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\newcommand{\Cerenkov}{\v{C}erenkov}
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\newcommand{\elm}{\mbox{$\mathrm{e}^-$}}
\newcommand{\red}{\mbox{$\mathrm{Re}(\delta)$}}
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\newcommand{\gs}{\mbox{$\Gamma_{\mathrm{S}}$}}



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\newcommand{\mita}{\mbox{$|\eta_{+-}|$}}
\newcommand{\itaoo}{\mbox{$\eta_{00}$}}
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\newcommand{\D}{\mbox{$\Delta$}}
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\begin{document}

\begin{flushright}
 \\
CERN-TH/99-80  \\
LPTENS-99-12 \\
\end{flushright}

\vspace{8mm}

\begin{center}
{\bf
DIRECT 
\mbox{\boldmath{${\cal T}$}}-VIOLATION MEASUREMENTS AND }\\
\vspace*{0.15 cm} 
{ \bf
\mbox{\boldmath{${\cal T}$}}-ODD EFFECTS  IN
DECAY EXPERIMENTS
}
\end{center}

\vspace*{0.25cm}

\begin{center}
{ \bf
L. Alvarez-Gaum\'e $^{a}$,
~C. Kounnas $^{a,b}$, ~S. Lola $^{a}$ ~and~
P. Pavlopoulos $^{c}$
} 



\vspace*{0.6 cm}
$^{a}$
CERN Theory Division, CH-1211 Geneva,
Switzerland \\
\vspace*{0.25 cm}
$^{b}$
Ecole Normale Sup\'erieure 
24 rue Lhommond, \\
F-75231, Paris Cedex 05, France \\
\vspace*{0.25 cm}
$^{c}$
Institut $~$f\"ur Physik, University of Basle CH-4056, and \\
CPLEAR Collaboration, CH-1211 Geneva
Switzerland 


\end{center}



\vspace*{0.4 cm}

\begin{center}
{\bf ABSTRACT}
\end{center}

\noindent
{\small 
Motivated by the recent experimental announcements
for direct measurements of time-reversal
non-invariance in the neutral kaon system, we make a comparative discussion
of the CPLEAR and KTeV measurements.
The most suitable way 
to consistently incorporate the mixing, the time evolution 
and the decays of kaons, is to describe the neutral kaon system
as a system with a non-Hermitean Hamiltonian.
In this framework, the physical (decaying) incoming and outgoing states
are distinct and belong to dual spaces.
Moreover, since they are 
eigenstates of the full Hamiltonian, 
they never oscillate.
This is directly manifest in the orthogonality conditions
of the physical states, which 
entirely determine the evolution of the kaon system.
Along these lines we conclude:
CPLEAR studies $K^0$-$\bar{K}^0$ oscillations,
a process where initial and final states
can be reversed,  the CPLEAR asymmetry being an effect
directly related to the definition
of time-reversal.
Conclusively, CPLEAR provides 
a direct measurement of  ${\cal T}$-violation 
without any assumption either on unitarity or on $CPT$-invariance.
The KTeV experiment studies in particular the process
$K_L \rightarrow \pi^+ \pi^- e^+ e^-$, 
where they measure a ${\cal T}$-odd effect.
However, using unitarity together with estimates of the
final state interactions, it should be
possible to determine whether this
effect can be identified with a genuine
${\cal T}$-reversal violation
.}


\vspace*{0.4 cm}

\begin{center}
{\small {\it { Talk given by S. Lola at the XXXIV$^{th}$ Rencontres 
de Moriond on \\
Electroweak Interactions and Unified Theories, Les Arcs, 13-20 March 1999}}}
\end{center}

\thispagestyle{empty}


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\vfill\eject

\newpage



\section{\large \sl Introduction}

Recently, the CPLEAR experiment at CERN, reported the
first direct observation of time-reversal violation
in the neutral kaon
 system~\cite{at_paper}. This observation is
made by comparing the probabilities
of a $\bar{K}^0$ state transforming into a
$K^0$ and vice-versa. 
Moreover, the KTeV experiment at Fermilab,
similarly reported evidence for
${\cal T}$-violation in the decay
$K_L \rightarrow \pi^+ \pi^- e^+ e^- $.~\cite{KTeV}
In the present note, we will discuss the
experimental asymmetries used by both collaborations
and interpret their measurements on 
${\cal CP}$, ${\cal T}$ and/or
${\cal CPT}$-violation.

The  discrete symmetry properties of the 
neutral kaon system
have been extensively studied 
in the literature \cite{rev}.
To analyse this issue, in a consistent way 
one needs to study a system with a non-hermitean Hamiltonian.
This is clear, because of the following:
Although the physical kaons at rest coincide with
the strong interaction (strangeness) eigenstates
$ |K^0> = |d \bar{s}>$ and  $|\bar{K}^0> = |\bar{d} s>$,
the latter are not the eigenstates of the the full
Hamiltonian. Since however weak interactions do not
conserve strangeness (but also allow $K^0$--$\bar{K}^0$ oscillations)
 the full Hamiltonian
eigenstates, denoted by
$|K_S>$ and $|K_L>$, are different from the 
strangeness eigenstates, and obey the relations
\bea
H~|K_S> & = & \lambda_S~|K_S>~, ~~~~~|K_S(t)> = e^{-i \lambda_S t} |K_S>~,
\nonumber \\
H~|K_L> & = & \lambda_L~|K_L>~, ~~~~~|K_L(t)> = e^{-i \lambda_L t} |K_L>~,
\eea
with $\lambda_L = m_L - i \Gamma_L / 2$
and $\lambda_S = m_S - i \Gamma_S /2 $,
where $m_{S,L}$ denotes the masses of the
physical kaons and 
$\Gamma_{S,L}$ their decay widths.
The complexity of the eigenvalues,
implies the non-hermiticity of the full
Hamiltonian of the neutral kaon system.

Non-hermiticity of $H$
implies that the physical incoming 
and outgoing states ($|K_{S,L}^{in}>$
and $|K_{S,L}^{out}> $ $\equiv $  $  {<K_{S,L}^{out}|}^{\dagger}
$ respectively), are not identical, 
but instead belong to two distinct (dual) spaces \cite{AKLP}.
In the Heisenberg representation (where the
states are time-independent), the
physical incoming and outgoing states coincide with
the left- and right-eigenstates of the full Hamiltonian:
\bea
H~|K_{S,L}^{in}>  ~=~  \lambda_{S,L}~|K_{S,L}^{in}>~,~~~~~
<K_{S,L}^{in}|~H^{\dagger} ~=~ <K_{S,L}^{in}|~\lambda^*_{S,L} ~,
\nonumber \\
H^{\dagger}|K_{S,L}^{out}> ~=~  \lambda_{S,L}^*~|K_{S,L}^{out}>~,~~~~~
{}~<K_{S,L}^{out}|~H  ~=~ <K_{S,L}^{out}|~\lambda_{S,L} ~,
\eea
where 
\bea
|K_{S,L}^{out}> ~~\equiv~~ {<K_{S,L}^{out}|}^{\dagger}  ~~\ne  ~~
|K_{S,L}^{in}>, ~~~~~
<K_{S,L}^{in}| ~~\equiv~~
{|K_{S,L}^{in}>}^{\dagger} ~~ \ne ~~
<K_{S,L}^{out}|~.
\eea
Notice that {\em only if }
$H=H^{\dagger}$, 
$\lambda_{S,L} ~=~ \lambda^*_{S,L}$ and 
$|K_{S,L}^{out}> ~=~ |K_{S,L}^{in}>$,
thus the incoming and outgoing states are identical.
In the generic case ($H \neq H^{\dagger}$),
the time  evolution  of the incoming and
outgoing
states $|\Psi_I^{in}(t_i)>$ and $|\Psi_I^{out}(t_f)>$  are
obtained
from $|\Psi_I^{in}>$ and $|\Psi_I^{out}>$, using the
evolution operators $e^{-iHt_i}$ and
$e^{-iH^{\dagger}t_f}$ respectively:
\bea
|K_{S,L}^{in}(t_i)> ~=~ e^{-iH\,t_i}~|K_{S,L}^{in}>, ~~~~~
|K_{S,L}^{out}{(t_f)}> ~=~ e^{-iH^{\dagger}\,t_f}~|K_{S,L}^{out}>~.
\eea
{}From the above equations, follows the evolution of the conjugate
states:
\bea
<K_{S,L}^{in}(t_i)| ~=~ <K_{S,L}^{in}|~ e^{iH^{\dagger}\,t_i}, ~~~~~
<K_{S,L}^{out}(t_f)| ~=~ <K_{S,L}^{out}|~e^{iH \,t_f}~. 
\eea
An important point to stress here, is
that the physical incoming and outgoing eigenstates
have to obey {\it at all times} the orthogonality conditions
\cite{AKLP}
\bea
<K_I^{out}(t_f)|K_J^{in}(t_i)>
=<K_I^{out}|e^{-iH\Delta t}|K_J^{in}>
=e^{-i\lambda_I~\Delta t}~\delta_{IJ}~,
\eea
and in particular for $ \Delta t = 0$
\bea
<K_L^{out}|K_S^{in}> = 0~, ~~~~~~~~ <K_S^{out}|K_L^{in}> = 0~,
\nonumber \\
<K_S^{out}|K_S^{in}> = 1~, ~~~~~~~~ <K_L^{out}|K_L^{in}> = 1 ~.
\label{ortho}
\eea
unlike what has been stated in a wide part of the
literature.
These conditions express the fact that the Hamiltonian
eigenstates cannot oscillate to each-other at any
time, and therefore an initial $|K_{S}^{in}>$ may not
be transformed to a final
$|K_{S}^{out}>$. Moreover, it follows that
the inner products among incoming
(outgoing) states {\em do not obey}
the usual  orthogonality conditions
\bea
<K_{I}^{in}|K_J^{in}> \ne \delta_{IJ}~~~~{\rm and}
{}~~~~ <K_I^{out}|K_J^{out}> \ne \delta_{IJ}~. 
\eea 
Finally, in the basis of the states $K_L$ and $K_S$,
$H$ can be expressed in terms of a diagonal  $2 \times  2$ matrix
\bea
H = |K_S^{in}> \lambda_S  < K_S^{out}| +
|K_L^{in}> \lambda_L <K_L^{out}| ~,
\label{HAMDIAG}
\eea
where the unity operator $\bf 1$ takes the form:
\bea
{\bf 1}= \sum_{I=S,L}~|K_{I}^{in}><K_{I}^{out}|~.
\eea

\section{\large \sl Study of discrete symmetries in the neutral kaon system}
\noindent

Having clarified our formalism, 
we may now proceed to study particle-antiparticle
mixing in the neutral kaon system.
A convenient representation to study
the action of ${\cal CP}$,
${\cal T}$ and ${\cal CPT}$, is the
$K^0, \bar{K}^0$ (particle-antiparticle) base.
In this representation
\begin{eqnarray}
CP \; |K_0^{in}> & = & |\bar{K}^{in}_0> ~, \nonumber \\
T \; |K_0^{in}> & = & <K_0^{out}|  ~, \nonumber \\
CPT \; |K_0^{in}> & = & <\bar{K}_0^{out}|~. 
\label{EQ1}
\end{eqnarray}


Without loss of generality,
we can express the physical incoming states in terms of
$|K_0^{in}>$ and  $|\bar{K}_0^{in}>$ as:
\begin{eqnarray}
|K_S^{in}> & = & \frac{1}{N} \left ( \; (1+\a) \; |K_0^{in}> + \;
            (1-\a) \; |\bar{K}_0^{in}>  \right ) ~, \nonumber \\
|K_L^{in}> & = & \frac{1}{N} \left ( \; (1+\b) \; |K_0^{in}> - \;
            (1-\b) \; |\bar{K}_0^{in}>  \right ) ~,
\label{EQ2}
\end{eqnarray}
where $\a$ and $\b$ are complex variables
associated with $CP$, $T$ and $CPT$-violation
(usually denoted by $\epsilon_S$ and 
$\epsilon_L$ respectively),
and $N$ a normalization factor. Then, the respective
equations for the outgoing states are not independent,
but are determined by the orthogonality conditions 
for the physical states \cite{AKLP}
\begin{eqnarray}
<K_S^{out}| & = & \frac{1}{\tilde{N}} \left ( \;
(1-\b) <K_0^{out}| + \; (1+\b) <\bar{K}_0^{out}|  \right ) ~, \nonumber \\
<K_L^{out}| & = & \frac{1}{\tilde{N}} \left ( \;
(1-\a) < {K}_0^{out}| - \; (1+\a)  <\bar{K}_0^{out}| \right ) ~.
\label{EQ3}
\end{eqnarray}
where the normalisation 
factor $N$ can always be chosen  equal to
$N=\sqrt{2 (1 - \alpha\beta)}$ \cite{AKLP}
Using the equations(\ref{HAMDIAG}, \ref{EQ2}, 
\ref{EQ3}) the Hamiltonian can be expressed 
in  the basis of
$K^0,\bar{K}^0$ as
\bea
H =  \frac{1}{2} \left(
\begin{array}{cc}
( \lL + \lS) -
\Delta \lambda \frac{ \a-\b}{1-\a\b} &
{}~~~~~~~~~~
\Delta \lambda   \;
\frac{1 + \a\b}{1-\a\b}
+ \Delta \lambda \frac{ \a+\b}{1-\a\b} \\
 & \\
\Delta \lambda  \;
\frac{1 + \a\b}{1-\a\b}
- \Delta \lambda \frac{\a+\b}{1-\a\b} &
{}~~~~~
( \lL + \lS) +
\Delta \lambda \frac{\a-\b}{1-\a\b}
\end{array}
\right )~, 
\label{eqH}
\eea
where $ \Delta \lambda = \lambda_L - \lambda_S$.

From eq.(\ref{eqH}), we can identify
the $T$-, $CP$- and $CPT$- violating parameters. Indeed:

$\bullet$ Under {\underline {$T$--transformations}},
 $$
<K_0^{out}|H |\bar{K}_0^{in}> \; \leftrightarrow \;
<\bar{K}_0^{out}|H|K_0^{in}> ~,
$$
thus, the off-diagonal elements of $H$
are interchanged.
This indicates that the parameter $\epsilon \equiv (\a+\b)/2$,
which is related to the difference of the
off-diagonal elements of $H$,
measures the magnitude of  the $T$-violation\footnote{
$~2/N^2 \approx 1$, in the linear approximation.}.
\begin{equation}
  \frac{2}{N^2}~\e = \frac {
<K_0^{out}|H|\bar{K}_0^{in}> - <\bar{K}_0^{out}|H|K_0^{in}>
}{2 \; \Delta \lambda} ~ .
\label{epsilon1}
\end{equation}



$\bullet$ Under {\underline {$CPT$--transformations}},
$$
<K_0^{out}|H |{ K}_0^{in}>
\; \leftrightarrow \;   <\bar{K}_0^{out}|H |\bar{K}_0^{in}> ~,
$$
and therefore, the parameter
$\delta \equiv (\a-\b)/2$, related to the
difference of the diagonal elements of $H$,
 measures the magnitude
of $CPT$-violation.
\bea
\frac{2}{N^2}~ \d = \frac{<\bar{K}_0^{out}|H|\bar{K}_0^{in}> -
<K_0^{out}|H|K_0^{in}>}
{2 \; \Delta \lambda}  ~.
\eea

$\bullet$ Under {\underline {$CP$--transformation}},
$$
<K_0^{out}|H |K_0^{in}> \; \leftrightarrow \;
<\bar{K}_0^{out}|H |{\bar
K}_0^{in}> ~,
$$
and simultaneously
$$
<K_0^{out}|H |\bar{K}_0^{in}>\; \leftrightarrow \;    <{\bar
K}_0^{out}|H|K_0^{in}> ~,
$$
thus, {\it both} the diagonal and the off-diagonal elements of
$H$
are interchanged.
Then, the parameters $\a=\e+\d$ and $\b=\e-\d$, are the ones
which measure the  magnitude of $CP$-violation in the decays of
$K_S$ and $K_L$  respectively.

\section{\large \sl 
CPLEAR direct measurement of time-reversibility}

Having identified the 
${\cal CP}$, ${\cal T}$ and  ${\cal CPT}$-violating
operations, one may construct asymmetries that measure
discrete symmetry-violations. For instance,
a time-reversal operation interchanges
 initial and final states,
with identical positions and opposite velocities:
\bea
T~[~<\bar{K}_0^{out} (t_f)|K_0^{in}(t_i)>~]~
=~<K_0^{out} (-t_i)|\bar{K}_0^{in}(-t_f)> ~.
\eea 
Assuming time-translation invariance
\bea
T~[~<\bar{K}_0^{out} (t_f)|K_0^{in}(t_i)>~]~
=~<K_0^{out} (t_f)|\bar{K}_0^{in}(t_i)>~.
\eea 
The time evolution from $t_i$ to $t_f$ implies
that 
\bea
<\bar{K}_0^{out} (t_f)|K_0^{in}(t_i)>
& =  &
\frac{1}{N^2} ~ (1-\a) (1-\b) ~
( e^{-i \lambda_S \Delta t} - e^{-i \lambda_L \Delta t} ) ~,
\eea 
\bea
<K_0^{out}(t_f)|\bar{K}_0^{in}(t_i)>
& = &
\frac{1}{N^2} ~ (1+\a) (1+\b) ~
( e^{-i \lambda_S \Delta t} - e^{-i \lambda_L \Delta t} ) ~.
\eea
Then, by definition,  the magnitude of 
${\cal T}$-violation is
directly related to 
the Kabir asymmetry \cite{Kabir}
\begin{eqnarray} 
 A_T 
&=&
\frac{
|<K_0^{out}(t_f)|\bar{K}_0^{in}(t_i)>|^2
-|<\bar{K}_0^{out} (t_f)|K_0^{in}(t_i)>|^2
} {
|<K_0^{out}(t_f)|\bar{K}_0^{in}(t_i)>|^2
+|<\bar{K}_0^{out} (t_f)|K_0^{in}(t_i)>|^2
}~, \nonumber \\
& & \nonumber \\
& = & \frac{
| (1+\a) (1+\b) |^2 - | (1-\a) (1-\b) |^2}
{| (1+\a) (1+\b) |^2 + | (1-\a) (1-\b) |^2} 
 ~\approx~ 4~Re[\epsilon] ~,
\end{eqnarray}
which is time-independent.
Any non-zero value for $A_T$ signals a direct
measurement of
$T$-violation without any assumption about $CPT$ invariance.
Here, we should note that
in linear order in $\epsilon$ and $\delta$, the
approximate equality
\bea
 <K_S^{in}|K_L^{in}>+<K_L^{in}|K_S^{in}>
{}~ \approx ~ 4 Re~[\epsilon] ~,
\label{Tviol}
\eea
holds. This follows directly 
from the non-orthogonality of
the adjoint states $<K_S^{in}|$ and $<K_L^{in}|$
that is manifest in
the equations 
\bea
<K_S^{in}|K_S^{in}>&=& \frac{1+|\alpha|^2}{|1-\alpha\beta|} ~,~~~~~~
<K_L^{in}|K_L^{in}> ~=~ \frac{1+|\beta|^2}{|1-\alpha\beta|} ~,
\nonumber\\
<K_S^{in}|K_L^{in}>&=& \frac{\alpha^*+\beta}{|1-\alpha\beta|}
{}~,~~~~~~
<K_L^{in}|K_S^{in}> ~=~ \frac{\alpha+\beta^*}{|1-\alpha\beta|} ~.
\eea
However, although the time-reversal asymmetry can
{\em in the linear approximation} be expressed
in terms of only incoming states, the conceptual
issue of reversing the time-arrow for
any ${\cal T}$-violation measurement is unambiguous.
For this reason, the CPLEAR
collaboration searched for
${\cal T}$-violation through $K^0$-$\bar{K}^0$ oscillations,
a process where initial and final states can be interchanged.

CPLEAR produces initial neutral kaons with defined strangeness from
proton-antiproton annihilations
at rest, via the reactions
\begin{equation} \ppb \longrightarrow
 \biggl \{\begin{array}{l}
           K^{-} \pip K^0  \\
           K^+ \pim \bar{K}^0 ~,
    \end{array}
\end{equation}
and tags the neutral kaon strangeness at the production time
by the charge of the accompanying charged kaon.
Since weak interactions do not conserve
strangeness,  the $K^0$ and $\bar{K}^0$ may subsequently transform
into each-other via oscillations with $\Delta S = 2$.
The final strangeness of the neutral kaon
is then tagged through the semi-leptonic
decays 
\bea
K^0 & \rightarrow  &  e^+ \pi^- \nu ~, ~~~
\bar{K}^0  \rightarrow    e^- \pi^+ \bar{\nu} ~,\nonumber \\
K^0 & \rightarrow  &  e^- \pi^+ \bar{\nu} ~, ~~~
\bar{K}^0  \rightarrow    e^+ \pi^- \nu ~.
\eea
Among them, the first two are characterized by
$\Delta S = \Delta Q$ while the other two
are characterized by
$\Delta S = -\Delta Q$ and would therefore indicate
either (i) explicit violations of the 
$\Delta S = \Delta Q$ rule, or 
(ii) oscillations between
$K^0$ and $\bar{K}^0$ that 
even if $\Delta S = \Delta Q$ holds,
would lead at a final state
similar to (i) (with the ``wrong-sign'' leptons).
The CPLEAR  experimental asymmetry  is given by
\bea
A_T^{exp}  =
\frac{\overline{R}_+ ~ (\Delta t) - R_{-} ~(\Delta t)}
{\overline{R}_+ ~(\Delta t) + R_{-} ~(\Delta t)} ~, \nonumber
\eea
 with 
\bea
             \overline{R}_+ {}~(\Delta t)
& = &
| <e^+ \pi^- \nu(t_f)~ |\bar{K}_0^{in}(t_i)> \nonumber \\
& + & 
 <e^+ \pi^- \nu(t_f)~ |K_0^{in}(t_f)>
<K_0^{out}(t_f)~ |\bar{K}_0^{in}(t_i)> |^2 ~,
\eea
\bea
 R_{-} ~(\Delta t) & = &
| <e^- \pi^+ \bar{\nu}(t_f)~ |{K}_0^{in}(t_i)> \nonumber \\
& + & <e^- \pi^+ \bar{\nu}(t_f)~ |\bar{K}_0^{in}(t_f)>
<\bar{K}_0^{out}(t_f)~ |K_0^{in}(t_i)> |^2 ~.
\eea 
where the first term in each sum stands for
(i) and the second for (ii) (thus containing
the kaon oscillations multiplied by the matrix element
for semileptonic decays through
$\Delta S = \Delta Q$.
The experimental asymmetry $A_T^{exp}$ therefore,
besides $\epsilon$, also contains the parameters
$ x_{-}$  and $y$, where $x_{-}$ 
measures $\Delta Q = -\Delta S$, while
$y$ stands for ${\cal CPT}$ violation in
the decays.
\bea
A_T^{exp}= 4 Re \; [\epsilon] ~  -  2 Re ~[x_{-}]  - 2 Re ~ [y]~.
\eea
In the CPLEAR experiment, with the proper experimental normalisations,
the measured asymptotic asymmetry is \cite{CPLEARlast}:
\bea
\tilde{A}_T^{exp}= 4 Re \; [\epsilon] ~  -  4 Re ~[x_{-}]  - 4 Re ~ [y]~.
\eea

The average value of $\tilde{A}_T^{exp}$  was found 
to be  $= (6.6 \pm 1.6) \times 10^{-3}$, which
 is to be compared to  the recent CPLEAR measurement of
$ (Re~[x_{-}] + Re~[y]) = (-2 \pm 3) \times 10^{-4}$,
indicating that the measured asymmetry 
is related to the violation of time-reversal invariance.
Conclusively, CPLEAR made a direct measurement
of time-reversal violation, as we had already
stated \cite{AKLP}.
Similar arguments have been presented \cite{JN}, using the density matrix 
formalism for the description of the kaon system.


An interesting question to ask at this stage,
is what information one could obtain from previous measurements
plus unitarity \cite{bell-stein}. Unitarity
implies the relations
\bea
<K_L^{in} |K_S^{in}> & = & \Sigma_{f}
<K_L^{in} |f^{in}> <f^{out} |K_S^{in}> ~,
\nonumber \\
<K_S^{in} |K_L^{in}> & = & \Sigma_{f}
<K_S^{in} |f^{in}> <f^{out}| K_L> ~,
\eea
where $f$ stands for {\it all}
possible decay channels.
Making the additional assumption
that the final decay modes satisfy the relation
\, $|f^{in}> = |f^{out}> \equiv <f^{out}|^\dagger$ \,
(which is equivalent to making use of
$CPT$-invariance of the final state interactions), it is
possible to calculate the sum
$ <K_L^{in}| K_S^{in}> + <K_S^{in} |K_L^{in}> $,
by {\it measuring only the branching ratios of kaon decays}.
This is what can be done in $K_L$, $K_S$ experiments, where only
the {\it incoming kaon states} are used.
(Note here, however, that in the next section we discuss a 
${\cal T}$-odd asymmetry that can be measured in a 
single decay channel).
In the linear approximation,
this sum is equal
to $4 ~Re~[\epsilon]$
(see eq. (\ref{Tviol})).
However, this is an {\it indirect}
determination of $T$-violation, and
would not have been possible if invisible decays were
present. This is to be contrasted with
the results of CPLEAR, which  
use only one out of the possible decaying channels,
and does not rely at all on unitarity and or the knowledge
of other decay channels than the one used
in the analysis \cite{AKLP}.


\section{ \large \sl ${\cal T}$-odd effects versus ${\cal T}$-reversal
violation}

The KTeV experiment looks at the rare decay
$K_L \rightarrow \pi^+ \pi^- e^+ e^-$
of which they have collected more than 2000 events.
In particular, they measure the asymmetry in 
the differential cross section, with respect
to the angle $\phi$ between the pion and electron
planes \cite{sehgal}. To give to the angle an unambiguous sign,
they define $\phi$ according to
\bea
\sin\phi \cos \phi = (\vec{n}_e + \vec{n}_\pi) \cdot \hat{z} ~,
\eea 
where 
$ \vec{n}_e (\vec{n}_\pi)$ is the unit vector
in the direction 
$ \vec{p}_{e^{-}} \times \vec{p}_{e^{+}} $
$ (\vec{p}_{\pi^{-}} \times \vec{p}_{\pi^{+}} $),
and $\hat{z}$ is the unit vector in the direction
of the sum of the two pion momenta \cite{sehgal}.
A ${\cal T}$-odd observable
is one that changes sign under the reversal of all
incoming and outgoing three-momenta and polarisations.
By construction, $\phi$ satisfies this property.
The operation of ${\cal T}$-reversal,
involves in addition to the operations mentioned,
a flip of the arrow of time (i.e. exchanging
initial and final states). The KTeV collaboration
observes an asymmetry of nearly $14 \%$ about $\phi = 0$,
thus identifying a ${\cal T}$-odd effect.

The important issue is to assess when such an effect can be
interpreted as a direct measurement of 
${\cal T}$-reversal violation, since 
nowhere have the initial and final states been interchanged
\cite{AdR}.
The key ingredient that effectively allows one to
invert the arrow of time in such a process,
is the hypothesis of the unitarity of the
$ S $-matrix: $S S^{\dagger} = 1$. The $S$-matrix
can be written in terms of the $T$-matrix
for a process $i \rightarrow f$, as
\bea
S_{if} = \delta_{if} + i T_{if}~,
\eea
where a $delta$-function for energy-momentum conservation
is included in $T_{if}$. Unitarity now implies:
\bea
T^*_{fi} = T_{if} - i A_{if}~,
\label{uni}
\eea
where $T_{fi}$ is the amplitude for a process $f \rightarrow i$
(i,e exchanging initial and final states), and $A_{if}$ is
the absorptive part of the $i \rightarrow f$ process:
\bea
A_{if} = \sum_{k} T_{ik} T_{fk}^* ~,
\eea
and the sum extends over all possible on-shell
intermediate states. Taking the absolute square of
(\ref{uni}):
\bea
|T_{fi}|^2 = |T_{if}|^2 + 2 Im (A_{if} T_{if}^*) + |A_{if}|^2 ~.
\label{squar}
\eea
If $\tilde{\imath}$,  $\tilde{f}$ denote the initial and
final states with three-momenta and polarisations reversed, 
${\cal T}$-reversal invariance would imply
\bea
|T_{fi}|^2 = |T_{\tilde{\imath} \tilde{f}}|^2 ~,
\eea
and from (\ref{squar}) we can construct
\bea
|T_{if}|^2 - |T_{\tilde{\imath}\tilde{f}}|^2 
& = & - 2 Im (A_{if} T_{if}^*) - |A_{if}|^2  \nonumber \\
& +  &  ( |T_{fi}|^2 - |T_{\tilde{\imath}\tilde{f}}|^2 ) ~.
\label{BASIC}
\eea
The left-hand side of (\ref{BASIC}) is precisely a
${\cal T}$-odd probability, for instance the one measured by
KTeV. However on the right-hand side we have two
contributions. The first contribution
arises from the terms in the first line corresponding
to final-state interactions (for instance
the exchange of a photon between the $\pi$'s
and $e$'s) which can affect the dependence on the angle
$\phi$ and generate a ${\cal T}$-odd effect through 
${\cal T}$-reversal conserving interactions.
The other contribution, the last line of (\ref{BASIC}),
is a genuine ${\cal T}$-reversal violating
contribution.
To identify a ${\cal T}$-odd effect with a violation
of ${\cal T}$-reversal,
it is thus necessary to estimate the effect of the
final state interactions for the process concerned
and to determine how big these contributions are
with respect to the measured ${\cal T}$-odd effect.
If these effects are small, then we can say that
using unitarity (and ${\cal CPT}$ invariance of the
final state interactions, which results in
$<\pi^+ \pi^- e^+ e^- |^{out} = 
(|\pi^+ \pi^- e^+ e^- >^{in})^\dagger $
), we are effectively interchanging the roles
of past and future and it is legitimate to
identify the ${\cal T}$-odd effect with a measurement
of ${\cal T}$-reversal violation.



\section{\large \sl Conclusions}

In the light of the recent data by
the CPLEAR and KTeV collaborations,
we discuss violations of discrete symmetries in the
neutral kaon system, with particular
emphasis to ${\cal T}$-reversal violation
versus ${\cal T}$-odd effects.
Since decaying kaons correspond mathematically
to a system with a non-hermitean Hamiltonian,
we use the dual space formalism, where
the physical (decaying) incoming and outgoing states
are distinct and dual of each-other.
This reflects the fact that the 
eigenstates of the full Hamiltonian 
may never oscillate to each-other and
have to be orthogonal at all times.
The orthogonality conditions
of the physical states, 
entirely determine the evolution of the kaon system.
In this framework, we study both the
asymmetries reported by CPLEAR and
KTeV and conclude the following:
CPLEAR, through $K^0$-$\bar{K}^0$ oscillations,
effectively reverses the arrow of time and thus
its measured asymmetry 
is directly related to the definition
of ${\cal T}$-reversal.
Having measured in the same experiment that additional
effects which enter in the experimental asymmetry
(arising by tagging the final kaon strangeness
by semileptonic decays, i.e.
violations of the $\Delta S = \Delta Q$ rule
and ${\cal CPT}$- invariance in the decays) are small,
it is concluded that CPLEAR indeed made the first
direct measurement of  ${\cal T}$-violation.
Since  the experiment
uses only one out of the possible decaying channels,
its results are also independent of any
unitarity assumption,
and the possible existence of invisible decay modes.


On the other hand, KTeV  studies the decay
$K_L \rightarrow \pi^+ \pi^- e^+ e^-$, which 
being an irreversible process
measures ${\cal T}$-odd effects.
These are not necessarily the same as
${\cal T}$-violating effects,
since they reverse momenta and polarisations but
not the time-arrow. It is straightforward to
demonstrate that ${\cal T}$-odd and ${\cal T}$-violating effects
are two different concepts.
Non-vanishing ${\cal T}$-odd effects
due to final state interactions,
may arise even if unitarity and ${\cal T}$-invariance hold.
However, since unitarity implies the inversion of the arrow of time,
a ${\cal T}$-odd effect could be interpreted as
time-reversal violation,  provided
${\cal CPT}$-invariance of the final states holds
and final state interactions are negligible.











\vspace*{0.3 cm}

\noindent
{\large \bf Acknowledgements: }
We would like to thank A. de Rujula, for very illuminating
discussions on ${\cal T}$-odd effects.
The work of C.K. is supported by the TMR contract
ERB-4061-PL-95-0789.

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


\begin{thebibliography}{99}


\bibitem{at_paper}
CPLEAR Collaboration, CERN-EP/98-153, Phys. Lett. B444 (1998) 43.

\bibitem{KTeV}
J. Belz, for the KTeV Collaboration,
.

\bibitem{AKLP}
L.Alvarez-Gaum\'e, C. Kounnas, S. Lola and 
P. Pavlopoulos,  .


\bibitem{rev}
For a review, see 
N.W. Tanner and R.H. Dalitz,
Ann. of Phys. 171 (1986) 463. \\
Some of the earlier references are: \\
T.D. Lee and C.S. Wu, Ann. Rev. Nucl. Sci. (1966) 511;
G. Sachs, Phys. Rev. 129 (1963) 2280;
Ann. Phys. 22 (1963) 239;
P. H. Eberhard, Phys. Rev. Lett. 16 (1966) 150;
K. R. Schubert et al., Phys. Lett. B31 (1970) 662.

\bibitem{Kabir}
P.K. Kabir, Phys. Rev. D2 (1970) 540.

\bibitem{CPLEARlast}
CPLEAR Collaboration, A. Angelopoulos et al.,
submitted to Phys. Lett. B.



\bibitem{JN}
J. Ellis and N.E. Mavromatos,
 contribution to
the Festschrift for L.B. Okun, to appear in a     
special issue of Physics Reports.

\bibitem{bell-stein}
S. Bell and J. Steinberger, in Proceedings of the Oxford
International Conference on Elementary Particles, Oxford, England,
1965, edited by R.G. Moorehouse et al., page 195.

\bibitem{sehgal} L.M. Sehgal and M. Wanninger, Phys. Rev. 
D46 (1992), 1035; {\it ibid.} 5209 (1992) (Erratum);
P. Heiliger and L.M. Sehgal, Phys. Rev. D48 (1993), 4146.


\bibitem{AdR}
A. De Rujula, J. Kaplan and E. de Rafael,
Nucl. Phys. B35 (1971) 365;
A. Bilal, E. Masso and A. De Rujula,
Nucl. Phys. B355 (1991) 549.





\end{thebibliography}



\end{document}








\begin{flushright}
 \\
CERN-TH/98-392 \\
LPTENS-98-47 \\
\end{flushright}
\vspace{8mm}

\begin{center}
{\large \bf
Violation  of  Time-Reversal Invariance
and CPLEAR Measurements} \\
\end{center}

\vspace*{0.45cm}

\begin{center}
{\bf Luis Alvarez-Gaum\'e~,~
Costas Kounnas$^{\dagger}$~,~ Smaragda Lola~} \\
\vspace*{0.1cm}
{\small CERN Theory Division, CH-1211 Geneva,
Switzerland} \\
\vspace*{0.2 cm}
{\bf  Panagiotis ~Pavlopoulos~}\\
{\small Institut $~$f\"ur Physik, University of Basle CH-4056,\\
and CPLEAR Collaboration, CH-1211 Geneva
Switzerland}
\end{center}

\vspace{0.6 cm}

\begin{center}
{\bf ABSTRACT}
\end{center}


{\small
Motivated by the recent CPLEAR measurement on the time-reversal
non-invariance, we review the situation concerning the
experimental measurements of charge conjugation,
parity violation and time reversibility,
in systems with non-Hermitean Hamiltonians.
This includes in particular neutral
meson systems, like $K^0-\bar{K}^0$, $D^0-\bar{D}^0$ and
$B^0-\bar{B}^0$.  We  discuss
the formalism that describes particle-antiparticle
mixing and time evolution of states,
paying particular emphasis to the
orthogonality conditions
of incoming and outgoing states.
As a result, we confirm that the
CPLEAR experiment makes a direct
measurement of  violation of time-reversal
without any assumption of unitarity and $CPT$-violation.
The asymmetry which signifies $T$-violation,
is found to be independent  of time and decay
processes.}

\vspace*{0.8 cm}

\begin{flushleft}
CERN-TH/98-392 \\
December 1998
\end{flushleft}

\vspace*{0.3 cm}

\noindent
\rule[.1in]{15.5cm}{.002in}
$^{\dagger}$ {\small On leave from Ecole Normale Sup\'erieure,
24 rue Lhommond, F-75231, Paris Cedex 05, France.} \\
{\small E-mail addresses: ~~alvarez@nxth04.cern.ch},$~~$
{\small kounnas@nxth04.cern.ch},$~~$ {\small magda@mail.cern.ch}, \\
\hspace*{3.0 cm}{\small Noulis.Pavlopoulos@cern.ch}

\thispagestyle{empty}


\setcounter{page}{0}
\vfill\eject




\section{Introduction}

Recently, the CPLEAR experiment at CERN, reported the
first direct observation of time-reversal violation
in the neutral kaon
 system~\cite{at_paper}. This observation is
made by comparing the probabilities
of a $\bar{K}^0$ state transforming into a
$K^0$ and vice-versa.
CPLEAR produces initial neutral kaons with defined strangeness from
proton-antiproton annihilations
at rest, via the reactions
\begin{equation*}
 \ppb \longrightarrow \biggl \{
           \begin{array}{l}
           \km \pip K^0  \\
           \kp \pim \bar{K}^0 ~,
    \end{array}
\end{equation*}
and tags the neutral kaon strangeness at the production time
by the charge of the accompanying charged kaon.
Since weak interactions do not conserve
strangeness,  the $K^0$ and $\bar{K}^0$ may subsequently transform
into each-other via oscillations with $\Delta S = 2$.
The final strangeness of the neutral kaon
is then tagged through the semi-leptonic
decays of the type
\bea
K^0 (\bar{K}^0) \rightarrow e^{\pm} \pi^{\mp} \bar{\nu}
({\nu}) ~, \nonumber
\eea
where, a positive (negative) lepton charge is associated with
a $K^0$ ($\bar{K}^0$).

In this way, among other quantities, CPLEAR
also measured the asymmetry
\begin{equation}
\label{at1}
A_T^{exp} =
\frac{R(\bar{K}^0~{(t=0)}
\rightarrow \elp\pim\net ~{(t=\tau)}) - R(K^0~{(t=0)}
\rightarrow\elm\pip\netb ~{(t=\tau)})}
     {R(\bar{K}^0~{(t=0)}
\rightarrow \elp\pim\net~{(t=\tau)}) + R(K^0~{(t=0)}
\rightarrow\elm\pip\netb~{(t=\tau)})} ~,
\end{equation}
which parametrizes
the difference of the probability that an initial
$\bar{K}^0(t_i)$ oscillates to a final $K^0(t_f)$,
from the probability that an initial $K^0(t_i)$
oscillates to a final $\bar{K}^0(t_f)$.
The average value of $A_T^{exp}$  was found over the  time interval
from $1\tau_S$ to $20\tau_S$ (where $\tau_S$ is the lifetime of the
short-lived kaon), to be different than zero
by $4 \sigma$ and this has been interpreted
by CPLEAR as the first direct measurement of time-reversal
non-invariance.



However, doubts have been expressed concerning
the interpretation of the CPLEAR result in
providing
such a direct
evidence for $T$-violation. The basic argument
is that decay processes enter in the observables,
making $CP$-violation manifest. The observed effect is
then attributed to these irreversible
processes, rather than $T$-violation.
It is also argued that this is only
a direct effect of the decaying states
being non-orthogonal.

The aim of this work is to clarify these
points. In order to do so, we
are going to re-discuss
the formalism that describes the particle-antiparticle
mixing and time evolution of states
in the kaon system. Since the
Hamiltonian $H$ of the system
is non-Hermitean, the various masses, widths and eigenstates
have to be  found by using two matrices
$V_L$ and $V_R$ such that   $V_L\, H \,V_R$
is diagonal.
The form of these  matrices is
found by  left- and right- independent diagonalizations,
while the  physical states
are defined by the $V_L ^{-1}$ and  $V_R^{-1}$ ``rotations'' of the
left- and right- states.
This is equivalent to  identifying the
form of the matrices and the eigenstates, by
looking consistently at the correct
orthogonality conditions for the outgoingand and incoming  states.
This analysis is done in section 2,
where we describe the states in
the vector  space of the system, its dual, as well
as the dual complex space.
In section 3, we are going to show that
the theoretical asymmetry which arises
directly from the {\it definition of  $T$-violation},
is independent  of time and decay
processes.  In section 4, we point out that
this is also true for the
experimental asymmetry that CPLEAR uses,
which differs from the theoretical one
due to the appearance of the semileptonic
decays in the process.
In the same section, we
 show that since the experiment uses a specific
search-channel, rather than summing
over all possible modes, {\it no unitarity
or $CPT$-invariance}
arguments enter in the analysis.
Finally, in section 5 we present a summary of the
basic points and conclude that the
CPLEAR experiment indeed makes a direct
measurement of  $T$-violation.

\section{Definition of states in the incoming ${\bf \rm \cal
H}^{in}$  and outgoing ${\bf \rm \cal  H}^{out}$  dual
spaces}

We denote by ${\bf \rm \cal  H}^{in}$ and ${\bf  \rm \cal H}^{out}$
the Hilbert
space of incoming and outgoing (dual) states,
respectively.
\bea
{\bf \rm \cal  H}^{in}~\equiv~
\left\{~|\Psi_I^{in}>~~,~I=1,2,...,n~\right\}~,~~~~~~~
{\bf \rm \cal H}^{out}~\equiv~
\left\{~<\Psi_I^{out}|~,~~I=1,2,...,n {}~\right\} ~,
\eea
$n$ is the dimension of the space and $|\Psi_I^{in}>$ and
$~<\Psi_I^{out}|$
are the  {\it right}- and {\it left}- eigenstates\footnote{
Technically, we assume that the Hamiltonian $H$
is an $ n \times n$ matrix with $n$ well-defined
left- and right- eigenvectors, to avoid some pathological
cases that are irrelevant in the
$K^0-\bar{K}^0$ system\cite{pila}.
.} of the effective
Hamiltonian $H$:
\bea
H~|\Psi_I^{in}> &=&\lambda_I~|\Psi_I^{in}>~~, \nonumber\\
{}~<\Psi_I^{out}|~H &=&<\Psi_I^{out}|~\lambda_I ~.
\eea
In this  basis,  the effective Hamiltonian is diagonal
and can be expressed in the
following form in terms of the incoming and outgoing states:
\bea
H=\sum~~|\Psi_I^{in}> \lambda_I <\Psi_I^{out}|~,
{}~~~{\rm with}
{}~~~<\Psi_I^{out}|\Psi_J^{in}>=\delta_{IJ}~,
\eea
where the unity operator $\bf 1$ takes the usual form:
\bea
{\bf 1}= \sum~|\Psi_J^{in}><\Psi_I^{out}|~.
\eea
Up to this point, we {\it do  not assume} that $H$ is  Hermitean;
$H\ne H^{\dagger}$. This implies that the conjugate states
${~<\Psi_I^{out}|}^{\dagger}$ and ${|\Psi_I^{in}>}^{\dagger}$ are not
isomorphic to their duals:
\bea
|\Psi_I^{out}> & \equiv & {<\Psi_I^{out}|}^{\dagger}  ~~\ne  ~~
|\Psi_I^{in}> ~, \nonumber\\
<\Psi_I^{in}| & \equiv  & ~~
{|\Psi_I^{in}>}^{\dagger} ~~ \ne ~~
|\Psi_I^{out}> .
\eea
The vectors, $|\Psi_I^{out}>$ and $~<\Psi_I^{in}|$ are
eigenstates of the $H^{\dagger}$ operator but they {\it are not
eigenstates} of $H$:
\bea
H^{\dagger}~|\Psi_I^{out}>&=&\lambda^*_I~ |\Psi_I^{out}>~,
\nonumber\\
<\Psi_I^{in}|~H^{\dagger}&=&<\Psi_I^{in}|~\lambda^*_I ~.
\eea
{\it Only if} the effective Hamiltonian is Hermitean,
(i.e. $H=H^{\dagger})$,
the conjugate outgoing states become isomorphic to the incoming
ones, $|\Psi_I^{out}>=|\Psi_I^{in}>$;  in this case the eigenvalues
$\lambda_I=\lambda^*_I$ are real.

When $H\ne H^{\dagger}$, the time  evolution  of the incoming and
outgoing
states $|\Psi_I^{in}(t_i)>~~$ and $|\Psi_I^{out}(t_f)>~$  are
obtained
from $|\Psi_I^{in}>$ and $|\Psi_I^{out}>$, using the
evolution operators $e^{-iHt_i}$ and
$e^{-iH^{\dagger}t_f}$ respectively:
\bea
|\Psi_I^{in}(t_i)> &=&e^{-iHt_i}~|\Psi_I^{in}> ~, \nonumber\\
|\Psi_I^{out}(t_f)>&=&e^{-iH^{\dagger}t_f}~|\Psi_I^{out}> ~.
\eea
{}From the above equations, follows the evolution of the conjugate
states:
\bea
<\Psi_I^{in}(t_i)| &=& <\Psi_I^{in}|~ e^{iH^{\dagger}t_i} ~,
\nonumber\\
<\Psi_I^{out}(t_f)|&=& <\Psi_I^{out}|~e^{iHt_f} ~.
\eea

In view of our later discussion, it is important to stress here that
the
inner products among incoming  and outgoing states {\it do not obey}
the usual orthogonality conditions. Indeed,
\bea
<\Psi_I^{out}|\Psi_J^{out}> \ne \delta_{IJ}~~~~{\rm and}
{}~~~~ <\Psi_I^{in}|\Psi_J^{in}> \ne \delta_{IJ} ~.
\eea
On the other hand, the physical incoming and outgoing eigenstates
obey {\it at all times} the orthogonality conditions
\bea
<\Psi_I^{out}(t_f)|\Psi_J^{in}(t_i)>
=<\Psi_I^{out}|e^{-iH\Delta t}|\Psi_J^{in}>
=e^{-i\lambda_I~\Delta t}~\delta_{IJ} ~.
\eea

We now proceed to discuss particle-antiparticle
mixing in the neutral kaon system.

\section{ Particle-antiparticle mixing in the neutral kaon system}
\noindent

The $K^0, \bar{K}^0$ states are produced under strong interactions
and are strangeness eigenstates.
Moreover, they obey the relations:
\begin{eqnarray}
CP \; |K_0^{in}> & = & |\bar{K}^{in}_0> ~, \nonumber \\
T \; |K_0^{in}> & = & <K_0^{out}|  ~, \nonumber \\
CPT \; |K_0^{in}> & = & <\bar{K}_0^{out}| ~.
\label{EQ1}
\end{eqnarray}


These states are admixtures of the physical
{\it incoming} ($|K_S^{in}>$ and $|K_L^{in}>$) and
{\it outgoing} ($<K_S^{out}|$ and $<K_L^{out}|$) states
of the full Hamiltonian and obey the following orthogonality
conditions:
\bea
<K_L^{out}|K_S^{in}> = 0 ~,~~~~~~ <K_S^{out}|K_L^{in}> = 0 ~,
\nonumber \\
<K_S^{out}|K_S^{in}> = 1 ~,~~~~~~ <K_L^{out}|K_L^{in}> = 1 ~.
\label{ortho}
\eea

The physical states, are the left- and right-
eigenvectors of the
effective Hamiltonian of the system, $H \equiv
M - i \Gamma /2 $:
\bea
& &  H ~|K_L^{in}>   =  \lambda_L ~|K_L^{in}> ~, \; \; \;
H ~|K_S^{in}> = \lambda_S ~|K_S^{in}>  ~,\nonumber \\
& & <K_L^{out}|~ H  = <K_L^{out}|~\lambda_L ~, \; \; \;
<K_S^{out}|~ H  = <K_S^{out}|~ \lambda_S ~.
\eea
Since $H$ is not Hermitean, this implies in general that the incoming
and outgoing eigenvectors in the $K^0$, $\bar{K}^0$ base
{\it are not related} simply by complex conjugation.

Without loss of generality,
we can express the physical incoming states in terms of
$|K_0^{in}>$ and  $|\bar{K}_0^{in}>$ as:
\begin{eqnarray}
|K_S^{in}> & = & \frac{1}{N_S} \left ( \; (1+\a)|K_0^{in}> + \;
            (1-\a) \; |\bar{K}_0^{in}>  \right ) ~, \nonumber \\
|K_L^{in}> & = & \frac{1}{N_L} \left ( \; (1+\b)|K_0^{in}> - \;
            (1-\b) \; |\bar{K}_0^{in}>  \right ) ~,
\label{EQ2}
\end{eqnarray}
where $\a$ and $\b$ are complex variables
associated with $CP,T$ and $CPT$-violation,
and $N_L, N_S$ are normalization factors
to be discussed below.
Similar relations exist for the dual outgoing states:
\begin{eqnarray}
<K_S^{out}| & = & \frac{1}{\tilde{N}_S} \left ( \;
(1+\ba) <K_0^{out}| + \;
            (1-\ba) \; <\bar{K}_0^{out}|  \right ) ~, \nonumber \\
<K_L^{out}| & = & \frac{1}{\tilde{N}_L} \left ( \;
(1+\bb) < {K}_0^{out}| - \;
            (1-\bb) \; <\bar{K}_0^{out}| \right ) ~.
\label{EQ3}
\end{eqnarray}

The parameters ( $\a, \b$) and ($\ba, \bb$)
that are associated with the
incoming and outgoing states respectively, are not independent
but are related through the orthogonality conditions (eqs.13) valid
for the physical states:
\bea
<K_L^{out} | K_S^{in}> = 0 & \Rightarrow & \bb = -\alpha  ~,
\nonumber \\
<K_S^{out} | K_L^{in}> = 0 & \Rightarrow & \ba = -\beta   ~,
\nonumber \\
<K_S^{out} | K_S^{in}> = 1 & \Rightarrow & N_{S} \tilde{N}_S = 2 (1
- \alpha
\beta)
  ~, \nonumber \\
<K_L^{out} | K_L^{in}> = 1 & \Rightarrow & N_{L} \tilde{N}_L = 2 (1
- \alpha
\beta)  ~.
\eea

The above relations indicate that, while the normalizations
$\tilde{N}_{S,L}$ can be expressed in terms of
$N_{S,L}$, the latter remain unspecified. This
ambiguity however will not affect any measurable
quantity. Thus we can always choose
\bea
N \equiv N_{S}= \tilde{N}_S= N_{L} =\tilde{N}_L =\sqrt{2 (1 - \alpha
\beta)} ~.
\eea

Let us write down for completeness the inverse
transformations that express the $K^0, \bar{K}^0$
states in terms of $K_S$ and $K_L$:
\begin{eqnarray}
|K_0^{in}> & = & \frac{1}{N}
            \left ( \; (1-\b)|K_S^{in}> + \;
            (1-\a) \; |K_L^{in}>  \right ) ~,  \nonumber \\
|\bar{K}_0^{in}> & = &  \frac{1}{N}
            \left ( \; (1+\b)|K_S^{in}> - \;
            (1+\a) \; |K_L^{in}>  \right ) ~,
\label{EQ4}
\end{eqnarray}
and
\begin{eqnarray}
<K_0^{out}| & = & \frac{1}{N}
            \left ( \; (1+\a)<K_S^{out}| + \;
            (1+\b) \; <K_L^{out}| \; \right ) ~,  \nonumber \\
<\bar{K}_0^{out}| & = & \frac{1}{N}
            \left ( \; (1-\a)<K_S^{out}| - \;
            (1-\b) \; <K_L^{out}|  \; \right ) ~.
\label{EQ5}
\end{eqnarray}




In the basis of the states $K_L, K_S$,
$H$ can be expressed in terms of a diagonal  $2 \times  2$ matrix
\bea
H = |K_S^{in}> \lambda_S  < K_S^{out}| +
|K_L^{in}> \lambda_L <K_L^{out}| ~,
\eea
while in  the basis of
$K^0,\bar{K}^0$, $H$ takes the following form:


\bea
H_{ij} =  \frac{1}{2} \left(
\begin{array}{cc}
( \lL + \lS) -
\Delta \lambda \frac{ (\a-\b)}{1-\a\b} &
{}~~~~~~~~~~
\Delta \lambda   \;
\frac{(1 + \a\b)}{1-\a\b}
+ \Delta \lambda \frac{ \a+\b}{1-\a\b} \\
 & \\
\Delta \lambda  \;
\frac{(1 + \a\b)}{1-\a\b}
- \Delta \lambda \frac{\a+\b}{1-\a\b} &
{}~~~~~
( \lL + \lS) +
\Delta \lambda \frac{\a-\b}{1-\a\b}
\end{array}
\right ) ~.
\label{eqH}
\eea



Here,
\[
 \Delta \lambda = \lambda_L - \lambda_S, \; \; \;
\lambda_L = m_L - i \frac{\Gamma_L}{2}, \; \; \;
\lambda_S = m_S - i \frac{\Gamma_S}{2}, \; \; \;
\]
where
$m_{\mathrm S}, m_{\mathrm L}$ are the $\ks, \kl$ masses
and $\Gamma_{\mathrm S},\Gamma_{\mathrm L}$,
the $\ks , \kl$ widths.
 From eq.(\ref{eqH}), we can identify
the $T$-, $CP$- and $CPT$- violating parameters. Indeed:


$\bullet$ Under {\underline {$T$--transformations}},
 $$
<K_0^{out}|H |\bar{K}_0^{in}> \; \leftrightarrow \;
<\bar{K}_0^{out}|H|K_0^{in}> ~,
$$
thus, the off-diagonal elements of $H$
are interchanged.
This indicates that the parameter $\epsilon \equiv (\a+\b)/2$,
which is related to the difference of the
off-diagonal elements of $H$,
measures the magnitude of  the $T$-violation\footnote{
$~2/N^2 \approx 1$, in the linear approximation.}.
\begin{equation}
  \frac{2}{N^2}~\e = \frac {
<K_0^{out}|H|\bar{K}_0^{in}> - <\bar{K}_0^{out}|H|K_0^{in}>
}{2 \; \Delta \lambda} ~ .
\label{epsilon1}
\end{equation}



$\bullet$ Under {\underline {$CPT$--transformations}},
$$
<K_0^{out}|H |{ K}_0^{in}>
\; \leftrightarrow \;   <\bar{K}_0^{out}|H |\bar{K}_0^{in}> ~,
$$
and therefore, the parameter
$\delta \equiv (\a-\b)/2$, related to the
difference of the diagonal elements of $H$,
 measures the magnitude
of $CPT$-violation.
\bea
\frac{2}{N^2}~ \d = \frac{<\bar{K}_0^{out}|H|\bar{K}_0^{in}> -
<K_0^{out}|H|K_0^{in}>}
{2 \; \Delta \lambda}  ~.
\eea

$\bullet$ Under {\underline {$CP$--transformation}},
$$
<K_0^{out}|H |K_0^{in}> \; \leftrightarrow \;
<\bar{K}_0^{out}|H |{\bar
K}_0^{in}> ~,
$$
and simultaneously
$$
<K_0^{out}|H |\bar{K}_0^{in}>\; \leftrightarrow \;    <{\bar
K}_0^{out}|H|K_0^{in}> ~,
$$
thus, {\it both} the diagonal and the off-diagonal elements of
$H$
are interchanged.
Then, the parameters $\a=\e+\d$ and $\b=\e-\d$, usually denoted as
$\epsilon_S$ and $\epsilon_L$, are the ones
which measure the  magnitude of $CP$-violation in the decays of
$\ks$ and $\kl$  respectively.









\section{Direct measurement testing time-reversibility}

The meaning of classical time-reversal invariance is unambiguous.
A system at a
final classical configuration
retraces its way back to some initial configuration by reversing the
velocities.  As a result of time-reversal invariance, initial and
final quantum mechanical  states
are interchanged with identical positions and opposite velocities:
\bea
T~[~<\Psi^{out} (t_f)|\Phi^{in}(t_i)>~]~
=~<\Phi^{out} (t_f)|\Psi^{in}(t_i)> ~.
\eea

In order to test time reversibility, one has to compare
the magnitude of the probability
$|<\Psi^{out} (t_f)|\Phi^{in}(t_i)>|^2$
with that of the time-reversed process
$|<\Phi^{out} (t_f)|\Psi^{in}(t_i)>|^2$.
Any possible difference in the two probabilities will signal
deviations of time-reversibility. In that case, the process is not
equivalent to its
time reversed one, resulting in time-reversal
violation.
In the neutral kaon system,  at a given time $t_i$ one has an initial
 strangeness eigenstate, such that $
|K_0^{in}(t_i)> = |\Psi^{in}(t_i)>$.
At some later time $t_f$, one  finds a final strangeness eigenstate
$<\bar{K}_0^{out}(t_f)| = <\Phi^{out}(t_f)|$.
According to time-reversibility, we may conclude that
the above process should have the same probability with the reversed
one, namely, an initial $|\bar{K}_0^{in}(t_i)>$ to be transformed
into a
final
$<K_0^{out} (t_f)|$. Then, for the kaon system we can write
for the case of time-reversal invariance:
 \begin{equation}
|<\bar{K}_0^{out} (t_f)|K_0^{in}(t_i)>|^2
=|<K_0^{out} (t_f)|\bar{K}_0^{in}(t_i)>|^2  ~.
\label{KK}
\end{equation}
Any deviation from the above equality will definitely signal
time-reversal
violation.
The comparison of the   probabilities of a
$\bar{K}^0$ transforming into $K^{0}$,
and  $K^0$ transforming into $\bar{K}^{0}$
can demonstrate  a  departure from time-reversal invariance.
More explicitly, such a departure is manifest in the
asymmetry
\begin{eqnarray}
A_T &=& \frac
{ P_{\bar{K} K} (\Delta t) - P_{K \bar{K}} (\Delta t)
}
{ P_{\bar{K} K} (\Delta t) + P_{K \bar{K}} (\Delta t)
} ~,
\nonumber \\
&=&
\frac{
|<K_0^{out}(t_f)|\bar{K}_0^{in}(t_i)>|^2
-|<\bar{K}_0^{out} (t_f)|K_0^{in}(t_i)>|^2
} {
|<K_0^{out}(t_f)|\bar{K}_0^{in}(t_i)>|^2
+|<\bar{K}_0^{out} (t_f)|K_0^{in}(t_i)>|^2
}
 ~, \label{Kabir}
\end{eqnarray}
known in the literature as the Kabir asymmetry \cite{Kabir}.

The time evolution from $t_i$ to $t_f$ is induced by the effective
Hamiltonian $H$:
\bea
A_{K_0 \rightarrow \bar{K}_0}
& = &
<\bar{K}_0^{out} (t_f)|K_0^{in}(t_i)>
{}~~=~~<\bar{K}_0^{out}|e^{-iH\Delta t}|K_0^{in}> ~,
\nonumber \\
A_{\bar{K}_0 \rightarrow K_0}
& = &
<K_0^{out} (t_f)|\bar{K}_0^{in}(t_i)>
{}~~=~~<K_0^{out}|e^{-iH\Delta t}|\bar{K}_0^{in}> ~.
\eea

Inserting the unity operator
\bea
{\bf 1} = |K_L^{in}><K_L^{out}| + |K_S^{in}> <K_S^{out}| ~,
\eea
to the right
of  the evolution operator $e^{-iH\Delta t}$ and using the fact that
$K_{L,S}$ are Hamiltonian eigenstates,
we obtain:
\bea
A_{K_0 \rightarrow \bar{K}_0}
& = &
<\bar{K}_0^{out}|K_L^{in}> <K_L^{out}|K_0^{in}>  e^{-i \lambda_L
\Delta t}  \nonumber \\
& + &
<\bar{K}_0^{out}|K_S^{in}> <K_S^{out}|K_0^{in}>  e^{-i \lambda_S
\Delta t}   \nonumber \\
& =  &
\frac{1}{N^2} ~ (1-\a) (1-\b) ~
( e^{-i \lambda_S \Delta t} - e^{-i \lambda_L \Delta t} ) ~,
\eea
and
\bea
A_{\bar{K}_0 \rightarrow K_0}
& = &
<K_0^{out}|K_L^{in}> <K_L^{out}|\bar{K}_0^{in}>  e^{-i \lambda_L
\Delta t} \nonumber \\
& + &
<K_0^{out}|K_S^{in}> <K_S^{out}|\bar{K}_0^{in}>  e^{-i \lambda_S
\Delta t}   \nonumber \\
& = &
\frac{1}{N^2} ~ (1+\a) (1+\b) ~
( e^{-i \lambda_S \Delta t} - e^{-i \lambda_L \Delta t} ) ~.
\eea

We see  that the time-dependent factor
$ g(\Delta t)  \equiv
( e^{-i \lambda_S \Delta t} - e^{-i \lambda_L \Delta t} )$, whose
absolute value square is given by
\bea
|g(\Delta t)|^2 =
e^{- \Gamma_S \Delta t} +
e^{- \Gamma_L \Delta t} - 2 cos (m_L-m_S) ~\Delta t ~
e^{-\frac{ \Gamma_S + \Gamma_L}{ 2} \Delta t} ~,
\eea
is common in both amplitudes and
therefore will cancel in the asymmetry $A_T$, which
becomes {\it time-independent}\cite{Kabir}. Thus
\bea
A_T =
\frac{
| (1+\a) (1+\b) |^2 - | (1-\a) (1-\b) |^2}
{| (1+\a) (1+\b) |^2 + | (1-\a) (1-\b) |^2} ~,
\eea

We note therefore that a non-zero value for $A_T$ signals a direct
measurement of
$T$-violation without any assumption about $CPT$ invariance.
Making the substitutions $\a = \epsilon + \delta$ and
$\b = \epsilon -\delta$, and keeping only linear
terms, one finds that
\bea
A_T \approx 4 Re \; [\epsilon] ~.
\eea

To make clear the misunderstandings in the
literature,\cite{lee-wu}--\cite{Maiani} (with the exception of
ref.\cite{CORRECT, corr})  we need to
introduce
the adjoint  outgoing states:
\bea
<K_S^{in}|&=&\frac{1}{N^*}\left((1+\alpha^*)<K_0^{in}| +(1-\alpha^*)
<{\bar K}_0^{in}| \right) ~,\nonumber\\
<K_L^{in}| &=& \frac{1}{N^*}\left((1+\beta^*)<K_0^{in}| -(1-\beta^*)
<{\bar K}_0^{in}|
\right) ~.
\eea

Notice that the adjoint states $<K_S^{in}|$ and $<K_L^{in}|$, are not
orthogonal
to $|K_S^{in}>$ and  $|K_L^{in}>$:
\bea
<K_S^{in}|K_S^{in}>&=& \frac{1+|\alpha|^2}{|1-\alpha\beta|} ~,~~~~~~
<K_L^{in}|K_L^{in}> ~=~ \frac{1+|\beta|^2}{|1-\alpha\beta|} ~,
\nonumber\\
<K_S^{in}|K_L^{in}>&=& \frac{\alpha^*+\beta}{|1-\alpha\beta|}
{}~,~~~~~~
<K_L^{in}|K_S^{in}> ~=~ \frac{\alpha+\beta^*}{|1-\alpha\beta|} ~,
\nonumber\\
\rightarrow ~~~
<K_S^{in}|K_L^{in}>&+&<K_L^{in}|K_S^{in}> ~=~
\frac{2 Re ~[(\alpha+\beta)]}{|1-\alpha\beta|} ~~=~~
\frac{4Re~ [\epsilon]}{|1-\alpha\beta|} ~.
\eea

In linear order in $\epsilon$ and $\delta$, the
approximate equality
\bea
A_T ~ \approx ~ <K_S^{in}|K_L^{in}>+<K_L^{in}|K_S^{in}>
{}~ \approx ~ 4 Re~[\epsilon] ~,
\label{Tviol}
\eea
holds. This relation resulted in
some misleading conclusion in the literature, namely that
$A_T \neq 0$ is not associated with $T$-violation,
but rather with the non-orthogonality of the physical
incoming states $K_L^{in}$
and $K_S^{in}$ states, and with the violation
of $CP$. However, as we already stressed, (i) the
relevant physical states
$<K_L^{out}|$ and $|K_S^{in}>$
are {\it always orthogonal} (see eq. (\ref{ortho} )
 and (ii) $A_T$ is  {\it by definition} the magnitude of
$T$-violation,
without any assumption about the validity of $CPT$ or even unitarity.

To better illustrate the misunderstanding,
let us imagine
that the $CP$-violating part
$\a$ of $K_S$ is zero. In this case $\epsilon = - \delta$, so
that $T$ is violated together
with $CPT$, with $CP$ invariance  in the $K_S$ decays.
Besides, if $CPT$ is assumed, then $\delta=0$ and
$\epsilon= \a =\b$.
In that case, clearly, $T$-violation is identical to $CP$-violation.

\section{The CPLEAR measurement}


Up to now, we described the behaviour of the
theoretical asymmetry that
stems directly from the definition of
$T$-reversal. However, as we mentioned
in the introduction, CPLEAR uses semi-leptonic
decays in order to tag the strangeness of the
final states and therefore the experimental
asymmetry of eq.(\ref{at1}) is:
\bea
A_T^{exp} =
\frac{\overline{R}_+ ~ (\Delta t) - R_{-} ~(\Delta t)}
{\overline{R}_+ ~(\Delta t) + R_{-} ~(\Delta t)}
\label{at2} ~,
\eea
where
\bea
\overline{R}_+
{}~(\Delta t)
& = &
| <e^+ \pi^- \nu(t_f)~ |K_0^{in}(t_f)>
<K_0^{out}(t_f)~ |\bar{K}_0^{in}(t_i)> |^2 ~,\nonumber \\
R_{-} ~(\Delta t)& = &
| <e^- \pi^+ \bar{\nu}(t_f)~ |\bar{K}_0^{in}(t_f)>
<\bar{K}_0^{out}(t_f)~ |K_0^{in}(t_i)> |^2 ~.
\eea

The basic idea here is the following:
There are in principle four semi-leptonic decays for neutral kaons:
\bea
K^0 & \rightarrow  &  e^+ \pi^- \nu ~, ~~~
\bar{K}^0  \rightarrow    e^- \pi^+ \bar{\nu} ~,\nonumber \\
K^0 & \rightarrow  &  e^- \pi^+ \bar{\nu} ~, ~~~
\bar{K}^0  \rightarrow    e^+ \pi^- \nu ~.
\eea
Among them, the first two are characterized by
$\Delta S = \Delta Q$ and are
allowed, while the others
are characterized by
$\Delta S = -\Delta Q$ and would be forbidden
if no oscillations between
$K^0$ and $\bar{K}^0$ were occurring. By looking
therefore at the ``wrong-sign'' leptons, one
studies $K^0-\bar{K}^0$ conversions.

As we see from the above expressions, the squared matrix elements
\bea
| <e^+ \pi^- \nu(t_f)~ |K_0 ^{in}(t_f)> |^{2}
& \equiv & |a|^2 ~|1-y|^2 ~,\nonumber \\
| <e^- \pi^+ \bar{\nu}(t_f)~ |\bar{K}_0^{in}(t_f)> |^2
& \equiv & |a|^2 ~|1+y|^2 ~,
\eea
enter in the calculation
and are parametrized by the
quantity $y$ \cite{rev,Maiani}, which describes
$CPT$-violation in semileptonic decays,
when the $\Delta S = \Delta Q$ rule holds.
Moreover, although the
$\Delta S = \Delta Q$ rule is expected from
the Standard Model to be valid up to
order $10^{-14}$, the experimental
limit before CPLEAR was much larger \cite{pdg}. For this
reason, two quantities (denoted by $x$ and $\bar{x}$
\cite{rev,Maiani}),
which are
 related to violation of
the $\Delta S = \Delta Q$ in the decays,
have been retained in the analysis \cite{at_paper}.
These parameters were
found to be very small, and will not
concern us further.

Even if $y$ is included in the calculation the time-independence of
the asymmetry still holds. However, $y$ does enter in the
asymmetry calculation:
\bea
A_T =
\frac{| (1+\a) (1+\b) |^2|1-y|^2 - | (1-\a) (1-\b) |^2|1+y|^2}
{| (1+\a) (1+\b) |^2|1-y|^2 + | (1-\a) (1-\b) |^2|1+y|^2} ~.
\eea
In particular for the
linear approximation one finds that
\bea
A_T^{exp} \approx 4 Re \; [\epsilon] ~ - 2 Re ~ [y] ~.
\eea
Since $y$ is compatible  with zero \cite{rev}
and the measured value of $A_T^{exp}$
is very near to the value of $4 Re~[\epsilon]$
\cite{pdg}, we
conclude that the non-zero value
of $A_T^{exp}$ is due to $T$-violation.

One basic point to emphasize here, is
that CPLEAR
uses only one out of the possible decaying channels,
and therefore  its measurements
are independent of {\it any unitarity assumption}
and the possible existence of invisible decay modes.
An interesting question to ask, however,
is what information one could obtain from previous measurements
plus unitarity \cite{bell-stein,phas,lee-book}. Unitarity
implies the relations
\bea
<K_L^{in} |K_S^{in}> & = & \Sigma_{f}
<K_L^{in} |f^{in}> <f^{out} |K_S^{in}> ~,
\nonumber \\
<K_S^{in} |K_L^{in}> & = & \Sigma_{f}
<K_S^{in} |f^{in}> <f^{out}| K_L> ~,
\eea
where $f$ stands for {\it all}
possible decay channels.
Making the additional assumption
that the final decay modes satisfy the relation
\, $|f^{in}> = |f^{out}> \equiv <f^{out}|^\dagger$ \,
(which is equivalent to making use of
$CPT$-invariance of the final state interactions), it is
possible to calculate the sum
$ <K_L^{in}| K_S^{in}> + <K_S^{in} |K_L^{in}> $,
by {\it measuring only the branching ratios of kaon decays}.
This is what is done in $K_L$, $K_S$ experiments, where only
the {\it incoming kaon states} are used. In the linear approximation,
this sum is equal
to $4 ~Re~[\epsilon]$
(see eq. (\ref{Tviol})).
However, this is an {\it indirect}
determination of $T$-violation, and
would not have been possible if invisible decays were
present. Also, it would have  to be reviewed in stochastic models
of $CPT$-violation \cite{ehns}--\cite{John}.
This is to be contrasted with
the results of CPLEAR, which  do not rely at
all on unitarity and thus on the knowledge
of other decay channels than the one used
in the analysis.


\section{Concluding comments}

Motivated by the recent CPLEAR report on the
first direct observation of time-reversal non-invariance,
we attempted to clarify
the situation on measurements of charge conjugation,
parity violation and time reversibility,
in systems with non-Hermitean Hamiltonians.
To do so, we re-discussed  the formalism of the neutral kaon system,
paying particular attention in the definition
of states in the vector space of the system,
but also in its dual and in the dual complex spaces.
This allows a consistent implementation
of the orthogonality conditions for the incoming and outgoing states,
used to describe particle-antiparticle
mixing and the time evolution of the system.

As a result, we confirm that the asymmetry measured
by CPLEAR, is directly related to the definition
of $T$-violation. In addition, it
does not get affected by time and decay
processes.  Finally, the experiment
uses only one out of the possible decaying channels,
therefore  its results
are independent of any $CPT$ or unitarity assumption,
and the possible existence of invisible decay modes.
We conclude therefore that, CPLEAR indeed made
the first direct measurement of $T$-violation.

\vspace*{0.4 cm}

{\bf \large Acknowledgements: }
We would like to thank the CPLEAR Collaboration
and in particular M. Fidecaro and P. Kokkas, for many fruitful
discussions on the CPLEAR results; we also thank A. Kehagias,
for illuminating discussions and comments. The work of C.K.
has been partially supported by the TMR contract
ERB-4061-PL-95-0789.

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


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