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%\begin{tabular}
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\noindent\hbox to\textwidth{February 1998 \hfill BROWN-HET-1110}\\
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\noindent\hbox to\textwidth{ \hfill BROWN-TA-555}\\
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\noindent\hbox to\textwidth{ \hfill KIAS-P98004}\\
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\begin{center}
{\bf Neutrino Mass Matrix, Mixing and Oscillation}\\
\vskip 0.2in
 Kyungsik Kang\footnote{supported in part by the 
USDOE contract DE-FG02-91ER 40688-Task A.} \\ {\small 
Department of Physics, Brown University,
Providence, RI 02912, USA}\\
%~\\
and\\
%~\\
Sin Kyu Kang \\ {\small
School of Physics, Korea Institute for Advanced Study, Seoul, Korea}\\
\vskip 0.2in
%\author{ Kyungsik Kang }
%\address{Department of Physics, Brown University,
%Providence, RI 02912, USA}
%\author{ Sin Kyu Kang }
%\address{School
%of Physics, Korea Institute for Advanced Study, Seoul, Korea}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
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%\maketitle
{\bf Abstracts}
\end{center}
\begin{quote}
We investigate the phenomenological consequences of lepton mass
matrices originated from the family permutation symmetry and its 
suitable breakings. Adopting the recently proposed new  mass matrix 
for the charged lepton mass matrix and the Majorana neutrino mass 
matrix considered before, we find that the resulting lepton flavor
mixing  matrix  is consistent with the current data on various type
of  neutrino oscillation experiments except the Chlorine data and
LSND measurement and that three neutrinos are almost degenerate
in mass. Numerically the three neutrinos can account for about $50\%$ 
of the hot dark matter(HDM) when the neutrino mass matrix is 
constrained by the small angle MSW solution for the solar neutrino 
problem, atmospheric neutrino data of the Super-Kamiokande group, 
and the limitation on the electron-type neutrino mass due to 
non-observation of neutrino-less double beta decays.
\end{quote}
\begin{center}
 (Presented by Kyungsik Kang\footnote{Presented at the APCTP Workshop: {\it Pacific Particle Physics Phenomenology}, Seoul, Korea, 31 October - 2 November, 1997})
\end{center}

\section{Introduction}
The flavor problem concerning the fermion masses, their hierarchical patterns,
as well as the flavor mixing
remains to be one of the most fundamental problems in particle 
physics.  Various attempts toward the understanding of the flavor mixing
with the quark mass matrix ansatz satisfying the calculability
condition are presented in another session \cite{skk}
Though the simple Fritzsch-type of mass matrix had attracted a great
deal of attention, it has been ruled out\cite{kangt} because
it  predicts the top quark mass to be below 
100 GeV.

Nevertheless, the Fritzsch-type mass matrix is very attractive
due to its simplicity.
Thus, the next move  is to generalize and modify the Fritzsch-type mass matrix
by maintaining the
calculability  property.   
Recently, we have proposed a generalized mass matrix ansatz and made
a systematic phenomenological study \cite{kang2}.
The general mass matrix ansatz leads to a hermitian mass matrix
\begin{eqnarray}
   M_H = \left( \begin{array}{ccc}
               0 & A & 0 \\
               A & D & B \\
               0 & B & C  \end{array} \right)
\end{eqnarray}
which is consistent with experiments for the range of $|w=B/D|=0.97 ~-~ 1.87$.
The case of $D=0$ reduces to the original Fritzsch type.
As shown in Refs. 1 and 3,
this form can be achieved by successive breaking of the ''democratic
mass matrix" that has the
maximal permutation symmetry, and is the generalization of
various specific forms of mass matrices proposed by others 
as special cases where $D$ and $B$ are related to each other 
in a particular way but mostly outside the range of $w$ that we found.
We  have shown that the matrix (1) with the finite range of 
$w$ and CP phase is consistent with experimental results including 
heavy top quark mass and the maximal CP violation.
%Regarding the phenomenological form of the mass matrix, a few years
%ago Ramond {\it et al.} \cite{ramond} narrowed down the possible 
%forms of mass matrices having texture zeros at the supersymmetric 
%unification scale.  Eq.(1) was, of course,  one of the mass matrix
%patterns.  
%While they constructed the different patterns of mass matrices for 
%the up- and down-quark sectors, Ref. \cite{kang2} assumed 
%the same form of mass matrices for both sectors.
%
We will assume the same form of mass matrices 
for the charged leptons sector.

On the other hand, all neutrino masses are zero and lepton numbers
are exactly conserved in the context of the standard model(SM).
Strictly speaking, it has not been established yet that the
neutrino masses are non-zero and hierarchical.  
However, the current experimental anomalies of 
solar \cite{homestake,gallex,sage,kamioka}
and atmospheric \cite{kamioka2,soudan,imb} neutrinos lead
us to speculate that neutrinos may be massive and have mixing,
as they can be interpreted as neutrino oscillations.
%The deficit of the solar neutrino flux is sometimes
%explained economically by the Mikheyev-Smirnov-Wolfenstein (MSW)
%effect \cite{msw}.
%The ``atmospheric neutrino anomaly" can be interpreted by
%the muon neutrino oscillations into other flavors.
%If we remove the Chlorine data, these neutrino problems
%can be solved by the simple vacuum oscillation.
If we assume that only two neutrino flavors 
participate in the oscillations,
the parameter space consists of one mixing angle and one
mass-squared difference.
On the other hand, oscillations among three-neutrino flavors 
may provide a simultaneous solution for solar and atmospheric
neutrino problems.
In this case, we have two degrees of freedom in the
choice of mass-squared difference.
Several authors have suggested \cite{kkkk,degenerate}
that almost degenerate neutrinos are needed to accommodate
the solar and atmospheric neutrino observations as well as
the cosmological constraint
that arises when we regard neutrinos as  candidates for
the hot dark matter (HDM) within the three-flavor framework.

We will show that almost degenerate scenario
among three flavors indeed follows from the standpoint
of mass matrix ansatz.
In order to do so, we will 
construct neutrino mass matrices that are constrained by
the solar and atmospheric neutrino deficits and 
the non-observation of the neutrinoless double beta decay.
We note that both charged 
and neutral lepton mass matrices assumed above can originate from the 
permutation symmetry and its suitable breakings.

\section{Mass matrix for charged lepton}
Let us start with the new class of mass matrix Eq.(1) which is obtained from
the ``democratic mass matrix" with the maximal
$S(3)_{L}\times S(3)_{R}$ symmetry which is broken successively down to
$S(2)_L\times S(2)_R$ and $S(1)_L \times S(1)_R$, followed by a
%\begin{eqnarray}
%   \frac{c}{3}~\left( \begin{array}{ccc}
%               1 & 1 & 1 \\
%               1 & 1 & 1 \\
%               1 & 1 & 1  \end{array} \right),
% \qquad
%               \left( \begin{array}{ccc}
%               0 & 0 & a \\
%               0 & 0 & a \\
%               a & a & b \end{array} \right),
% \qquad
%         d ~\left( \begin{array}{ccc}
%               1 & 0 & -1 \\
%               0 & -1 & 1 \\
%              -1 & 1 & 0  \end{array} \right),
%\end{eqnarray}
unitary transformation with  
$ U=(u_1^{T}, u_2^{T}, u_3^{T})$, where $u_1=(\frac{1}{\sqrt{2}},
\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{3}}), 
u_2=(-\frac{1}{\sqrt{2}},  \frac{1}{\sqrt{6}}, 
\frac{1}{\sqrt{3}}), u_3=(0, -\frac{2}{\sqrt{6}}, \frac{1}{\sqrt{3}} )$.
As discussed in Refs. 1 and 3, the (2,2) element is related to (2,3) element
in Eq. (1) by $w\equiv B/D = (k+1)/\sqrt{2}(2k-1) $ in the
hierarchical mass eigenstate.

Because of the quark-charged lepton symmetry, 
the mass matrix for the charged lepton sector is assumed to be exactly the same as
the quark mass matrix (1).
The mass matrix $M_H$ for $K=U^l_LU^{l\dagger}_R=diag[-1,1,1]$ 
can then be written as \cite{skk,kang2}
\begin{eqnarray}
   M_H = \left( \begin{array}{ccc}
               0 & \sqrt{\frac{m_1 m_2 m_3}{m_3-\epsilon}} & 0 \\
               \sqrt{\frac{m_1 m_2 m_3}{m_3-\epsilon}} & 
               m_2-m_1+\epsilon & w(m_2-m_1+\epsilon) \\
               0 & w(m_2-m_1+\epsilon ) &
               m_3-\epsilon  \end{array} \right),
\end{eqnarray}
in which the small parameter $\epsilon$ is related to $w$, i.e.,
$w \simeq \pm \frac{\sqrt{\epsilon m_3}}{m_2}
\left(1+\frac{m_1}{m_2}-\frac{m_2}{2m_3}\right)$,
whose range is to be determined from the experiments.
%Note the sign of $B$ is undetermined from the characteristic
%equation but the KM matrix elements are independent of the sign
%of $B$.

Then, the diagonalizing matrix $U_L^{l}$ can be written as
%({\it i.e.}, $\epsilon \rightarrow 0$) 
\begin{equation}
U_L^{l} = U_{23}(\theta_{23}) \cdot U_{12}(\theta_{12})
\end{equation}
where
\begin{eqnarray}
%R_{12} \approx \left( \begin{array}{ccc}
%               1-\frac{m_1}{2m_2} &
%               \sqrt{\frac{m_1}{m_2}} &
%               w\sqrt{\frac{m_1}{m_2}}\frac{m_2}{m_3} \\
%               -\sqrt{\frac{m_1}{m_2}} &
%               1-\frac{m_1}{m_2} & -w\frac{m_2}{m_3} \\
%               w\frac{m_2}{m_3} &
%               -w\frac{m_2}{m_3} & 1 \end{array} \right).
U_{12} =\left( \begin{array}{ccc}
              \cos \theta_{12} & \sin \theta_{12} & 0 \\
              -\sin \theta_{12} & \cos \theta_{12} & 0 \\
              0 & 0 & 1 \end{array} \right),
 \qquad
U_{23} =\left( \begin{array}{ccc}
              1 & 0 & 0 \\
              0 & \cos \theta_{23} & \sin \theta_{23}  \\
              0 & -\sin \theta_{23} & \cos \theta_{23} \end{array} \right)
\end{eqnarray}
Since (1,1), (1,3) and (3,1) elements of $M_H$ are zero, 
we may put $U_{13}(\theta_{13})=1$ 
without loss of generality.
The mixing angles $\theta_{12}$ and $\theta_{23}$ can be
written to a very good approximation as
%\begin{equation}
$\tan \theta_{12}=\sqrt{\frac{m_1}{m_2}}$
%\end{equation}
and
\begin{equation}
\tan \theta_{23}=\frac{1}{2w}\left[\left(1+\frac{m_1-m_2}{m_3}\right)
-\sqrt{\left(1+\frac{m_1-m_2}{m_3}\right)^2+4w^2\left(\frac{m_1-m_2}
 {m_3}\right)}\right]
\end{equation}

%{\it The Case II {\rm with} $K=diag[1, -1, 1]$}:  
For a negative $D$, 
the real symmetric  matrix $M_H$ can be diagonalized as 
$U_L^{l}M_H U_L^{l^{\dagger}}
= diag[m_{1}, -m_{2}, m_{3}]$, 
thus reversing the signs of both $m_1$ and $m_{2}$ in the above equations.
%As we noted, a positive $D$ in this case is excluded for the reasons of 
%naturalness due to the charged lepton mass hierarchy and
%{\it calculability}.

In both cases above
%{\it Case I} and {\it Case II} above,
it turns out that the experimentally allowed range of $w$ for the quark
sector is $0.97 \leq  |w| \leq 1.87 $ in the leading
approximation \cite{kang2}.  
Thus, the same range of $w$ is assumed also for the charged lepton sector.
However, physical observables such as survival and transition 
probabilities for $\nu_{\alpha}$'s turn out to be
insensitive to the precise value of 
$w$ in the allowed range, as discussed 
in the following.  

\section{Mass matrix for neutrinos}
It is evident that the mass matrix of the charged lepton sector is not 
appropriate for the neutrino sector.
The neutrino oscillation experiments do not seem to support such 
hierarchical pattern for neutrino masses unlike the quark and charged 
lepton masses 
but rather nearly degenerate neutrinos
within the three-flavor framework \cite{kkkk,degenerate}.
In this section, we will show how such an almost degenerate neutrino scenario
follows from the neutrino mass matrix,
which is clearly different from the approach
used by others \cite{degenerate}.
We proposed for the neutrino sector the permutation symmetry among
three family indices rather than the flavor democracy.
It can be represented by the following matrix on the {\it symmetry basis}
of $S(3)$\cite{kkkk},
\begin{eqnarray}
   {\widetilde M}_{\nu}^{(0)} = c_{\nu}\left( \begin{array}{ccc}
               1 & r & r \\
               r & 1 & r \\
               r & r & 1  \end{array} \right).
%& ~~ \qquad
\end{eqnarray}
After the unitary transformation with $V=(v^T_1, v^T_2, v^T_3)$ where
$v_1=(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{2}})$,
$v_2=(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{6}}, -\frac{1}{\sqrt{2}})$
and
$v_3=(\frac{1}{\sqrt{3}}, -\frac{2}{\sqrt{6}}, 0),$
 the matrix becomes (6)
\begin{eqnarray}
    M_{\nu}^{(0)} = c_{\nu}\left( \begin{array}{ccc}
               1+2r & 0 & 0 \\
               0 & 1-r & 0 \\
               0 & 0 & 1-r  \end{array} \right).
%& ~~ \qquad
\end{eqnarray}
Note that all three neutrinos are degenerate either for $r=0$ or
$r=-2$, while two neutrinos are always degenerate for any $r$ \cite{kkk}, 
so that only one $\Delta m^2_{\nu} $ is available.

In order to confront the above mass matrix ansatz with the experimental
and cosmological observations, we make three observations for neutrinos 
which may be accounted for by assuming massive neutrinos: \\
%\begin{itemize}
%\item{} 
(1) solar neutrino data from four different experiments,
the HOMESTAKE \cite{homestake}, GALLEX \cite{gallex}, SAGE \cite{sage},
and the KAMIOKANDE II-III \cite{kamioka}, (2)
%\item{} 
atmospheric neutrino data measured by three experiments,
the Super-KAMIOKANDE \cite{kamioka2}, SOUDAN2 \cite{soudan}, and IMB \cite{imb},
%\item{} 
(3) the likely need of neutrinos as a candidate of dark matter \cite{dark}.
%\end{itemize}
As is well known, the solar neutrino deficit can be explained through
the MSW mechanism if $\Delta m_{solar}^{2} \simeq
6\times 10^{-6}~{\rm eV}^2$ and $\sin^{2} 2\theta_{solar} \simeq 7 \times 10^{-3}$
(small angle case) , or $\Delta m_{solar}^2 \simeq 9\times 10^{-6}~{\rm eV}^2$
and $\sin^{2} 2\theta_{solar} \simeq 0.6$ (large angle case)
or through the just-so vacuum oscillations if $\Delta m_{solar}^{2} \simeq
10^{-10}~{\rm eV}^2$ and $\sin^{2} 2\theta_{solar} \simeq 0.9$ \cite{PDG}.
The atmospheric neutrino  data of the Super-Kamiokande is consistent with
$\Delta m_{32}^2 \approx (2-8)\times 10^{-3} eV^2$ and 
$\sin^2\theta_{32}=0.85-1,$ at $90\%$CL.
If the light neutrinos account for the HDM of
the universe, one has to require \cite{dark}
%\begin{equation}
$\sum_{i=1,2,3} | m_{\nu_i} | \sim 6~{\rm eV}.$
%\end{equation}
Thus we see that all three neutrinos may be almost 
degenerate in their masses, with $m_{\nu_i} \sim $~ a few eV, rather than
$m_{\nu_1} \ll m_{\nu_2} \ll m_{\nu_3}$, as assumed sometimes in the 
three-neutrino mixing scenarios.
As discussed above, in order to account for the solar and atmospheric
neutrino oscillations simultaneously, two separate $\Delta m^2_{ij}
=m^2_{\nu_i}-m^2_{\nu_j}$ 
scales are required.  
The two $\Delta m^2_{ij}$ scales can be generated from our neutrino mass 
matrix (7) in such a way to lift the degeneracy 
between the second and third neutrinos, i.e., by breaking $S(3)$ to $S(2)$.
It can be achieved by allowing (3,2) and (2,3) elements of matrix (7)
to be nonzero.
Then, the resulting matrix for the neutrinos in the flavor basis becomes
\begin{eqnarray}
    M_{\nu} = c_{\nu}\left( \begin{array}{ccc}
               1+2r & 0 & 0 \\
               0 & 1-r & \epsilon_{\nu} \\
               0 & \epsilon_{\nu} & 1-r  \end{array} \right),
%& ~~ \qquad
\end{eqnarray}
which can be diagonalized by 
$ U_{\nu}=(u_1^{T}, u_2^{T}, u_3^{T})$, where 
$u_1=(1, 0, 0 ),
u_2=(0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}), 
u_3=(0,\frac{1}{\sqrt{2}},  -\frac{1}{\sqrt{2}})$,
%\begin{eqnarray}
%    U_{\nu} = \left( \begin{array}{ccc}
%               -\frac{1}{\sqrt{2}} &\frac{1}{\sqrt{2}} &0 \\
%               \frac{1}{\sqrt{2}} &\frac{1}{\sqrt{2}} &0 \\
%               0 & 0 & 1  \end{array} \right)
%%& ~~ \qquad
%\end{eqnarray}
with the resulting eigenvalues
%\begin{equation}
$m_{\nu_{i}} / c_{\nu} = (1+2r), (1-r \mp \epsilon_{\nu}).$
%\end{equation}
%Note that we have shown only one possibility for labeling three neutrino
%mass eigenstates.  Furthermore, 
Note that $U_{\nu}$ is independent of neutrino mass 
eigenvalues, or equivalently, of $c_{\nu}, \epsilon_{\nu}$ and $r$. 
This is because the mass matrix $M_{\nu}$ given by (8) 
has a residual
$S(2)$ symmetry acting upon the first and the second family indices.  
Alternatively, one may take nonzero (2,1) and (1,2) elements of the
matrix (7), but such case turns out to be inconsistent with the
atmospheric data from Super-Kamiokande.

One can solve for $c_{\nu}, r$ and $\epsilon_{\nu}$
by requiring three conditions, $\Delta m_{solar}^2 = 10^{-5}~
{\rm eV}^2$, $\Delta m_{atmos}^2 = 10^{-2} ~{\rm eV}^2$ 
and the neutrino mass bound due to non-observation of the neutrino-less
double beta decay\cite{doublebeta}, $<m_{\nu}>=|\sum^{3}_{i=1}
\eta_iV^2_{\nu_1i}m_i| \leq 1\sim 2$ eV, where $\eta_i=\pm1$ depending
on the CP property of $\nu_i$.
Note that we solve the solar
neutrino problem by adopting the MSW mechanism.
Then we get $r\simeq -1.9925, c_{\nu} \simeq 0.3345$ and 
$\epsilon_{\nu} \simeq 0.0075$ leading
to $|m_1|\simeq 0.998301 ~\mbox{eV}, |m_2| \simeq 0.998305 ~\mbox{eV}$ and 
$|m_3|\simeq 1.0033 ~\mbox{eV}$.
We note that the breaking parameter $\epsilon_{\nu}$
turns out to be much smaller than the parameter $r$

\section{Neutrino mixing matrix and predictions}
Combining  the $U^{l}_L$  given by Eq.(3) with $U_{\nu}$ diagonalize the
neutrino mass matrix (8), 
we get the flavor mixing matrix in the lepton sector,
\begin{eqnarray}
V_{\nu} \equiv U_{L}^{l^{\dagger}} ~U_{\nu}
=\frac{1}{\sqrt{2}}\left( \begin{array}{ccc}
               \sqrt{2}c_{12} &
                s_{12}(-c_{23}+s_{23} &
                s_{12}(c_{23}+s_{23}) \\
               \sqrt{2}s_{12} &
                c_{12}(c_{23}-s_{23}) &
                c_{12}(-c_{23}-s_{23}) \\
               0 & s_{23}+c_{23} & -s_{23}+c_{23}
                \end{array} \right),
\end{eqnarray}
where we have abbreviated $\cos\theta_{ij}$ and $\sin\theta_{ij}$ as
$c_{ij}$ and $s_{ij}$ respectively.
%We note that the mixing matrix is independent of neutrino masses, 
%although it depends on the charged lepton masses.
For the whole range of $0.97 \leq |w| \leq 1.87 $, 
the neutrino mixing matrix is given by
\begin{eqnarray}
    |V_{\nu}| = \left( \begin{array}{ccc}
               0.9952 & -0.0453 & 0.0521  \\
               0.0692 & 0.6521 & -0.7494 \\
               0.0000 & 0.7531 & 0.6551
          \end{array} \right) \leftarrow 
 \qquad
      \rightarrow
     \left( \begin{array}{ccc}
               0.9952 & -0.0440 & 0.0506  \\
               0.0692 & 0.6326 & -0.7271 \\
               0.0000 & 0.7307 & 0.6356
          \end{array} \right).
\end{eqnarray}
%where $s_{ij}=\sin \theta_{ij}$ and $c_{ij}=\cos \theta_{ij}$.
Note that our lepton mixing matrix predicts zero for $(V_{\nu})_{31}$ 
element, i.e., the tau-$\nu_{e}$ coupling is forbidden.
Because of the smallness of the (1,2) and (1,3) elements in (10),
$<m_{\nu}> \simeq m_1 \simeq 1$ eV.
>From this mixing matrix, we obtain $\sin^2{2\theta_{12}} \simeq 0.0054 ~-~
0.0061$ which is consistent with the small angle MSW solution
 and $\sin^2{2\theta_{23}} \simeq 0.85 ~-~ 0.96$ consistent with the 
Super-Kamiokande result.

The predicted probability of $\nu_{e} ~-~ \nu_{\mu}$ oscillation
is given by $P(\nu_{e}-\nu_{\mu})\sim 4|V_{e3}|^2|V_{\mu 3}|^2
sin^2\left(\frac{1.27\Delta m^2_{32}L}{E}\right) \sim 
0.006~sin^2\left(\frac{1.27\Delta m^2_{32}L}{E}\right)$.
In addition, we see that three neutrino can account for about
$50\%$ of the HDM becaus we get $\sum|m_{\nu_i}| \sim 3 $ eV whereas
HDM mass limit is 6 eV.

%Now, we check if the 
%solutions of three neutrino mass eigenvalues satisfy the constraint coming
%from the neutrino-less double $\beta-$decay,
%as well as other data from neutrino oscillation experiments.   
%Since we are considering Majorana neutrinos, 
%there is a constraint from
%non-observation of neutrino-less double $\beta-$decays \cite{doublebeta} :
%%\begin{equation} 
%$\langle m_{\nu_e} \rangle \equiv | \sum_{i=1}^{3} ~ \eta_i V_{ei}^{2} m_{i} |
%\approx 1\sim 2 ~{\rm eV}$  with $\eta_i=
%\pm1$ depending on the CP property of $\nu_i$.
%%\end{equation}
%The lepton flavor mixing matrix Eq.(10) and neutrino mass eigenvalues $m_{\nu_i}$
%lead to
%$\langle m_{\nu_e} \rangle \approx m_{\nu_e} $ because of smallness of
%$(V_{\nu})_{12}$ and $(V_{\nu})_{13}$  for the considered range of $w$.
%
%Next, we study the transition and survival probabilities of the neutrinos.
%In order to calculate the transition probabilities, the mass differences
%$\Delta m^2_{ij}$ should be identified 
%with $\Delta m^2_{solar}$ or $\Delta m^2_{atmos}$.
%Among the possibilities, it turns out that only the case for
%$\Delta m^2_{solar}=\Delta m^2_{21}$ and $\Delta m^2_{atmos}=\Delta m^2_{32}$
%can fit the available data quite well, and thus we will consider 
%henceforth only this  case.
%In Fig.~1, the  survival probability 
%$P(\nu_e \rightarrow \nu_e)$ for $w=0.97$ is shown by the solid line along 
%with various types of neutrino oscillation data including  the $\nu_e$ 
%disappearance experiments at reactors \cite{karmen,bugey,ill,kras}  
%and the solar neutrino experiments \cite{homestake,gallex,sage,kamioka}.
%Averaging the oscillation probabilities, we obtain two plateaus for the
%survival probability for $\nu_e \rightarrow \nu_e$.
%Since the plateau for the large $L/E$ turns out to be about 0.508, the solar
%neutrino deficit is solved in terms of vacuum oscillation, except for the 
%HOMESTAKE Chlorine data.    
%
%In Fig. 2, we represent the survival probability  $P(\nu_{\mu} \rightarrow
%\nu_{\mu})$ for $ w=0.97$ in the solid line along with the $\nu_{\mu}$ disappearance 
%experiment data \cite{cdhs,charmps}.
%Two plateaus are shown again and the survival probability for $\nu_{\mu}$
%at large $L/E$ is about 0.505.
%
%It also turns out that the survival probability $P(\nu_e \rightarrow
%\nu_e)$ and the plateau of $P(\nu_{\mu} \rightarrow \nu_{\mu})$
%at small $L/E$ are hardly changed with the value of $w$, 
%while the plateau of $P(\nu_{\mu}
%\rightarrow \nu_{\mu}) $ at large $L/E$ gets a little bit lower as
%$w$ is increased, for example,
%$0.477 \leq P(\nu_{\mu}\rightarrow \nu_{\mu})(L/E\approx 10^{16}
%{\rm km/GeV})
%\leq 0.505 $ for the considered range of $w$.
%%The change of parameter $R$ for the atmospheric neutrino,
%%$\Delta R\simeq 0.004 $ for the range of $|w|$.
%In particular, we find that the probability
%$P(\nu_{e}\rightarrow \nu_{\mu})$ at large $L/E$ is hardly changed
%with the value of $w$, while
%$P(\nu_{\mu}\rightarrow \nu_{\tau})$ at large $L/E$ can be changed up to
%order of one within the considered range of $ w$, for example,
%$0.006 \leq P(\nu_{\mu}\rightarrow \nu_{\tau})(L/E\approx 10^{16} 
%{\rm km/GeV})
%\leq 0.049 $.
%
%Further test of our ansatz is provided with the long baseline experiments
%searching for $\nu_{\mu} \rightarrow \nu_{\tau}$ oscillation in the range 
%of  $\Delta m_{\mu\tau}^2 \simeq 10^{-2}~{\rm eV}^2$.  Our prediction is that 
%%\begin{equation}
%$P ( \nu_{\mu} \rightarrow \nu_{\tau} ) \leq 0.049,$
%%\end{equation}
%with $\Delta m_{\mu\tau}^2 = 0.72 \times 10^{-2}~~{\rm eV}^2$ 
%for the allowed range of $w$. 
%This is still below the current upper limit 0.08 as well as the planned 
%search for the $\nu_{\mu} \rightarrow \nu_{\tau}$ oscillations at 
%CHORUS, NOMAD, FNAL P803, CERN/ICARUS and FNAL/SOUDAN2 \cite{test11}.
%In the future, the measurement of the $\nu_{\mu} \rightarrow \nu_{\tau}$
%oscillation would play a crucial role in testing our prediction and
%in constraining the range of $w$.
%%An observation of $\nu_{\mu} \leftrightarrow \nu_{\tau}$ oscillation 
%%larger than Eq.(21) will exclude our model for charged lepton and neutrino mass matrices. 
%%
%%For the atmospheric neutrino data, the so-called $R$ defined by
%%\begin{equation}
%%R\equiv \frac{(N_{\mu}/N_e)_{Data}}{(N_{\mu}/N_e)_{MC}}
%%\end{equation}
%%along with our prediction \cite{kkk}
%%\begin{equation}
%%R=\frac{P_{\mu \mu} + P_{e\mu}/r}{P_{ee}+rP_{\mu e}}
%%\end{equation}
%% where $r$ is the incident $(\mu/e)$ ratio,
%% for different
%%$L/E$'s  are given in Table~I along with the 
%%current data from KAMIOKANDE \cite{kamioka2}, IMB \cite{imb}, 
%%FREJUS \cite{prejus}, NUSEX \cite{nusex} and SOUDAN \cite{soudan}. From
%%Table ~I, we see that our prediction based on the lepton mass matrix ansatz
%%Eqs. (5) and (13) reproduce all the known data on the neutrino oscillation 
%%experiments except the Chlorine and LSND data.
%%The variation of $R$ due to the uncertainty of $w$ is at most 
%%0.004, and thus completely negligible.
%%
%%From these considerations, we may say that
%%our mass matrix ansatzs not only can describe 
%%such available data on neutrino oscillations, but also predict
%%the mixing for $\nu_{\mu} \rightarrow \nu_{\tau}$ in an interesting 
%%range which is below the current and planned experimental limit.
%%%CHORUS, NOMAD, FNAL P803, CERN/ICARUS and FNAL/SOUDAN2.
%%
%%
%%\vspace{.3in}
%
\section{Conclusion}
In conclusion, we investigated in this paper phenomenological consequences of 
the lepton mass matrix ansatzs with the minimal 
number of parameters, three each in the charged lepton 
and Majorana neutrino mass matrices $M_{H}$ and $M_{\nu}$, 
with permutation symmetries among three 
generations $S(3)_L \times S(3)_R$ and $S(3)$ that are broken down to
$S(2)_L \times S(2)_R$ and $S(1)_L \times S(1)_R$ and $S(2)$ and $S(1)$, 
respectively.
We find the mass matrix ansatz Eqs.(1) and (8) lead to a lepton mixing matrix which can
accommodate the solar and atmospheric neutrino observations as well as
the constraint from
non-observation of neutrino-less beta decays.
Three light Majorana neutrinos can  provide about $50\%$ of the 
HDM, with $\Sigma | m_{\nu_i} | \sim 6~$ eV. 

{\it Note added }: After the presentation of this talk,  
the results of the long-baseline
reactor experiment CHOOZ appeared \cite{chooz}, which are consistent with
our results.
%Our prediciton for P(\nu_{e}\rightarrow
%\nu_{\mu}) \sim 0.003$ and $\sin^2{2\theta_{21}}=0.006$ are consistent with
%the experiment.
%\acknowledgements
%Two of us (SKK,JEK) would like to thank 
%the members of the
%High Energy Theory Group for the warm hospitality extended to them
%at Brown University.
%This work is supported in part by the 
%%Korea Science and Engineering Foundation 
%KOSEF Postdoctoral fellowship(SKK), the KOSEF grant through Center for Theoretical Physics at Seoul National
%University (PK), SNU-Brown Exchange Program (KK, JEK, PK),
%the Ministry of Education through the Basic Science Research Institute,
%Contract No. BSRI-96-2418 (JEK,PK), Hoam Foundation (JEK), Distinguished
%Scholar Exchange Program of Korea Research Foundation(JEK,PK),
%and also the US DOE Contract DE-FG-02-91ER40688 - Task A (KK).
%\begin{references}
%\section*{References}
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     .
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\bibitem{PDG} Particle data Group, R.M.Barnett {\it et al.}, Phys Rev. D{\bf 54}, 1 (1996).
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% R3697 (1993).
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Prog. Part. Nucl. Phys. {\bf 32}, 261 (1994), and references therein. 
The 130Te experiment with more uncertain nuclear matrix element calculations
could favor $\langle m_{\nu}\rangle \sim 4$eV.
See E. Garcia, in TAUP93 Workshop, GranSasso, Italy, 1993.
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\bibitem{bugey} Y. Declais, LAPP-EXP-94-08; 
 Y. Declais, et al., Phys. Lett. B{\bf 338}, 383 (1994);
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{\bf 55}, 206 (1992); {\it ibid.} {\bf 59}, 391 (1994).
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CERN-TH-7541/94 (1994).
\bibitem{chooz}M. Apollonio et al., .
%\bibitem{prejus} C. Berger {\it et al.}, 
%Phys. Lett. B{\bf 227}, 489 (1989) ; {\it ibid.}{\bf 245}, 305 (1990).
%\bibitem{nusex} M. Aglietta {\it et al.}, Europhys. Lett. {\bf 15}, 
%559 (1991).
\end{thebibliography}
%\begin{table}
%\caption{The atmospheric neutrino data $R$ for various $L/E$'s along with
%our predictions for $\Delta m_{21}^{2} = 0.72 \times 10^{-2}~{\rm eV}^2$, 
%$\Delta m_{32}^{2} = 
%  10^{-10}~{\rm eV}^2$
%We show the $r = (\mu/e)_{\rm incident}$ 
%values for each data point also.}
%\label{table1}
%\begin{tabular}{ccccc}
%Experiments & $r$ & $L/E$ (km/GeV) & Measured & Prediction 
%\\   \tableline
%KAMIOKA \cite{kamioka2} & $4.5/1$ & 5  & $1.27^{+0.61}_{-0.38}$ & 
%0.99
%\\
%(Multi-GeV)            & $3.2/1$ & 10 & $0.63^{+0.21}_{-0.16}$ & 
%0.97
%\\
%                       & $2.2/1$ & 100& $0.51^{+0.15}_{-0.12}$ & 
%0.42
%\\
%                       & $3.2/1$ & 1000 & $0.46^{+0.18}_{-0.12}$ & 
%0.31
%\\
%                       & $4.5/1$ & 2000 & $0.28^{+0.10}_{-0.07}$ & 
%0.23
%\\
%KAMIOKA \cite{kamioka2} & $2.1/1$ & 80   & $0.59 \pm 0.10$ & 
%0.51
%\\
%(Sub-GeV)              & $2.1/1$ & 12800 & $0.62 \pm 0.10$  &
%0.48
%\\
%IMB \cite{imb}         & $2.1/1$ & 1000 &  $0.54 \pm 0.13$ & 
%0.48
%\\
%FREJUS \cite{prejus}   & $2.1/1$ & 500  &  $0.87 \pm 0.18$ &   
%0.47
%\\
%NUSEX \cite{nusex}     & $2.1/1$ & 500  &  $0.99 \pm 0.32$ & 
%0.47
%\\
%SOUDAN \cite{soudan}   & $2.1/1$ & 1000 & $0.69 \pm 0.21$  & 
%0.48
%\end{tabular}
%\end{table}
%
%\begin{figure}
%\caption
%{The survival probabilities $P(\nu_{e}\rightarrow \nu_{e})$
%for $w=0.97$ using our lepton mass matrices Eqs.(5) and (13),
%along with the reactor experiment data from KARMEN, ILL/GOSGEN,
%BUGEY, KRASNOYARSK, and the solar neutrino data from
%KAMIOKA, HOMESTAKE, SAGE and GALLEX.}
%\end{figure}
%\begin{figure}
%\caption
%{The survival probabilities $P(\nu_{\mu}\rightarrow \nu_{\mu})$
%for $w=0.97$  using our lepton mass matrices Eqs.(5) and (13),
%along with the reactor experiments from CDHS-SPS and
%CHARM-PS.}
%\end{figure}
\end{document}
\end






















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\caption{Experimental Data bearing on $\Gamma(K
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\begin{center}\begin{tabular}
{|c|c|c|l|}\hline
& & & \\
&$\Gamma(\pi^- \pi^0)\; s^{-1}$
&$\Gamma(\pi^- \pi^0 \gamma)\; s^{-1}$ &\\ \hline
\mco{2}{|c|}{Process for
Decay} & & \\ \cline{1-2}
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&\begin{minipage}{1in}
$2.22 \times 10^4$ \\
(DE $ 1.46 \times10^3)$
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&\begin{minipage}{1.5in}
No (IB)-E1 interference seen but data shows excess
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to $100MeV$\end{minipage} \\
& & &  \\ \hline
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The following may be (and has been) described as
`dangerously irrelevant' physics. The
Lorentz-invariant phase space integral for a
general n-body decay from a particle with momentum
$P$ and mass $M$ is given by:
\begin{equation}
I((P - k_i)^2, m^2_i, M) = \frac{1}{(2
\pi)^5}\!\int\!\frac{d^3 k_i}{2 \omega_i} \! \delta^4(P -
k_i).
\label{eq:murnf}
\end{equation}
The only experiment on $K
^{\pm} \ra \pi^{\pm} \pi^0 \gamma$ since 1976 is that
of Bolotov {\it et al}.~\cite{bu}
There are two necessary conditions required for any
acceptable parametrization of the quark mixing matrix.
The first is that the matrix must be unitary, and the
second is that it should contain a CP violating phase
$\delta$. In Sec.~\ref{subsec:wpp} the connection between
invariants (of form similar to J) and unitarity relations
will be examined further for the more general
$ n\times n $ case.The reason is that such a
matrix is not a faithful representation of the group,
i.e. it does not cover all of the parameter space
available.
\begin{equation}
\begin{array}{rcl}\bf{K} & = &  Im[V_{j, \alpha}
{V_{j,\alpha + 1}}^*{V_{j + 1,\alpha }}^*
V_{j + 1, \alpha + 1} ] \\
& & + Im[V_{k, \alpha + 2} {V_{k,\alpha + 3}}^*
{V_{k + 1,\alpha + 2 }}^*
V_{k + 1, \alpha + 3} ]  \\
&   & + Im[V_{j + 2, \beta} {V_{j +
2,\beta + 1}}^*{V_{j + 3,\beta }}^*
V_{j + 3, \beta + 1} ]  \\
&   &
+ Im[V_{k + 2, \beta + 2} {V_{k + 2,
\beta + 3}}^*{V_{k + 3,\beta + 2 }}^*
V_{k + 3, \beta + 3}] \\
& & \\
\bf{M}
& = &  Im[{V_{j, \alpha}}^* V_{j,\alpha + 1}
V_{j + 1,\alpha } {V_{j + 1,
\alpha + 1}}^* ]  \\
&   & + Im[V_{k, \alpha + 2} {V_{k,\alpha +
3}}^*{V_{k + 1,\alpha + 2 }}^* V_{k
+ 1, \alpha + 3} ]  \\
&   & + Im[{V_{j + 2, \beta}}^* V_{j + 2,\beta
+ 1}V_{j + 3,\beta } {V_{j + 3, \beta + 1}}^* ]  \\
&   & + Im[V_{k +
2, \beta + 2} {V_{k + 2,\beta + 3}}^*
{V_{k + 3,\beta + 2 }}^* V_{k + 3,
\beta + 3}],\\
& &\end{array}
\label{eq:spa}
\end{equation}
where $ k = j$ or $j+1$ and $\beta = \alpha$ or
$\alpha+1$, but if $k = j + 1$, then $\beta \neq \alpha + 1$
and similarly, if $\beta = \alpha + 1$ then
$ k \neq j + 1$.\footnote{An example of a matrix
which has elements containing the phase variable
$e^{i \delta}$ to second order, i.e. elements with a phase
variable $e^{2i \delta}$ is given at the end of this section.}
There are only 162 quark mixing matrices using these parameters
which are to first order in the phase variable
$e^{i\delta}$ as is the case for the Jarlskog parametrizations,
and for which J is not identically zero. It should be noted that
these are physically identical and form just one true
parametrization.
\bea
T & = & Im[V_{11}
{V_{12}}^* {V_{21}}^* V_{22}]  \nonumber \\
&  & + Im[V_{12} {V_{13}}^* {V_{22}}^* V_{23}]   \nonumber \\
&  & -
Im[V_{33} {V_{31}}^* {V_{13}}^*
V_{11}].
\label{eq:sp}
\eea\\
\begin{figure}\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}\vskip
2.5cm\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%\psfig{figure=filename.ps,height=1.5in}
\caption{Radiative (off-shell, off-page and out-to-lunch)
SUSY Higglets.
\label{fig:radish}}
\end{figure}

\section*{Acknowledgments}
This is where one places acknowledgments for funding bodies etc.
Note that there are no section numbers for the Acknowledgments,
Appendix or References.

\section*{Appendix}
We can insert an appendix here and place equations so that they are
given numbers such as Eq.~\ref{eq:app}.
\be
x =y.
\label{eq:app}
\ee

\section*{References}
\begin{thebibliography}{99}
\bibitem{ja}C Jarlskog in {\em CP Violation},
ed. C Jarlskog (World Scientific, Singapore, 1988).
\bibitem{ma}L. Maiani, \Journal{\PLB}{62}{183}{1976}.
\bibitem{bu}J.D. Bjorken and I. Dunietz, \Journal{\PRD}{36}{2109}{1987}.
\bibitem{bd}C.D. Buchanan {\it et al}, \Journal{\PRD}{45}{4088}{1992}.
\end{thebibliography}
\end{document}
%%%%%%%%%%%%%%%%%%
%%%%% End of sprocl.tex
%%%%%%%%%%%%%%%%%%%%%%%


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%\begin{tabular}
\preprint{\begin{tabular}{c}
\hbox to\textwidth{February 1998 \hfill BROWN-HET-1110}\\
[-10pt]
\hbox to\textwidth{ \hfill BROWN-TA-555}\\
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%\vskip 0.6in
\title{ Neutrino Mass Matrix, Mixing and Oscillation\footnote{Supported in part by the 
USDOE Contract DE-FG02-91ER 40688-Task A and presented by one of us (KK) at the APCTP Workshop: {\it Pacific Particle Physics Phenomenology}, Seoul, Korea, 31 October - 2 November, 1997.}}
%\vskip 0.2in
\author{ Kyungsik Kang} 
\address {\it
Department of Physics, Brown University,
Providence, RI 02912, USA}
%~\\
%and\\
%~\\
\author{Sin Kyu Kang} 
\address{\it
School of Physics, Korea Institute for Advanced Study, Seoul, Korea}
%\vskip 0.1in
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% You may repeat \author \address
% as often as necessary
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{-.1in}
\begin{abstract}
We investigate the phenomenological consequences of lepton mass
matrices that are originated from the family permutation
symmetry and its suitable breakings.
Adopting the recently proposed new  mass matrix for the charged lepton mass
matrix and the Majorana neutrino mass matrix  considered
before, we find that the resulting lepton flavor
mixing  matrix  is consistent with the current data on various type
of  neutrino oscillation experiments except the Chlorine data and
LSND measurement and that three neutrinos are almost degenerate
in mass.
%The $\nu_{\mu} \leftrightarrow
%\nu_{\tau}$   probability is predicted to be smaller than  
% 0.049 
%with $\Delta m_{\mu\tau}^2 = 0.72 \times 10^{-2}~{\rm eV}^2$,  which still
%lies beyond the scope of the planned CHORUS and other experiments searching
%for $\nu_{\mu} \leftrightarrow \nu_{\tau}$ oscillation. 
Three neutrinos 
can account for about $50\%$ of the hot dark matter(HDM) when neutrino
mass matrix is constrained by the small angle MSW solution
for the solar neutrino problem, atmospheric neutrino data of the 
Super-Kamiokande group, and the limitation on the electron-type
neutrino mass due to non-observation of neutrino-less
double beta decays.
\end{abstract}
%\begin{center}
% ()
%\end{center}

\section{Introduction}
The flavor problem concerning the fermion masses, their hierarchical patterns,
as well as the flavor mixing
remains to be one of the most fundamental problems in particle 
physics.  Various attempts toward the understanding of the flavor mixing
with the quark mass matrix ansatz satisfying the calculability
condition are presented in another session \cite{skk}
Though the simple Fritzsch-type of mass matrix had attracted a great
deal of attention, it has been ruled out\cite{kangt} because
it  predicts the top quark mass to be below 
100 GeV.

Nevertheless, the Fritzsch-type mass matrix is very attractive
due to its simplicity.
Thus, the next move  is to generalize and modify the Fritzsch-type mass matrix
by maintaining the
calculability  property.   
Recently, we have proposed a generalized mass matrix ansatz and made
a systematic phenomenological study \cite{kang2}.
The general mass matrix ansatz leads to a hermitian mass matrix
\begin{eqnarray}
   M_H = \left( \begin{array}{ccc}
               0 & A & 0 \\
               A & D & B \\
               0 & B & C  \end{array} \right)
\end{eqnarray}
which is consistent with experiments for the range of $|w=B/D|=0.97 ~-~ 1.87$.
The case of $D=0$ reduces to the original Fritzsch type.
As shown in Refs. 1 and 3,
this form can be achieved by successive breaking of the ''democratic
mass matrix" that has the
maximal permutation symmetry, and is the generalization of
various specific forms of mass matrices proposed by others 
as special cases where $D$ and $B$ are related to each other 
in a particular way but mostly outside the range of $w$ that we found.
We  have shown that the matrix (1) with the finite range of 
$w$ and CP phase is consistent with experimental results including 
heavy top quark mass and the maximal CP violation.
%Regarding the phenomenological form of the mass matrix, a few years
%ago Ramond {\it et al.} \cite{ramond} narrowed down the possible 
%forms of mass matrices having texture zeros at the supersymmetric 
%unification scale.  Eq.(1) was, of course,  one of the mass matrix
%patterns.  
%While they constructed the different patterns of mass matrices for 
%the up- and down-quark sectors, Ref. \cite{kang2} assumed 
%the same form of mass matrices for both sectors.
%
We will assume the same form of mass matrices 
for the charged leptons sector.

On the other hand, all neutrino masses are zero and lepton numbers
are exactly conserved in the context of the standard model(SM).
Strictly speaking, it has not been established yet that the
neutrino masses are non-zero and hierarchical.  
However, the current experimental anomalies of 
solar \cite{homestake,gallex,sage,kamioka}
and atmospheric \cite{kamioka2,soudan,imb} neutrinos lead
us to speculate that neutrinos may be massive and have mixing,
as they can be interpreted as neutrino oscillations.
%The deficit of the solar neutrino flux is sometimes
%explained economically by the Mikheyev-Smirnov-Wolfenstein (MSW)
%effect \cite{msw}.
%The ``atmospheric neutrino anomaly" can be interpreted by
%the muon neutrino oscillations into other flavors.
%If we remove the Chlorine data, these neutrino problems
%can be solved by the simple vacuum oscillation.
If we assume that only two neutrino flavors 
participate in the oscillations,
the parameter space consists of one mixing angle and one
mass-squared difference.
On the other hand, oscillations among three-neutrino flavors 
may provide a simultaneous solution for solar and atmospheric
neutrino problems.
In this case, we have two degrees of freedom in the
choice of mass-squared difference.
Several authors have suggested \cite{kkkk,degenerate}
that almost degenerate neutrinos are needed to accommodate
the solar and atmospheric neutrino observations as well as
the cosmological constraint
that arises when we regard neutrinos as  candidates for
the hot dark matter (HDM) within the three-flavor framework.

We will show that almost degenerate scenario
among three flavors indeed follows from the standpoint
of mass matrix ansatz.
In order to do so, we will 
construct neutrino mass matrices that are constrained by
the solar and atmospheric neutrino deficits and 
the non-observation of the neutrinoless double beta decay.
We note that both charged 
and neutral lepton mass matrices assumed above can originate from the 
permutation symmetry and its suitable breakings.

\section{Mass matrix for charged lepton}
Let us start with the new class of mass matrix Eq.(1) which is obtained from
the ``democratic mass matrix" with the maximal
$S(3)_{L}\times S(3)_{R}$ symmetry which is broken successively down to
$S(2)_L\times S(2)_R$ and $S(1)_L \times S(1)_R$, followed by a
%\begin{eqnarray}
%   \frac{c}{3}~\left( \begin{array}{ccc}
%               1 & 1 & 1 \\
%               1 & 1 & 1 \\
%               1 & 1 & 1  \end{array} \right),
% \qquad
%               \left( \begin{array}{ccc}
%               0 & 0 & a \\
%               0 & 0 & a \\
%               a & a & b \end{array} \right),
% \qquad
%         d ~\left( \begin{array}{ccc}
%               1 & 0 & -1 \\
%               0 & -1 & 1 \\
%              -1 & 1 & 0  \end{array} \right),
%\end{eqnarray}
unitary transformation with  
$ U=(u_1^{T}, u_2^{T}, u_3^{T})$, where $u_1=(\frac{1}{\sqrt{2}},
\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{3}}), 
u_2=(-\frac{1}{\sqrt{2}},  \frac{1}{\sqrt{6}}, 
\frac{1}{\sqrt{3}}), u_3=(0, -\frac{2}{\sqrt{6}}, \frac{1}{\sqrt{3}} )$.
As discussed in Refs. 1 and 3, the (2,2) element is related to (2,3) element
in Eq. (1) by $w\equiv B/D = (k+1)/\sqrt{2}(2k-1) $ in the
hierarchical mass eigenstate.

Because of the quark-charged lepton symmetry, 
the mass matrix for the charged lepton sector is assumed to be exactly the same as
the quark mass matrix (1).
The mass matrix $M_H$ for $K=U^l_LU^{l\dagger}_R=diag[-1,1,1]$ 
can then be written as \cite{skk,kang2}
\begin{eqnarray}
   M_H = \left( \begin{array}{ccc}
               0 & \sqrt{\frac{m_1 m_2 m_3}{m_3-\epsilon}} & 0 \\
               \sqrt{\frac{m_1 m_2 m_3}{m_3-\epsilon}} & 
               m_2-m_1+\epsilon & w(m_2-m_1+\epsilon) \\
               0 & w(m_2-m_1+\epsilon ) &
               m_3-\epsilon  \end{array} \right),
\end{eqnarray}
in which the small parameter $\epsilon$ is related to $w$, i.e.,
$w \simeq \pm \frac{\sqrt{\epsilon m_3}}{m_2}
\left(1+\frac{m_1}{m_2}-\frac{m_2}{2m_3}\right)$,
whose range is to be determined from the experiments.
%Note the sign of $B$ is undetermined from the characteristic
%equation but the KM matrix elements are independent of the sign
%of $B$.

Then, the diagonalizing matrix $U_L^{l}$ can be written as
%({\it i.e.}, $\epsilon \rightarrow 0$) 
\begin{equation}
U_L^{l} = U_{23}(\theta_{23}) \cdot U_{12}(\theta_{12})
\end{equation}
where
\begin{eqnarray}
%R_{12} \approx \left( \begin{array}{ccc}
%               1-\frac{m_1}{2m_2} &
%               \sqrt{\frac{m_1}{m_2}} &
%               w\sqrt{\frac{m_1}{m_2}}\frac{m_2}{m_3} \\
%               -\sqrt{\frac{m_1}{m_2}} &
%               1-\frac{m_1}{m_2} & -w\frac{m_2}{m_3} \\
%               w\frac{m_2}{m_3} &
%               -w\frac{m_2}{m_3} & 1 \end{array} \right).
U_{12} =\left( \begin{array}{ccc}
              \cos \theta_{12} & \sin \theta_{12} & 0 \\
              -\sin \theta_{12} & \cos \theta_{12} & 0 \\
              0 & 0 & 1 \end{array} \right),
 \qquad
U_{23} =\left( \begin{array}{ccc}
              1 & 0 & 0 \\
              0 & \cos \theta_{23} & \sin \theta_{23}  \\
              0 & -\sin \theta_{23} & \cos \theta_{23} \end{array} \right)
\end{eqnarray}
Since (1,1), (1,3) and (3,1) elements of $M_H$ are zero, 
we may put $U_{13}(\theta_{13})=1$ 
without loss of generality.
The mixing angles $\theta_{12}$ and $\theta_{23}$ can be
written to a very good approximation as
%\begin{equation}
$\tan \theta_{12}=\sqrt{\frac{m_1}{m_2}}$
%\end{equation}
and
\begin{equation}
\tan \theta_{23}=\frac{1}{2w}\left[\left(1+\frac{m_1-m_2}{m_3}\right)
-\sqrt{\left(1+\frac{m_1-m_2}{m_3}\right)^2+4w^2\left(\frac{m_1-m_2}
 {m_3}\right)}\right]
\end{equation}

%{\it The Case II {\rm with} $K=diag[1, -1, 1]$}:  
For a negative $D$, 
the real symmetric  matrix $M_H$ can be diagonalized as 
$U_L^{l}M_H U_L^{l^{\dagger}}
= diag[m_{1}, -m_{2}, m_{3}]$, 
thus reversing the signs of both $m_1$ and $m_{2}$ in the above equations.
%As we noted, a positive $D$ in this case is excluded for the reasons of 
%naturalness due to the charged lepton mass hierarchy and
%{\it calculability}.

In both cases above
%{\it Case I} and {\it Case II} above,
it turns out that the experimentally allowed range of $w$ for the quark
sector is $0.97 \leq  |w| \leq 1.87 $ in the leading
approximation \cite{kang2}.  
Thus, the same range of $w$ is assumed also for the charged lepton sector.
However, physical observables such as survival and transition 
probabilities for $\nu_{\alpha}$'s turn out to be
insensitive to the precise value of 
$w$ in the allowed range, as discussed 
in the following.  

\section{Mass matrix for neutrinos}
It is evident that the mass matrix of the charged lepton sector is not 
appropriate for the neutrino sector.
The neutrino oscillation experiments do not seem to support such 
hierarchical pattern for neutrino masses unlike the quark and charged 
lepton masses 
but rather nearly degenerate neutrinos
within the three-flavor framework \cite{kkkk,degenerate}.
In this section, we will show how such an almost degenerate neutrino scenario
follows from the neutrino mass matrix,
which is clearly different from the approach
used by others \cite{degenerate}.
We proposed for the neutrino sector the permutation symmetry among
three family indices rather than the flavor democracy.
It can be represented by the following matrix on the {\it symmetry basis}
of $S(3)$\cite{kkkk},
\begin{eqnarray}
   {\widetilde M}_{\nu}^{(0)} = c_{\nu}\left( \begin{array}{ccc}
               1 & r & r \\
               r & 1 & r \\
               r & r & 1  \end{array} \right).
%& ~~ \qquad
\end{eqnarray}
After the unitary transformation with $V=(v^T_1, v^T_2, v^T_3)$ where
$v_1=(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{2}})$,
$v_2=(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{6}}, -\frac{1}{\sqrt{2}})$
and
$v_3=(\frac{1}{\sqrt{3}}, -\frac{2}{\sqrt{6}}, 0),$
 the matrix becomes (6)
\begin{eqnarray}
    M_{\nu}^{(0)} = c_{\nu}\left( \begin{array}{ccc}
               1+2r & 0 & 0 \\
               0 & 1-r & 0 \\
               0 & 0 & 1-r  \end{array} \right).
%& ~~ \qquad
\end{eqnarray}
Note that all three neutrinos are degenerate either for $r=0$ or
$r=-2$, while two neutrinos are always degenerate for any $r$ \cite{kkk}, 
so that only one $\Delta m^2_{\nu} $ is available.

In order to confront the above mass matrix ansatz with the experimental
and cosmological observations, we make three observations for neutrinos 
which may be accounted for by assuming massive neutrinos: \\
%\begin{itemize}
%\item{} 
(1) solar neutrino data from four different experiments,
the HOMESTAKE \cite{homestake}, GALLEX \cite{gallex}, SAGE \cite{sage},
and the KAMIOKANDE II-III \cite{kamioka}, (2)
%\item{} 
atmospheric neutrino data measured by three experiments,
the Super-KAMIOKANDE \cite{kamioka2}, SOUDAN2 \cite{soudan}, and IMB \cite{imb},
%\item{} 
(3) the likely need of neutrinos as a candidate of dark matter \cite{dark}.
%\end{itemize}
As is well known, the solar neutrino deficit can be explained through
the MSW mechanism if $\Delta m_{solar}^{2} \simeq
6\times 10^{-6}~{\rm eV}^2$ and $\sin^{2} 2\theta_{solar} \simeq 7 \times 10^{-3}$
(small angle case) , or $\Delta m_{solar}^2 \simeq 9\times 10^{-6}~{\rm eV}^2$
and $\sin^{2} 2\theta_{solar} \simeq 0.6$ (large angle case)
or through the just-so vacuum oscillations if $\Delta m_{solar}^{2} \simeq
10^{-10}~{\rm eV}^2$ and $\sin^{2} 2\theta_{solar} \simeq 0.9$ \cite{PDG}.
The atmospheric neutrino  data of the Super-Kamiokande is consistent with
$\Delta m_{32}^2 \approx (2-8)\times 10^{-3} eV^2$ and 
$\sin^2\theta_{32}=0.85-1,$ at $90\%$CL.
If the light neutrinos account for the HDM of
the universe, one has to require \cite{dark}
%\begin{equation}
$\sum_{i=1,2,3} | m_{\nu_i} | \sim 6~{\rm eV}.$
%\end{equation}
Thus we see that all three neutrinos may be almost 
degenerate in their masses, with $m_{\nu_i} \sim $~ a few eV, rather than
$m_{\nu_1} \ll m_{\nu_2} \ll m_{\nu_3}$, as assumed sometimes in the 
three-neutrino mixing scenarios.
As discussed above, in order to account for the solar and atmospheric
neutrino oscillations simultaneously, two separate $\Delta m^2_{ij}
=m^2_{\nu_i}-m^2_{\nu_j}$ 
scales are required.  
The two $\Delta m^2_{ij}$ scales can be generated from our neutrino mass 
matrix (7) in such a way to lift the degeneracy 
between the second and third neutrinos, i.e., by breaking $S(3)$ to $S(2)$.
It can be achieved by allowing (3,2) and (2,3) elements of matrix (7)
to be nonzero.
Then, the resulting matrix for the neutrinos in the flavor basis becomes
\begin{eqnarray}
    M_{\nu} = c_{\nu}\left( \begin{array}{ccc}
               1+2r & 0 & 0 \\
               0 & 1-r & \epsilon_{\nu} \\
               0 & \epsilon_{\nu} & 1-r  \end{array} \right),
%& ~~ \qquad
\end{eqnarray}
which can be diagonalized by 
$ U_{\nu}=(u_1^{T}, u_2^{T}, u_3^{T})$, where 
$u_1=(1, 0, 0 ),
u_2=(0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}), 
u_3=(0,\frac{1}{\sqrt{2}},  -\frac{1}{\sqrt{2}})$,
%\begin{eqnarray}
%    U_{\nu} = \left( \begin{array}{ccc}
%               -\frac{1}{\sqrt{2}} &\frac{1}{\sqrt{2}} &0 \\
%               \frac{1}{\sqrt{2}} &\frac{1}{\sqrt{2}} &0 \\
%               0 & 0 & 1  \end{array} \right)
%%& ~~ \qquad
%\end{eqnarray}
with the resulting eigenvalues
%\begin{equation}
$m_{\nu_{i}} / c_{\nu} = (1+2r), (1-r \mp \epsilon_{\nu}).$
%\end{equation}
%Note that we have shown only one possibility for labeling three neutrino
%mass eigenstates.  Furthermore, 
Note that $U_{\nu}$ is independent of neutrino mass 
eigenvalues, or equivalently, of $c_{\nu}, \epsilon_{\nu}$ and $r$. 
This is because the mass matrix $M_{\nu}$ given by (8) 
has a residual
$S(2)$ symmetry acting upon the first and the second family indices.  
Alternatively, one may take nonzero (2,1) and (1,2) elements of the
matrix (7), but such case turns out to be inconsistent with the
atmospheric data from Super-Kamiokande.

One can solve for $c_{\nu}, r$ and $\epsilon_{\nu}$
by requiring three conditions, $\Delta m_{solar}^2 = 10^{-5}~
{\rm eV}^2$, $\Delta m_{atmos}^2 = 10^{-2} ~{\rm eV}^2$ 
and the neutrino mass bound due to non-observation of the neutrino-less
double beta decay\cite{doublebeta}, $<m_{\nu}>=|\sum^{3}_{i=1}
\eta_iV^2_{\nu_1i}m_i| \leq 1\sim 2$ eV, where $\eta_i=\pm1$ depending
on the CP property of $\nu_i$.
Note that we solve the solar
neutrino problem by adopting the MSW mechanism.
Then we get $r\simeq -1.9925, c_{\nu} \simeq 0.3345$ and 
$\epsilon_{\nu} \simeq 0.0075$ leading
to $|m_1|\simeq 0.998301 ~\mbox{eV}, |m_2| \simeq 0.998305 ~\mbox{eV}$ and 
$|m_3|\simeq 1.0033 ~\mbox{eV}$.
We note that the breaking parameter $\epsilon_{\nu}$
turns out to be much smaller than the parameter $r$

\section{Neutrino mixing matrix and predictions}
Combining  the $U^{l}_L$  given by Eq.(3) with $U_{\nu}$ diagonalize the
neutrino mass matrix (8), 
we get the flavor mixing matrix in the lepton sector,
\begin{eqnarray}
V_{\nu} \equiv U_{L}^{l^{\dagger}} ~U_{\nu}
=\frac{1}{\sqrt{2}}\left( \begin{array}{ccc}
               \sqrt{2}c_{12} &
                s_{12}(-c_{23}+s_{23} &
                s_{12}(c_{23}+s_{23}) \\
               \sqrt{2}s_{12} &
                c_{12}(c_{23}-s_{23}) &
                c_{12}(-c_{23}-s_{23}) \\
               0 & s_{23}+c_{23} & -s_{23}+c_{23}
                \end{array} \right),
\end{eqnarray}
where we have abbreviated $\cos\theta_{ij}$ and $\sin\theta_{ij}$ as
$c_{ij}$ and $s_{ij}$ respectively.
%We note that the mixing matrix is independent of neutrino masses, 
%although it depends on the charged lepton masses.
For the whole range of $0.97 \leq |w| \leq 1.87 $, 
the neutrino mixing matrix is given by
\begin{eqnarray}
    |V_{\nu}| = \left( \begin{array}{ccc}
               0.9952 & -0.0453 & 0.0521  \\
               0.0692 & 0.6521 & -0.7494 \\
               0.0000 & 0.7531 & 0.6551
          \end{array} \right) \leftarrow 
 \qquad
      \rightarrow
     \left( \begin{array}{ccc}
               0.9952 & -0.0440 & 0.0506  \\
               0.0692 & 0.6326 & -0.7271 \\
               0.0000 & 0.7307 & 0.6356
          \end{array} \right).
\end{eqnarray}
%where $s_{ij}=\sin \theta_{ij}$ and $c_{ij}=\cos \theta_{ij}$.
Note that our lepton mixing matrix predicts zero for $(V_{\nu})_{31}$ 
element, i.e., the tau-$\nu_{e}$ coupling is forbidden.
Because of the smallness of the (1,2) and (1,3) elements in (10),
$<m_{\nu}> \simeq m_1 \simeq 1$ eV.
>From this mixing matrix, we obtain $\sin^2{2\theta_{12}} \simeq 0.0054 ~-~
0.0061$ which is consistent with the small angle MSW solution
 and $\sin^2{2\theta_{23}} \simeq 0.85 ~-~ 0.96$ consistent with the 
Super-Kamiokande result.

The predicted probability of $\nu_{e} ~-~ \nu_{\mu}$ oscillation
is given by $P(\nu_{e}-\nu_{\mu})\sim 4|V_{e3}|^2|V_{\mu 3}|^2
sin^2\left(\frac{1.27\Delta m^2_{32}L}{E}\right) \sim 
0.006~sin^2\left(\frac{1.27\Delta m^2_{32}L}{E}\right)$.
In addition, we see that three neutrino can account for about
$50\%$ of the HDM becaus we get $\sum|m_{\nu_i}| \sim 3 $ eV whereas
HDM mass limit is 6 eV.

%Now, we check if the 
%solutions of three neutrino mass eigenvalues satisfy the constraint coming
%from the neutrino-less double $\beta-$decay,
%as well as other data from neutrino oscillation experiments.   
%Since we are considering Majorana neutrinos, 
%there is a constraint from
%non-observation of neutrino-less double $\beta-$decays \cite{doublebeta} :
%%\begin{equation} 
%$\langle m_{\nu_e} \rangle \equiv | \sum_{i=1}^{3} ~ \eta_i V_{ei}^{2} m_{i} |
%\approx 1\sim 2 ~{\rm eV}$  with $\eta_i=
%\pm1$ depending on the CP property of $\nu_i$.
%%\end{equation}
%The lepton flavor mixing matrix Eq.(10) and neutrino mass eigenvalues $m_{\nu_i}$
%lead to
%$\langle m_{\nu_e} \rangle \approx m_{\nu_e} $ because of smallness of
%$(V_{\nu})_{12}$ and $(V_{\nu})_{13}$  for the considered range of $w$.
%
%Next, we study the transition and survival probabilities of the neutrinos.
%In order to calculate the transition probabilities, the mass differences
%$\Delta m^2_{ij}$ should be identified 
%with $\Delta m^2_{solar}$ or $\Delta m^2_{atmos}$.
%Among the possibilities, it turns out that only the case for
%$\Delta m^2_{solar}=\Delta m^2_{21}$ and $\Delta m^2_{atmos}=\Delta m^2_{32}$
%can fit the available data quite well, and thus we will consider 
%henceforth only this  case.
%In Fig.~1, the  survival probability 
%$P(\nu_e \rightarrow \nu_e)$ for $w=0.97$ is shown by the solid line along 
%with various types of neutrino oscillation data including  the $\nu_e$ 
%disappearance experiments at reactors \cite{karmen,bugey,ill,kras}  
%and the solar neutrino experiments \cite{homestake,gallex,sage,kamioka}.
%Averaging the oscillation probabilities, we obtain two plateaus for the
%survival probability for $\nu_e \rightarrow \nu_e$.
%Since the plateau for the large $L/E$ turns out to be about 0.508, the solar
%neutrino deficit is solved in terms of vacuum oscillation, except for the 
%HOMESTAKE Chlorine data.    
%
%In Fig. 2, we represent the survival probability  $P(\nu_{\mu} \rightarrow
%\nu_{\mu})$ for $ w=0.97$ in the solid line along with the $\nu_{\mu}$ disappearance 
%experiment data \cite{cdhs,charmps}.
%Two plateaus are shown again and the survival probability for $\nu_{\mu}$
%at large $L/E$ is about 0.505.
%
%It also turns out that the survival probability $P(\nu_e \rightarrow
%\nu_e)$ and the plateau of $P(\nu_{\mu} \rightarrow \nu_{\mu})$
%at small $L/E$ are hardly changed with the value of $w$, 
%while the plateau of $P(\nu_{\mu}
%\rightarrow \nu_{\mu}) $ at large $L/E$ gets a little bit lower as
%$w$ is increased, for example,
%$0.477 \leq P(\nu_{\mu}\rightarrow \nu_{\mu})(L/E\approx 10^{16}
%{\rm km/GeV})
%\leq 0.505 $ for the considered range of $w$.
%%The change of parameter $R$ for the atmospheric neutrino,
%%$\Delta R\simeq 0.004 $ for the range of $|w|$.
%In particular, we find that the probability
%$P(\nu_{e}\rightarrow \nu_{\mu})$ at large $L/E$ is hardly changed
%with the value of $w$, while
%$P(\nu_{\mu}\rightarrow \nu_{\tau})$ at large $L/E$ can be changed up to
%order of one within the considered range of $ w$, for example,
%$0.006 \leq P(\nu_{\mu}\rightarrow \nu_{\tau})(L/E\approx 10^{16} 
%{\rm km/GeV})
%\leq 0.049 $.
%
%Further test of our ansatz is provided with the long baseline experiments
%searching for $\nu_{\mu} \rightarrow \nu_{\tau}$ oscillation in the range 
%of  $\Delta m_{\mu\tau}^2 \simeq 10^{-2}~{\rm eV}^2$.  Our prediction is that 
%%\begin{equation}
%$P ( \nu_{\mu} \rightarrow \nu_{\tau} ) \leq 0.049,$
%%\end{equation}
%with $\Delta m_{\mu\tau}^2 = 0.72 \times 10^{-2}~~{\rm eV}^2$ 
%for the allowed range of $w$. 
%This is still below the current upper limit 0.08 as well as the planned 
%search for the $\nu_{\mu} \rightarrow \nu_{\tau}$ oscillations at 
%CHORUS, NOMAD, FNAL P803, CERN/ICARUS and FNAL/SOUDAN2 \cite{test11}.
%In the future, the measurement of the $\nu_{\mu} \rightarrow \nu_{\tau}$
%oscillation would play a crucial role in testing our prediction and
%in constraining the range of $w$.
%%An observation of $\nu_{\mu} \leftrightarrow \nu_{\tau}$ oscillation 
%%larger than Eq.(21) will exclude our model for charged lepton and neutrino mass matrices. 
%%
%%For the atmospheric neutrino data, the so-called $R$ defined by
%%\begin{equation}
%%R\equiv \frac{(N_{\mu}/N_e)_{Data}}{(N_{\mu}/N_e)_{MC}}
%%\end{equation}
%%along with our prediction \cite{kkk}
%%\begin{equation}
%%R=\frac{P_{\mu \mu} + P_{e\mu}/r}{P_{ee}+rP_{\mu e}}
%%\end{equation}
%% where $r$ is the incident $(\mu/e)$ ratio,
%% for different
%%$L/E$'s  are given in Table~I along with the 
%%current data from KAMIOKANDE \cite{kamioka2}, IMB \cite{imb}, 
%%FREJUS \cite{prejus}, NUSEX \cite{nusex} and SOUDAN \cite{soudan}. From
%%Table ~I, we see that our prediction based on the lepton mass matrix ansatz
%%Eqs. (5) and (13) reproduce all the known data on the neutrino oscillation 
%%experiments except the Chlorine and LSND data.
%%The variation of $R$ due to the uncertainty of $w$ is at most 
%%0.004, and thus completely negligible.
%%
%%From these considerations, we may say that
%%our mass matrix ansatzs not only can describe 
%%such available data on neutrino oscillations, but also predict
%%the mixing for $\nu_{\mu} \rightarrow \nu_{\tau}$ in an interesting 
%%range which is below the current and planned experimental limit.
%%%CHORUS, NOMAD, FNAL P803, CERN/ICARUS and FNAL/SOUDAN2.
%%
%%
%%\vspace{.3in}
%
\section{Conclusion}
In conclusion, we investigated in this paper phenomenological consequences of 
the lepton mass matrix ansatzs with the minimal 
number of parameters, three each in the charged lepton 
and Majorana neutrino mass matrices $M_{H}$ and $M_{\nu}$, 
with permutation symmetries among three 
generations $S(3)_L \times S(3)_R$ and $S(3)$ that are broken down to
$S(2)_L \times S(2)_R$ and $S(1)_L \times S(1)_R$ and $S(2)$ and $S(1)$, 
respectively.
We find the mass matrix ansatz Eqs.(1) and (8) lead to a lepton mixing matrix which can
accommodate the solar and atmospheric neutrino observations as well as
the constraint from
non-observation of neutrino-less beta decays.
Three light Majorana neutrinos can  provide about $50\%$ of the 
HDM, with $\Sigma | m_{\nu_i} | \sim 6~$ eV. 

{\it Note added }: After the presentation of this talk,  
the results of the long-baseline
reactor experiment CHOOZ appeared \cite{chooz}, which are consistent with
our results.
%Our prediciton for P(\nu_{e}\rightarrow
%\nu_{\mu}) \sim 0.003$ and $\sin^2{2\theta_{21}}=0.006$ are consistent with
%the experiment.
%\acknowledgements
%Two of us (SKK,JEK) would like to thank 
%the members of the
%High Energy Theory Group for the warm hospitality extended to them
%at Brown University.
%This work is supported in part by the 
%%Korea Science and Engineering Foundation 
%KOSEF Postdoctoral fellowship(SKK), the KOSEF grant through Center for Theoretical Physics at Seoul National
%University (PK), SNU-Brown Exchange Program (KK, JEK, PK),
%the Ministry of Education through the Basic Science Research Institute,
%Contract No. BSRI-96-2418 (JEK,PK), Hoam Foundation (JEK), Distinguished
%Scholar Exchange Program of Korea Research Foundation(JEK,PK),
%and also the US DOE Contract DE-FG-02-91ER40688 - Task A (KK).
%\begin{references}
%\section*{References}
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\end{thebibliography}
%\begin{table}
%\caption{The atmospheric neutrino data $R$ for various $L/E$'s along with
%our predictions for $\Delta m_{21}^{2} = 0.72 \times 10^{-2}~{\rm eV}^2$, 
%$\Delta m_{32}^{2} = 
%  10^{-10}~{\rm eV}^2$
%We show the $r = (\mu/e)_{\rm incident}$ 
%values for each data point also.}
%\label{table1}
%\begin{tabular}{ccccc}
%Experiments & $r$ & $L/E$ (km/GeV) & Measured & Prediction 
%\\   \tableline
%KAMIOKA \cite{kamioka2} & $4.5/1$ & 5  & $1.27^{+0.61}_{-0.38}$ & 
%0.99
%\\
%(Multi-GeV)            & $3.2/1$ & 10 & $0.63^{+0.21}_{-0.16}$ & 
%0.97
%\\
%                       & $2.2/1$ & 100& $0.51^{+0.15}_{-0.12}$ & 
%0.42
%\\
%                       & $3.2/1$ & 1000 & $0.46^{+0.18}_{-0.12}$ & 
%0.31
%\\
%                       & $4.5/1$ & 2000 & $0.28^{+0.10}_{-0.07}$ & 
%0.23
%\\
%KAMIOKA \cite{kamioka2} & $2.1/1$ & 80   & $0.59 \pm 0.10$ & 
%0.51
%\\
%(Sub-GeV)              & $2.1/1$ & 12800 & $0.62 \pm 0.10$  &
%0.48
%\\
%IMB \cite{imb}         & $2.1/1$ & 1000 &  $0.54 \pm 0.13$ & 
%0.48
%\\
%FREJUS \cite{prejus}   & $2.1/1$ & 500  &  $0.87 \pm 0.18$ &   
%0.47
%\\
%NUSEX \cite{nusex}     & $2.1/1$ & 500  &  $0.99 \pm 0.32$ & 
%0.47
%\\
%SOUDAN \cite{soudan}   & $2.1/1$ & 1000 & $0.69 \pm 0.21$  & 
%0.48
%\end{tabular}
%\end{table}
%
%\begin{figure}
%\caption
%{The survival probabilities $P(\nu_{e}\rightarrow \nu_{e})$
%for $w=0.97$ using our lepton mass matrices Eqs.(5) and (13),
%along with the reactor experiment data from KARMEN, ILL/GOSGEN,
%BUGEY, KRASNOYARSK, and the solar neutrino data from
%KAMIOKA, HOMESTAKE, SAGE and GALLEX.}
%\end{figure}
%\begin{figure}
%\caption
%{The survival probabilities $P(\nu_{\mu}\rightarrow \nu_{\mu})$
%for $w=0.97$  using our lepton mass matrices Eqs.(5) and (13),
%along with the reactor experiments from CDHS-SPS and
%CHARM-PS.}
%\end{figure}
\end{document}
\end






















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The following may be (and has been) described as
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\label{eq:murnf}
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{V_{j,\alpha + 1}}^*{V_{j + 1,\alpha }}^*
V_{j + 1, \alpha + 1} ] \\
& & + Im[V_{k, \alpha + 2} {V_{k,\alpha + 3}}^*
{V_{k + 1,\alpha + 2 }}^*
V_{k + 1, \alpha + 3} ]  \\
&   & + Im[V_{j + 2, \beta} {V_{j +
2,\beta + 1}}^*{V_{j + 3,\beta }}^*
V_{j + 3, \beta + 1} ]  \\
&   &
+ Im[V_{k + 2, \beta + 2} {V_{k + 2,
\beta + 3}}^*{V_{k + 3,\beta + 2 }}^*
V_{k + 3, \beta + 3}] \\
& & \\
\bf{M}
& = &  Im[{V_{j, \alpha}}^* V_{j,\alpha + 1}
V_{j + 1,\alpha } {V_{j + 1,
\alpha + 1}}^* ]  \\
&   & + Im[V_{k, \alpha + 2} {V_{k,\alpha +
3}}^*{V_{k + 1,\alpha + 2 }}^* V_{k
+ 1, \alpha + 3} ]  \\
&   & + Im[{V_{j + 2, \beta}}^* V_{j + 2,\beta
+ 1}V_{j + 3,\beta } {V_{j + 3, \beta + 1}}^* ]  \\
&   & + Im[V_{k +
2, \beta + 2} {V_{k + 2,\beta + 3}}^*
{V_{k + 3,\beta + 2 }}^* V_{k + 3,
\beta + 3}],\\
& &\end{array}
\label{eq:spa}
\end{equation}
where $ k = j$ or $j+1$ and $\beta = \alpha$ or
$\alpha+1$, but if $k = j + 1$, then $\beta \neq \alpha + 1$
and similarly, if $\beta = \alpha + 1$ then
$ k \neq j + 1$.\footnote{An example of a matrix
which has elements containing the phase variable
$e^{i \delta}$ to second order, i.e. elements with a phase
variable $e^{2i \delta}$ is given at the end of this section.}
There are only 162 quark mixing matrices using these parameters
which are to first order in the phase variable
$e^{i\delta}$ as is the case for the Jarlskog parametrizations,
and for which J is not identically zero. It should be noted that
these are physically identical and form just one true
parametrization.
\bea
T & = & Im[V_{11}
{V_{12}}^* {V_{21}}^* V_{22}]  \nonumber \\
&  & + Im[V_{12} {V_{13}}^* {V_{22}}^* V_{23}]   \nonumber \\
&  & -
Im[V_{33} {V_{31}}^* {V_{13}}^*
V_{11}].
\label{eq:sp}
\eea\\
\begin{figure}\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}\vskip
2.5cm\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%\psfig{figure=filename.ps,height=1.5in}
\caption{Radiative (off-shell, off-page and out-to-lunch)
SUSY Higglets.
\label{fig:radish}}
\end{figure}

\section*{Acknowledgments}
This is where one places acknowledgments for funding bodies etc.
Note that there are no section numbers for the Acknowledgments,
Appendix or References.

\section*{Appendix}
We can insert an appendix here and place equations so that they are
given numbers such as Eq.~\ref{eq:app}.
\be
x =y.
\label{eq:app}
\ee

\section*{References}
\begin{thebibliography}{99}
\bibitem{ja}C Jarlskog in {\em CP Violation},
ed. C Jarlskog (World Scientific, Singapore, 1988).
\bibitem{ma}L. Maiani, \Journal{\PLB}{62}{183}{1976}.
\bibitem{bu}J.D. Bjorken and I. Dunietz, \Journal{\PRD}{36}{2109}{1987}.
\bibitem{bd}C.D. Buchanan {\it et al}, \Journal{\PRD}{45}{4088}{1992}.
\end{thebibliography}
\end{document}
%%%%%%%%%%%%%%%%%%
%%%%% End of sprocl.tex
%%%%%%%%%%%%%%%%%%%%%%%


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From kang@het.brown.edu Fri Feb 13 01:10 KST 1998
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Date: Thu, 12 Feb 1998 11:15:09 -0500
From: kang@het.brown.edu (Kyungsik Kang)
Message-Id: <199802121615.LAA19408@het.brown.edu>
To: skkang@chep6.kaist.ac.kr
Subject: Re: proceeding2 and 1
X-Sun-Charset: US-ASCII
Status: RO

Dear Sin Kyu,
In the second talk, HET-1110, can you fix the title page so that the second
footnote (Presended by....) can appear in the same first page. For the legal
US size papers, it goes over to the beginning of the second page. The corrected
file that I sent you last time was fixed to do this correctly.
Also, in the first Reference of this talk, we may want to add the HET number of the first talk: [1] S. K. Kang's talk in the Proceedings, BROWN-HET-1109 and 
KIAS-P98003.
Cheers,
Kyungsik Kang


