cancel
each other. It is because $2 \pi$ and $3 \pi$ decays get the
contributions from both $P^{0}$ and $\overline{P}^{0}$. The more
fundamental reason is that neutral meson is not an elementary
particle and consists of particle and antiparticle. Though we call
$P^{0}$ and $\overline{P}^{0}$ particle and antiparticle, none of
both is usual particle and antiparticle in the sense that usual
particle interacts only as a particle not as an antiparticle or as
both of them. In semileptonic decay, only one of quark is
dominating the interactions in a way that the final states
inherits the property of the initial particles in decay amplitude.
Therefore, if we have only semileptonic decays $f$ and
$\overline{f}$, there is no interference terms when we sum over
the decay rates since $f$ and $\overline{f}$ inherits the
properties of initial states $P^{0}$ and $\overline{P}^{0}$
respectively. However, in $2\pi$ and $3\pi$ decays, quark and
antiquark involves in a way that the final states of $P^{0}$ and
$\overline{P}^{0}$ are quantum mechanically indistinguishable.
Since these unusual interactions are not considered in Eq.
(\ref{leftBS}) and it is also assumed that every final state
inherits the properties of the initial particles while not in
$2\pi$ and $3\pi$ decays, it is not Eq. (\ref{leftBS}) but Eq.
(\ref{rightBS}) that is the appropriate equation describing the
decays of neutral mesons. Also, it is far fetched to attribute
interference to the nonorthogonality of the initial states, since
it is due to the existence of $2 \pi$ type decays, which is the
consequence of unusual interactions of neutral mesons.

\section{CP violation of effective mass.}
In order to avoid nonorthogonality of physical particles, I
introduce an idea of effective mass in neutral meson with an
assumption of CPT symmetry \cite{CPTAssumption}. It is presumably
assumed that the two states of neutral mesons are the same whether
we start with initial states of $P^{0}$ and $\overline{P}^{0}$.
However, it is possible to have different masses in each
considering a meson is only approximately described as two free
quarks while gluon fields may play an important role in
determining the mass of meson. Let us consider a ground state of
$P^{0}$. Later, $\overline{P}^{0}$ is introduced by the
interactions exchanging quark to antiquark and vice versa. Since
the gluon field was also satisfied in the ground state, just
flipping quark and antiquark will introduce a different
interactions between quark and gluon field and this may cause the
mass difference. This also can be interpreted as an effective mass
when quark fields are exchanged. Since the mesons in the decay
amplitudes still can be understood as free quarks neglecting the
gluon field interactions, the decay amplitude will be the same as
before. As long as we have the same mass for the initial states of
$P^{0}$ and $\overline{P}^{0}$, CPT symmetry is not violated.

\section{CP Violation Mechanism.}
Let us consider two close quantum states of neutral mesons
ignoring other possible states. The Shr$\rm{\ddot{o}}$dinger
equation of the neutral meson is given by
\begin{equation}
i{d \over {d\tau}} \Psi =  M\Psi
\end{equation}
where hermitian matrix M in the basis of $P^0$, $\overline{P}^{0}$
is
\begin{eqnarray}
M = \left( \begin{array}{c c} M_{11} & M^{*}_{12} \\ M_{12} &
M_{22}
\end{array} \right) = \left( \begin{array}{c c} M_{0} - \Delta E/2
& M^{*}_{12} \\ M_{12} & M_{0} + \Delta E/2
\end{array} \right) \nonumber
\end{eqnarray}
for the initial state(the ground state) is $P^0$.
\begin{eqnarray}
\overline{M} = \left( \begin{array}{c c} M_{11} & M_{12} \\
M^{*}_{12} & M_{22}
\end{array} \right) = \left( \begin{array}{c c} M_{0} + \Delta E/2
& M_{12} \\ M^{*}_{12} & M_{0} - \Delta E/2
\end{array} \right) \nonumber
\end{eqnarray}
for the initial state(the ground state) is $\overline{P}^0$. The
matrix $M$ and $\overline{M}$ has eigenvalues
\begin{eqnarray}
M_{S,L} &=& M_{\overline{S},\overline{L}} = M_{0} \pm \sqrt{\Delta
E^{2}/4 + |M_{12}|^{2}}.
\end{eqnarray}
In general, the corresponding eigenstates are given by
\begin{eqnarray}
|S\rangle & = & [(1 - \Delta_P)|{P^0}\rangle +(1 +
\Delta_{P})|{\overline{P}^{0}}\rangle]/{\sqrt {2%(1 +|\Delta_{P}|^{2})
}}
\quad  \nonumber\\
|L \rangle & = & [(1 + \Delta^{*}_P)|{P^0}\rangle -(1 -
\Delta^{*}_P)|{\overline{P}^{0}}\rangle]/{\sqrt {2%(1 +|\Delta_{P}|^{2})
}}~~~~~
\end{eqnarray}
for the initial state of $P^0$ and
\begin{eqnarray}
|\overline{S}\rangle & = & [(1 + \Delta^{*}_P)|{P^0}\rangle +(1 -
\Delta^{*}_{P})|{\overline{P}^{0}}\rangle]/{\sqrt {2%(1 +|\Delta_{P}|^{2})
}}
\quad  \nonumber\\
|\overline{L} \rangle & = & [-(1 - \Delta_P)|{P^0}\rangle +(1 +
\Delta_P)|{\overline{P}^{0}}\rangle]/{\sqrt {2%(1 +|\Delta_{P}|^{2})
}}~~~~~
\end{eqnarray}
for the initial state of $\overline{P}^0$  where the complex
parameter $\Delta_P$ represents a CP violation, the real part
represents CPT violation and the imaginary part corresponds T
violation.
\begin{eqnarray}
\Delta_{P} = {{\Delta E/2} + i{\rm{Im}M_{12}} \over
{{\rm{Re}}M_{12} + \sqrt{ \Delta E^{2}/4 + |M_{12}|^{2}} }}
\end{eqnarray}
From the eigenstates we obtained, we can set up the effective
Hamiltonian
\begin{equation}
i{d \over {d\tau}} \Psi
  =  (M -  {i \over 2} \Gamma)\Psi
\end{equation}
where two hermitian matrices M and $\Gamma$ in the basis of
$P^0_{S}$, ${P}^{0}_{L}$ are
\begin{equation}
M = \left( \begin{array}{c c} M_{S} & 0 \\ 0 & M_{L}
\end{array} \right) ~~~\mathrm{and}~~~
\Gamma = \left( \begin{array}{c c} \Gamma_{S} & 0 \\ 0 &
\Gamma_{L}
\end{array} \right)
\end{equation}
This effective Hamiltonian is not a fundamental equation governing
the physics since the quantum states will evolve in time without
$\Gamma_{S,L}$, but it is induced from the fact that the decay
rates will be exponential. Therefore, it only has a proper
physical meaning in the decay rates including interactions.
$\Gamma_{S,L}$ should be obtained from the decay rates and
explains the length of the lifetime as it will be shown. The
following calculation of this CP violation mechanism will be
similar to the usual calculations except the physical meaning of
the parameters are significantly different.

\section{The Decay Rates.}
The decays of the neutral meson can be classified into two types,
one is where the transition amplitude are contributed by both
$P^{0}$ and $\overline{P}^{0}$ such as $2\pi$ and $3 \pi$ decays,
the other is where the decays are contributed by either $P^{0}$ or
$\overline{P}^{0}$ such as semileptonic decays $f$ and
$\overline{f}$. Let us consider the decays of $K^{0}$ and
$\overline{K}^{0}$. In semileptonics decays, assuming no direct CP
violation for initial states of $K^{0}$ and $\overline{K}^{0}$,
the initial state $K^{0}$ will evolve in time to be
\begin{eqnarray}
|K(t) \rangle & = & (1 - \Delta^{*}_{P})e^{-im_{S}t-\gamma_{S}t/2}
|S \rangle + (1 + \Delta_{P}) e^{-im_{L}t-\gamma_{L}t/2}| L\rangle
\nonumber
\end{eqnarray}
where $\Delta m = m_{S} - m_{L}$ and the decay rates for the
initial state of $K^{0}$ will be
\begin{eqnarray}
{dN \over dt} \Big{|}_{f}
&=&{|F_{f}|^2 \over 4}\Big((1 - 4 {\rm{Re}}
\Delta)e^{-\gamma_{S}t} + (1 +
4{\rm{Re}}\Delta)e^{-\gamma_{L}t}\nonumber \\
&&{}+ 2 \cos \Delta m e^{-\gamma t/2} \Big) \nonumber \\
{dN \over dt} \Big{|}_{\overline{f}}
&=& {|F_{f}|^2 \over 4}\Big(e^{-\gamma_{S}t} + e^{-\gamma_{L}t} -
2 \cos \Delta m e^{-\gamma t/2} \Big)
\end{eqnarray}
to the first order of CP violation parameters. The first will be
referred to as the right sign decay and the later as the wrong
sign decay. The initial state $\overline{K}^{0}$ will evolve in
time to be
\begin{eqnarray}
|\overline{K}(t) \rangle & = & (1 -
\Delta_{P})e^{-im_{S}t-\gamma_{S}t/2} |\overline{S} \rangle + (1 +
\Delta^{*}_{P}) e^{-im_{L}t-\gamma_{L}t/2}| \overline{L}\rangle
\nonumber
\end{eqnarray}
where $\Delta \overline{m} = \Delta m = m_{\overline{S}} -
m_{\overline{L}}$, $m_{S,L} = m_{\overline{S},\overline{L}}$ and
$\gamma_{S,L} = \gamma_{\overline{S},\overline{L}}$ and the decay
rates for the initial state of $\overline{K}^{0}$ will be the same
respectively for the right and wrong decays to the first order of
CP violation parameters. In $2 \pi$ decays, assuming the decay
amplitudes are the same for $K^{0}$ and $\overline{K}^{0}$, the
decay rates for the initial state of $K^{0}$ will be
\begin{eqnarray}
{dN \over dt} \Big{|}_{2\pi}
&=&{|F_{2\pi}|^2} \Big((1 - 2 {\rm{Re}} \Delta)e^{-\gamma_{S}t} +
|\Delta|^{2} e^{-\gamma_{L}t} \nonumber \\
&&{} + 2[ {\rm{Re}}\Delta \cos \Delta m + {\rm{Im}}\Delta \sin
\Delta m] e^{-\gamma t/2} \Big)~~~~~
\end{eqnarray}
The decay rates for the initial state of $\overline{K}^{0}$ will
be
\begin{eqnarray}
{d\overline{N} \over dt} \Big{|}_{2\pi}
&=&{|F_{2\pi}|^2} \Big((1 - 2 {\rm{Re}} \Delta)e^{-\gamma_{S}t} +
|\Delta|^{2} e^{-\gamma_{L}t} \nonumber \\
&&{} + 2[ {\rm{Re}}\Delta \cos \Delta m - {\rm{Im}}\Delta \sin
\Delta m] e^{-\gamma t/2} \Big)~~~~~
\end{eqnarray}
In $3 \pi$ decays, assuming the decay amplitudes have the opposite
sign for $K^{0}$ and $\overline{K}^{0}$, the decay rates for the
initial state of $K^{0}$ and $\overline{K}^{0}$ can be obtained
with a simple calculation. These decay rates are similar to those
of usual CP violation theory except for the physical meaning of
the parameters. Therefore, the most of experimental observations
will be successfully explained by these decay rates.

\section{The Difference of The Lifetimes.} The difference between
the lifetimes of $P_S$ and $P_L$ should be explained by the
difference of the decay rates as we have observed in simple
exponential decays since the property of the exponential is
originated from the decay amplitude. This should also be how we
find $\Gamma_{S}$ and $\Gamma_{L}$. Let us consider only
semileptonic decays and $2 \pi$ and $3 \pi$ decays.
\begin{eqnarray}
\Gamma_{S} &=& |F_{2\pi}|^{2}(1 - 2{\rm{Re}}\Delta_{P})
+|F_{3\pi}|^{2}|\Delta_{P}|^{2} + {|F_{f}|^{2} \over 4} (2 -
4{\rm{Re}}\Delta_{P})
 \nonumber \\
\Gamma_{L} &=& |F_{3\pi}|^{2}(1 +
2{\rm{Re}}\Delta_{P})+|F_{2\pi}|^{2}|\Delta_{P}|^{2} +
{|F_{f}|^{2} \over 4} (2 + 4{\rm{Re}}\Delta_{P})
\end{eqnarray}
In $K^{0}$ meson, $2\pi$ decays should be dominating over other
decays since $\Gamma_{S}$ is roughly $10^3$ times bigger than
$\Gamma_{L}$. Considering the branching ratios of semileptonic
decays and $3\pi$ are of the same order in $K_{L}$,
$|F_{2\pi}|^{2}$ is $10^{3}$ times bigger than $|F_{3\pi}|^{2}$
and $|F_{f}|^{2}$. Since $\Delta_{K}$ is about $10^{-3}$, we
expect the branching ratio of $2\pi$ in $K_{L}$ to be order of
$10^{-3}$ and that of $3\pi$ in $K_{S}$ to be order of $10^{-9}$.
They are in a good agreement with experiments except for $3\pi$ in
$K_{S}$ is a bit smaller than the experimental value (order of
$10^{-7}$) \cite{PDG}. However, this may be due to a simple
estimation or we might have other effects at this level of
accuracy.

\section{Experimental Test.}
One of the differences between CP violation theories is in $2 \pi$
decay rates. While one has the interference term like $\pm \cos
\Delta m$, the other has $+ {\rm{Re}} \Delta_{P} \cos \Delta m \pm
{\rm{Im}} \Delta_{P} \sin \Delta m$. Though the interference term
has been believed to be like the first and the experimental data
has been analyzed based on this, as long as we have nonzero
${\rm{Im}} \Delta_{P}$, the second interference term would be
similar to the first.

The conclusive test would be the charge asymmetry of
$\overline{K}^{0}$, since CP violation with effective mass predict
$\overline{f}$(the right sign decay of $\overline{K}^{0}$) to be
dominating over $f$, i.e. the charge asymmetry of $+2 {\rm{Re}}
\Delta_{K}$ and the plots of asymmetries to be the same, while the
charge asymmetries are expected to be $-2 {\rm{Re}} \epsilon_{K}$
and the plots are to be different in the tails in other CP
violations.

The difference of asymmetries in $K^{0}$ and $\overline{K}^{0}$
implies symmetry violation in the sense that the decay of neutral
meson prefers one type of final states in long decay whether it is
those of $K^{0}$ or $\overline{K}^{0}$, while it is expected to
the same, i.e. the right sign dominating, with CPT symmetry.
Though we have been tested the asymmetry of $K^{0}$, we still need
to test that of $\overline{K}^{0}$.

\section{Conclusion.}
It is argued that nonorthogonality of physical particles are not
desirable. In order to avoid this, the idea of CP violation with
effective mass is introduced and the charge asymmetry of
$\overline{K}^{0}$ is suggested as a possible experimental test.

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\begin{thebibliography}
{}
\bibitem{WW} V.F.\ Weisskopf and E.P.\ Wignaer, Z. Physik {\bf 65},
(1930) 18;{\bf 63}, (1930) 54.


\bibitem{LeeYangOehme}
T.D. Lee, R. Oehme and C.N. Yang, Phys. Rev. {\bf 106} (1957) 340.

\bibitem{CPTAssumption}
It is possible to describe CP violation with othogonal physical
partilces without an introduction of effective mass, but it leads
to conclude CPT violation. In this work, CPT symmetry is assumed.

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

\end{thebibliography}

\end{document}
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