%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                                                                  %%
%%                     LATEX final Version                          %%
%%                                                                  %%
%%               The renormalization group analysis of              %%
%%        the large lepton flavor mixing and the neutrino mass      %%
%%                                                                  %%
%%                N. Haba, N. Okamura, and M. Sugiura               %%
%%                           Oct, 1998                              %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 98/08/29 Ver.0.0   Original by N. Haba
% 98/08/31 Ver.0.1   Revised  by M. Sugiura
% 98/09/02 Ver.0.2   Revised  by M. Sugiura
% 98/09/04 Ver.0.2a  Revised  by N. Okamura
% 98/09/04 Ver.0.3   Revised  by M. Sugiura
% 98/09/18 Ver.0.4   Revised  by N. Haba
% 98/09/22 Ver.0.6   Revised  by N. Haba
% 98/09/24 Ver.0.6a  Revised  by N. Okamura  
% 98/09/24 Ver.0.7   Revised  by N. Haba
% 98/09/25 Ver 0.8   Revised  by N. Okamura  
% 98/10/01 Ver 0.8.1 Revised  by N. Okamura  
% 98/10/01 Ver 0.9   Revised  by N. Haba
% 98/10/08 Ver 0.9a  Revised  by M. Sugiura
% 98/10/08 Ver 1.0   Revised  by N. Haba
% 98/10/09 Ver 1.0a  Revised  by M. Sugiura
% 98/10/11 Ver.1.1   Revised  by N. Haba / M. Sugiura
% 98/10/11 Ver 1.2   Revised  by N. Haba
% 98/10/13 Ver 1.3   Revised  by M. Sugiura
% 98/10/18 Ver 1.4   Revised  by M. Sugiura
% 98/10/22 Ver 1.5   Revised  by M. Sugiura
% 98/10/22 Ver 1.6X  Revised  by N. Haba
% 98/10/23 Ver 1.7   Revised  by M. Sugiura
% 98/10/23 Ver 1.8   Revised  by N. Haba
% 98/10/23 Ver 1.9   Revised  by M. Sugiura
% 98/10/24 Ver 2.0   Revised  by N. Haba
% 98/10/24 Final Version      by N. Haba



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% ------------------------------------------------------------ %
\def\VKM{V_{\rm CKM}}
\def\Su{\rm u}
\def\Sd{\rm d}
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\def\Sn{\rm n}
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% ======================================================================== %
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%%%%%%%%%%%%%%%%  START  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\setlength{\baselineskip}{8mm}
\begin{titlepage}
\begin{flushright}
\begin{tabular}{c c}
& {\normalsize   \\
& {\normalsize  DPNU-98-36} \\
& {\normalsize  KEK-TH-597} \\
& {\normalsize  OHSTPY-HEP-T-98-022} \\
& {\normalsize Oct. 1998}
\end{tabular}
\end{flushright}
%
\vspace{5mm}
%
\begin{center}
%{\large \bf Does the degeneracy of neutrino masses imply
% the large enhancement of the mixing angle ? (tentative)}\\
%{\large \bf Is the large neutrino mixing the result of 
%eV neutrino mass? (tentative)}\\
%{\large Does the large neutrino mixing imply eV-order neutrino mass? }\\
{\large The renormalization group analysis of
 the large lepton flavor mixing and the neutrino mass
}\\



\vspace{15mm} 
%
{N.~Haba}$^1$,\footnote{E-mail: haba@pacific.mps.ohio-state.edu} 
\hspace{2mm}
{N.~Okamura}$^2$,\footnote{E-mail: okamura@theory.kek.jp} 
\hspace{2mm}
{M.~Sugiura}$^3$ \footnote{E-mail: sugiura@eken.phys.nagoya-u.ac.jp} \\

\vspace{5mm}
$^1${\it Department of Physics, The Ohio State University,
Columbus, Ohio 43210, USA
} \\
$^2${\it Theory Group, KEK, Oho 1-1, Tsukuba 305-0801, Japan
} \\
% and \\
$^3${\it Department of Physics, Nagoya University, 
Nagoya, 464-8602, Japan
} \\

\end{center}

\vspace{10mm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%% ABSTRACT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract}

The Superkamiokande experiment suggests 
the large flavor mixing between 
$\nu_{\mu}$ and $\nu_{\tau}$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v1.4 -> v1.5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Change following sentence.
%In this paper 
We show that 
 the mixing angle receives significant corrections
 from the 
 renormalization group equation (RGE) 
 when both the second and the third generation neutrino masses 
 are larger than $O(0.1\mbox{eV})$. 
This means that 
 the mixing angle must be small
 at the decoupling scale of right-handed neutrinos 
 in the model containing a sterile neutrino $\nu_s$
 with the mass spectrum of 
 $m_{\nu_s} \approx m_{\nu_e} \ll 
  m_{\nu_{\mu}} \approx m_{\nu_{\tau}}$. 



%This result is independent of the 
%mixing angles at the 
%decoupling scale of right-handed neutrinos. 
%Neutrinos with eV masses 
%are particle candidates of the cosmological
%dark matter.

\end{abstract}
\end{titlepage}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%% INTRODUCTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Introduction}
%


The recent Superkamiokande data suggests the 
large neutrino flavor mixing between 
$\nu_{\mu}$ and $\nu_{\tau}$\cite{SK1}. 
According to this experimental result,
there have been a lot of theoretical attempts
to explain why 
the large flavor mixing is realized in 
the lepton sector\cite{ex1}. 
One of the interesting approaches 
is concentrating on the effects 
of the the renormalization group 
equation (RGE). 
The RGE effects
cause the enhancement of the neutrino flavor
mixing in some situations\cite{Babu}\cite{Tanimoto}\cite{MHE}. 


\par
In this paper we analyze the RGE 
of the neutrino flavor mixing 
between $\nu_{\mu}$ and $\nu_{\tau}$
in the minimal supersymmetric standard model (MSSM) 
with right-handed neutrinos. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v1.6X -> v1.7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Change the followings.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  but too verbose..
%Here we stand the following position; % that \\
%
%\noindent
% (1): the neutrino mass smallness is explained 
% by the see-saw mechanism, \\
% ``Firstly'' HA AMARI TUKAWANAI MITAI.
%First, the smallness of neutrino masses is explained 
% by the seesaw mechanism.
%
%\noindent
% (2): the order of $m_{\nu_{e}}$ 
Here we stand the position that 
the smallness of neutrino masses is explained 
 by the seesaw mechanism\cite{seesaw}.
We consider the situation that 
 $m_{\nu_e}$
 is much smaller than $m_{\nu_{\mu}}$ and $m_{\nu_{\tau}}$, 
 and expect the solar neutrino problem\cite{2}
 is solved by the oscillation between
 $\nu_e$ and a sterile neutrino $\nu_s$ \cite{Hata}.
This situation is so-called
 four light neutrino scenario\cite{4Neutrino}\cite{4NeutrinoGUT} 
\footnote{
The four neutrino scenario might also explain 
 the results of the LSND\cite{LSND}. 
The LSND results suggest the small mixing
 between $\overline{\nu}_{\mu}$ and
 $\overline{\nu}_{e}$ with 
 $m^2_{\nu_{\mu}} - m^2_{\nu_{e}} \sim 1\mbox{eV}$. 
However, the confirmation of this 
 result still awaits future experiments.
Recent measurements in the KARMEN
 detector exclude part of the 
 LSND allowed region\cite{KARMEN}.
}  with the mass spectrum of  
 $m_{\nu_s} \approx m_{\nu_e} \ll 
  m_{\nu_{\mu}} \approx m_{\nu_{\tau}}$.
In this neutrino mass hierarchy, 
 we find that 
 the mixing angle between $\nu_{\mu}$ 
 and $\nu_{\tau}$  
 receives significant corrections
 from the 
 renormalization group equation (RGE). 
% the large mixing at the weak scale is  
% derived % 
% by the RGE effects 
% when $m_{\nu_{\mu}}( \sim m_{\nu_{\tau}}) \geq O(0.1\mbox{eV})$. 
%the neutrino masses of the second 
% and the third generation are 
% more than $O(0.1)$ eV.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v1.4 -> v1.5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Change the following sentence.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v1.4 -> v1.5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Change the following sentences.
%This result does not depend on 
%the mixing angle 
%at the decoupling scale of right-handed neutrinos at all,
%Taking the RGE effects into account, 
In this case 
 the mixing angle at the high energy  
 must be small 
 as long as the mixing at the low energy 
 is maximal. 
{}From the view point of the model building, 
 we must find the fundamental theory which induces
 the small mixing angle at the high energy 
 in the four neutrino scenario with the 
 above mass spectrum. 


%A lot of people are trying to 
%find the fundamental theory with neutrino
%large mixing at the high energy scale,
%e.g. GUT scale.
% which results in the four neutrino scenario.\cite{4NeutrinoGUT}
% based on the four neutrino scenario.\cite{4NeutrinoGUT}
%However,
%our analysis shows that 
%the mixing must be rather small at the 
%scale of right-handed neutrinos 
%in the four light neutrino scenario
%where the atmospheric neutrino
%deficit is explained by 
%$\nu_{\mu}$ and $\nu_{\tau}$
%oscillation.\cite{4NeutrinoGUT}




%In this case the large mixing is always derived 
% by the RGE effects in the low energy 
% whatever the high energy theory
% above the scale of the right-handed neutrino is. 
%i.e., the large mixing angle at low energy
%can be derived whatever the high energy theory
%above the scale of the right-handed neutrino is.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v1.4 -> v1.5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Deleted the following two sentences.
%We know that the neutrinos with eV masses are 
%particle candidates of the cosmological 
%dark matter\cite{DM}. 
%We might say that
%the large neutrino mixing might
%be the result that the second 
%and the third generation neutrinos are 
%the cosmological dark matter.
\par
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% Section 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The RGE effects of the neutrino flavor mixing}

\subsection{The RGEs of the Yukawa couplings}
In this section
 we show the RGEs of the MSSM with right-handed neutrinos.
The superpotential of the MSSM is given by
%
\beq
{\cal W}
   = \hu_{ij} {Q_i} H_{\Su} {\bar U_j} 
     + \hd_{ij} {Q_i} H_{\Sd} {\bar D_j}
     + \hn_{ij} {L_i} H_{\Su} {\bar N_j}
     + \he_{ij} {L_i} H_{\Sd} {\bar E_j}
     + \mu_H H_{\Su} H_{\Sd}
     + \frac12 M_{R\;ij} {\bar N_{i}} {\bar N_{j}},
\label{SuperPotMSSMH}
\eeq
%
 where the index $i,j$ stands for the generation 
number $(i,j = 2,3)$.
In this paper we neglect Yukawa couplings
of the first generation since
we consider the case where
$m_{\nu_e}$ is much smaller than 
$m_{\nu_{\mu}}$ and $m_{\nu_{\tau}}$.
$Q_i$, $L_i$, ${\bar U}_i$, ${\bar D}_i$, ${\bar E}_i$, ${\bar N}_i$
 and $H_{\Sud}$ are quark doublet, lepton doublet, 
 right-handed up-sector, right-handed down-sector, 
 right-handed charged lepton, right-handed neutrino
 and Higgs fields, respectively. 
$M_{R\;ij}$ is the Majorana mass matrix of 
the right-handed neutrinos, which is symmetric
 under the generation indices $i,j$. 
$\mu_H$ is the supersymmetric mass parameter of Higgs particles. 
\par

In this model the RGEs of Yukawa couplings are given by
\bea
\dfrac{d}{dt}\hu & = & \dfrac{1}{\l( 4\pi \r)^2}
     \l[\tr \l( 3 \hu \hu^{\dagger} + \hn \hn^{\dagger} \r)
        + 3 \hu \hu^{\dagger} + \hd \hd^{\dagger} 
     -4\pi\l(\dfrac{16}{3}\alpha_3+3\alpha_2+\dfrac{13}{15} \alpha_1\r)
     \r] \hu , 
\nn
\\
\dfrac{d}{dt}\hd & = & \dfrac{1}{\l( 4\pi \r)^2}
      \l[\tr \l( 3 \hd \hd^{\dagger} + \he \he^{\dagger} \r)
       + 3 \hd \hd^{\dagger} + \hu \hu^{\dagger} 
     -4\pi\l(\dfrac{16}{3}\alpha_3+3\alpha_2+\dfrac{7}{15} \alpha_1\r)
     \r] \hd, 
\nn
\\
\dfrac{d}{dt}\he & = & \dfrac{1}{\l(4\pi\r)^2}
      \l[\tr \l( 3 \hd \hd^{\dagger} + \he \he^{\dagger} \r)
       + 3 \he \he^{\dagger} + \hn \hn^{\dagger}
     -4\pi\l(3\alpha_2+\dfrac{9}{5} \alpha_1\r) \r] \he,
\nn
\\
\dfrac{d}{dt}\hn & = & \dfrac{1}{\l(4\pi\r)^2}
      \l[\tr \l( 3 \hu \hu^{\dagger} + \hn \hn^{\dagger} \r)
       + 3 \hn \hn^{\dagger} + \he \he^{\dagger} 
     -4\pi\l(3\alpha_2+\dfrac{3}{5} \alpha_1\r)\r] \hn,
\eqlab{RGE-high}
\eea
 where $t = \ln \mu$ and $\mu$ is a renormalization point, respectively.
These equations are available for the energy region of 
$\mu > {\cal M_R}$,
where ${\cal M_R}$ denotes
the energy scale of the Majorana mass.
\par

Below the scale of ${\cal M_R}$, 
we should take the decoupling effects 
of heavy neutrinos into account.
The effective theory is described without heavy neutrinos.
The superpotential of Eq.(\ref{SuperPotMSSMH}) is modified as
%
\beq
{\cal W}
   = \hu_{ij} {Q_i} H_{\Su} {\bar U_j} 
     + \hd_{ij} {Q_i} H_{\Sd} {\bar D_j}
     + \he_{ij} {L_i} H_{\Sd} {\bar E_j}
     - \frac12 \kappa_{ij}{\nu_{i}} {\nu_{j}} H_{\Su} H_{\Su}.
\label{SuperPotMSSML}
\eeq
Here $\nu_i$s are the light modes of neutrinos 
 which remain after integrating out the heavy ones.
The coupling constant 
$\kappa_{ij}$ is defined as
\beq
  \kappa_{ij} = ( \hn \; M_R^{-1}\; \hn^{T})_{ij}\; .
\eeq
It relates to the mass matrix of the light neutrinos as
\newcommand{\VEV}[1]{{\langle {#1} \rangle}}
\beq
  m^\nu_{ij} = \dfrac{v_{\Su}^2}{2} \kappa_{ij}
             = \dfrac{v^2 \sin^2 \beta}{2}  \kappa_{ij}\; ,
  \eqlab{NeutrinoMass} 
\eeq
where 
\beq
\tan\beta \equiv \dfrac{v_{\Su}}{v_{\Sd}},~~~~~~~
 v^2 = v_{\Su}^2 + v_{\Sd}^2,
\eeq
 with $\langle H_{\Su}\rangle =v_{\Su}$ and 
 $\langle H_{\Sd}\rangle = v_{\Sd}$. 
The value of $v$ is given by 
\beq
 v = M_Z \dfrac{\sin{2\theta_W}}{2}
         \sqrt{\dfrac{\alpha}{\pi}} = 245.4 \l(\mbox{GeV}\r),
 \eqlab{vev}
\eeq
% here we take 
 with $M_Z=91.187$ GeV,
 $\alpha=127.9$, 
 and $\sin^2{\theta_W}=0.23$\cite{PDG}.
\par

%Below the heavy neutrinos' decoupling scale,
% which we denote ${\cal M_R}$,
In $\mu < {\cal M_R}$,
 the RGEs of the Yukawa couplings
 Eqs.(\ref{eqn:RGE-high}) are modified as 
\bea
\dfrac{d}{dt}\hu & = & \dfrac{1}{\l(4\pi\r)^2}
     \l[\tr\l(3\hu \hu^{\dagger}\r) + 
     3\hu\hu^{\dagger} + \hd\hd^{\dagger}
    -4\pi\l(\dfrac{16}{3}\alpha_3+3\alpha_2+\dfrac{13}{15}\alpha_1\r)
      \r] \hu, 
\nn
\\
%
\dfrac{d}{dt}\hd & = & \dfrac{1}{\l(4\pi\r)^2}
      \l[\tr\l(3\hd\hd^{\dagger}+\he\he^{\dagger}\r)
       + 3\hd\hd^{\dagger} + \hu\hu^{\dagger}
     -4\pi\l(\dfrac{16}{3}\alpha_3+3\alpha_2+\dfrac{7}{15}\alpha_1\r) \r] \hd,
\nn
\\
%
\dfrac{d}{dt}\he & = & \dfrac{1}{\l(4\pi\r)^2}
      \l[\tr\l(3\hd\hd^{\dagger}+\he\he^{\dagger}\r)
       + 3\he\he^{\dagger}
     -4\pi\l(3\alpha_2+\dfrac{9}{5} \alpha_1\r) \r] \he,
%
\eqlab{RGE-low}
\eea
%
and 
\beq
 \dfrac{d}{dt}\kappa = {1 \over 8\pi^2}
    \l[
      \l\{ \tr \l( 3\hu\hu^{\dagger} \r) 
       - 4\pi \l( 3\alpha_2 + \dfrac{3}{5} \alpha_1 \r) \r\} \kappa
+\dfrac{1}{2} \l\{ \l( \he \he^{\dagger}\r) \kappa
                 + \kappa\l( \he \he^{\dagger}\r)^{T} \r\}
    \r].
\eqlab{RGE_kappa}
\eeq
%
{}From Eqs.(\ref{eqn:RGE-low}), we can see that 
 the RGEs of quark and charged lepton
 do not include the neutrino Yukawa couplings 
 contrary to the case of $\mu >{\cal M_R}$ as Eqs.(\ref{eqn:RGE-high}). 
Hence below ${\cal M_R}$ the running of the Yukawa couplings
 of quark and charged lepton can be determined
 independently of that of the neutrinos.
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The RGEs of neutrinos in the effective theory}
%
{}From now on we are concentrating on 
 the RGE effects below the scale of ${\cal M_R}$,
 which are given by Eqs.(\ref{eqn:RGE-low}) and (\ref{eqn:RGE_kappa}). 
Since RGEs of the Yukawa couplings for quark and charged lepton of
 Eqs.(\ref{eqn:RGE-low}) 
 can be solved without information about the neutrino sector
 as mentioned before,
 the renormalization point dependences of $y_\mu$ and $y_\tau$
 are completely determined
 by the RGEs' boundary conditions which we take 
 the masses of quark and charged lepton\footnote{
We use $m_t = 174.5$ GeV, $m_c = 0.657$ GeV, 
$m_b = 3.02$ GeV, $m_s = 9.935 \times 10^{-2}$ GeV, 
$m_{\tau} = 1.746$ GeV, and $m_{\mu} = 1.0273 \times 10^{-3}$ GeV
at $\mu = M_Z$\cite{PDG}. 
},
 the Cabibbo-Kobayashi-Maskawa matrix \cite{CKM}
 and $\tan\beta$ at the weak scale. 
Then we have only to concentrate on Eq.(\ref{eqn:RGE_kappa}).
Here we neglect $CP$ phases in the flavor mixing matrices
 of the quark and the lepton sector for simplicity. 
\par

{}For convenience, 
 we take three independent parameters 
 $\kappa_r \equiv \kappa_{22}/\kappa_{33}$,
 $\sin^2 2\theta_{23}$ and $\delta \kappa^2$ 
 instead of $\kappa_{ij}$ 
 [$\kappa_{22}$, $\kappa_{33}$ and 
  $\kappa_{23} (= \kappa_{32})$]. 
Here $\sin\theta_{23}$ and $\delta \kappa^2$
 are determined from $\kappa_{ij}$ by the following equations:
\bea
  \kappa &=&
    \left(
      \begin{array}{cc}
          \cos \theta_{23} & \sin \theta_{23} \\
        - \sin \theta_{23} & \cos \theta_{23}
      \end{array}
    \right)
    \left(
      \begin{array}{cc}
        \kappa_2 & 0 \\
        0 & \kappa_3
      \end{array}
    \right)
    \left(
      \begin{array}{cc}
        \cos \theta_{23} & - \sin \theta_{23} \\
        \sin \theta_{23} &   \cos \theta_{23}
      \end{array}
    \right),
    \eqlab{KappaToMixing} \\
  \delta \kappa^2 &\equiv&
    \kappa_3^2 - \kappa_2^2,
    \eqlab{DeltaKappa}
\eea
 where 
\bea
    \kappa_3 =
    \dfrac{\sqrt{\delta \kappa^2}}{2}
    \left(
      \sqrt{\alpha} + \dfrac{1}{\sqrt{\alpha}}
    \right), \;\;\;
    \kappa_2 =
    \dfrac{\sqrt{\delta \kappa^2}}{2}
    \left(
      \sqrt{\alpha} - \dfrac{1}{\sqrt{\alpha}}
    \right),
  \eqlab{EVal2KrSin}
\eea
 with
\beq
   \alpha \equiv
   \left|
     {\dfrac{1 + \kappa_r}{1 - \kappa_r}} \cos 2\theta_{23}
   \right| .
\eeq
By using this relation, 
 the RGE of Eq.(\ref{eqn:RGE_kappa}) can be rewritten into
 the following three equations\footnote{
Equation (\ref{eqn:RGEkappa2b}) was firstly 
 derived by Babu, Leung and Pantaleone
 in Ref.~\cite{Babu}.
}:
\bea\
  \dfrac{d}{dt} \kappa_r &=&
      - \dfrac{1}{8\pi^2}(y_\tau^2 - y_\mu^2)\kappa_r,
  \eqlab{RGEkappa2a}\\
  \dfrac{d}{dt} \sin^2 2\theta_{23} &=&
      - \dfrac{1}{8\pi^2}\sin^22\theta_{23}(1-\sin^22\theta_{23})
      (y_\tau^2 - y_\mu^2)
      \dfrac{1 + \kappa_r}{1 - \kappa_r},
  \eqlab{RGEkappa2b}\\
  \dfrac{d}{dt} \delta \kappa^2 &=&
      \dfrac{1}{8\pi^2}
      \Bigg[
        2 \left\{
          \tr \l( 3\hu\hu^{\dagger} \r) 
          - 4\pi \l( 3\alpha_2 + \dfrac{3}{5} \alpha_1 \r)
        \right\}
        + y_\tau^2 + y_\mu^2 \nonumber \\
 & & +  \left( y_\tau^2 - y_\mu^2 \right)
        \left(
          \dfrac{1 + \kappa_r^2}{1 - \kappa_r^2}
          - \frac12 \cdot 
          \dfrac{1 + \kappa_r}{1 - \kappa_r} \sin^2 2 \theta_{23}
        \right)
      \Bigg] \delta \kappa^2.
\eqlab{RGEkappa2c}
\eea
%
Both $\sin^2 2\theta_{23}$ and $\delta \kappa^2$ 
 directly relate to the observed quantities
 in neutrino oscillation experiments%
\footnote{%
Since we take a diagonal base of the charged lepton,
$\sin^2 2\theta_{23}$ of this paper 
is an observable quantity. 
%parameter.
}.
The mass squared difference can be written as
 $\delta m^2 = v^4\sin^4\!\beta\; \delta\kappa^2/4$
 by using Eq.(\ref{eqn:NeutrinoMass}).
\par

As shown before, 
 $y_\mu(\mu)$ and $y_\tau(\mu)$ are determined 
 without knowing the neutrino Yukawa coupling,
 thus we can obtain values of 
 $\sin 2 \theta_{23}$ and $\kappa_r$ 
 at the weak scale by using only 
 Eqs.(\ref{eqn:RGEkappa2a}) and (\ref{eqn:RGEkappa2b}). 
We analyze RGEs of Eqs.(\ref{eqn:RGEkappa2a}) and (\ref{eqn:RGEkappa2b})
 by inputting various values of $\sin \theta_{23}$ and $\kappa_r$ for 
 the initial conditions at $\mu ={\cal M_R}$\footnote{
As shown later, we do not need to
 calculate Eq.(\ref{eqn:RGEkappa2c}) in our analysis.
}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Numerical results of the RGEs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Now we show the numerical results of the RGEs. 
%%%%%%%%%%%%%%%%%%%% Figure 1 %%%%%%%%%%%%%%%%%%%%%%
{}Figure \ref{fig:Yem} shows the 
 energy dependence of the values of $[y_\tau^2 - y_\mu^2]$,
 which is the coefficients of the RGEs of 
 $\kappa_r$ in \eqref{RGEkappa2a} and $\sin^2 2 \theta_{23}$
 in \eqref{RGEkappa2b}.
{}Four lines
 correspond to the various values of 
 $\tan\beta$, which we take 5, 20, 35, and 50.
The values of $[y_\tau^2 - y_\mu^2]$ do not receive 
 the significant RGE corrections when $\tan \beta$ is small. 
\par

%%%%%%%%%%%%%%%%%%%% Figure 2 %%%%%%%%%%%%%%%%%%%%%%
{}Figures \ref{fig:Kr}
 show the energy dependence of $\kappa_r$ with $\tan\beta = 50$.
We show 
 the two cases of (a):${\cal M_R}= 10^{14}$ GeV and 
 (b):$10^{16}$ GeV 
 with various initial conditions at ${\cal M_R}$.
%and fix the initial condition at
% $\kappa_r({\cal M_R}) = 0.8$, $0.9$ and $0.99$. 
{}From the high energy to the low energy,
$\kappa_r$ simply increases.
If we take the small $\tan \beta$, 
 the slope tends to be flat because
%decreases since 
 the value of $[y_{\tau}^2-y_\mu^2]$ in \eqref{RGEkappa2a}
 decreases as Fig.~\ref{fig:Yem}.
\par

%%%%%%%%%%%%%%%%%%%% Figure 3 %%%%%%%%%%%%%%%%%%%%%%
Figures~\ref{fig:EngDepOfSin}
 show the energy dependence of the mixing angle
 with the same values of ${\cal M_R}$ and $\kappa_r({\cal M_R})$
 as Figs.~\ref{fig:Kr}.
There are three curves corresponding to
 the values of $\kappa_r({\cal M_R})$,
% $0.8$(solid line), $0.9$(dashed line) and $0.99$(dotted line),
 and all of them have the same boundary condition
 of $\sin^2 2\theta_{23}({\cal M_R}) = 0.1$.
{}From these figures we can see that
 the mixing angle at the weak scale changes
%is sorely different
 depending on the value of $\kappa_r({\cal M_R})$.
% even the same value of $\sin^2 2\theta_{23}({\cal M_R})$.
This difference can be easily understood
 by comparing Figs.~\ref{fig:EngDepOfSin} with Figs.~\ref{fig:Kr}
 and by the existence of the factor $[(1+ \kappa_r)/(1- \kappa_r)]$
 in the R.H.S. of the RGE of Eq.(\ref{eqn:RGEkappa2b}).
%We use Fig.~\ref{fig:Kr}(a) and Fig.~\ref{fig:EngDepOfSin}(a),
Let us see Fig.~\ref{fig:Kr}(a) and Fig.~\ref{fig:EngDepOfSin}(a),
 for example.
%In the case of $\kappa_r({\cal M_R})=0.8$,
% where $\kappa_r < 1$ for all energy scale,
% the mixing angle does not receive the significant RGE corrections. 
In the case of $\kappa_r({\cal M_R}) = 0.8$,
 where $\kappa_r$ does not exceed one at all the energy scales,
 the mixing angle does not receive the significant RGE corrections.
It is because the R.H.S. of Eq.(\ref{eqn:RGEkappa2b})
 does not become so large as to enhance the mixing angle.
On the other hand,
 in the case of $\kappa_r({\cal M_R}) = 0.9$
 $\kappa_r$ exceeds one near the weak scale,
 which results in the significant enhancement
 of the mixing angle by the factor $[(1+ \kappa_r)/(1- \kappa_r)]$.
Finally in the case of $\kappa_r({\cal M_R}) = 0.99$,
 $\kappa_r$ exceeds one above the weak scale.
%far from the weak scale.
%% Stupid English below!!
Then the mixing angle once becomes maximal at high energy,
 however, after there it decreases rapidly
 since the sign of $[(1+ \kappa_r)/(1- \kappa_r)]$ changes.

%Figure \ref{fig:EngDepOfSin}(a) shows that
%the mixing angle does not receive the significant
%RGE corrections
%in the case of $\kappa_r({\cal M_R})=0.8$.
%In the case of $\kappa_r({\cal M_R})=0.9$,
%the mixing angle increases and realizes the maximal mixing 
%at the weak scale. 
%It is because $\kappa_r$ is closed to one,
%which increases the coefficient in
%Eq.(\ref{eqn:RGEkappa2b}),
%at the weak scale. 
%Finally in the case of $\kappa_r({\cal M_R})=0.99$,
% in which $\kappa_r$ exceeds one at the high energy region,
% $\sin^2 2\theta$ becomes maximal at the high energy
% and reduces to zero at the low energy.
%It is because $\kappa_r$ becomes larger than one,
%and the sign of \eqref{RGEkappa2b} changes.
%
%Since the absolute value of $[(1+ \kappa_r)/(1- \kappa_r)]$
% in the R.H.S. of Eq.(\ref{eqn:RGEkappa2b})
% becomes large when $\kappa_r$ is close to one,
% the mixing angle drastically changes
% around at $\kappa_r \simeq 1$.
\par

%%%%%%%%%%%%%%%%%%%% Figure 4 %%%%%%%%%%%%%%%%%%%%%%
Figures \ref{fig:sin} are 
the contour plots of the mixing angle $\sin^2 2\theta_{23}$ 
at the weak scale.
These are obtained by solving Eq.(\ref{eqn:RGEkappa2b})
 with the various $\kappa_r({\cal M_R})$(horizontal axis)
 and $\sin^2 2\theta_{23}({\cal M_R})$(vertical axis).
% Stupid, Stupid, Stupid...
Here $\kappa_c$ in Figs.~\ref{fig:sin} 
 is the value at ${\cal M_R}$ that induces  
%determined to satisfy
 $\kappa_r(\mbox{weak}) = 1$.
% when $\kappa_r({\cal M_R}) = \kappa_c$.
%Similar to Figs.~\ref{fig:EngDepOfSin},
% Figs.~\ref{fig:sin} also lead to the following results.
In the parameter region of 
 $\kappa_r({\cal M_R}) < \kappa_c$, 
 there are no significant RGE corrections  
 and then the mixing angle does not change drastically. 
The energy dependence of the mixing angle in this case
 is similar to the solid lines in Figs.~\ref{fig:EngDepOfSin}.
%Moreover 
 Next in the case of $\kappa_r({\cal M_R}) \simeq \kappa_c$,
 which means $\kappa_r(\mbox{weak}) \simeq 1$,
 the mixing angle at the weak scale is strongly enhanced
 near the weak scale,
 as the dashed lines in Figs.~\ref{fig:EngDepOfSin}.
Then the mixing angle at the weak scale becomes maximal
 independently of the mixing angle at ${\cal M_R}$.
Finally in the case of $\kappa_r({\cal M_R}) > \kappa_c$,
 the mixing angle at the weak scale becomes small
 even for the large mixing angle at ${\cal M_R}$,
 where the energy dependence of the mixing angle is
 similar to the dotted lines in Figs.~\ref{fig:EngDepOfSin}.
{}From these arguments we can easily see that 
 there is 
 the large enhancement of the mixing angle from the RGE 
 around $\kappa_r \simeq 1$. 


% the condition of $\kappa_r \simeq 1$ at low energy is necessary.

%{}Four figures show 
%(a): $\tan \beta =35$, ${\cal M_R}=10^{14}$ GeV,
%(b): $\tan \beta =50$, ${\cal M_R}=10^{14}$ GeV,
%(c): $\tan \beta =35$, ${\cal M_R}=10^{16}$ GeV,
%(d): $\tan \beta =50$, ${\cal M_R}=10^{16}$ GeV\footnote{
%Although we pick up 
% the large $\tan \beta$ case here,
% the situation does not change drastically
% even in the small $\tan \beta$ case.
%In the small $\tan \beta$ case
% the ``bottom point'' just moves
% to close to one.
%}. 
%The horizontal axes shows the value of $\kappa_r$
%at $\mu ={\cal M_R}$, 
%and the vertical axes shows 
%the mixing angle at $\mu ={\cal M_R}$. 
%The values of $\sin^2 2\theta_{23}$ at the weak scale 
%are determined by the RGEs of Eqs.(\ref{eqn:RGEkappa2a})
%and (\ref{eqn:RGEkappa2b}),
%corresponding to the input parameters
%of $\sin^2 2\theta_{23}({\cal M_R})$ and $\kappa_r({\cal M_R})$. 
%We can see the ``bottom point'' of 
%$\kappa_r$ in Figs.~\ref{fig:sin}, 
%which we denote
%$\kappa_c$ from now on. 
%The region of $\kappa_r < \kappa_c$
% corresponds to the case of the solid line
% in Figs.~\ref{fig:EngDepOfSin},
% where the RGE effect does not cause the significant change
% of the mixing angle.
%In the region of $\kappa_r < \kappa_c$, 
%which region corresponds to 
%$\kappa_r$(weak)$<1$ as 
%$\kappa_r({\cal M_R})=0.8$
% in Fig.~\ref{fig:sin}(a),
% the RGE effect does not cause the significant
% change of the mixing angle. 
%The value of $\sin 2 \theta_{23}$(weak) 
%is almost same as 
%that of $\sin 2 \theta_{23}({\cal M_R})$. 
%It is because the factor 
% $(1/8 \pi^2)[(1+ \kappa_r)/(1- \kappa_r)]$ 
% in \eqref{RGEkappa2b} 
% is too small to cause the significant 
% change in the mixing angle by the 
% RGE effects. 
%However around $\kappa_r \sim \kappa_c$, 
% which region corresponds to 
% $\kappa_r$(weak)$\sim 1$ as 
% $\kappa_r({\cal M_R})=0.9$
% in Fig.~\ref{fig:sin}(a), 
% the situation is drastically changed. 
%Around this point, 
% the neutrino mixing at the weak scale always becomes large
% independently of the mixing angle at $\mu = {\cal M_R}$. 
%Even if the mixing angle is small at the ${\cal M_R}$ scale, 
% the mixing angle at the weak scale is enhanced 
% as $\sin^2 2 \theta_{23}$(weak) $> 0.9$. 
%In the region of $\kappa_r > \kappa_c$,
% which region corresponds to 
% $\kappa_r$(weak)$>1$ as 
% $\kappa_r({\cal M_R})=0.99$
% in Fig.~\ref{fig:sin}(a), 
% the mixing angle at the weak scale 
% is always small independently of the mixing angle at 
% ${\cal M_R}$. 
\par

%%%%%%%%%%%%%%%%%%%% Figure 5 %%%%%%%%%%%%%%%%%%%%%%
Figures \ref{fig:m3} are
 the contour plots of the heaviest neutrino mass $m_3$ 
 at the weak scale.
The horizontal and vertical axes are the same as 
 those of Figs.~\ref{fig:sin}. 
We determine the masses of neutrino
 by substituting the parameters in \eqref{EVal2KrSin}
 with the result of Figs.~\ref{fig:sin}
 and the experimental value
 $\delta m^2_{23} \simeq 1.3 \times 10^{-3}\;\mbox{eV}^2$.
Since we use the experimental value of $\delta m^2_{23}$
 instead of evaluating the RGE of \eqref{RGEkappa2c},
 we can determine the masses without any additional input parameters.
%Figures \ref{fig:m3} are the ones shown from another 
% view point of Figs.~\ref{fig:sin}. 
%{}Figures \ref{fig:m3} and Figs.~\ref{fig:sin}
% are related with each other through \eqref{EVal2KrSin}. 
%The horizontal and vertical axes are as same as 
% those of Figs.~\ref{fig:sin}. 
As $m_3$ becomes large, the region 
 of $\kappa_r({\cal M_R})$ is limited
 around $\kappa_c$. 
Compared with Figs.~\ref{fig:sin},
%This means 
 it is found
 that the region where the heaviest mass $m_3$ is larger 
 than $O(0.1\mbox{eV})$ corresponds to the region
 where the mixing angle at the weak scale 
 is always larger than 0.9
 despite the small mixing at ${\cal M_R}$ scale. %(maximal enhancement).
% by  
% comparing Figs.~\ref{fig:sin} with Figs.~\ref{fig:m3}.
% is larger than 0.9 despite the small mixing at ${\cal M_R}$ scale.
%
% we obtain the surprising results. 
%If the heaviest mass $m_3$ is larger than $O(0.1)$ eV,
% where $m_2$ is also large as $m_3$
% since $\delta m_{23}^2 = O(10^{-3}{\rm eV}^2)$,
% the mixing angle at the weak scale is always larger than 0.9. 
%This result is completely 
% independent of the mixing angle at the decoupling scale of 
% right-handed neutrinos. 
%The reason of this RGE enhancement can be understood 
% by the factor $[(1+ \kappa_r)/(1- \kappa_r)]$ in \eqref{RGEkappa2b}. 
%This factor enhances as neutrino masses increase 
% with fixed value of $\delta m_{23}^2$. 



Figures~\ref{fig:sin35} and Figures~\ref{fig:m335}
 correspond to Figures~\ref{fig:sin} and Figures~\ref{fig:m3}
 with another value of $\tan\beta=35$.
%%%%%%%%%%%%%%%%%%% v1.6X -> v1.7
%%%%%%%%%%%%%%%%%%% Change the following sentence.
%%%%%%%%%%%%%%%%%%% but, verbose..stupid...
%%%%%%%%%%%%%%%%%%% TOKUNI ``and on which''... TSUKAWANE-YONA, HUTSUU.
Compared with the case of $\tan\beta=50$,
 the value of $\kappa_c$ are just shifted to the right 
 in the case of $\tan\beta=35$.
Around $\kappa_r({\cal M_R}) \simeq \kappa_c$,
 the value of 
 $m_3$ is slightly larger than that of $\tan\beta=50$
 at the same value of $\sin^2 2\theta_{23}$.
%In general, the more $\tan\beta$ becomes small,
%the more $\kappa_c$ approaches one.
%Then the qualitative results does not change
% except for $\kappa_c$ are shifted to right.
%
%%%%%%%%%%%%%%%%%%% v1.6X -> v1.7
%%%%%%%%%%%%%%%%%%% Change following two sentences.
%The situation is similar to the case of 
% $\tan \beta=50$. 
%Similar to the case of $\tan\beta=50$,
%There is also the region
% where the mixing angle is maximally enhanced 
% in $\tan\beta=35$, 
% where the mass $m_3$ is larger than that of $\tan\beta=50$.
% while the value of $m_3$ 
% where the heaviest mass $m_3$ is larger than $O(0.1)$ eV
% where the largest mass $m_3$ is more than $O(0.1)$ eV
% corresponding to the region
% where the mixing angle at the weak scale 
% is always larger than 0.9.
% by
% comparing Figs.~\ref{fig:sin35} with Figs.~\ref{fig:m335}. 
%
%\par
%{\it kokode dousitatte Figs7 ni senwo
%kuwaete tanb=50 no tokito onaji hanasiwo
%suruhituyouga aru.m3 ha kc ni
%tikazuku none.
%}\\
%%%%%%%%%%%%%%%%%%% v1.6X -> v1.7
%%%%%%%%%%%%%%%%%%% Add the following sentences.
In general,
 the smaller $\tan\beta$ becomes, 
 the more $\kappa_c$ approaches one, 
 and the value of $m_3$ around $\kappa_r({\cal M_R}) \simeq \kappa_c$
 becomes larger.
%Moreover, 
The region where 
 the maximal enhancement of the mixing angle is derived by 
 the RGE exists even in the case of 
 small $\tan \beta$. 
The value of $m_3$ around $\kappa_c$ becomes larger 
 corresponding to  
 the smaller value of $\tan\beta$.

%At closing this section,
We stress here that the large enhancement factor
 at the weak scale induced by
 $[(1+ \kappa_r)/(1- \kappa_r)]$ in \eqref{RGEkappa2b}
 is not the fine-tuning.
This factor must be inevitably large with 
$\kappa_r(\mbox{weak})\simeq 1$ if $m_2$ and $m_3$ are
 larger than  
%enough to satisfy 
%%%%%%%%%%%%%%%%%%%%%%%%%%%% v1.6X -> v1.7
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Change below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%% delete ``order''
 $O(0.1\mbox{eV})$. 
% with $\delta m_{23}^2 \simeq 10^{-3}\;\mbox{eV}^2$.
%Then the another fine-tuning might be suspected
% when each of neutrino masses are relatively large
% compared with $\delta m^2_{23}$,
% however,
% this fine-tuning does not occur in this case
% since 
%It is note worthing that 
It is worth noting that
 the enough enhancement of the mixing angle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v1.4 -> v1.5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Change the following.
 can be obtained even in the case of 
 $O(0.1\mbox{eV})$ neutrino masses.
Even in the small $\tan \beta$, 
 $O(1\mbox{eV})$ mass is enough
 to obtain this RGE enhancement.




%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%  Summary    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Summary and Discussion}
%

In this paper we analyzed the RGE effects
of the neutrino flavor mixing between $\nu_\mu$ and $\nu_\tau$
in the MSSM
with right-handed neutrinos.
%%%%%%%%%%%%%% SuperK results are about \delta m^2 and the mixing,
%%%%%%%%%%%%%% but we use only the data of \delta m^2.
The experimental result of the Superkamiokande 
 suggests the 
 large neutrino flavor mixing between 
 $\nu_{\mu}$ and $\nu_{\tau}$ with 
 $\delta m^2_{23} \simeq 10^{-3}$ eV. 
Then we found that 
 the mixing angle between $\nu_{\mu}$ 
 and $\nu_{\tau}$  
 receives significant corrections
 from the RGE
 in the case of $m_{\nu_s} \approx m_{\nu_e} \ll 
  m_{\nu_{\mu}} \approx m_{\nu_{\tau}}$. 
In this mass spectrum, 
 the mixing angle at the decoupling 
 scale of right-handed neutrinos must be small 
 when $O(0.1\mbox{eV}) \leq m_{\nu_{\tau}} 
 (\approx m_{\nu_{\mu}})$. 

%Using the result of the Superkamiokande
% As long as the Superkamiokande result
%of $\delta m^2_{23} \simeq 10^{-3}$ eV, % is true,we found that 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v1.4 -> v1.5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Change the following.
%the large mixing at the weak scale is possibly % always 
%derived % at the weak scale 
%by the RGE effects 
%if neutrino masses of the second 
%and the third generation are % of order 
%more than $O(0.1\mbox{eV})$.

Finally we give a brief comment 
 about the LSND experimental result. 
In the above mass spectrum, 
 both $m_{\nu_{\mu}}$ and $m_{\nu_{\tau}}$
 must be of order 1 eV
 to explain the LSND result.
In this case, the RGE analysis
 shows that
 the mixing angle at the decoupling 
 scale of right-handed neutrinos must be 
 negligibly small\footnote{
Although we have not shown the lines of 1 eV in 
 Figs.~\ref{fig:m3} and Figs.~\ref{fig:m335}, 
 there exist
 these lines. 
They are too small to be drawn apparently 
 in the large $\tan \beta$.
}.
{}From the view point of the model building,
 we must find the fundamental theory which induces
 the small mixing angle at the high energy scale
 in the four neutrino scenario 
 with the mass spectrum 
 of $m_{\nu_s} \approx m_{\nu_e} \ll 
  m_{\nu_{\mu}} \approx m_{\nu_{\tau}}$.
%In this scenario,
%both of neutrino masses % of the second and the third generation
%concerning with the atmospheric neutrino deficit
%must be around 1 eV.
%This is just the case that the maximal enhancement of the mixing angle
% is induced in our analysis.\footnote{
%We have not shown the lines of 1 eV in 
% Figs.~\ref{fig:m3} and Figs.~\ref{fig:m335}, but there exist
% these lines. 
%They are too small to be drawn apparently 
% in the large $\tan \beta$.
%}
%Therefore
%%Including LSND result,
% we can conclude 
% that the mixing angle at the scale of
% right-handed neutrino must be
% negligible small in that scenario. 

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%  Acknowledgements  %%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\vspace{2em}\par
\noindent
{\bf Acknowledgments}\par
We would like to thank M.~Tanimoto 
 for useful discussions. 
NO would like to thank
 K.~Hagiwara, J.~Hisano and Y.~Okada
 for useful discussions and comments.
NH would like to thank
 S.~Raby and K.~Tobe for helpful discussions and 
 comments. 
The work of NO is financially supported by the JSPS
 Research Fellowships for young scientist, No.2996.
NH is partially supposed by DOE grant DOE/ER/01545-753. 


%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%  References  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%J. R. Primack, J. Holtzman, A. Klypin, and 
%D. O. Caldwell, Phys. Rev. Lett. {\bf 74} (1995) 2160.
%K. S. Babu, R. K. Schaefer, and Q. Shafi, 
%Phys. Rev. {\bf D53} (1996) 606. 


\bibitem{LSND}
LSND Collaboration, C.~Athanassopoulos et al., 
Phys. Rev. Lett. {\bf 75} (1995) 2650;
{\it ibid}. {\bf 77} (1996) 3082;
{\it ibid}. {\bf 81} (1998) 1774.


\bibitem{KARMEN}
The LSND results will be tested by the KARMEN experiment, 
talk by B.~Armbruster at 33rd Rencontres de Moriond :
 Electroweak Interactions and Unified Theories, 
Les Arcs, France, March 1998, and talk by 
B.~Zeitnitz at `Neutrino 98', Takayama, Japan, June 1998, 
and also by the BooNE experiment, E.~Church et al., . 


\bibitem{PDG}
C.~Caso et al.,
Eur. Phys. J. {\bf C 3} (1998) 1.


\bibitem{CKM}
N.~Cabibbo,
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M.~Kobayashi and T.~Maskawa,
Prog. Theor. Phys. {\bf 49} (1973) 652.


\end{thebibliography}

\newpage
%
%%%%%%% Fig.1 %%%%%%%%%%
%
\begin{figure}[htbp]
 \begin{center}
 \resizebox{.8\textwidth}{!}{\includegraphics{yem.eps}}
 \end{center}
 \vspace{-1em}
 \caption{
The energy dependence of the values of 
$y_\tau^2 - y_\mu^2$ with various values of 
$\tan\beta$. 
These values are the coefficients of the RGEs of 
$\kappa_r$ and 
$\sin^2 2 \theta_{23}$. 
}
 \label{fig:Yem}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%
%
%\vspace{1em}
\newpage
%
%%%%%% Fig.2 %%%%%%%%%%%
\begin{figure}[htbp]
 \begin{center}
 \resizebox{.75\textwidth}{!}{\includegraphics{kr14.eps}}\\
 Figure~\ref{fig:Kr}(a): ${\cal M_R}=10^{14}$ GeV \\
 \vspace{1em}\par\noindent
 \resizebox{.75\textwidth}{!}{\includegraphics{kr16.eps}}\\
 Figure~\ref{fig:Kr}(b): ${\cal M_R}=10^{16}$ GeV
 \end{center}
 \vspace{-1em}
 \caption{
The energy dependence of $\kappa_r$ with 
$\tan\beta =50$. 
(a): The initial condition is fixed as 
$\kappa_r({\cal M_R}) = 0.8, 0.9$, and $0.99$ 
at ${\cal M_R}=10^{14}$ GeV.
(b): The initial condition is fixed as 
%%%%%%%%%%%%%%%%%%%%%%%%%%%% v1.6X -> v1.7
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Change below.
$\kappa_r({\cal M_R}) = 0.8, 0.88$, and $0.96$ 
at ${\cal M_R}=10^{16}$ GeV. 
}
 \label{fig:Kr}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%
%
%\vspace{1em}
%
%%%%%%% Fig.3 %%%%%%%%%%
\begin{figure}[htbp]
 \begin{center}
  \resizebox{.75\textwidth}{!}{\includegraphics{sin14.eps}}\\
  Figure~\ref{fig:EngDepOfSin}(a): ${\cal M_R}=10^{14}$ GeV \\
  \vspace{1em}\par\noindent
  \resizebox{.75\textwidth}{!}{\includegraphics{sin16.eps}}\\
  Figure~\ref{fig:EngDepOfSin}(b): ${\cal M_R}=10^{16}$ GeV
 \end{center}
\caption{%
The energy dependence of $\sin^2 2 \theta$ 
with $\tan \beta =50$. 
We show two cases of (a): 
${\cal M_R}=10^{14}$ GeV and 
 (b): $10^{16}$ GeV. 
}
 \label{fig:EngDepOfSin}
\end{figure}

%%%%%%% Fig.4 %%%%%%%%%%
\begin{figure}[htbp]
 \begin{center}
 \resizebox{.75\textwidth}{!}{\includegraphics{sinTB50-14.eps}}\\
 Figure~\ref{fig:sin}(a): $\tan\beta=50$, ${\cal M_R}=10^{14}$ GeV\\
 \vspace{.5em}\par\noindent
 \resizebox{.75\textwidth}{!}{\includegraphics{sinTB50-16.eps}}\\
 Figure~\ref{fig:sin}(b): $\tan\beta=50$, ${\cal M_R}=10^{16}$ GeV
 \end{center}
 \vspace{-1.5em}
\caption{%
The contour plots of the mixing angle $\sin^2 2\theta_{23}$ 
 at the weak scale with $\tan \beta =50$.
We show two cases of
(a):${\cal M_R}=10^{14}$ GeV and
(b):$10^{16}$ GeV.
The horizontal axes show the value of $\kappa_r$
at $\mu ={\cal M_R}$ scale, 
and the vertical axes show 
the mixing angle at $\mu ={\cal M_R}$
The values of $\sin^2 2\theta_{23}$ at the weak scale 
are determined by inputting the initial values of 
$\sin^2 2\theta_{23}({\cal M_R})$ and 
$\kappa_r({\cal M_R})$. 
Around $\kappa_c$, the RGE effects make 
the weak scale mixing be large at any initial conditions.
}
 \label{fig:sin}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%
%
%\vspace{1em}
%
%%%%%% Fig.5 %%%%%%%%%%%
\begin{figure}[htbp]
 \begin{center}
 \resizebox{.75\textwidth}{!}{\includegraphics{m3TB50-14.eps}}\\
 Figure~\ref{fig:m3}(a): $\tan\beta=50$, ${\cal M_R}=10^{14}$ GeV \\
 \vspace{.5em}\par\noindent
 \resizebox{.75\textwidth}{!}{\includegraphics{m3TB50-16.eps}}\\
 Figure~\ref{fig:m3}(b): $\tan\beta=50$, ${\cal M_R}=10^{16}$ GeV
 \end{center}
 \vspace{-1.5em}
 \caption{
The contour plots of the heaviest neutrino mass $m_3$ 
at the weak scale with $\tan\beta = 50$.
The horizontal and vertical axes are the same as Figs.~\ref{fig:sin}.
The values of $m_3$ at the weak scale 
are determined by inputting the initial values of 
$\sin^2 2\theta_{23}({\cal M_R})$ 
and $\kappa_r({\cal M_R})$, and 
the experimental value of 
$\delta m_{23}^2 = 1.3 \times 10^{-3} {\rm eV}^2$. 
As the mass becomes heavy, 
the region of $\kappa_r({\cal M_R})$ is limited 
around $\kappa_c$.
}
 \label{fig:m3}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%
%
%\vspace{1em}
%
%%%%%% Fig.6 %%%%%%%%%%%
\begin{figure}[htbp]
 \begin{center}
 \resizebox{.75\textwidth}{!}{\includegraphics{sinTB35-14.eps}}\\
 Figure~\ref{fig:sin35}(a): $\tan\beta=35$, ${\cal M_R}=10^{14}$ GeV \\
 \vspace{.5em}\par\noindent
 \resizebox{.75\textwidth}{!}{\includegraphics{sinTB35-16.eps}}\\
 Figure~\ref{fig:sin35}(b): $\tan\beta=35$, ${\cal M_R}=10^{16}$ GeV
 \end{center} 
 \vspace{-1.5em}
\caption{%
The contour plots of the mixing angle $\sin^2 2\theta_{23}$ 
 at the weak scale with $\tan \beta =35$.
In this case $\kappa_c$ is larger than that of Figs.~\ref{fig:sin}.
}
 \label{fig:sin35}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%
%
%\vspace{1em}
%
%%%%%% Fig.7 %%%%%%%%%%%
\begin{figure}[htbp]
 \begin{center}
 \resizebox{.75\textwidth}{!}{\includegraphics{m3TB35-14.eps}}\\
 Figure~\ref{fig:m335}(a): $\tan\beta=35$, ${\cal M_R}=10^{14}$ GeV \\
 \vspace{.5em}\par\noindent
 \resizebox{.75\textwidth}{!}{\includegraphics{m3TB35-16.eps}}\\
 Figure~\ref{fig:m335}(b): $\tan\beta=35$, ${\cal M_R}=10^{16}$ GeV
 \end{center} 
 \vspace{-1.5em}
\caption{%
The contour plots of the heaviest neutrino mass
 at the weak scale with $\tan \beta =35$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%% v1.6X -> v1.7
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Change below.
The region where $m_3$ is larger than $O(0.1\mbox{eV})$
 corresponds to the region 
 in which the maximal mixing angle is realized 
 at low energy. 
}
 \label{fig:m335}
\end{figure}

\end{document}




