Fully supersymmetric CP violations \\ in the kaon system
S. Baek
We show that, on the contrary to the usual claims, fully supersymmetric CP violations in the kaon system are possible through the gluino mediated flavor changing interactions. Both $\epsilon_K$ and ${Re} (\epsilon' / \epsilon_K)$ can be accommodated for relatively large $\tan\beta$ without any fine tunings or contradictions to the FCNC and EDM constraints.
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\begin{document}

\title{Fully  supersymmetric CP violations \\ in the kaon system }

%\author{S. Baek}

%\address{Theory Division, KEK, 
%1-1 Oho, Tsukuba, Ibaraki 305-0801
%JAPAN \\ E-mail: swbaek@ccthmail.kek.jp} 

\author{S. Baek,~J.-H. Jang, ~~P. Ko$^*$ ~~and ~~J. H. Park} 
%\author{J.-H. Jang, ~~P. Ko$^*$ ~~and ~~J. H. Park} 
\address{Dep. of Physics, KAIST, Taejon 305-701, KOREA
\\ E-mail: $*$pko@charm.kaist.ac.kr}

\twocolumn[\maketitle\abstract{
We show that, on the contrary to the usual claims, fully supersymmetric 
CP violations in the kaon system are possible through the gluino 
mediated flavor changing interactions. Both $\epsilon_K$ and ${\rm Re} 
(\epsilon' / \epsilon_K)$ can be accommodated for relatively 
large $\tan\beta$ without any fine tunings or contradictions 
to the FCNC and EDM constraints. 
}]

%\section{Introduction}%1

%\subsection{Producing the Hard Copy}\label{subsec:prod}%1.1

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

%Until this year, the only CP violation observed was in $K_{L} \rightarrow 
%2 \pi$ \cite{cronin}, which could be attributed to $\Delta S = 2$ $K^0 - 
%\overline{K^0}$ mixing. 
%The mixing parameter $\epsilon_K$ is accurately measured by now :
%$\epsilon_K = e^{i \pi/4} ~( 2.280 \pm 0.013 ) \times 10^{-3}$ \cite{pdg}. 
Recent observation of ${\rm Re} (\epsilon' / \epsilon_K )$ by KTeV 
collaboration, 
%\cite{ktev} 
%\[
$
{\rm Re} ( \epsilon' / \epsilon_K ) = (28 \pm 4) \times 10^{-4}
$
%\] 
\cite{ktev}, 
nicely confirms the earlier NA31 experiment \cite{na31}
%\[
$
{\rm Re} ( \epsilon' / \epsilon_K ) = (23 \pm 7) \times 10^{-4}.
$
%\]
This nonvanishing number indicates unambiguously the existence of  CP 
violation in the decay amplitude $(\Delta S = 1)$. Along with another 
CP violating parameter known for long time,  
$\epsilon_K = e^{i \pi/4} ~( 2.280 \pm 0.013 ) \times 10^{-3}$ \cite{pdg},
these two parameters quantifying CP violations in the kaon system can be 
accommodated by the KM phase in the %Glashow-Salam-Weinberg's 
standard model (SM). 
The SM prediction for ${\rm Re} ( \epsilon' / \epsilon_K )  $ 
is about $5 \times 10^{-4}$ and lies in the lower side of the data, 
although theoretical uncertainties 
%from nonperturbative matrix elements and the strange quark mass 
are rather large \cite{buras_1}.  

However, it would be interesting to consider a possibility that %these CP 
%violations 
both $\epsilon_K$ and ${\rm Re} ( \epsilon' / \epsilon_K )  $
have their origins entirely different from the KM phase 
in the SM, in particular in supersymmetric models.   % \cite{recent}. 
%In the minimal supersymmetric standard model 
%(MSSM), there are many new CP violating phases that 
%fall into two categories : phases with flavor 
%preserving (FP) and flavor changing (FC), each of which is constrained by 
%electron/neutron electric dipole moments (EDM's) and the $\epsilon_K$,
%respectively. Recently, it was shown that the FP and CP %
%violating phases in $A_t$ and $\mu$ in the more minimal SUSY model do not 
%generate enough $\epsilon_K$ \cite{abel} \cite{baek2} or new phase shift in 
%$B^0 - \overline{B^0}$ \cite{abel} \cite{baek1}, 
%although they can lead to a large 
%direct CP asymmetry in $B\rightarrow X_s \gamma$ upto $\sim\pm 16\%$ if 
%charginos and stops are light enough \cite{baek1}. However, another class 
%of CP violating phases in the flavor changing quark-squark-gluino vertices 
%in the MSSM may be relevant to CP violations in the $K$ meson system. 
In this talk, we argue that all the observed CP violating phenomena in the 
kaon system
%, namely $\epsilon_K$ and ${\rm Re} (\epsilon^{'} / \epsilon_K)$, 
in fact can be accommodated in terms of a {\it single} complex number 
$( \delta_{12}^d )_{LL}$ that parameterizes the squark mass mixings in the 
chirality and flavor spaces 
%Namely, we can accommodate both $\epsilon_K$ 
%and ${\rm Re} (\epsilon^{'} / \epsilon_K)$ in terms of a FC and CP violating 
%$( \delta_{12}^d )_{LL}$  
for relatively large $\tan\beta$ without any fine tuning or any 
contradictions with experimental data on FCNC, even if we assume
 $\delta_{\rm KM}=0$ as in this talk.
%we assume to be the case in this talk.  
This talk is based on Ref.~\cite{ours}.
%We assume that CKM matrix is real in the most of this letter for simplicity 
%and maximizing the effect of our mechanism. %we propose. 
%The case where the KM phase is nonzero is discussed in brief, and the details
%will be given elsewhere \cite{future}. 

In order to study the gluino  mediated flavor changing phenomena 
in the quark sector,
it is convenient to use the so-called mass insertion approximation (MIA) 
\cite{mia}.
%The quark-squark-gluino vertex is flavor diagonal in the MIA, and
%the flavor/chirality mixing occur through the insertion of 
%$( \delta_{ij}^d )_{AB}$, where $i,j=1,2,3$ and $A,B=L,R$ denote the flavors 
%of the squark under consideration and the chiralities of its superpartner. 
%The superscript denotes that the down type squark mass matrix is involved.
The parameters $( \delta_{ij}^d )_{AB}$ characterize the size of the 
gluino-mediated flavor ($i,j$) and chirality ($A,B$) changing amplitudes. 
They may be also CP violating  complex numbers. 

Now, if one saturates $\Delta m_K$ and $\epsilon_K$ with 
$( \delta_{12}^d )_{LL}$ alone, the resulting ${\rm Re} ( \epsilon^{'} / 
\epsilon_K )$ is too small by more than an order of magnitude,
unless one invokes some finetuning \cite{gabriel}.   
On the other hand, if one saturates ${\rm Re} ( \epsilon^{'} / \epsilon_K )$ 
by $| {\rm Im} ( \delta_{12}^d )_{LR} | \sim 10^{-5}$, the resulting 
$\epsilon_K$ is too small by more than an order of magnitude, unless one 
invokes some finetuning.   % \cite{finetune}.  
Therefore the folklore was that the supersymmetric contributions to 
${\rm Re} ( \epsilon^{'} / \epsilon_K )$ is small.  Recently, Masiero and 
Murayama showed that this conclusion can be evaded in generalized SUSY models 
\cite{mm} with a few reasonable assumptions on the size of the 
$(\delta_{12}^d)_{LR}$.  % and the relations between the down quark Yukawa 
%couplings and the CKM mixings. 
But in their model, one has to introduce a new 
% But they did not consider possibility to generate 
CP violating parameter $( \delta_{12}^d )_{LL} $  
in order to generate $\epsilon_K$ and also
%should be generated by another parameter 
%from a supersymmetric phase in Ref.~\cite{mm}, 
predict too large neutron EDM which is very close to the current upper limit.

In the following, we show that there is another generic way to evade 
this folklore in supersymmetric models if $|\mu \tan\beta|$ is relatively 
large, say  $\sim 10 - 20$ TeV.  Moreover, both $\epsilon_K$ and 
${\rm Re} ( \epsilon^{'} / \epsilon_K )$ can be generated by a single 
CP violating complex parameter in the MSSM. In other words, fully 
supersymmetric CP violations are possible in the kaon system. 
%We will assume that the KM phase is zero in most part of 
%this work in order to simplfy the analysis and also maximize the effects of
%our mechanism. Thus all the CP violations in the kaon system are assumed to 
%be originated from a single complex parameter $( \delta_{12}^d )_{LL}$. Then, 
The argument goes as follows : 
if $| ( \delta_{12}^d )_{LL} | \sim O(10^{-3} - 10^{-2})$ with the phase 
$\sim O(1)$ saturates $\epsilon_K$, this same parameter can
lead to a sizable ${\rm Re} (\epsilon' / \epsilon_K )$ through the 
$( \delta_{12}^d )_{LL}$ insertion followed by the FP $(LR)$ mass 
insertion, which is proportional to 
%\begin{eqnarray*}
\[
(\delta_{22}^d)_{LR} \equiv m_s ( A_s^* - \mu \tan\beta) / \tilde{m}^2 
\sim O(10^{-2}), 
\]
%\\
%( \delta_{12}^d )_{LR}^{\rm ind} 
%\equiv ( \delta_{12}^d )_{LL} \times (\delta_{22}^d)_{LR} 
%\sim O(10^{-5} - 10^{-4}). 
%\end{eqnarray*}
where $\tilde{m} $ denotes the common squark mass in the MIA.
%In doing this, it is crucial to observe that the
%$| ( \delta_{12}^d )_{LL} | \sim O(10^{-3}-10^{-2})$ with 
%$\sim O(1)$ phase can saturate the $\epsilon_K$, whereas 
%${\rm Re} (\epsilon' / \epsilon_K )$ is dominated by the induced $LR$ mixing 
%$| ( \delta_{12}^d )_{LR}^{\rm ind} | \sim 10^{-5}$. 
%(through the flavor conserving $LR$ mixing proportional to 
%$m_s ( A_s^* - \mu \tan\beta) / \tilde{m}^2 \sim O(10^{-2})$). 
It should be emphasized that the induced $( \delta_{12}^d )_{LR}^{\rm ind}
\equiv ( \delta_{12}^d )_{LL} \times (\delta_{22}^d)_{LR} $ 
is different from the conventional  $ ( \delta_{12}^d )_{LR}$ in the 
literature. 
%The loop functions for these two $LR$ insertions  are different 
%with each other in general. 
The $LR$ mixing $( \delta_{12} )_{LR}^{\rm ind}$ 
induced by $( \delta_{12}^d )_{LL}$  is typically very small in size 
$\sim O(10^{-5})$, but this is enough to generate the full size of 
${\rm Re} (\epsilon' / \epsilon_K)$ as shown below. 
Thus the usual folklore can be simply evaded.
Our spirit to generate supersymmetric ${\rm Re} 
(\epsilon^{'} / \epsilon_K)$ is different from  Ref.~\cite{mm}, 
where the $LR$ mass matrix form is assumed to be similar to
the Yukawa matrix so that they predict the neutron EDM to be close to the 
current upper limit. On the other hand, we do not assume any specific 
flavor structure in trilinear $A$ couplings. 
Also our model does not suffer from the
EDM constraint at all.%, as shown below.  

Let us first consider the gluino-squark contributions to the 
$K^0 - \overline{K^0}$ mixing due to two insertions of 
$( \delta_{12}^d )_{LL}$. The corresponding $\Delta S = 2$ effective 
Hamiltonian is given by 
\[
{\cal H}_{\rm eff} (\Delta S =2) 
= C_{1} \ol{d}_L^\al \ga_\mu s_L^\al ~\ol{d}_L^\be \ga^\mu s_L^\be
\]
%\cite{gabbiani:96}, 
%where the relevant operator $Q_1$ is defined as  
%for this $\Delta S=2$ %process are 
%\begin{equation}
%$  Q_1 = \ol{d}_L^\al \ga_\mu s_L^\al ~\ol{d}_L^\be \ga^\mu s_L^\be.$ %\nl
%  Q_2 &=& \ol{d}_R^\al  s_L^\al \ol{d}_R^\be s_L^\be, \nl
%  Q_3 &=& \ol{d}_R^\al  s_L^\be \ol{d}_R^\be s_L^\al, \nl
%  Q_4 &=& \ol{d}_R^\al  s_L^\al \ol{d}_L^\be s_R^\be, \nl
%  Q_5 &=& \ol{d}_R^\al  s_L^\be \ol{d}_L^\be s_R^\al, \nl
%  \wt{Q}_1 &=& \ol{d}_R^\al \ga_\mu s_R^\al \ol{d}_R^\be \ga^\mu s_R^\be,\nl
%  \wt{Q}_2 &=& \ol{d}_L^\al  s_R^\al \ol{d}_L^\be s_R^\be, \nl
%  \wt{Q}_3 &=& \ol{d}_L^\al  s_R^\be \ol{d}_L^\be s_R^\al .
%\end{equation}
%The Wilson coefficient $C_{1}$ corresponding to $Q_1$ is given by 
%can be obtained from
%the calculation of gluino mediated box digrams with FC $LL$ 
%insertion~\cite{gabbiani:96} in the squark propagators.
with the Wilson coefficient $C_{1}$ being 
\bea
  C_1 = -{\al_s^2 \over 216 \wt{m}^2} 
         (\de^d_{12})^2_{LL} ~ f_1 (x) %+ 66 \wt{f}_6(x) \right] .
\eea
Here, $x=m^2_{\wt{g}}/\tilde{m}^2$ and the loop function $f_1 (x)$ 
is given in~\cite{mia}.
%The double mass insertion diagrams of FC $LL$ followed by the FP $LR$ 
%generate another operators : 
%$ \wt{Q}_2 = \ol{d}_L^\al  s_R^\al \ol{d}_L^\be s_R^\be$ and 
%$ \wt{Q}_3 = \ol{d}_L^\al  s_R^\be \ol{d}_L^\be s_R^\al$, whose 
%\begin{equation}
% \wt{Q}_2 = \ol{d}_L^\al  s_R^\al \ol{d}_L^\be s_R^\be, ~~~%\nl
%   \wt{Q}_3 = \ol{d}_L^\al  s_R^\be \ol{d}_L^\be s_R^\al .
%\end{equation}
%If we define $(\de_{22}^d)_{LR}$ parametrizing the FP $LR$ mass insertion by 
%\bea
%\label{inducedLR}
%  (\de_{22}^d)_{LR} \equiv \frac{m_s (A_s^*-\mu \tan\be)}{\wt{m}^2},
%\eea
%The corresponding 
%Wilson coefficients %of $\wt{Q}_2$ and $\wt{Q}_3$ are 
%are proportional to $\left[ (\de_{12}^d)_{LR}^{\rm ind} \right]^2$. 
%Since their effects on $\Delta m_K$ and $\epsilon_K$ are negligible, 
%we do not show them here explicitly although their effects 
%have been included  
%in the numerical analyses. 

Now we turn to the $\Delta S=1$ effective Hamiltonian 
${\cal H}_{\rm eff} ( \Delta S = 1 ) = \sum_{i=3}^8 C_i \O_i$.
The $sdg$ operator $O_8$ which is relevant to 
${\rm Re} (\epsilon^{'}/\epsilon_K )$ is defined as 
\begin{equation}
\O_8 = {g_s \over 4 \pi}  m_s \ol{d}_L^\al \si^{\mu\nu} T^a s_R^\al 
                G^a_{\mu\nu}, 
\end{equation}
and other four quark operators $O_{i=3,...,6}$ and the corresponding Wilson 
coefficients from $C_3$ to $C_8$ with a single mass
insertion are available in the literature~\cite{mia}. 
%One has to remind that $C_{3,...,6}$'s are proportional to 
%$(\delta_{12}^d )_{LL}$, whereas 
%$C_8$ is given by a linear combination of
%$(\delta_{12}^d )_{LL}$ and $(\delta_{12}^d )_{LR}$. 
%This $(\delta_{12}^d )_{LR}$ dependent part in $C_8$ is proportional to 
%$m_{\tilde{g}}/ m_s$, and thus is very important for generating
%${\rm Re} (\epsilon^{'} / \epsilon_K)$ even if 
%$(\delta_{12}^d )_{LR}$ is fairly small.
If we consider the penguin diagram Fig.~\ref{fig1} with the double mass 
insertion, the Wilson coefficient $C_8$ is given by 
\begin{equation}
\label{newdel1}
   C_8^{(2)} = {\al_s \over \wt{m}^2} \; {m_{\wt{g}} \over m_s} 
           (\de_{12}^d)_{LR}^{\rm ind} ~
%           (\de_{22}^d)_{LR}
%            \l[ \C_1 
M_8 (x) 
%_1^{(2)}(x) +\C_2 M_2^{(2)}(x) \r],
\end{equation}
where the explicit form of the $\sim O(1)$ function %$\C_1=3/2, \C_2=-1/6$ and 
$M_8 (x)$ can be found in Ref.~\cite{ours}. 
Since $C_8^{(2)}$ is proportional to 
$m_{\tilde{g}}/ m_s$, it is very important for generating
${\rm Re} (\epsilon^{'} / \epsilon_K)$ even if 
$(\delta_{12}^d )_{LR}^{\rm ind}$ is fairly small.
%\begin{eqnarray}
%  M_1^{(2)}(x) &=& \frac{2(3-3x^2+(1+4x+x^2)\log x)}{(x-1)^5} \nl
%  M_2^{(2)}(x) &=& \frac{1+9x-9x^2-x^3+6x(1+x)\log x}{(x-1)^5} .
%\end{eqnarray}
%The contributions of photon penguin and $Z$ penguin diagrams with the double 
%mass insertion are negligible as in the case of the single mass insertion 
%~\cite{gabbiani:96}.
%
%Now we are ready to calculate the SUSY contributions to 
%$\Delta m_K, \epsilon_K$ and $\epsilon' / \epsilon_K$ using the 
%$\Delta S = 1,2$ effective Hamiltonians obtained above and the following 
%expressions \cite{buras}:
%\begin{eqnarray}
%\Delta m_K ( {\rm SUSY} )  & = & 2 {\rm Re} M_{12} ,
%\nonumber  
%\\
%\epsilon_K ({\rm SUSY})    & = & 
%\frac{\mbox{exp}(i\pi/4)}{\sqrt{2}\Delta m_K ({\rm exp})}~{\rm Im} M_{12},
%\\
%{\rm Re} ( \epsilon' / \epsilon_K )  & = & 
%\frac{\omega}{\sqrt{2} |\epsilon_K| {\rm Re} A_0}
%\sum_{i} {\rm Im} ( C_i) \langle O_i \rangle_0 ( 1- \Omega_{\eta+\eta'})
%\nonumber
%\end{eqnarray} 
%where $2 m_K M_{12}^* \equiv \langle \bar{K}^0 | H_{eff}^{\Delta S=2} 
%| K^0 \rangle$ and $A_I$'s are the isospin amplitudes defined as
%$A_I e^{i \delta_I} \equiv \langle ( \pi\pi)_I | {\cal H}_{\rm eff}^{\Delta
%S = 1} | K^0 \rangle$.  
%

%\psfrag{XX1}[][][0.85]{{\Large $s_R$}}
%\psfrag{XX2}[][][0.85]{{\Large $\wt{g}$}}
%\psfrag{XX3}[][][0.85]{{\Large $d_L$}}
%\psfrag{XX4}[][][0.85]{{\Large $\wt{s}_R$}}
%\psfrag{XX5}[][][0.85]{{\Large $\wt{s}_L$}}
%\psfrag{XX6}[][][0.85]{{\Large $\wt{d}_L$}}
%\psfrag{XX7}[][][0.85]{{\Large $g$}}
\begin{figure}[h]
\centerline{\epsfxsize=5.0cm \epsfbox{fig1_new.ps}}
\vspace{.5cm}
%\centerline{\epsfxsize=5.0cm \epsfbox{fig1_new.epsi}}
%\centerline{\epsfxsize=7.3cm \epsfbox{fig1.eps}}
\caption{ Feynman diagram for $\Delta S=1$ process. The cross denotes the 
flavor changing $(LL)$ and the flavor preserving $(LR)$ mixings, 
respectively.}
\label{fig1}
\end{figure}

It is straightforward to calculate $\epsilon_K$ and 
$\epsilon^{'} / \epsilon_K$ using the same parameters as in Ref.~\cite{buras}
with $m_s (2~ {\rm GeV}) = 130~ {\rm MeV}$. 
The corresponding SM prediction for ${\rm Re} ( \epsilon^{'} / \epsilon_K )
= 5.7 \times 10^{-4}$.  
%We vary the modulus and the phase of $( \delta_{12}^d )_{LL}$ as indepent 
%parameters, and select 
For those points which satisfy 
$\Delta m_K ( {\rm SUSY} ) \leq \Delta m_{K} ({\rm exp})$ 
and $| \epsilon_K ({\rm SUSY}) - \epsilon_K ({\rm exp}) | < 1 \sigma $, 
%Then, for these points, 
we plot $\epsilon' / \epsilon_K$ in Fig.~2 
as functions of the modulus $r$  and the phase $\varphi$ 
of the parameter $(\delta_{12}^d)_{LL} \equiv r e^{i \varphi}$  
for the common squark mass $\tilde{m}= 500$ GeV. 
The upper (lower) rows correspond to 
$\wt{A_s} \equiv (A_s - \mu^* \tan\beta ) = -10 (20)$ TeV. 
%Different $x$'s (= $0.3,~1.0,~2.0$)  are represented 
%by the solid, the dashed and the dotted curves, respectively.
%If we choose the opposite sign for $\wt{A_s}$, the phase of the relevant 
%$(\delta_{12}^d )_{LL}$ should be shifted by 180$^{\circ}$ in order that we
%have correct sign for $\epsilon_K$.  
It is clear that both $\epsilon_K$ and 
${\rm Re} (\epsilon' / \epsilon_K)$ can be nicely 
accommodated with a single complex number $(\delta_{12}^{d})_{LL}$ with 
$\sim O(1)$ phase in our 
model without any difficulty, if $| \mu| $ and $\tan\beta$ is relatively 
large so that $|\wt{A_s}| $ becomes a few tens of TeV.  
%If the common squark 
%mass $\tilde{m}$ differs from 500 GeV, the $\wt{A_t}$ should be multiplied by
%$(\tilde{m} {\rm ~in ~GeV} / 500 )^2$ for the fixed $x$. %Therefore the 
\begin{figure}[h]
\centerline{\epsfxsize=8.0cm \epsfbox{fig2.eps}}
%\centerline{\epsfxsize=9.3cm \epsfbox{fig2.eps}}
\caption{ $Re ( \epsilon' / \epsilon_K )$ as a function 
of the modulus $r$ [(a) and (c)] and 
the phase $\varphi$ [(b) and (d)] of the parameter $(\delta_{LL}^d )_{12}$ 
with $\wt{A_S}$ to be $-10~TeV$ ((a),(b)) and 
$-20~TeV$ ((c),(d)).  
The common squark mass is chosen to be $\wt{m} = 500$ GeV, and the solid, 
the  dashed and the dotted curves correspond to $x = 0.3, ~1.0, ~2.0$, 
respectively.
}
\label{fig2}
\end{figure}
It is important to realize that in our model there is no conflict with 
neutron EDM from $( \delta_{12}^d )_{LL}$, since the EDM is generated by  
another parameter $\l(\de^d_{11} \r)_{LR}$, which can be taken as real 
independent of $( \delta_{12}^d )_{LL}$.  
%Let us consider the neutron EDM constraint. 
%The FP $LR$ mass insertion in the gluino-squark diagram 
%contributes to the neutron EDM. 
%We found that the EDM constraint is very strong. 
%If the universality of the trilinear couplings are 
%assumed ($A_d = A_s$), then $\l(\de^d_{11} \r)_{LR}$ should be essentially 
%real. Still a single CP violating phase in $( \delta_{12}^d )_{LL}$ can 
%generate right amounts of $\epsilon_K$ and ${\rm Re} (\epsilon' / \epsilon_K)$
%without any fine tuning.  
This is in sharp contrast with the case of Ref.~\cite{mm} where the 
neutron edm is inevitably close to the current upper limit.
%Even if we relax the universality condition $A_d =  A_s$, the result is 
%basically the same, since we are in the regime of large $\mu \tan\beta$ and
%its phase is constrained by the neutron EDM irrespective of $A_d = A_s$ as 
%long  as $| A_{d,s} | \leq 1$ TeV. 
Note that we are not assuming any 
specific flavor structures in the $A$ terms at all, unlike many other recent 
models in the literatures. %in Ref.~\cite{recent}.


%The branching ratio for the decay $K_L \rightarrow \pi^0 \nu\bar{\nu}$ is 
%essentially zero in our model, since it is purely CP violating but there is 
%no appreciable CP violation in the $s \rightarrow d \nu \bar{\nu}$ amplitude 
%through gluino loop diagram \cite{gabbiani:96}.  On the contrary, 
%$K^+ \rightarrow \pi^+ \nu\bar{\nu}$ involves both CP conserving and CP 
%violating amplitudes, and the corresponding branching ratio in our model 
%ranges over $(8.9 \pm 7.2) \times 10^{-11}$, compared to the SM prediction : 
%$B ( K^+ \rightarrow \pi^+ \nu\bar{\nu} )_{\rm SM} = (7.7 \pm 3.0) \times 
%10^{-11}$ \cite{kpinn}. Our predictions can be changed by two ways, however. 
%The chargino-upsquark loop contributions contribute to 
%$K\rightarrow \pi \nu\bar{\nu}$ through the enhanced $sdZ$ penguin vertex, 
%but one still expects the branching ratio for 
%$K_L \rightarrow \pi^0 \nu\bar{\nu}$ to be smaller than the SM predictions 
%\cite{kpinn1}. 
%Also, if our assumption on the real CKM matrix is relaxed, the KM phase will
%contribute to the $K\rightarrow \pi \nu\bar{\nu}$, but the predictions
%will differ from the SM, since there is additional contribution to 
%$\epsilon_K$ so that the CKM elements are less constrained \cite{future}.  
%Our predictions for these decays also differ from those in the models with 
%enhanced $Z$ penguin vertex \cite{chanowitz}, which were
%suggested as another possibility for having larger ${\rm Re} 
%(\epsilon^{'}/\epsilon_K)$.  
%If we stick to $\delta_{KM} =0$, the CP violations in $B$ system will be very
%different from the SM predictions. 
%For example, it can be $0 - 0.2$ depending 
%on $\delta_{KM}$, if we ignore gluino-squark loop contributions to the 
%$B$ system which would governed by new parameters, 
%$(\delta_{13}^d)_{AB}$ and $(\delta_{23}^d)_{AB}$.
%. However, the gluino-squark loop 
%contributions to the $B$ system is governed by new parameters, 
%$(\delta_{13}^d)_{AB}$ and $(\delta_{23}^d)_{AB}$, which are independent of 
%$(\delta_{12}^d)_{AB}$ we considered 
%here. Therefore we cannot make defninte predictions for $B$ decays.  
%Generically the situation could be very different from the SM case \cite{kkl}.

In conclusion, we showed that both $\epsilon_K$ and 
${\rm Re} (\epsilon' / \epsilon_K )$ 
can be accommodated with a single CP violating and flavor changing 
down-squark mass matrix elements $[ (\delta_{12}^d )_{LL} \sim 10^{-3} ]$ 
without any fine tuning or any conflict with the data on FCNC processes,
if $| \mu \tan\beta| \sim O(10)$ TeV.
Our mechanism utilizes this FC $LL$ mass insertion along with the FP $LR$ 
mass insertion propotional to $(\delta_{22}^d )_{LR} \sim 10^{-2}$.  
The latter is {\it generically present in any SUSY models} including the MSSM, 
and thus there is no fine tuning in our model for accommodating both 
$\epsilon_K$ and ${\rm Re} (\epsilon' / \epsilon_K)$ in terms of a single
$(\delta_{12}^d )_{LL} $. Phrasing differently, {\it the SUSY $\epsilon_K$ 
problem implies SUSY $\epsilon^{'}$ problem if 
$| \mu \tan \beta | ( \sim O(10-20)$  TeV) is relatively large}.  
One can also consider our mechanism in the more minimal SUSY model, where 
$(\delta_{22})^d_{LR}$ is proportional to $m_b ( A_b - \mu \tan\beta)$ 
so that $A_b - \mu \tan\beta$ may be lowered significantly.
The best discriminant between our model and the SM model would be probably
the branching ratios for $K\rightarrow \pi \nu\nu$ and CP violations in 
$B$ decays.  %, and will be discussed elsewhere \cite{future}. 
For example, the time dependent asymmetry in $B^0 \rightarrow J/\psi K_S$,
$\sin 2\beta$, can be completely different from the SM prediction. 
Also if the KM phase is nonzero, there will 
be additional constributions to $\epsilon_K$ and ${\rm Re} (\epsilon^{'} / 
\epsilon_K)$ from the  SM and other SUSY loop diagrams.
%If we assume that only the gluino-squark contribution considered above is 
%comparable with the SM contribution, it would be possible that the 
%$\epsilon_K$ is mainly dominated by the KM phase contributions, but the 
%${\rm Re} (\epsilon^{'} / \epsilon_K)$ has  significant contributions from 
%the induced $(\delta_{12}^d )_{LR}$ as disucssed in this letter 
%\cite{future}. 
All these finer details %including $B( K \rightarrow \pi \nu \bar{\nu} )$ 
will be discussed elsewhere in the forthcoming publication \cite{future}. 
%One can perform similar analyses for the $B_{(d,s)}$ meson system 
%($B_{d,s}^0 - \overline{B_{d,s}^0}$ mixing, a new data on $a_{J/\psi K}$ in 
%$B_d^0$ decays and $B\rightarrow X_s \gamma$, etc.), which will be the 
%subject for the future publication.

%\newpage
%\psfrag{XX1}[][][0.85]{{\Large $s_R$}}
%\psfrag{XX2}[][][0.85]{{\Large $\wt{g}$}}
%\psfrag{XX3}[][][0.85]{{\Large $d_L$}}
%\psfrag{XX4}[][][0.85]{{\Large $\wt{s}_R$}}
%\psfrag{XX5}[][][0.85]{{\Large $\wt{s}_L$}}
%\psfrag{XX6}[][][0.85]{{\Large $\wt{d}_L$}}
%\psfrag{XX7}[][][0.85]{{\Large $g$}}




 
%\section{Sample Text}
 
%\begin{figure}%1
%\epsfxsize120pt
%\figurebox{120pt}{160pt}{}
%\caption{Caption for the figure will come here.}
%\label{fig:radk}
%\end{figure}

%\begin{figure*}
%\epsfxsize30pc
%
%\figurebox{16pc}{32pc}{}
%\caption{Dynamics of two limit cycle oscillators. The repulsive 
%case $(K<0)$ and attractive case $(K>0)$ are shown. The rate of 
%phase separation, $f$, is plotted versus the phase separation 
%between the oscillators. \label{fig:radish}}
%\end{figure*}

 
%\subsection{Parametrizations of the CKM Matrix}
 
%\subsection{Four and N-Generation Mixing Matrix} \label{sec:four}
 
\section*{Acknowledgments}
This work is supported by BK21 program of Ministry of Education.

%\begin{appendix}
%\section*{Appendix}
%We can insert an appendix here and place equations so that they are
%given numbers such as Eq.~(\ref{eq:app}). 
%\begin{equation}
%x = y.
%\label{eq:app}
%\end{equation}
%\end{appendix}
 
\begin{thebibliography}{99}
%\bibitem{cronin}
%J.H.Christenson, J.W.Cronin, V.L.Fitch and R.Turlay, {\it Phys. Rev. Lett.}
%{\bf 13},138 (1964).
%\bibitem{pdg} Particl%e Data Group, {\it Eur. Phys. J.} C {\bf 3}, 1 (1998).
\bibitem{ktev} A. Alavi-Harati et al., {\it Phys. Rev. Lett.}
 {\bf 83}, 22 (1999).
\bibitem{na31} G.D.Barr et al., NA31 Collaboration, 
{\it Phys. Lett.} B {\bf 317}, 233 (1993). 
\bibitem{pdg} Particle Data Group, {\it Eur. Phys. J.} C {\bf 3}, 1 (1998).
%\bibitem{keum} Y.Keum, U. Nierste and A.I. Sanda, hep-ph/9903230.
\bibitem{buras_1} S. Bosch {\it et al.},  
%A.J. Buras, M. Gorbahn, S. Jager, M. Jamin, M.E. Lautenbacher, 
%L. Silvestrini, 
TUM-HEP-347-99, hep-ph/9904408. 
%\bibitem{recent} 
%J.-M. Frere and M. Gavela, {\it Phys. Lett.} B {\bf 132}, 107 (1983) ;
%S. A. Abel and J.-M. Frere, {\it Phys. Rev.} D {\bf  55}, 1623 (1997) ;
%Y.Keum, U. Nierste and A.I. Sanda, {\it Phys. Lett. } B
%{\bf 457}, 157 (1999) ; 
%X.-G. He, {\it Phys. Lett.} B {\bf  460}, 405 (1999) ; 
%A. Masiero and H. Murayama,  {\it Phys. Rev. Lett.} {\bf 83}, 907 (1999) ;
%S. Khalil,  T. Kobayashi and  A. Masiero, {\it Phys. Rev.} D {\bf 60}, 075003 
%(1999) ; K.S. Babu, B. Dutta and R.N. Mohapatra, hep-ph/9905464 ; 
%S. Khalil and T. Kobayashi, {\it Phys. Lett.} B {\bf 460}, 341 (1999).
%M. S. Chanowitz, LBNL-43248, hep-ph/9905478 ;
%L. Silvestrini, presented at XXXIV Rencontres de Moriond 
%Les Arc, March, 1999, to be published in the proc., hep-ph/9906202.
%\bibitem{abel} 
%S.A. Abel and J.-M. Frere in Ref.~\cite{recent}.
%J.-M. Frere and M. Gavela, Phys. Lett. {\bf B 132}, 107 (1983) ;
%S. A. Abel and J.-M. Frere, Phys. Rev. {\bf D 55}, 1623 (1997).
%\bibitem{baek2} 
%S. Baek and P. Ko, {\it Phys. Lett.} B {\bf 462}, 95 (1999) ;
%S. Pokorski, J. Rosiek, C. A. Savoy,
%IFT-99-10A, hep-ph/9906206.
%\bibitem{baek1} S. Baek and P. Ko, 
%{\it Phys. Rev. Lett.} {\bf 83}, 488 (1999).
%\bibitem{future} S. Baek, J.-H. Jang, P. Ko and J.H. Park, in preparation.
\bibitem{ours} S. Baek, J.-H. Jang, P. Ko and J. H. Park,
hep-ph/9907572, KAIST-TH 1999/04, to appear in Phys. Rev. D.
\bibitem{mia} 
%L.J. Hall, V.A. Kostelecky and S. Raby, {\it Nucl. Phys.} B
%{\bf 267}, 415 (1986) ; J.S. Hagelin, S. Kelley and T. Tanaka, 
%{\it Nucl. Phys.} B {\bf 415}, 293 (1994) ; E. Gabrielli, A. Masiero and 
%L. Silvestrini, {\it Phys. Lett.} B {\bf 374}, 80 (1996) ;
F. Gabbiani, E.Gabrielli, A. Masiero, L. Silvestrini, {\it Nucl. Phys.} B 
{\bf 477}, 321 (1996) and references therein.
%; Y.G. Kim, P. Ko, K.Y. Lee and J.S. Lee, 
%{\it Phys. Rev.} D {\bf 59}, 055018 (1999).
\bibitem{gabriel} E. Gabrielli {\it et al.},
  {\it Phys. Lett.} B {\bf 374}, 80 (1996).%in Ref.~\cite{mia}.
%\bibitem{mm} A. Masiero and H. Murayama in Ref.~\cite{recent}.   
%\bibitem{finetune} See, for example, Ref.~[15] of Ref.~\cite{mm}. 
\bibitem{mm} A. Masiero and H. Murayama,  
{\it Phys. Rev. Lett.} {\bf 83}, 907 (1999). 
%\bibitem{mm} A. Masiero and H. Murayama in Ref.~\cite{recent}.
%\bibitem{gabbiani:96} F.~Gabbiani {\it et al.} in Ref.~\cite{mia}.
%, E.~Gabrielli, A.~Masiero,~L.~Silvestrini,
%   Nucl. Phys. {\bf B 477}, 321 (1996).
\bibitem{buras} G. Buchalla, A.J.Buras, M.E. Lautenbacher, 
{\it Rev. Mod. Phys.} {\bf 68}, 1125 (1996).
%\bibitem{brhlik:98} M.~Brhlik, G.J.~Good and G.L.~Kane,
%{\it Phys. Rev.} D {\bf 59}, 115004 (1999), and references therein.
\bibitem{future} S. Baek, J.-H. Jang, P. Ko and J.H. Park, in preparation.
%\bibitem{kpinn} A.J. Buras and L. Silvestrini, 
%{\it Nucl. Phys.} B {\bf 546},  299 (1999).
%\bibitem{kpinn1} A.J. Buras, A. Romanino and L. Silvestrini, 
%{\it Nucl. Phys.} B {\bf 520}, 3 (1998) ;
%G. Colangelo and G. Isidori,  {\it JHEP} 9809:009 (1998).   
%\bibitem{kkl} Y.-G. Kim, P. Ko and J.S. Lee, {\it Nucl. Phys.} B {\bf 544}, 
%64 (1999).
%\bibitem{kchoi} A. Brignole, L.E. Ibanez, C. Munoz and C. Scheich, 
%{\it Z. Phys.} C {\bf  74}, 157 (1997) ;
%K. Choi, J.S. Lee and C. Munoz, {\it Phys. Rev. Lett.}
%{\bf 80}, 3686 (1998) ; K. Choi, H.B. Kim and C. Munoz,
%{\it Phys. Rev.} D {\bf 57}, 7521 (1998). 
%\bibitem{comment} For this $(\delta_{12}^d )_{LR}$ generated from the high 
%energy scale $(\delta_{12}^d )_{LL} (M_{\rm GUT})$ via renormalization 
%effects, one should use the loop functions given in Ref.~\cite{gabbiani:96}.
%However, there is a typo in Eq.~(14) of Ref.~\cite{gabbiani:96}, 
%which should read as $M_2 (x) = - x^{-1} B_2 (x)$, instead of 
%$M_2 (x) = -x B_2 (x)$. %We thank F. Gabbiani for confirming this.
\end{thebibliography}
 
\end{document}


%%%%%%%%%%%%%% End of ws-p10x7.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





hep-ph0009105.extracted_bib
\begin{thebibliography}{99}
%\bibitem{cronin}
%J.H.Christenson, J.W.Cronin, V.L.Fitch and R.Turlay, {\it Phys. Rev. Lett.}
%{\bf 13},138 (1964).
%\bibitem{pdg} Particl%e Data Group, {\it Eur. Phys. J.} C {\bf 3}, 1 (1998).
\bibitem{ktev} A. Alavi-Harati et al., {\it Phys. Rev. Lett.}
 {\bf 83}, 22 (1999).
\bibitem{na31} G.D.Barr et al., NA31 Collaboration, 
{\it Phys. Lett.} B {\bf 317}, 233 (1993). 
\bibitem{pdg} Particle Data Group, {\it Eur. Phys. J.} C {\bf 3}, 1 (1998).
%\bibitem{keum} Y.Keum, U. Nierste and A.I. Sanda, hep-ph/9903230.
\bibitem{buras_1} S. Bosch {\it et al.},  
%A.J. Buras, M. Gorbahn, S. Jager, M. Jamin, M.E. Lautenbacher, 
%L. Silvestrini, 
TUM-HEP-347-99, hep-ph/9904408. 
%\bibitem{recent} 
%J.-M. Frere and M. Gavela, {\it Phys. Lett.} B {\bf 132}, 107 (1983) ;
%S. A. Abel and J.-M. Frere, {\it Phys. Rev.} D {\bf  55}, 1623 (1997) ;
%Y.Keum, U. Nierste and A.I. Sanda, {\it Phys. Lett. } B
%{\bf 457}, 157 (1999) ; 
%X.-G. He, {\it Phys. Lett.} B {\bf  460}, 405 (1999) ; 
%A. Masiero and H. Murayama,  {\it Phys. Rev. Lett.} {\bf 83}, 907 (1999) ;
%S. Khalil,  T. Kobayashi and  A. Masiero, {\it Phys. Rev.} D {\bf 60}, 075003 
%(1999) ; K.S. Babu, B. Dutta and R.N. Mohapatra, hep-ph/9905464 ; 
%S. Khalil and T. Kobayashi, {\it Phys. Lett.} B {\bf 460}, 341 (1999).
%M. S. Chanowitz, LBNL-43248, hep-ph/9905478 ;
%L. Silvestrini, presented at XXXIV Rencontres de Moriond 
%Les Arc, March, 1999, to be published in the proc., hep-ph/9906202.
%\bibitem{abel} 
%S.A. Abel and J.-M. Frere in Ref.~\cite{recent}.
%J.-M. Frere and M. Gavela, Phys. Lett. {\bf B 132}, 107 (1983) ;
%S. A. Abel and J.-M. Frere, Phys. Rev. {\bf D 55}, 1623 (1997).
%\bibitem{baek2} 
%S. Baek and P. Ko, {\it Phys. Lett.} B {\bf 462}, 95 (1999) ;
%S. Pokorski, J. Rosiek, C. A. Savoy,
%IFT-99-10A, hep-ph/9906206.
%\bibitem{baek1} S. Baek and P. Ko, 
%{\it Phys. Rev. Lett.} {\bf 83}, 488 (1999).
%\bibitem{future} S. Baek, J.-H. Jang, P. Ko and J.H. Park, in preparation.
\bibitem{ours} S. Baek, J.-H. Jang, P. Ko and J. H. Park,
hep-ph/9907572, KAIST-TH 1999/04, to appear in Phys. Rev. D.
\bibitem{mia} 
%L.J. Hall, V.A. Kostelecky and S. Raby, {\it Nucl. Phys.} B
%{\bf 267}, 415 (1986) ; J.S. Hagelin, S. Kelley and T. Tanaka, 
%{\it Nucl. Phys.} B {\bf 415}, 293 (1994) ; E. Gabrielli, A. Masiero and 
%L. Silvestrini, {\it Phys. Lett.} B {\bf 374}, 80 (1996) ;
F. Gabbiani, E.Gabrielli, A. Masiero, L. Silvestrini, {\it Nucl. Phys.} B 
{\bf 477}, 321 (1996) and references therein.
%; Y.G. Kim, P. Ko, K.Y. Lee and J.S. Lee, 
%{\it Phys. Rev.} D {\bf 59}, 055018 (1999).
\bibitem{gabriel} E. Gabrielli {\it et al.},
  {\it Phys. Lett.} B {\bf 374}, 80 (1996).%in Ref.~\cite{mia}.
%\bibitem{mm} A. Masiero and H. Murayama in Ref.~\cite{recent}.   
%\bibitem{finetune} See, for example, Ref.~[15] of Ref.~\cite{mm}. 
\bibitem{mm} A. Masiero and H. Murayama,  
{\it Phys. Rev. Lett.} {\bf 83}, 907 (1999). 
%\bibitem{mm} A. Masiero and H. Murayama in Ref.~\cite{recent}.
%\bibitem{gabbiani:96} F.~Gabbiani {\it et al.} in Ref.~\cite{mia}.
%, E.~Gabrielli, A.~Masiero,~L.~Silvestrini,
%   Nucl. Phys. {\bf B 477}, 321 (1996).
\bibitem{buras} G. Buchalla, A.J.Buras, M.E. Lautenbacher, 
{\it Rev. Mod. Phys.} {\bf 68}, 1125 (1996).
%\bibitem{brhlik:98} M.~Brhlik, G.J.~Good and G.L.~Kane,
%{\it Phys. Rev.} D {\bf 59}, 115004 (1999), and references therein.
\bibitem{future} S. Baek, J.-H. Jang, P. Ko and J.H. Park, in preparation.
%\bibitem{kpinn} A.J. Buras and L. Silvestrini, 
%{\it Nucl. Phys.} B {\bf 546},  299 (1999).
%\bibitem{kpinn1} A.J. Buras, A. Romanino and L. Silvestrini, 
%{\it Nucl. Phys.} B {\bf 520}, 3 (1998) ;
%G. Colangelo and G. Isidori,  {\it JHEP} 9809:009 (1998).   
%\bibitem{kkl} Y.-G. Kim, P. Ko and J.S. Lee, {\it Nucl. Phys.} B {\bf 544}, 
%64 (1999).
%\bibitem{kchoi} A. Brignole, L.E. Ibanez, C. Munoz and C. Scheich, 
%{\it Z. Phys.} C {\bf  74}, 157 (1997) ;
%K. Choi, J.S. Lee and C. Munoz, {\it Phys. Rev. Lett.}
%{\bf 80}, 3686 (1998) ; K. Choi, H.B. Kim and C. Munoz,
%{\it Phys. Rev.} D {\bf 57}, 7521 (1998). 
%\bibitem{comment} For this $(\delta_{12}^d )_{LR}$ generated from the high 
%energy scale $(\delta_{12}^d )_{LL} (M_{\rm GUT})$ via renormalization 
%effects, one should use the loop functions given in Ref.~\cite{gabbiani:96}.
%However, there is a typo in Eq.~(14) of Ref.~\cite{gabbiani:96}, 
%which should read as $M_2 (x) = - x^{-1} B_2 (x)$, instead of 
%$M_2 (x) = -x B_2 (x)$. %We thank F. Gabbiani for confirming this.
\end{thebibliography}