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\hbox to\hsize{1998 October \hfill SNUTP 97-067}\\[-3mm]
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%\title{ Lepton Mass Matrices and Neutrino Oscillations}
\title{ Mass Matrix Ansatz for Degenerate Neutrinos Consistent with
Solar and Atmospheric neutrino Data}
\draft
\author{Kyungsik Kang$^{(a),(b)}$, Sin Kyu Kang$^{(b)}$, Jihn E. Kim$^{(b),(c)}$
       and Pyungwon Ko$^{(d)}$ }
\address{$^{(a)}$ Department of Physics, Brown University, Providence,
Rhode Island 02912, USA}
\address{$^{(b)}$ School of Physics, Korea Institute for Advanced Study, Seoul 130-012, 
Korea}
\address{$^{(c)}$ Departmen of Physics, Seoul National University, Seoul , Korea}
\address{$^{(d)}$ Department of Physics, KAIST, Taejon 305-701, Korea}
\date{\today}
\maketitle
\tighten
%\mabstract{
\begin{abstract}
We  suggest mass matrices for neutrinos and charged leptons that
can explain solar and atmospheric neutrino data.
The resulting flavor mixing matrix $V_{\nu}$ has a property that
$(V_{\nu})_{13}=0$, thus making $\nu_e \leftrightarrow \nu_{\mu}$
and $\nu_{\mu} \leftrightarrow \nu_{\tau}$ oscillations to be
effectively a two-channel problem.
Phenomenological consequences of the lepton mass matrix 
ansatze are  consistent with the current data 
on various type of  neutrino oscillation experiments except the LSND 
measurement. 
%and that three neutrinos are almost degenerate in mass.
Three neutrinos, being almost degenerate with $\sum | m_{\nu_i} | \lesssim 1 $ 
eV, can be a part of hot dark matter without any conflict with the 
constraint from neutrinoless double beta decay experiments.
The $\nu_{\mu} \leftrightarrow \nu_{\tau}$ oscillation,
$\sin^2{2\theta_{\mu \tau}}$, is predicted to be 0.86-0.97 with
$\Delta m^2_{\mu \tau}\simeq 2 \times 10^{-3}~\mbox{eV}^2$, which
is consistent with the atmospheric neutrino data and can be tested
further at the planned MINOS and K2K experiments searching for
$\nu_{\mu} \rightarrow \nu_{\tau}$ oscillation.
\end{abstract}
\pacs{PACS number(s): 12.15Ff, 14.60Pq, 11.30Er}
%\pacs{12.15Ff, 14.60Pq, 11.30Er}
%
%\mmaketitle
\newpage
\narrowtext
% \tighten

%{rbf 1.} 

\section{Introduction}
In this paper, we suggest a specific form of neutrino and charged lepton
mass matrices
that follow from permutation symmetry and the quark-charged lepton symmetry,
and show that an almost degenerate scenario among three 
flavor neutrinos \cite{kkkk,degenerate} can explain solar and atmospheric 
neutrino data
following from the standpoint of the mass matrix ansatze. 
The parameters of the neutrino mass matrix are constrained by 
the solar and the atmospheric neutrino deficits and the neutrinoless double 
beta decay experiment. Then the 
$\nu_{\mu} \leftrightarrow \nu_{\tau}$ oscillation,
$\sin^2 2\theta_{\mu \tau}$, is predicted to be 0.86--0.97
with $\Delta m_{\mu\tau}^2 \simeq 2\times 10^{-3}~{\rm eV}^2$, 
which is consistent with the atmospheric neutrino data \cite{superk2}.
Three neutrinos, being almost degenerate with $\sum |m_{\nu_i}|\lesssim 1$ eV,
can be a part of hot dark matter of the universe. 

The flavor mixing, the fermion masses, and their hierarchical patterns
still remain to be one of the most fundamental problems in particle 
physics.  As an attempt toward the understanding of the quark mass hierarchy
and the flavor mixing, the quark mass matrix ansatz was introduced 
by Weinberg \cite{wein}.
The key idea is to make the number
of parameters in the mass matrix to be less than the total number of 
flavor mixing parameters, so that there result relations between
mixing parameters and mass eigenvalues.  Some
call it  {\it calculability}.  In particular, 
the Cabibbo angle is calculable in terms of the quark masses 
in this scheme.  Weinberg's idea of {\it calculability} was extended 
for three and more generations by Fritzsch \cite{fritz} and Kang 
{\it et al.} \cite{kang1}.
Since then, the Fritzsch-type mass matrix had attracted a great
deal of attention until the top quark was discovered.
But the Fritzsch texture predicts the top quark mass to be at most about
100 GeV and thus was ruled out \cite{kangt}.

Nevertheless, the Fritzsch type mass matrix is attractive
due to its simplicity. Though its original form is phenomenologically
ruled out, one may want to generalize the Fritzsch texture
by introducing just one more parameter but by maintaining the
calculability  property.    The obvious next move is to increase a nonvanishing 
entry in the Fritzsch-type mass matrix at the $(2,2)$ element. 
Recently, a systematic phenomenological study of such generalized  
mass matrix has been studied by Kang and Kang \cite{kang2}, which is
parametrized by
\begin{eqnarray}
   M_H = \left( \begin{array}{ccc}
               0 & A & 0 \\
               A & D & B \\
               0 & B & C  \end{array} \right).
\end{eqnarray}
The case of $D=0$ reduces to the original Fritzsch type.
As shown in Ref. \cite{kang2}, 
this form can be achieved by successive breaking of the maximal permutation 
symmetry in the mass matrix. Various mass matrix ansatze proposed by 
others \cite{others1} can be identified as special cases of the above form 
by appropriately relating $D$ to $B$.
It has also been shown in \cite{kang2} that the mass matrix (1) with a finite 
range of non-zero relation of $D$  to $B$ can be consistent with experimental 
results including heavy top quark mass, while ruling out several special
$D/B$ ratios considered in \cite{others1}.

Regarding the phenomenological form of the mass matrix
Ramond {\it et al.} \cite{ramond} narrowed down a few years ago possible forms of 
mass matrices having texture zeros at the supersymmetric unification scale.  
Eq.(1) was, of course,  one of the mass matrix patterns considered in Ref.~
\cite{ramond}.  However, they constructed the different patterns of mass 
matrices for the up- and down-quark sectors, whereas Ref. \cite{kang2} 
assumed the same form of mass matrices for both sectors. 
In this paper, we assume the same form of mass matrix for the  charged
lepton sector because  the charged leptons  exhibit a similar hierarchy in mass.

On the other hand, all neutrino masses are zero and lepton numbers
are exactly conserved in the context of the standard model(SM).
%Moreover, it has not been established that the neutrino masses are 
%hierarchical.  
However, the current experimental anomalies of 
solar \cite{homestake,gallex,sage,kamioka} and atmospheric 
\cite{kamioka2,superk,soudan,imb} neutrinos lead
us to speculate nearly degenerate but non-zero neutrino masses and mixing
among the three flavors,
as they can be interpreted as originating from the 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 oscillation into other neutrino, possibly, of tau flavor.
The recent CHOOZ experiment \cite{chooz} which is a long baseline experiment 
indicates that one has to invoke a large mixing between the $\nu_{\mu} 
\leftrightarrow \nu_{\tau}$,
which is supported by the more recent result from the Super-Kamiokande
Collaborations \cite{superk2}.
It has been suggested by several authors \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 within 
the three-flavor framework.

\section{Mass matrix for charged lepton}
Let us start with the new class of mass matrix  Eq.(1) for charged lepton.
%Eq.(1) based on successive breaking of the maximal permutation symmetry 
%$S(3)_L \times S(3)_R$  to $S(2)_L \times S(2)_R$ and to $S(1)_L \times S(1)_R$
%as suggested in Ref.  \cite{kang2}.
Since the matrix $M_H$ contains four independent parameters, one might
think that  the ``{\it calculability}" \cite{kang3} might have been lost.
However, one can make additional ansatz to relate $B$ and $D$ via 
$B=wD$ with  the same ratio parameter $w$ for both the up- and down-quark 
sectors, to maintain the ``{\it calculability}" \cite{kang3}. 

In this paper, we assume the quark-charged lepton symmetry for the mass matrix
so that the matrix form of charged lepton sector is exactly the same as
the new type of quark mass matrix Eq.(1).
Let us diagonalize the mass matrix of the charged lepton sector.
In general, a hermitian matrix $M_H$ can be diagonalized by a single unitary
transformation $U_{L(R)} M_H U^{\dagger}_{L(R)}, $ while
a mass matrix needs a biunitary transformation
$U_LM_HU^{\dagger}_{R}=diag[m_1,m_2,m_3]$ in general. 
Then, we can write 
$U_LM_HU^{\dagger}_L=K\cdot diag[m_1, m_2, m_3]$ where
$K=U_LU^{\dagger}_R$ is a diagonal matrix having the diagonal elements
$\pm 1$ or a phase factor $e^{i\phi}$ in general. 
Since we deal with the real mass matrix in our
problem, $U_L$ is a real matrix and $K$ is real too.
%Since we deal with the real mass matrix, it can be diagonalized by a
%real orthogonal matrix $R_l$ so that
%$R_lM_H R_l^{\dagger} =K\cdot diag[m_1, m_2, m_3].$
%In general, $K$ should be identity matrix, but if one requires
%that $U_LU^{\dagger}_R\equiv K$ where $U_L$ and $U_R$ are unitary
%matrices, it is enough to take $K$ to be diagonal.
As discussed in Ref.\cite{kang2}, 
because of the empirical mass hierarchy $m_1 \ll m_2 \ll m_3$,
$K=diag[1,-1,1]$ irrespective
of the sign of $D$ and $K=diag[-1,1,1]$ only for positive $D$. 
In view of the hierarchical pattern of the charged lepton masses, it is also
natural to expect that $A < |D| \ll C $, and the case of $K=diag[1,-1,1]$
for positive $D$ can be excluded if the same ratio parameter $w$ 
as that for the quark mass matrices is required.
Then, the parameters $A, B, C$ and $D$ can be expressed in terms of
the charged lepton masses and $w$.
For the other case $K = diag[1,1,-1]$,
the characteristic equation for the mass matrix does not admit any
solution.

{\it The Case I {\rm with} $K=diag[-1, 1, 1]$}: From 
the characteristic equation for the $M_H$, 
the mass matrix $M_H$ can be written by
\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 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 \cite{others2}
%({\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 Eqs. (2),(5) and (6).
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 discussed above,
it turns out that the experimentally allowed range of $w$ in the quark
mass matrices is $0.97 \lesssim  |w| \lesssim 1.87 $ in the leading
approximation \cite{kang2}.  
Thus, we will assign the value of $w$ for the charged lepton sector to
the above range. However, physical observables such as survival and transition 
probabilities for $\nu_{\alpha}$'s are insensitive to the precise value of 
$w$ in the allowed range, as discussed 
in the following.  

\section{Mass matrix for neutrinos}
It is likely that the mass matrix of the charged lepton sector is not 
appropriate for the neutrino sector, since the neutrino oscillation 
experiments do not seem to support such hierarchical pattern for neutrino 
masses as that of quark or charged lepton masses but rather {\it nearly 
degenerate neutrinos} within the three-flavor framework \cite{kkkk,degenerate}.
We will show that such an almost degenerate neutrino scenario
can follow from a neutrino mass matrix, which is clearly different from the 
approach used by others \cite{degenerate}.
We assume that three light neutrinos are Majorana particles.
This can be partly motivated by the fact that there is a dimension-5 operator
which can generate the Majorana masses for SM neutrinos after electroweak
symmetry is spontaneously broken, if one considers the SM as an effective 
field theory of more fundamental theories \cite{eff}.  
%Then the neutrino mass matrix should be symmetric in the family indices.  
%Secondly, we propose the permutation symmetry among three family indices 
%rather than the flavor democracy. It can be represented by the following 
%matrix in the {\it symmetry basis} \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}
%In this symmetry basis,  the permutation symmetry $S(3)$ acting upon the 
%family indices is manifest. Note that the flavor democratic form is a special
%case of the above $S(3)$ symmetric form if we set $r=1$. After the unitary 
%transformation with
%\begin{eqnarray}
%U=\left(\begin{array}{ccc}
%          \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\
%          \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & -\frac{2}{\sqrt{6}} \\
%          \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0
%       \end{array} \right),
%\end{eqnarray}
%the above matrix becomes
%\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{kkkk}, 
%so that only one $\Delta m^2_{\nu} $ is available. In order to solve both
%the solar and the atmospheric neutrino problems, we need two different 
%$\Delta m_{ij}^2$'s. Therefore the  $S(3)$ symmetry should be broken into 
%$S(2)$ or $S(1)$. 

In order to construct a neutrino mass matrix so as to be consistent with
the experiments, we consider three observations for neutrinos 
which may be accounted for by assuming massive neutrinos: \\
\begin{itemize}
\item{} solar neutrino data from four different experiments,
the HOMESTAKE \cite{homestake}, GALLEX \cite{gallex}, SAGE \cite{sage},
and the KAMIOKANDE II-III \cite{kamioka}. 
\item{} atmospheric neutrino data measured by four experiments,
the KAMIOKANDE \cite{kamioka2}, Super-Kamiokande \cite{superk},
SOUDAN2 \cite{soudan} and IMB \cite{imb}
\item constraint from the neutrinoless double beta decay experiments 
\cite{doublebeta}
\begin{equation}
\langle m_{\nu_e} \rangle \equiv \left| \sum_{i=1}^3 \eta_i V_{ei}^2 m_i \right|^2 
\le 0.45~{\rm eV}
\end{equation}
where $\eta_i = \pm 1$ depending on the CP property of $\nu_i$.
\item{} the likely need for neutrinos as a candidate of hot dark matter 
\cite{dark}.
\end{itemize}
As is well known, the solar neutrino deficit can be explained through
the MSW mechanism if $\Delta m_{solar}^{2} \approx
6\times 10^{-6}~{\rm eV}^2$ and $\sin^{2} 2\theta_{solar} \approx 7 \times 
10^{-3}$ (small angle case) , or $\Delta m_{solar}^2 \approx 9\times 10^{-6}
~{\rm eV}^2$ and $\sin^{2} 2\theta_{solar} \approx 0.6$ (large angle case)
and through the just-so vacuum oscillations if $\Delta m_{solar}^{2} \approx
10^{-10}~{\rm eV}^2$ and $\sin^{2} 2\theta_{solar} \approx 0.9$ \cite{PDG}.
The atmospheric neutrino problem can be accommodated if $\Delta m_{atmos}^{2} 
\approx 2\times 10^{-3}~{\rm eV}^2$ and $\sin^{2} \theta_{atmos} \approx 1.0$. 
Especially the recent results from CHOOZ and Super-Kamiokande seem to
disfavor  the $\nu_{\mu} \rightarrow 
\nu_{e}$ oscillation as a possible solution to the atmospheric neutrino 
problem \cite{chooz,superk2}. 
So one has to invoke for large mixing between $\nu_{\mu}
\leftrightarrow \nu_{\tau}$.  
If the light neutrinos account for the hot dark matter of the universe, 
one has to require \cite{dark}
\begin{equation}
\sum_{i=1,2,3} | m_{\nu_i} | \lesssim 6~{\rm eV}.
\end{equation}
Thus we see that all three neutrinos may be almost 
degenerate in their masses, with $m_{\nu_i} \lesssim O(1)$~  eV, rather than
$m_{\nu_1} \ll m_{\nu_2} \ll m_{\nu_3}$, as sometimes assumed in the 
three-neutrino mixing scenarios \footnote{The recent LSND data \cite{lsnd1}, 
if confirmed, indicates $\Delta m_{LSND}^{2} \sim 1~{\rm eV}^2$ and $\sin^{2}
\theta_{LSND} \sim 10^{-3}$. Since the conclusions of two different analyses 
\cite{lsnd1,lsnd2} do not agree with each other, we do not consider the 
possibility alluded by the LSND data \cite{lsnd1} in this work.
See, however, Ref.~\cite{kkk} for a discussion when the LSND data is included.
}. 

In this paper, we would like to account for the solar ans atmospheric
neutrino deficits as $\nu_{e} - \nu_{\mu}$ and  $\nu_{\mu} - \nu_{\tau}$
oscillations, respectively.
Once $\nu_{\mu}$ and $\nu_{\tau}$ mixing is taken to be maximal, the
corresponding $2\times2$ mass matrix can be given by
\begin{eqnarray}
\left(\begin{array}{cc}
        A & B \\
        B & A
\end{array} \right).
\end{eqnarray}
After diagonalizing,  we get the eigenvalues $m_{\nu_i} = A\pm B$.
Note that if the parameter $B$ is taken to be small, the atmospheric data of
$\Delta m^2_{atmos}$ can be accommodated.
In addition, $\Delta m^2_{solar}$ can be accommodated by allowing
non-zero (1,1) element of the $3\times3$ mass matrix as follows
\begin{eqnarray}
\left(\begin{array}{ccc}
        C & 0 & 0 \\
        0 & A & B \\
        0 & B & A 
\end{array} \right)
\end{eqnarray}
which can be diagonalized by
\begin{eqnarray}
   U_{\nu} = \left( \begin{array}{ccc}
              1 & 0 & 0 \\
              0 &  \frac{1}{\sqrt{2}} &-\frac{1}{\sqrt{2}}  \\
              0 &   \frac{1}{\sqrt{2}} &\frac{1}{\sqrt{2}} 
              \end{array} \right).
\end{eqnarray}
One can solve for $A, B$ and $C$
by requiring three conditions, $\Delta m_{solar}^2 = 10^{-5}~
{\rm eV}^2$, $\Delta m_{atmos}^2 = 2\times 10^{-3} ~{\rm eV}^2$ 
and Eq.~(7).
%Note that we remove the Chlorine data in order to solve the solar
%neutrino problem by the just-so vacuum oscillation.
%\footnote{In this work, we solve the solar neutrino problem 
%in terms of vacuum oscillations. The results would remain the same even 
%if we invoke the MSW mechanism.}. 
Then, the set of parameters $(A, B, C)$ is given by:
\begin{equation}
(A, B, C) = 
(0.33383, 0.001498, 0.3323) ~~(\mbox{eV})
\end{equation}
for which three light neutrinos are almost degenerate with mass around 0.34
eV \footnote{
If $\Delta m^2_{atmos}=10^{-2}~~\mbox{eV}^2$ is used instead, the three
neutrinos are almost degenerate with mass around 1 eV \cite{apctp}.}.
%We note that our model can accomodate three light (almost degenerate) 
%Majorana neutrinos as a hot dark matter candidates only if 
%\begin{equation}
%\sum_{i=1,2,3} |m_{\nu_i}| \leq O(1) {\rm eV}. 
%\end{equation}
%Also as shown in Eq.~(21) below, it is hard to satisfy the constraint from 
%the neutrinoless double beta decay experiments if the above bound 
%Eq.~(18) is  violated.  
  
\section{Neutrino mixing matrix and predictions}

Combining  the $U^{l}_L$  given by Eq.~(3) with $U_{\nu}$ of Eq.~(11), 
we get the neutrino mixing matrix,
\begin{eqnarray}
V_{\nu} \equiv U_{\nu}^{\dagger} ~U_{L}^{l}
%=\frac{1}{\sqrt{2}}\left( \begin{array}{ccc}
%               -c_{12}-s_{12}c_{23} &
%                c_{12}-s_{12}c_{23} &
%                \sqrt{2}(s_{12}s_{23}) \\
%               -s_{12}+c_{12}c_{23} &
%                s_{12}+c_{12}c_{23} &
%               -\sqrt{2}(c_{12}s_{23}) \\
%               s_{23} & s_{23} &
%               \sqrt{2}(c_{23}) 
=\frac{1}{\sqrt{2}}\left( \begin{array}{ccc}
               \sqrt{2}c_{12} & \sqrt{2}s_{12} & 0 \\
                s_{12}(-c_{23}+s_{23}) &
                c_{12}(c_{23}-s_{23}) &
               s_{23}+c_{23} \\
                s_{12}(c_{23}+s_{23}) &
                -c_{12}(c_{23}+s_{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 \lesssim |w| \lesssim 1.87 $, 
the neutrino mixing matrix is given by
\begin{eqnarray}
    |V_{\nu}| = \left( \begin{array}{ccc}
               0.9952 & 0.0692 & 0.0  \\
               0.0453 & 0.6520 & 0.7531 \\
               0.052 & 0.7947 & 0.6551
          \end{array} \right) \sim
% \leftarrow 
% \qquad
%      \rightarrow
     \left( \begin{array}{ccc}
               0.9952 & 0.0692 & 0.0  \\
               0.0440 & 0.6326 & 0.7307 \\
               0.0506 & 0.7271 & 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 {\it predicts} zero for $(V_{\nu})_{13}$ element,
i.e., the $\nu_e$-tau coupling is forbidden,
which makes $\nu_{e} \leftrightarrow \nu_{\mu}$ and 
$\nu_{\mu} \leftrightarrow \nu_{\tau}$ oscillations to be effectively
a two-channel problem.

Now, we check if the solutions of three neutrino mass eigenvalues satisfy 
the constraint coming from the neutrinoless 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} ~ V_{ei}^{2} m_{i} |
%\lesssim 0.47~{\rm eV}.
%\end{equation}
The neutrino mixing matrix Eq.~(13) and neutrino mass eigenvalues 
lead to
\begin{equation}
\langle m_{\nu_e} \rangle \simeq 0.33 ~{\rm eV}
\end{equation}
for $w=0.97-1.87$. All of these solutions  are well below the current upper 
limit given in Eq.~(7). If we begin to increase the neutrino masses in order 
to make it dominant hot dark matter candidates, we cease to satisfy the 
$(\beta\beta)_{\nu 0}$ constraint, Eq.~(7).

Next, we study the transition and survival probabilities of the neutrinos.
In order to calculate the transition probabilities, the mass differences
$\Delta m^2_{ij}=m^2_{\nu_i}-m^2_{\nu_j}$ should be identified 
with $\Delta m^2_{solar}$ or $\Delta m^2_{atmos}$.
Among the possibilities, it turns out that only the case for
$\Delta m^2_{solar}=\Delta m^2_{12}$ and $\Delta m^2_{atmos}=\Delta m^2_{23}$
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 \lesssim P(\nu_{\mu}\rightarrow \nu_{\mu})(L/E\approx 10^{16}
%{\rm km/GeV})
%\lesssim 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
$P(\nu_{e}\rightarrow \nu_{\mu})$ and $P(\nu_{\mu}\rightarrow \nu_{\tau})$
%at large $L/E$ 
is  changed up to about $10\%$
with the value of $w$.
%$P(\nu_{\mu}\rightarrow \nu_{\tau})$ at large $L/E$ can be changed up to
%order of one within the considered range of $ w$, for example,
%$0.006 \lesssim P(\nu_{\mu}\rightarrow \nu_{\tau})(L/E\approx 10^{16} 
%{\rm km/GeV})
%\lesssim 0.05 $.

Further test of our ansatz is provided with the long baseline experiments
searching for $\nu_{\mu} \rightarrow \nu_{\tau}$ oscillation in the range 
of  $\Delta m_{\mu\tau}^2 \simeq 10^{-3}~{\rm eV}^2$. 
The MINOS \cite{minos} and K2K \cite{k2k} sensitivities to $\Delta m^2$
at $90\%$ CL can go down to $\Delta m^2 = 1.2 \times 
10^{-3}~\mbox{eV}^2$ and $2.0\times 10^{-3}~\mbox{eV}^2$, respectively,
while the ICARUS \cite{ica} sensitivity is achieved at $\Delta m^2=3.0\times 10^{-3}~
\mbox{eV}^2$.
 Our prediction is that 
\begin{equation}
%P ( \nu_{\mu} \rightarrow \nu_{\tau} ) \lesssim 0.049,
\sin^2 {2\theta_{\mu \tau}} \simeq 0.86 - 0.97
\end{equation}
with $\Delta m_{\mu\tau}^2 = 2\times 10^{-3}~~{\rm eV}^2$ 
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 $w$???).
This can be tested at the MINOS and K2K experiments searching for the
$\nu_{\mu} \rightarrow \nu_{\tau}$ oscillations in the foreseeable future,
but is beyond the sensitivity to $\Delta m^2$ at $90\%$ CL being achieved at ICARUS.
Future experiment on the $\nu_{\mu} \leftrightarrow \nu_{\tau}$ oscillation
from the MINOS and K2K will exclude our model for charged lepton and neutrino
mass matrices.
%An observation of $\nu_{\mu} \leftrightarrow \nu_{\tau}$ oscillation 
%larger than Eq.(22) 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 (14) reproduce all the known data on the atmospheric neutrino 
%experiments. 
%%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.
\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  
%with a permutation symmetry among three generations $S(3)_L \times S(3)_R$ and 
%its suitable breaking into $S(2)_L \times S(2)_R$ and $S(1)_L \times S(1)_R$  
and Majorana neutrino mass matrices 
We find the 
ansatze Eqs.~(1) and (10) lead to a lepton mixing matrix which is
consistent with the current data on various types of neutrino oscillation 
experiments except the controversial LSND data.
Three light Majorana neutrinos can constitute a part of hot dark matter, 
with $\Sigma | m_{\nu_i} | \sim 1~$ eV without contradicting the 
constraint from  neutrinoless beta decay experiments.  The resulting
amplitude of
$\nu_{\mu} \leftrightarrow \nu_{\tau}$ oscillation $\sin^2{2\theta_{\mu \tau}}$
is $0.86 - 0.97$
with $\Delta m_{\mu\tau}^2 = 2\times 10^{-3}~{\rm eV}^2$ for the range of
$w$ under consideration,  which is
consistent with the atmospheric neutrino oscillation and will be  
tested at the MINOS and K2K experiments. Finally, three neutrinos 
%lies beyond the scope of the planned CHORUS and other experiments searching 
%for $\nu_{\mu} \leftrightarrow \nu_{\tau}$ oscillation. 
being  almost degenerate, we expect that the lepton family number
breaking effects in  $\mu \rightarrow e \gamma$ and $\mu
\rightarrow 3 e$ and analogous tau decays will be very small.  


\acknowledgements   
We would like to thank Soo Bong Kim for valuable informations on the long
base line experiments.
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 through  Center for Theoretical 
Physics at Seoul National University and by KOSEF Contract No. 971-0201-002-2
(PK), SNU-Brown Exchange Program (KK, JEK, PK), the Ministry of Education 
through the Basic Science Research Institute, Contract No. BSRI-97-2418 (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).
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%\begin{table}
%\caption{The atmospheric neutrino data $R$ for various $L/E$'s along with
%our predictions for $\Delta m_{23}^{2} = 10^{-2}~{\rm eV}^2$, 
%$\Delta m_{12}^{2} = 
%  10^{-5}~{\rm eV}^2$
%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 & $0.63^{+0.21}_{-0.16}$ & 
%0.98
%\\
%                       & $2.2/1$ & 100& $0.51^{+0.15}_{-0.12}$ & 
%0.42
%\\
%                       & $3.2/1$ & 1000 & $0.46^{+0.18}_{-0.12}$ & 
%0.51
%\\
%                       & $4.5/1$ & 2000 & $0.28^{+0.10}_{-0.07}$ & 
%0.51
%\\
%KAMIOKA \cite{kamioka2} & $2.1/1$ & 80   & $0.59 \pm 0.10$ & 
%0.41
%\\
%(Sub-GeV)              & $2.1/1$ & 12800 & $0.62 \pm 0.10$  &
%0.51
%\\
%IMB \cite{imb}         & $2.1/1$ & 1000 &  $0.54 \pm 0.13$ & 
%0.51
%\\
%FREJUS \cite{prejus}   & $2.1/1$ & 500  &  $0.87 \pm 0.18$ &   
%0.51
%\\
%NUSEX \cite{nusex}     & $2.1/1$ & 500  &  $0.99 \pm 0.32$ & 
%0.51
%\\
%SOUDAN \cite{soudan}   & $2.1/1$ & 1000 & $0.69 \pm 0.21$  & 
%0.51
%\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}




