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\begin{document}
\title
{A new interpretation \\ of one CPT violation test\\
for $K_{0} - {\overline{K}}_{0}$ system.}
\author{K. Urbanowski\footnote{e--mail:
K.Urbanowski@if.uz.zgora.pl; K.Urbanowski@proton.if.uz.zgora.pl}
 \\ University of Zielona Gora, Institute of Physics,\\ ul. Podgorna
50, 65--246 Zielona Gora, Poland.} \maketitle
\noindent PACS
numbers:
11.30.Er, 13.20.Eb, 13.25.Es, 14.40.Aq\\
Key words: CPT symmetry, CPT symmetry tests, superweak interactions.\\

\begin{abstract}
Using a more accurate approximation than that  applied by
Lee--Oehme--Yang we show that the interpretation of the tests
measuring the difference between the $K_{0}$ mass and the
${\overline K}_{0}$ mass as the CPT--symmetry test is wrong. We
find that in fact such tests are tests for the existence of the
hypothetical interaction allowing the first order $|\Delta S| = 2$
transitions $K_{0} \rightleftharpoons {\overline K}_{0}$.
\end{abstract}

\section{Introduction}

CPT symmetry is a fundamental theorem of axiomatic quantum field
theory which follows from locality, Lorentz invariance, and
unitarity \cite{cpt}. Many tests of CPT--invariance consist in
searching for decay of neutral kaons. The proper interpretation of
them is crucial. All known CP--  and hypothetically possible
CPT--violation effects in neutral kaon complex are described by
solving the Schr\"{o}dinger--like evolution equation \cite{LOY}
--- \cite{chiu} (we use $\hbar = c = 1$ units)
\begin{equation}
i \frac{\partial}{\partial t} |\psi ; t >_{\parallel} =
H_{\parallel} |\psi ; t >_{\parallel} \label{l1}
\end{equation}
for $|\psi ; t >_{\parallel}$ belonging to the subspace ${\cal
H}_{\parallel} \subset {\cal H}$ (where ${\cal H}$ is the state
space of the physical system under investigation), e.g., spanned
by orthonormal neutral  kaons states $|K_{0}> \stackrel{\rm
def}{=} |{\bf 1}>, \; |{\overline{K}}_{0}> \stackrel{\rm def}{=}
|{\bf 2}> $, and so on, (then states corresponding to the decay
products belong to ${\cal H} \ominus {\cal H}_{\parallel}
\stackrel{\rm def}{=} {\cal H}_{\perp}$). The  nonhermitean
effective Hamiltonian $H_{\parallel}$ acts in ${\cal H}_{||}$ and

\begin{equation}
H_{\parallel} \equiv M - \frac{i}{2} \Gamma, \label{new1}
\end{equation}
where
\begin{equation}
M = M^{+}, \; \; \Gamma = {\Gamma}^{+}, \label{new1a}
\end{equation}
are $(2 \times 2)$ matrices.

Relations between matrix elements of  $H_{\parallel}$  implied  by
CPT--transformation properties of the Hamiltonian $H$ of the total
system, containing  neutral  kaon  complex  as  a  subsystem,  are
crucial for designing CPT--invariance and CP--violation tests  and
for proper interpretation of their results.

The eigenstates of  $H_{\parallel}$,  $|K_{l}>$  and  $|K_{s}>$,
for the eigenvalues ${\mu}_{l}$ and ${\mu}_{s}$ respectively
\cite{LOY}
--- \cite{chiu}, \cite{ur92} --- \cite{ur97-hep-ph}
\begin{equation}
{\mu}_{l(s)} = h_{0} - (+) h \equiv m_{l(s)} - \frac{i}{2}
{\gamma}_{l(s)},  \label{r5}
\end{equation}
where $m_{l(s)}, {\gamma}_{l(s)}$ are real, and
\begin{eqnarray}
h_{0} & = & \frac{1}{2}(h_{11} + h_{22}), \label{r5a} \\
h & \equiv & \sqrt{ h_{z}^{2} + h_{12} h_{21} }, \label{r5b} \\
h_{z} & = & \frac{1}{2} (h_{11} - h_{22}), \label{r5c} \\
h_{jk} & = & <{\bf j}|H_{\parallel}|{\bf k}>, \; (j,k=1,2),
\end{eqnarray}
correspond to the long (the vector $|K_{l}>$) and  short (the
vector $|K_{s}>$) living superpositions of $K_{0}$ and
$\overline{K_{0}}$.

Using the eigenvectors
\begin{equation}
|K_{1(2)}> \stackrel{\rm def}{=} 2^{-1/2} (|{\bf 1}> + (-) |{\bf
2}>) , \label{K-1,2}
\end{equation}
of the CP--transformation for the eigenvalues $\pm  1$ (we define
${\cal C}{\cal P} |{\bf 1}> = - |{\bf 2}>$, \linebreak ${\cal
C}{\cal P} | {\bf 2}> = - |{\bf 1}>$), vectors  $|K_{l}>$ and
$|K_{s}>$ can be expressed as follows
\cite{LOY-etc,Comins,{ur97-hep-ph}}

\begin{equation}
|K_{l(s)}> \equiv (1 + |{\varepsilon}_{l(s)}|^{2})^{- 1/2}
[|K_{2(1)}
> + {\varepsilon}_{l(s)} |K_{1(2)} > ] , \label{K-ls}
\end{equation}
where
\begin{eqnarray}
{\varepsilon}_{l} & = & \frac{h_{12} - h_{11} + {\mu}_{l}}{h_{12}
+ h_{11} - {\mu}_{l}} \equiv - \frac{h_{21} - h_{22} +
{\mu}_{l}}{h_{21} + h_{22} - {\mu}_{l}},
\label{e-l} \\
{\varepsilon}_{s} & = & \frac{h_{12} + h_{11} - {\mu}_{s}}{h_{12}
- h_{11} + {\mu}_{l}} \equiv - \frac{h_{21} + h_{22} -
{\mu}_{s}}{h_{21} - h_{22} + {\mu}_{s}}, \label{e-s}
\end{eqnarray}
This form of $|K_{l}>$ and $|K_{s}>$ is used in many papers when
possible departures from CP-- or CPT--symmetry in  the system
considered are discussed.

One can easily  verify that if ${\mu}_{l} \neq {\mu}_{s} $ then:
\begin{equation}
h_{11} = h_{22} \; \Longleftrightarrow \;{\varepsilon}_{l} =
{\varepsilon}_{s}. \label{e-l=s}
\end{equation}

Experimentally measured values of parameters ${\varepsilon}_{l},
{\varepsilon}_{s}$ are very small for neutral kaons. Assuming
\begin{equation}
|{\varepsilon}_{l}| \ll 1, \; \; |{\varepsilon}_{s}| \ll 1,
\label{e<1}
\end{equation}
one can, e.g. find:
\begin{equation}
h_{11} - h_{22} \simeq ({\mu}_{s} - {\mu}_{l}) ({\varepsilon}_{s}
- {\varepsilon}_{l} ). \label{h-h-simeq}
\end{equation}

Keeping in mind that $h_{jk} = M_{jk} - \frac{i}{2} {\Gamma}_{jk},
M_{kj} = M_{jk}^{\ast},{\Gamma}_{kj} = {\Gamma}_{jk}^{\ast}$
(where $M_{jj} \stackrel{\rm def}{=} M_{j}, \; (j =1,2)$ and
usually it is assumed that $M_{j}, \, (j =1,2)$ denotes the masses
of the particle "1" and its antiparticle "2" \cite{LOY}---
\cite{data}) and using (\ref{h-h-simeq}) one can find, among
others that \cite{ur97-hep-ph,dafne}
\begin{eqnarray}
\Re \, (h_{11} - h_{22})  \equiv M_{1} - M_{2} & \simeq & -
({\gamma}_{s} - {\gamma}_{l}) [ \tan {\phi}_{SW}\,  \Re \,
(\frac{{\varepsilon}_{s}
 - {\varepsilon}_{l}}{2})  \nonumber \\
& \; & - \Im \,(\frac{{\varepsilon}_{s} - {\varepsilon}_{l}}{2})
\, ], \label{test}
\end{eqnarray}
where $\Re \, (z)$ denotes the real part, $\Im \, (z)$ - the
imaginary part of a complex number $(z)$, and
\begin{equation}
\tan {\phi}_{SW} \stackrel{\rm def}{=} \frac{2(m_{l} -
m_{s})}{{\gamma}_{s} - {\gamma}_{l}} . \label{tan-phi}
\end{equation}

One should remember that relations (\ref{h-h-simeq}), (\ref{test})
are valid only if condition (\ref{e<1}) holds. On the other hand
it is appropriate to emphasize at this point that all relations
(\ref{K-ls}) --- (\ref{test}) do not depend on the specific form
of the effective Hamiltonian $H_{\parallel}$ and that they do not
depend on the approximation used to calculate $H_{||}$ . They are
induced by geometric relations between various base vectors in
two--dimensional subspace ${\cal H}_{\parallel}$ but the
interpretation of the relation, eg., (\ref{test}), depends on
properties of the matrix elements $h_{jk}$ of the effective
Hamiltonian $H_{\parallel}$. This means that, if for example,
$H_{\parallel} \neq H_{LOY}$, where $H_{LOY}$ is the
Lee--Oehme--Yang effective Hamiltonian, then the interpretation of
the relation (\ref{test}), e.g., as the CPT symmetry test, and
other similar relations need not be the same for $H_{\parallel}$
and for $H_{LOY}$.

The aim of this note is to analyze the interpretation of the test
based on the relation (\ref{test}) which is commonly considered as
the CPT violation test \cite{LOY1} --- \cite{Comins},
\cite{ur97-hep-ph,dafne}. Such an interpretation follows from the
properties of the matrix elements of $H_{LOY}$.

In the kaon rest frame, the time evolution is governed by the
Schr\"{o}dinger equation
\begin{equation}
i \frac{\partial}{\partial t} U(t)|\phi > = H U(t)|\phi >,  \; \;
U(0) = I, \label{Sr-eq}
\end{equation}
where $I$ is the unit operator in $\cal H$, $|\phi > \equiv |\phi
; t_{0} = 0> \in {\cal H}$  is  the  initial  state  of  the
system:
\begin{equation}
|\phi  >  \equiv  |\psi  >_{\parallel} \equiv q_{1}|{\bf 1}> +
q_{2}|{\bf 2}> \, \in {\cal H} \label{init}
\end{equation}
(in  our case  $|\phi ;t> = U(t) |\phi >$), $H$ is the total
(selfadjoint) Hamiltonian, acting in $\cal H$. Thus the  total
evolution  operator $U(t)$ is unitary.

Starting from the Schro\"{o}dinger equation and using the
Weisskopf--Wigner method Lee, Oehme and Yang derived the following
formula for the matrix elements $h_{jk}^{LOY}, \, (j,k =1,2)$ of
their effective Hamiltonian $H_{LOY}$ (see
\cite{LOY}---\cite{Comins}, \cite{ur-pi-00}):
\begin{eqnarray}
h_{jk}^{LOY} & = &  m_{0}\, {\delta}_{j,k} - {\Sigma}_{jk} (m_{0}
) , \; \; \;
(j,k = 1,2) ,  \label{h-jk-LOY} \\
& = & M_{jk}^{LOY} - \frac{i}{2} {\Gamma}_{jk}^{LOY}
\label{M-jk-LOY}
\end{eqnarray}
where ${\Sigma}_{jk}(\epsilon) = <{\bf j}|\Sigma (\epsilon)|{\bf
k}>, \, (j,k = 1,2)$, and
\begin{equation}
\Sigma ( \epsilon ) = PHQ \frac{1}{QHQ - \epsilon - i 0} QHP =
{\Sigma}^{R}(\epsilon) + i {\Sigma}^{I}(\epsilon), \label{Sigma}
\end{equation}
and ${\Sigma}^{R}(\epsilon = {\epsilon}^{\ast}) =
{\Sigma}^{R}(\epsilon = {\epsilon}^{\ast})^{+}, \,
{\Sigma}^{I}(\epsilon = {\epsilon}^{\ast}) = {\Sigma}^{I}(\epsilon
= {\epsilon}^{\ast})^{+}$, $P$ is the projector operator onto the
subspace ${\cal H}_{||}$:
\begin{equation}
P \equiv |{\bf 1}><{\bf 1}| + |{\bf 2}><{\bf 2}|, \label{P}
\end{equation}
$Q$ is the projection operator onto the subspace of decay products
${\cal H}_{\perp}$:
\begin{equation}
Q \equiv I - P, \label{Q}
\end{equation}
and vectors $|{\bf 1}>$, $|{\bf 2}>$ considered above are
eigenstates of the free Hamiltonian, $H^{(0)}$, ( here $H \equiv
H^{(0)} + H^{I}$), for 2-fold degenerate eigenvalue $m_{0}$:
\begin{equation}
H^{(0)} |{\bf j} > = m_{0} |{\bf j }>, \; \;  (j = 1,2),
\label{H-0-j}
\end{equation}
$H^{I}$ denotes the interaction which is responsible for the decay
process.

This means that
\begin{equation}
[P, H^{(0)}] = 0. \label{P-H-0}
\end{equation}
The condition guaranteeing the occurrence of transitions  between
subspaces ${\cal H}_{\parallel}$ and ${\cal H}_{\perp}$, i.e.,  a
decay process of states in ${\cal H}_{\parallel}$, can  be written
as follows
\begin{equation}
[P,H^{I}] \neq 0, \label{P-H-I-neq-0}
\end{equation}
that is
\begin{equation}
[P,H] \neq 0, \label{[P,H]}
\end{equation}

The operator $H_{LOY}$ has the following form
\begin{equation}
H_{LOY} = m_{0} P  - \Sigma (m_{0}) \equiv PHP -  \Sigma (m_{0}).
\label{H-LOY}
\end{equation}
Within the LOY approach it is assumed that
\begin{equation}
PH^{I}P \equiv 0, \label{LOY=ds=1-3}
\end{equation}
which means that in this case
\begin{equation}
PHP \equiv PH^{(0)}P = m_{0} P. \label{LOY=m0P}
\end{equation}

Following the method used by LOY one can discard the assumption
(\ref{LOY=ds=1-3}) or not. This has no effect on the properties of
the matrix elements of $H_{LOY}$. The assumption
(\ref{LOY=ds=1-3}) is a reflection of the opinion of physicists
deriving and applying $H_{LOY}$ that matrix elements of $H^{I}$
are too small in comparison with $m_{0}$ in order to have any
effect on time evolution in ${\cal H}_{||}$ (see
\cite{LOY,LOY1,LOY-etc,ur-pi-00,Lee-qft}). In other words, the
approximation applied by LOY leads to the operator $H_{LOY}$ whose
properties do not depend on whether $H_{12} \stackrel{\rm def}{=}
<{\bf 1}|H|{\bf 2}> \neq 0$ or not. (Note that within the LOY
assumptions  $H_{12} \equiv  <{\bf 1}|H^{I}|{\bf 2}>$). Thus the
theory of the evolution in neutral $K$ meson complex based on the
LOY approach, that is using $H_{LOY}$, is not sensitive to the
presence (or absence) of the hypothetical interactions causing the
first order $| \Delta S |= 2$ transitions in the system considered
\cite{Wolfenstein,Jarlskog,Lee-qft}.

The relation (\ref{LOY=ds=1-3}) is the mathematical expression of
the assumption used in the Standard Model of $K_{0}
\rightleftharpoons {\overline{K}}_{0}$ mixing and CP violation
that there are not direct, first order $K_{0} \rightleftharpoons
{\overline{K}}_{0}$ transitions \cite{Bigi,Buras}, that is that
there are no first order $| \Delta S |= 2$ transitions:  $H_{12}
\equiv 0$.

Usually, in LOY and related approaches, it is assumed that
\begin{equation}
{\Theta}H^{(0)}{\Theta}^{-1} = {H^{(0)}}^{+} \equiv H^{(0)} ,
\label{Theta-H-0}
\end{equation}
where $\Theta$ is the antiunitary operator
\cite{messiah,bohm,Wigner}:
\begin{equation}
\Theta \stackrel{\rm def}{=} {\cal C}{\cal P}{\cal T}.
\label{Theta=CPT}
\end{equation}

The subspace of neutral kaons ${\cal H}_{\parallel}$ is assumed to
be invariant under $\Theta$:
\begin{equation}
{\Theta} P {\Theta}^{-1} = P^{+} \equiv P. \label{Theta-P}
\end{equation}

Now, if additional to (\ref{Theta-H-0})
${\Theta}H^{I}{\Theta}^{-1} = H^{I}$, that is if
\begin{equation}
[ \Theta , H] = 0, \label{CPT-H}
\end{equation}
then using, e.g., the following phase convention \cite{LOY1}
--- \cite{Yu-V}
\begin{equation}
\Theta |{\bf 1}> \stackrel{\rm def}{=} - |{\bf 2}>, \;\;
\Theta|{\bf 2}> \stackrel{\rm def}{=} - |{\bf 1}>, \label{cpt1}
\end{equation}
and taking into account that $< \psi | \varphi > =
<{\Theta}{\varphi}|{\Theta}{\psi}>$, one easily finds from
(\ref{h-jk-LOY}), (\ref{Sigma})  that
\begin{equation}
h_{11}^{LOY} - h_{22}^{LOY} = 0  \label{LOYh11=h22}
\end{equation}
in the CPT--invariant system. This  is  the standard result of the
LOY approach and this is the picture  which one meets in  the
literature \cite{LOY}  ---  \cite{chiu}, \cite{dafne,Lee-qft}.
Property (\ref{LOYh11=h22}) leads to the conclusion that (see
(\ref{e-l=s}))
\begin{equation}
{\varepsilon}_{l} - {\varepsilon}_{s} = 0. \label{LOY-e-l=s}
\end{equation}
Therefore the tests based on the relation (\ref{test}) are
considered as the test of CPT--invariance and the results of such
tests are interpreted that the masses of the particle "1" (the
$K_{0}$ meson) and its antiparticle "2" (the ${\overline{K}}_{0}$
meson) must be equal if CPT--symmetry holds. Parameters appearing
in the right side of the relation (\ref{test}) can be extracted
from experiments in such tests and then these parameters can be
used to estimate of the right side of this relation. Such obtained
estimation for the mass difference according to the recent data
\cite{data} reads
\begin{equation}
\frac{|M_{1} - M_{2}|}{m_{K_{0}}} = \frac{|m_{K_{0}} -
m_{{\overline{K}}_{0}}|}{m_{K_{0}}} \leq 10^{-18}, \label{mk-mk}
\end{equation}
and this estimation is considered as no indication of
CPT--violation effect. This interpretation follows from the
properties of the $H_{LOY}$.

\section{Beyond the LOY approximation}

The more exact approximate formulae for $H_{||}$ than those
obtained within the LOY approach (i.e. than $H_{LOY}$) can be
derived using  the  Krolikowski--Rzewuski (KR) equation for the
projection of a state vector \cite{KR,ur-pra-94}, \cite{ur92}
--- \cite{ur95},  \cite{Pi00}. KR equation results from
the Schr\"{o}dinger  equation  for  the  total system under
consideration \cite{KR} and leads to $H_{||}$ which is very
sensitive to the properties of the matrix elements of $PHP$.
Within this approach \cite{KR,horwitz} in general $H_{||} =
H_{||}(t)$ and
\begin{equation}
H_{\parallel} (t) \equiv PHP + V_{\parallel}(t). \label{H||(t)}
\end{equation}
In the case (\ref{[P,H]}), to the lowest nontrivial order the
following formula for $V_{||}(t)$ has been found in
\cite{ur93,{ur-pra-94}}
\begin{equation}
V_{\parallel}(t) \cong V_{\parallel}^{(1)} (t) \stackrel{\rm
def}{=} -i \int_{0}^{\infty} K(t - \tau ) \exp {[} i ( t - \tau )
PHP {]} d \tau . \label{V||(t)}
\end{equation}
where:
\begin{eqnarray}
K(t) & = & {\mit \Theta} (t) PHQ \exp (-itQHQ)QHP, \label{K(t)} \\
{\mit \Theta} (t) & = & { \{ } 1 \;{\rm for} \; t \geq 0, \; \; 0
\; {\rm for} \; t < 0 { \} } . \nonumber
\end{eqnarray}

We are rather interested in the properties of the system at long
time period, at the same for which the LOY approximation was
calculated,  and therefore we will consider the properties of
\begin{equation}
V_{||} \stackrel{\rm def}{=}\lim_{t \rightarrow \infty}
V_{||}^{(1)}(t), \label{V||-infty}
\end{equation}
instead of the general case $V_{||}(t) \cong V_{||}^{(1)}(t)$.

For simplicity we assume that the CPT--symmetry is conserved in
our system, that is that the condition (\ref{CPT-H}) holds. The
consequence of this  assumption is that
\begin{equation}
H_{11} = H_{22} \stackrel{\rm def}{=} H_{0}, \label{H_0}
\end{equation}
where
\begin{equation}
 H_{jk} = <{\bf j}|H|{\bf k}>,  \label{H-jk}
\end{equation}
and $(j,k= 1,2)$.

So, in the case of the projector $P$ given by the formula
(\ref{P}) for
\begin{equation}
PHP \equiv H_{0}\, P, \label{P-H12=0}
\end{equation}
that is for
\begin{equation}
H_{12} = H_{21} = 0,  \label{H12=0}
\end{equation}
one finds that
\begin{equation}
V_{||} = - \Sigma (H_{0}). \label{V-H-0}
\end{equation}
This means that in the case (\ref{P-H12=0})
\begin{equation}
H_{||} = H_{0} \, P - \,\Sigma (H_{0}), \label{H||-H12=0}
\end{equation}
(where $H_{||} = \lim_{t \rightarrow \infty} H_{||}(t)$), that is
exactly as in the LOY approach (see (\ref{H-LOY})). This also
means that in such a case simply $(h_{11} - h_{22}) = 0$ when CPT
symmetry is conserved.

On the other hand, in the case
\begin{equation}
H_{12} = H_{21}^{\ast} \neq 0, \label{H12n0}
\end{equation}
and conserved CPT, one obtains \cite{Pi00}
\begin{eqnarray}
V_{||} & = & - \frac{1}{2} \Sigma (H_{0} + |H_{12}|)\, \Big[ \Big(
1 - \frac{H_{0}}{|H{_{12}|}} \Big)P + \frac{1}{|H_{12}|} PHP \Big]
\nonumber \\
& &  - \frac{1}{2} \Sigma (H_{0} - |H_{12}|)\, \Big[ \Big( 1 +
\frac{H_{0}}{|H{_{12}|}} \Big)P - \frac{1}{|H_{12}|} PHP \Big].
\label{V-H12n0}
\end{eqnarray}

In the case of preserved  CPT--symmetry  matrix elements of
$\Sigma (\epsilon)$ have the following properties
\cite{ur93,ur95,ur98}
\begin{equation}
{\Sigma}_{11} ( \epsilon = {\epsilon}^{\ast} ) \equiv
{\Sigma}_{22} ( \epsilon = {\epsilon}^{\ast} ) \stackrel{\rm
def}{=} {\Sigma}_{0} ( \epsilon = {\epsilon}^{\ast} ) .
\label{sigma-jk-0}
\end{equation}
Therefore  matrix  elements  $v_{jk} = <{\bf j}|V_{||}|{\bf k}>,
\, (j,k =1,2)$ we are interested in take the following form
\begin{eqnarray}
v_{j1} = & - & \frac{1}{2} {\Big\{ } {\Sigma}_{j1} (H_{0} + |
H_{12} |)
+ {\Sigma}_{j1} (H_{0} - | H_{12} |)  \nonumber  \\
& + & \frac{H_{21}}{|H_{12}|} {\Sigma}_{j2} (H_{0} + | H_{12} |) -
\frac{H_{21}}{|H_{12}|} {\Sigma}_{j2} (H_{0} - | H_{12} |) {\Big\}
} ,
\nonumber \\
& \; &   \label{v-jk} \\
v_{j2} = & - & \frac{1}{2} {\Big\{ } {\Sigma}_{j2} (H_{0} + |
H_{12} |)
+ {\Sigma}_{j2} (H_{0} - | H_{12} |)  \nonumber  \\
& + & \frac{H_{12}}{|H_{12}|} {\Sigma}_{j1} (H_{0} + | H_{12} |) -
\frac{H_{12}}{|H_{12}|} {\Sigma}_{j1} (H_{0} - | H_{12} |) {\Big\}
}, \nonumber
\end{eqnarray}

The form of these matrix elements is rather inconvenient for
searching for their properties depending on the matrix elements
$H_{12}$ of $PHP$. It can be done relatively simple assuming
\cite{ur95,ur98}
\begin{equation}
|H_{12}| \ll |H_{0} | . \label{H12<H0}
\end{equation}
Within such assumption one finds \cite{ur95,ur98}
\begin{equation}
v_{j1} \simeq - {\Sigma}_{j1} (H_{0} ) - H_{21} \frac{ \partial
{\Sigma}_{j2} (x) }{\partial x}
\begin{array}[t]{l} \vline \, \\ \vline \,
{\scriptstyle x = H_{0} } \end{array} , \label{vj1-H12<}
\end{equation}
\begin{equation}
v_{j2} \simeq - {\Sigma}_{j2} (H_{0} ) - H_{12} \frac{ \partial
{\Sigma}_{j1} (x) }{\partial x}
\begin{array}[t]{l} \vline \, \\ \vline \,
{\scriptstyle x = H_{0} } \end{array} , \label{vj2-H12<}
\end{equation}
where $j = 1,2$.  One  should  stress  that  due  to  a presence
of resonance terms, derivatives $\frac{\partial}{\partial x}
{\Sigma}_{jk} (x)$ need not  be  small,  and  so  the  products
$H_{jk} \frac{\partial}{\partial x} {\Sigma}_{jk}  (x)$  in
(\ref{vj1-H12<}), (\ref{vj2-H12<}).

From this formulae we  conclude  that, e.g., the  difference
between  diagonal   matrix   elements   which plays  an important
role in designing tests of type (\ref{test}) for  the neutral
kaons system, equals to the lowest order of  $|H_{12}|$,

\begin{equation}
h_{11} - h_{22} \simeq H_{12} \frac{ \partial {\Sigma}_{21} (x) }
{\partial x}
\begin{array}[t]{l} \vline \, \\ \vline \,
{\scriptstyle x = H_{0} } \end{array} - H_{21} \frac{ \partial
{\Sigma}_{12} (x) }{\partial x}
\begin{array}[t]{l} \vline \, \\ \vline \,
{\scriptstyle x = H_{0} }\end{array} \neq 0. \label{h11-h22-H12<}
\end{equation}

So, in a general case, in  contradistinction  to  the property
(\ref{LOYh11=h22})  obtained within the LOY theory, one finds for
diagonal matrix elements  of $H_{\parallel}$ calculated within the
above described approximation  that  in  CPT--invariant system the
nonzero matrix elements, $H_{12} \neq 0$, of $PHP$ cause that
$(h_{11} - h_{22}) \neq 0$.

From the formula (\ref{h11-h22-H12<}) it follows that the left
side of the relation (\ref{test}) takes the following form in the
case of very weak interactions allowing for the nonzero first
order transitions $|{\bf 1}> \rightleftharpoons |{\bf 2}>$
\begin{equation}
M_{1} - M_{2} \, = \, \Re \, (h_{11} - h_{22}) = 2\, \Im \, \Big(
H_{21} { \frac{\partial {\Sigma}_{12}^{I}(x)}{\partial x} \,
\vrule \,}_{x=H_{0}} \, \Big) \, + \, \ldots  \, \neq \, 0.
\label{Re-h11-h22}
\end{equation}
(Note that as a matter of fact assuming  (\ref{H-0-j}) one has
$H_{21} \equiv <{\bf 2}|H^{I}|{\bf 1}>$ in (\ref{Re-h11-h22})).
Thus taking into account this result and the implications of the
assumptions (\ref{P-H12=0}), (\ref{H12=0}) one can conclude that
in the considered approximation
\begin{equation}
\Re \, (h_{11} - h_{22})\, = 0 \, \Leftrightarrow \, |H_{12}| \, =
\, 0. \label{Re=0}
\end{equation}
Finally, using result (\ref{Re-h11-h22}) one can replace the
relation (\ref{test}) by the following one:
\begin{eqnarray}
2\, \Im \, \Big( <{\bf 2}|H^{I}|{\bf 1}> { \frac{\partial
{\Sigma}_{12}^{I}(x)}{\partial x} \, \vrule \,}_{x=H_{0}} \, \Big)
& \simeq & - ({\gamma}_{s} - {\gamma}_{l}) [ \tan {\phi}_{SW}\,
\Re \, (\frac{{\varepsilon}_{s}
 - {\varepsilon}_{l}}{2})   \nonumber \\
& \; & - \Im \,(\frac{{\varepsilon}_{s} - {\varepsilon}_{l}}{2})
\, ]. \label{test-a}
\end{eqnarray}


\section{Discussion}

The result (\ref{Re-h11-h22}) is in  full agreement with the
conclusions drawn in \cite{ur02-hep-ph} on the ground of basic
assumptions of quantum theory. In \cite{ur02-hep-ph} it has been
shown that the diagonal matrix elements of the exact effective
Hamiltonian governing the time evolution in the subspace of states
of an unstable particle and its antiparticle can not be equal at
$t > t_{0} =0$ ($t_{0}$ is the instant of  creation of the pair)
when the total system under consideration is CPT invariant but CP
noninvariant. The proof of this property is rigorous. The unusual
consequence of this result is that in such a case, contrary to the
properties of stable particles, the masses of the unstable
particle "1" and its antiparticle "2" need not be equal for $t \gg
t_{0}$. In fact there is nothing strange in this conclusion. From
(\ref{CPT-H}) (or from the CPT theorem \cite{cpt} of axiomatic
quantum field theory) it only follows that the masses of particle
and antiparticle eigenstates of $H$ (i.e., masses of stationary
states for $H$) should be the same in the CPT invariant system.
Such a conclusion can not be drawn from (\ref{CPT-H}) for particle
"1" and its antiparticle "2" if they are unstable, ie., if states
$|{\bf 1}>, |{\bf 2}>$ are not eigenstates of $H$. There is no
axiomatic quantum field theory of unstable particles.

In this place one should explain that the property $H_{12}=H_{21}
= 0$, which according to (\ref{h11-h22-H12<}) implies that
$(h_{11} - h_{22}) = 0$ in the considered approximation, does not
mean that the predictions following from the use of the exact
effective Hamiltonian (or the more accurate effective Hamiltonian
than the LOY theory) should lead to the the same masses for
$K_{0}$ and for ${\overline{K}}_{0}$. This does not contradict the
above mentioned conclusion  about masses of unstable particles
drawn in \cite{ur02-hep-ph} for the exact $H_{||}$: Simply, in the
case $H_{12}=H_{21} = 0$ the mass difference is very, very small
and should arise at higher orders of the more accurate
approximation considered in Sec.2.

Using the above, briefly described formalism, one can find
$(h_{11} - h_{22})$ for generalized Fridrichs--Lee model
\cite{chiu,ur93}. Within this model one finds \cite{ur98}
\begin{eqnarray}
\Re \, (h_{11}  -  h_{22}) \stackrel{\rm df}{=} \Re \,
(h_{11}^{FL} - h_{22}^{FL}) & \simeq & i \frac{
m_{21}{\Gamma}_{12} - m_{12}{\Gamma}_{21} }{4(m_{0} - \mu )}
\nonumber \\
& \equiv & \frac{{\Im \,}(m_{12}{\Gamma}_{21})}{2(m_{0}- \mu )}.
\label{FL1}
\end{eqnarray}
This estimation has been obtained in the case of conserved
CPT--symmetry for $|m_{12}| \ll (m_{0}- \mu)$, which corresponds
to (\ref{H12<H0}). In (\ref{FL1}) ${\Gamma}_{12}, {\Gamma}_{21}$
can be identified with those appearing in  the LOY theory, $m_{0}
\equiv H_{11} = H_{22}$ can be considered  as  the kaon mass
\cite{chiu}, $m_{jk} \equiv H_{jk} \, (j,k =1,2)$, $\mu$ can be
treated as the mass of the decay products of the neutral  kaon
\cite{chiu}.

For neutral $K$-system, to evaluate $(h_{11}^{FL} - h_{22}^{FL})$
one can follow, e.g., \cite{chiu,dafne} and one can take
$\frac{1}{2}{\Gamma}_{21} = \frac{1}{2}{\Gamma}_{12}^{\ast} \sim
\frac{1}{2}{\Gamma}_{s} \sim 5 \times 10^{10} {\rm sec}^{-1}$ and
$(m_{0} - \mu ) = m_{K} - 2m_{\pi} \sim 200$ MeV $\sim 3 \times
10^{23} {\rm sec}^{-1}$ \cite{data}. Thus
\begin{equation}
\Re \, (h_{11}^{FL} - h_{22}^{FL}) \sim  1,7 \times 10^{-13}
|m_{12}|\equiv 1,7 \times 10^{-13} |H_{12}| . \label{FL2}
\end{equation}

For the Fridrichs--Lee model it has been found in \cite{ur93} that
$h_{jk}(t) \simeq h_{jk}$ practically for $t \geq T_{as} \simeq
\frac{10^{2}}{\pi (m_{0} - |m_{12}| - \mu )}$. This leads to the
following estimation for the neutral $K$--system  $T_{as} \sim
10^{-22}$ sec.

Dividing both sides of (\ref{FL2}) by $m_{0}$ one arrives at the
relation corresponding to (\ref{mk-mk}):
\begin{equation}
\frac{\Re \, (h_{11}^{FL} - h_{22}^{FL})}{m_{0}} \sim  1,7 \times
10^{-13} \frac{|m_{12}|}{m_{0}}\equiv 1,7 \times 10^{-13}
\frac{|H_{12}|}{m_{0}}. \label{FL3}
\end{equation}
So, it is sufficient for $\frac{|H_{12}|}{m_{0}}$ to be
$\frac{|H_{12}|}{m_{0}} < 10^{-5}$ in order to fulfil  the
experimentally obtained estimation (\ref{mk-mk}). The required
order of $\frac{|H_{12}|}{m_{0}}$ is easily reached by the
hypothetical Wolfenstein superweak interactions
\cite{Wolfenstein}, which admits first order $|\Delta S| = 2$
transitions $K_{0} \rightleftharpoons {\overline K}_{0}$, that is,
which assumes a non--vanishing first order transition matrix
$H_{12} = <{\bf 1}|H^{I}|{\bf 2}> \sim g G_{F} \neq 0$ with $ g
\ll G_{F}$.

Note that contrary to the approximation described in Sec. 2, the
LOY approximation, as well as the similar approximation leading to
the Bell--Strinberger unitary relations \cite{ Bell} are unable to
detect effects caused by the existence (or absence) of the
superweak interactions (the interactions for which $H_{12} \neq
0$) in the system.

Let us analyze some important observations following from
(\ref{Re-h11-h22}), (\ref{test-a}) and from the rigorous result
obtained in \cite{ur02-hep-ph}. The non--vanishing of the right
hand side of the relation (\ref{test}) can not be considered as
the proof that the CPT--symmetry is violated. So, there are two
general conclusions following from (\ref{Re-h11-h22}),
(\ref{Re=0}), (\ref{test-a}) and \cite{ur02-hep-ph}. The first
one: the tests based on the relation (\ref{test}) can not be
considered as CPT--symmetry tests. The second one: such tests
should be considered as the tests for the existence of new
hypothetical (superweak) interactions allowing for the first order
$| \Delta S |= 2$ transitions.

On the other hand, one should remember that the non--vanishing
right hand side of the relations (\ref{test}), (\ref{test-a}) can
be considered as the conclusive proof that new interactions
allowing for the first order $|\Delta S |= 2$ transitions $K_{0}
\rightleftharpoons {\overline K}_{0}$ exist only if another
experiment, based on other principles, definitively confirms that
the CPT--symmetry is not violated in $K_{0} - {\overline K}_{0}$
system.

Unfortunately the accuracy of the today's experiments is not
sufficient to improve the estimation (\ref{mk-mk}) to the order
required by (\ref{FL3}). In the light of the above estimations,
keeping in mind (\ref{Re-h11-h22}), only much more accurate tests
based on the relation (\ref{test}) can give the answer whether the
superweak interactions exists or not.

Last remark, other results \cite{ur93,ur95,ur98} obtained  within
the approximation described in Sec. 2  suggest  also that the form
of other parameters usually used to describe properties of $K_{0}
- {\overline K}_{0}$ system is different for the case $H_{12} \neq
0$ and for the case $H_{12} = 0$. This can be used as the basis
for designing other tests for the hypothetical superweak
interactions.

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\end{document}


