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%             THEORETICAL STATUS OF EPSILON'/EPSILON               %
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\begin{document}

\runninghead{The role of final state interactions in $\eps'/\eps$}{
The role of final state interactions in $\eps'/\eps$}

\normalsize\textlineskip
\thispagestyle{empty}
\setcounter{page}{1}

\copyrightheading{}			%{Vol. 0, No. 0 (1993) 000--000}

\vspace*{0.88truein}

\fpage{1}
\centerline{\bf THE ROLE OF FINAL STATE INTERACTIONS IN
      $\varepsilon'/\varepsilon$
   %\footnote{
   %Work supported in part by the ECC, TMR Network $EURODAPHNE$
   %(ERBFMX-CT98-0169), and by
   %DGESIC (Spain) under grant No. PB97-1261.  }
   %
   %Invited talk at DPF2000 (Columbus, Ohio, August 2000).}
 }
%\vspace*{0.035truein}
%\centerline{\bf More title\footnote{}}
\vspace*{0.37truein}
\centerline{\footnotesize ELISABETTA PALLANTE}
\vspace*{0.015truein}
\centerline{\footnotesize\it SISSA, Via Beirut 2-4, I-34013 Trieste, Italy}
%}
%\baselineskip=10pt
%\centerline{\footnotesize\it }
\vspace*{10pt}
\centerline{\footnotesize ANTONIO PICH, \  IGNAZIO SCIMEMI}
%
\vspace*{0.015truein}
\centerline{\footnotesize\it
   %Departament de F\'{\i}sica Te\`orica, 
         IFIC, Universitat de Val\`encia -- CSIC,  %}
%\baselineskip=10pt
%\centerline{\footnotesize\it
  Apt. Correus 22085, E--46071 Val\`encia, Spain}
\vspace*{0.225truein}
%\publisher{(received date)}{(revised date)}

\vspace*{0.21truein}
\abstracts{
The Standard Model prediction for $\varepsilon'/\varepsilon$ is updated,
taking into account the chiral loop corrections induced by final state
interactions. The resulting value,
$\varepsilon'/\varepsilon = (17\pm 6)\times 10^{-4}$,
is in good agreement with present measurements.}{}{}

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                           %) A SECTION HEADING
\section{Introduction}	\label{sec:introduction}
\vspace*{-0.5pt}
\noindent
The CP--violating ratio  $\varepsilon'/\varepsilon$  constitutes
a fundamental test for our understanding of flavour--changing
phenomena.
The present experimental world average,\cite{ktev:99}
${\rm Re} \left(\varepsilon'/\varepsilon\right) =
(19.3 \pm 2.4) \cdot 10^{-4}$,
provides clear evidence for a non-zero value and,
therefore, the existence of direct CP violation.

The theoretical prediction has been rather controversial since
different groups, using different models or approximations,
have obtained different
results.\cite{PP:00a,PP:00b,PPS:00,munich,rome,trieste,dortmund,BP:00}
In terms of the $K\to\pi\pi$ isospin amplitudes,
$\cA_I = A_I \, e^{i\delta_I}$ ($I=0,2$),
%
\be
{\varepsilon^\prime\over\varepsilon} =
\; e^{i\Phi}\; {\omega\over \sqrt{2}\vert\eps\vert}\;\left[
{\mbox{Im}A_2\over\mbox{Re} A_2} - {\mbox{Im}A_0\over \mbox{Re} A_0}
 \right] \, ,
\qquad\qquad
\Phi \approx \delta_2-\delta_0+\frac{\pi}{4}\approx 0 \, ,
\ee
%
where
$\omega = \mbox{Re} A_2/\mbox{Re} A_0 \approx 1/22$.
The CP--conserving amplitudes $\mbox{Re} A_I$, their ratio
$\omega$ and $\eps$ are usually set to their experimentally
determined values. A theoretical calculation is then only needed
for the quantities $\mbox{Im} A_I$.

Since $M_W\gg M_K$, there are large short--distance logarithmic
contributions which can be summed up using the Operator Product
Expansion and the renormalization group.\cite{buras1,ciuc1}
To predict the physical amplitudes one also needs to
compute long--distance hadronic matrix elements of light
four--quark operators $Q_i$. They are usually parameterized
in terms of the so-called bag parameters $B_i$, which measure them
in units of their vacuum insertion approximation values.

To a very good approximation, the Standard Model prediction for
$\varepsilon'/\varepsilon$ can be written (up to global factors)
as\cite{munich}
%
\be
{\varepsilon'\over\varepsilon} \sim
\left [ B_6^{(1/2)}(1-\Omega_{IB}) - 0.4 \, B_8^{(3/2)}
 \right ]\, , \qquad\quad
\Omega_{IB} = {1\over \omega}
{(\mbox{Im}A_2)_{IB}
\over \mbox{Im}A_0} \, .
\label{EPSNUM}
\ee
%
Thus, only two operators are numerically relevant:
the QCD penguin operator $Q_6$ governs $\mbox{Im}A_0$
($\Delta I=1/2$), while $\mbox{Im}A_2$ ($\Delta I=3/2$)
is dominated by the electroweak penguin operator $Q_8$.
The parameter $\Omega_{IB}$
takes into account isospin breaking corrections;
the value $\Omega_{IB}=0.25$
was usually adopted in all calculations.\cite{Omega}
Together with $B_i\sim 1$, this produces a numerical cancellation
leading to values of $\eps'/\eps\sim 7\times 10^{-4}$.
This number has been slightly increased by a
recent Chiral Perturbation Theory ($\chi$PT) calculation at $O(p^4)$
which finds $\Omega_{IB}= 0.16\pm 0.03$.\cite{EMNP:00}


\section{Chiral Loop Corrections}
\label{sec:ChPT}
\noindent
Chiral symmetry determines the low--energy hadronic realization of the
operators $Q_i$,
through a perturbative expansion in powers of momenta and quark masses.
The corresponding chiral couplings can be calculated in the
large--$N_C$ limit of QCD. The usual input values
$B_8^{(3/2)}\approx B_6^{(1/2)}=1$ correspond
to the lowest--order approximation in both the $1/N_C$ and
$\chi$PT expansions.

The lowest--order calculation does not provide any strong phases
$\delta_I$. Those phases originate in the
final rescattering of the two pions and, therefore, are generated by
higher--order chiral loops.
Analyticity and unitarity require the presence of a corresponding
dispersive effect in the moduli of the isospin amplitudes.
Since the S--wave strong phases are quite large,
specially in the isospin--zero case,
one should expect large unitarity corrections.

The one--loop analyses of $K\to 2 \pi$ show in fact that pion
loop diagrams provide an important enhancement of the $\cA_0$
amplitude.\cite{KA91}
This chiral loop correction destroys the accidental numerical
cancellation in eq.~\eqn{EPSNUM}, generating a sizeable enhancement
of the $\eps'/\eps$ prediction.\cite{PP:00a}
The large one--loop correction to $\cA_0$ has its origin in the
strong final state interaction (FSI) of the two pions in S--wave,
which generates large infrared logarithms involving the light
pion mass.\cite{PP:00b}
Using analyticity and unitarity constraints,
these logarithms can be exponentiated to all orders in
the chiral expansion.\cite{PP:00a,PP:00b}
%
For the CP--conserving amplitudes, the result can be written as
%
\be\label{eq:OMNES_WA}
\cA_I \, =\,  \left(M_K^2-M_\pi^2\right) \; a_I(M_K^2) \, =\,
\left(M_K^2-M_\pi^2\right) \; \Omega_I(M_K^2,s_0) \; a_I(s_0)\, ,
\ee
%
where $a_I(s)$ denote reduced off-shell amplitudes with
$s\equiv \left(p_{\pi_1}+p_{\pi_2}\right)^2$ and
%
\be\label{eq:omega}
\Omega_I(s,s_0) \,\equiv\, e^{i\delta_I(s)}\; \Re_I(s,s_0) \, =\,
 \exp{\left\{ {(s-s_0)\over\pi}\int
{dz\over (z-s_0)} {\delta_I(z)\over (z-s-i\epsilon)}\right\}}
\ee
%
provides an evolution of $a_I(s)$ from an arbitrary
low--energy point $s_0$ to $s=M_K^2$.
The physical amplitude $a_I(M_K^2)$ is of course independent of $s_0$.

Taking the chiral prediction for $\delta_I(z)$ and expanding
the exponential to first order,
one just reproduces the one--loop $\chi$PT result.
Eq.~\eqn{eq:omega} allows us to get a much more accurate
prediction, by taking $s_0$ low enough that the $\chi$PT corrections
to $a_I(s_0)$ are
small and exponentiating the large logarithms with the
Omn\`es factor $\Omega_I(M_K^2,s_0)$.
Moreover, using the experimental phase-shifts in the dispersive
integral one achieves an all--order resummation of FSI effects.
The numerical accuracy of this exponentiation has been successfully
tested through an analysis of the scalar pion form factor,\cite{PP:00b}
which has identical FSI than $\cA_0$.


\section{Numerical Predictions}
\label{sec:numerics}
\noindent
At $s_0 =0$, the chiral corrections are rather small.
To a very good approximation,\cite{PPS:00} we can just multiply
the tree--level $\chi$PT result for $a_I(0)$
with the experimentally determined Omn\`es exponentials:\cite{PP:00b}
%
\be
\Re_0\equiv\Re_0(M_K^2,0) =  1.55 \pm 0.10\, ,
\qquad\qquad
\Re_2\equiv\Re_2(M_K^2,0) =  0.92 \pm 0.03\, .
\ee
%
Thus, \
$B_6^{(1/2)} \approx \Re_0\times
\left. B_6^{(1/2)}\right|_{N_C\to\infty} = 1.55$, \
%
$B_8^{(3/2)} \approx\Re_2\times
\left. B_8^{(3/2)}\right|_{N_C\to\infty} \approx  0.92$\
%
and\
$\Omega_{IB} \approx 0.16 \times\Re_2/\Re_0= 0.09$.
%\Re_2(M_K^2,0)/\Re_0(M_K^2,0) = 0.09$.
This agrees with the result \ $\Omega_{IB} = 0.08\pm 0.05$,
obtained recently with an explicit chiral loop calculation.\cite{MW:00}

The large FSI correction to the $I=0$ amplitude gets reinforced
by the mild suppression of the $I=2$ contributions. The net effect
is a large enhancement of $\eps'/\eps$ by a factor 2.4,
pushing the predicted central value from\cite{munich,rome}
$7\times 10^{-4}$ to\cite{PP:00b} $17\times 10^{-4}$.
A more careful analysis, taking into account all hadronic and
quark--mixing inputs gives the Standard Model prediction:\cite{PPS:00}
%
\be\label{eq:SMpred}
\varepsilon'/\varepsilon = (17\pm 6) \times 10^{-4}\, ,
\ee
%
which compares well with the present experimental world average.


\nonumsection{Acknowledgements}
\noindent
This work has been supported by the ECC, TMR Network
$EURODAPHNE$ (ERBFMX-CT98-0169), and by
DGESIC (Spain) under grant No. PB97-1261.


\nonumsection{References}
 
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\end{document}

