\documentclass{ws-p8-50x6-00}
\newcommand{\epsp}{\ensuremath{\varepsilon^\prime_K}}
\newcommand{\eps}{\ensuremath{\varepsilon_K}}
\newcommand{\re}{\mbox{Re}}
\newcommand{\im}{\mbox{Im}}
\newcommand{\ba}{\begin{eqnarray}}
\newcommand{\ea}{\end{eqnarray}}

\begin{document}

\title{ \epsp/\eps\ IN THE CHIRAL LIMIT USING LARGE $N_c$}
%Talk presented at Chiral Dynamics 2000, Jefferson Lab, Newport News,
%Virginia, USA, July 17-22, 2000.

\author{Johan Bijnens}

\address{Department of Theoretical Physics 2, Lund University\\
S\"olvegatan 14A, S 22362 Lund, Sweden}

\author{Joaquim Prades}

\address{Departamento de F\'\i sica Te\'orica y del Cosmos,
Universidad de Granada,\\
Campus de Fuente Nueva, E-18002 Granada, Spain}

\maketitle

\abstracts{
The $K\to\pi\pi$ system is analyzed in the chiral limit
and at NLO in $1/N_c$.
The $\Delta I=1/2$ rule is reproduced
and we obtain a
large value for $\varepsilon_K^\prime/\varepsilon_K$.}

\section{Introduction}

We present our calculation of \epsp/\eps\ in
the chiral limit.\cite{BP1} 
Its experimental observation
is the first new major CP-violation
result since the original discovery.\cite{Cenci}

This calculation uses the large $N_c$
approach pioneered by Bardeen et al.\cite{Bardeen}.
We  used this method previously to calculate $B_K$\cite{BP2},
the $\Delta I=1/2$-rule\cite{BP3}, the muon $g-2$
and electromagnetic mass differences.\cite{BPP}
Similar recent work is \cite{Hambye,LMD}

\section{Short-distance and $X$-boson or fictitious gauge boson scheme}

$\Delta S=1$ weak decays are produced by the
exchange of $W$-bosons. The resulting effective action, resumming
$\alpha_S\log(M_X/\nu)$ to two-loops, is now standard.
An introduction and references can be found in the
lectures by Buras.\cite{Buras} The resulting effective
action, $\Gamma_{\Delta S=1}\sim 
{ \sum_{i=1}^{10}}\, C_i(\nu)  \int {\rm d}^4 x \, Q_i(x)
+ {\rm h.c.} $
with  $Q_i(x)$ four-quark operators,
is dependent on the various schemes chosen in its definition.
This dependence should disappear in the matrix-elements of this effective
action which are usually calculated in a different way.

Because currents and densities can be matched more easily across theories,
we first replace the effective four-quark operator action by an equivalent one.
For instance, the effective action reproducing
$
Q_1(x)=\left[\overline s \gamma^\mu (1-\gamma_5)  d \right] \,
 \left[\overline u \gamma_\mu (1-\gamma_5)  u \right](x)
$ by the exchange of the heavy fictitious $X_1$
is 
\be
\Gamma_{X}\equiv g_1(\mu_c,\cdots) 
\int {\rm d}^4 y 
\, X_1^\mu \left\{ \left[\overline s \gamma_\mu (1-\gamma_5)  
d \right](x) +
 \left[\overline u \gamma_\mu (1-\gamma_5)  u \right](x) \right\}
\, .
\ee
The constants, like $g_1(\mu_c,\cdots)$,
are determined unambiguously from
matching conditions in the perturbative regime,\cite{BP1,BP2}
removing the scheme-dependence.

\section{Long-distance and results}
We calculate the matrix-elements of $X$-boson
exchange by integrating over its momentum $p_X$ 
in the Euclidean. The short-distance contribution,
$p^2_X > \mu^2$, is,
to NLO in $1/N_c$, calculable using perturbative QCD and factorization.
The long-distance part, small $p^2_X$, can be calculated
with Chiral Perturbation Theory, also done by\cite{Hambye}. 
This fails at fairly low $p_X^2$
and we have chosen a reasonable hadronic model, the ENJL model,
to extend the range to higher values. We find good matching
for the isospin zero amplitude $a_0$ and for all the imaginary parts.
Another improvement route is
the LMD method\cite{LMD}.

We obtain, to NLO in $1/N_c$ and in the chiral limit,\cite{BP1}
$a_0$ within errors and
\ba
&
B_{6\chi}^{(1/2)NDR}(2 \, {\rm GeV})= 2.5 \, \pm 0.4 \quad\quad
B_{8\chi}^{(3/2)NDR}(2 \, {\rm GeV})= 1.35 \, \pm 0.20
& \nonumber\\
&
\left|{\varepsilon_K'}/{\varepsilon_K}\right|
= (60 \pm 30 )\cdot 10^{-4}\,.  
&\ea
Including the main isospin breaking effect and final state interactions
gives\cite{BP1}
\ba
\left|{\varepsilon_K'}/{\varepsilon_K}\right|
&=& (34 \pm 18 )\cdot 10^{-4}
\ea
in reasonable agreement with the experimental value\cite{Cenci}
$(19.3\pm2.4)~10^{-4}$.


\section*{Acknowledgments}
Work partially supported  by NFR, Sweden,
EU TMR Network
EURODAPHNE (Contract No. ERBFMX-CT98-0169),
CICYT, Spain (Grant No. AEN-96/1672) and the
Junta de Andaluc\'{\i}a (Grant No. FQM-101).

\begin{thebibliography}{99}
\bibitem{BP1}
J.~Bijnens and J.~Prades,
%``epsilon'(K)/epsilon (K) in the chiral limit,''
JHEP {\bf 0006} (2000) 035.
%[hep-ph/0005189].
%%CITATION = HEP-PH 0005189;%%

\bibitem{Cenci}
P. Cenci, these proceedings; M. Arenton, these proceedings.

\bibitem{Bardeen}
W.~A.~Bardeen,
%``Weak matrix elements in the large N(c) limit,''
FERMILAB-CONF-99-255-T,
%{\it To be published in the proceedings of 1999 
%Chicago Conference
%on Kaon Physics (K 99), Chicago, IL, 21-26 Jun 1999
 and references therein.

\bibitem{BP2}
J.~Bijnens and J.~Prades,
%``Scheme dependence of weak matrix elements in the 1/N(c) expansion,''
JHEP {\bf 0001} (2000) 002:
%[hep-ph/9911392];
%%CITATION = HEP-PH 9911392;%%
Nucl.\ Phys.\  {\bf B444} (1995) 523;
%[hep-ph/9502363];
%%CITATION = HEP-PH 9502363;%%
Phys.\ Lett.\  {\bf B342} (1995) 331.
%[hep-ph/9409255].
%%CITATION = HEP-PH 9409255;%%

\bibitem{BP3}
J.~Bijnens and J.~Prades,
JHEP {\bf 9901} (1999) 023;
%[hep-ph/9811472];
%%CITATION = HEP-PH 9811472;%%
J.~Bijnens et al.,% E.~Pallante and J.~Prades,
Nucl.\ Phys.\  {\bf B521} (1998) 305.
%[hep-ph/9801326].
%%CITATION = HEP-PH 9801326;%%

\bibitem{BPP}
J.~Bijnens et al.,%, E.~Pallante and J.~Prades,
Nucl.\ Phys.\  {\bf B474} (1996) 379;
%[hep-ph/9511388];
%%CITATION = HEP-PH 9511388;%%
Phys.\ Rev.\ Lett.\  {\bf 75} (1995) 1447;
%[hep-ph/9505251].
%%CITATION = HEP-PH 9505251;%%
J.~Bijnens and J.~Prades,
Nucl.\ Phys.\  {\bf B490} (1997) 239;
%[hep-ph/9610360];
%%CITATION = HEP-PH 9610360;%%

\bibitem{Hambye}
T.~Hambye and P.~H.~Soldan,
%``1/N(c) and epsilon'/epsilon,''
hep-ph/0009073.
%%CITATION = HEP-PH 0009073;%%

\bibitem{LMD}
S.~Peris and E.~de Rafael,
%``K0 anti-K0 mixing in the 1/N(c) expansion,''
hep-ph/0006146;
%%CITATION = HEP-PH 0006146;%%
M.~Knecht et al.,% S.~Peris and E.~de Rafael,
%``Matrix elements of electroweak penguin operators in the 1/N(c)  expansion,''
Phys.\ Lett.\  {\bf B457} (1999) 227.
%[hep-ph/9812471].
%%CITATION = HEP-PH 9812471;%%

\bibitem{Buras}
A.~J.~Buras,
%``Weak Hamiltonian, CP violation and rare decays,''
%to appear in 'Probing the Standard Model of Particle
%Interactions', F.David and R. Gupta, eds., 1998, Elsevier Science B.V. 
hep-ph/9806471.
%%CITATION = HEP-PH 9806471;%%

\end{thebibliography}
\end{document}

