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%%%%%   Neutrino Bi-Large Mixing in MSSM with Right-Handed Neutrino  %%%%%   
%%%%%                 and Lepton Flavor Violation                    %%%%%
%%%%%                                                                %%%%%
%%%%%               by A. Kageyama, S. Kaneko,                       %%%%%
%%%%%                      N. Shimoyama, M. Tanimoto                 %%%%%
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\begin{document}
\title[Neutrino Bi-Large Mixing in MSSM with RHN and LFV]
{Neutrino Bi-Large Mixing in MSSM with Right-Handed Neutrino and 
Lepton Flavor Violation}
\author{
Atsushi Kageyama
\footnote[1]{E-mail address: atsushi@muse.sc.niigata-u.ac.jp}
, Satoru Kaneko
\footnote[2]{E-mail address: kaneko@muse.sc.niigata-u.ac.jp}
, \underline{Noriyuki Shimoyama}
\footnote[3]{E-mail address: simoyama@muse.sc.niigata-u.ac.jp}
and Morimitsu Tanimoto
\footnote[4]{E-mail address: tanimoto@muse.sc.niigata-u.ac.jp}
}
\address{Department of Physics, Niigata University, 
Ikarashi 2-8050, 950-2181 Niigata, JAPAN}

\begin{abstract}
We investigate the lepton flavor violation in the framework of the MSSM with
right-handed Majorana neutrinos 
taking the large mixing angle MSW solution in the 
hierarchical, degenerate and  the inverted neutrino masses.
We predict the branching ratio of $\m \ra e+\r$ process 
assuming the degenerate right-handed Majorana neutrino masses.  
We find that the branching ratio in the degenerate neutrino 
mass spectrum is about 100 times smaller than the one in the 
inverted and the hierarchical neutrino spectra.
We emphasize  that  the Majorana mass scale 
and the magnitude of  $U_{e3}$ are one of important
ingredients to predict BR($\m \ra e+\r $).
\end{abstract}

Super-Kamiokande  has almost confirmed  the neutrino oscillation in 
the atmospheric neutrinos, which favors the $\n_\m \ra \n_\tau$ process.
For the solar neutrinos, the recent data of 
the Super-Kamiokande and the SNO also favor the neutrino oscillation
$\n_\m\ra \n_e$ with the large mixing angle MSW solution. 
These  results  mean that neutrinos are  massive, moreover,
the flavor mixings are bi-large  in the lepton sector. 
 
If neutrinos are massive and flavor-mixed in the SM, 
there exists a source of the lepton flavor violation (LFV) through the
off-diagonal elements of the neutrino Yukawa coupling matrix. 
However, due to the smallness of the neutrino masses, 
the predicted branching ratios for these processes are too small 
to be observable.

On the other hand,
in the supersymmetric framework  the situation is quite different.
Many authors have already studied the LFV  in the minimal supersymmetric standard model (MSSM) with right-handed neutrinos assuming the relevant neutrino 
mass matrix \cite{Hisano}.
In the MSSM with soft breaking terms, 
there exist  lepton flavor violating terms such as 
off-diagonal elements of slepton mass matrices 
$\left({\bf m}^2_{\rm\t \sL}\right)_{ij}$,
$({\bf m}_{\rm \t e_\sR}^2)_{ij}$ 
and trilinear couplings ${\bf A}^{\rm e}_{ij}$.
Then the  decay rate can be given as 
\begin{eqnarray}
\Gamma(\m \ra e +\r)\propto
\frac{\a_{2}^{2}}{16\pi^{2}}
\frac{|(\bigtriangleup {\bf m}_{\t{L}}^{2})_{21}|^{2}}{m_{S}^{8}}\tan^{2}
\b \ ,
\end{eqnarray}
approximately.
Where $\a_{2}$ is  the couplng constant of $SU(2)_{W}$,  
$m_{S}$ is a common SUSY particle mass.
Strong bounds on these matrix elements come from requiring branching ratios 
for LFV processes to be below observed ratios.  
For the present, the most stringent bound comes from  the 
$\m \ra e +\r$ decay 
(${\rm BR(\m\ra e+\r)}<1.2\times 10^{-11}$).
However, if the LFV  occurs at tree level in the soft breaking terms, 
the branching ratio of this process exceeds the experimental bound too much.
Therefore  one assumes that the LFV does not occur at tree level 
in the soft parameters. 
This is realized by taking the assumption that soft parameters 
such as $\left({\bf m}^2_{\rm \t \sL}\right)_{ij}$,
$({\bf m}_{\rm\t e_\sR}^2 )_{ij}$,  ${\bf A}^{\rm e}_{ij}$,
are proportional to the unit matrix. 
This assumption follows from the  m-SUGRA.  
However, even though there are no flavor violation at tree level, 
it is generated by the effect of
the renormalization group equations (RGE's) via neutrino Yukawa couplings. 
Suppose that  neutrino masses are produced by the  see-saw mechanism, 
there are  the right-handed neutrinos 
above a scale $M_{\sR}$ ( $\> 10^{12}$GeV ).
Then neutrinos 
have  the Yukawa coupling matrix ${\bf Y}_\n$  with  off-diagonal entries
in the basis of the diagonal charged-lepton Yukawa couplings.
 The  off-diagonal elements of ${\bf Y}_\n$ drive off-diagonal ones 
in the $\left({\bf m}_{\rm\t \sL}^2\right)_{ij}$ and 
${\bf A}^{\rm e}_{ij}$ matrices through the RGE's running.
These terms at the low energy are approximated as 
\begin{eqnarray}
(\bigtriangleup {\bf m}_{\t{L}}^{2})_{21}
&\simeq&
-\frac{(6+2a_{0}^2)m_{0}^2}{16\pi^2}({\bf Y_{\n}^{\dag}Y_{\n}})_{21}
\ln
\frac{M_{\rm X}}{M_{\rm R}}\ ,
\end{eqnarray} 
where $M_{\rm X}$ is the GUT scale.
Therefore disscussing $({\bf Y_{\n}^{\dag}Y_{\n}})_{21}$ is crucial
for estimating ${\rm BR(\m\ra e+\r)}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
One can construct  ${\bf Y}_\n$ by the recent data of neutrino oscillations.
Assuming that oscillations need only accounting for the solar 
and the atmospheric neutrino data, we take  the LMA-MSW solution. 
Since the data of neutrino oscillations only indicate
the differences of the mass square  $\Delta m^2_{ij}$,
the neutrinos have three possible mass spectra. 
The quasi-degenerate type ($m_{\n 1}\sim m_{\n 2}\sim m_{\n 3}$)
gives the neutrino mass spectrum as
\begin{equation}
 m_{\n 1}\equiv m_\n \ ,  \quad
 m_{\n 2}=m_\nu + \frac{\Delta m^2_{\odot}\ }{2 m_\n},\quad
 m_{\n 3}=m_\nu +  \frac{ \Delta m_{\rm atm}^2}{2 m_\n} \ , 
\label{eqn:degeneratetype}
\end{equation}
and the inverted ($m_{\n 1}\sim m_{\n 2}\gg m_{\n 3}$) type gives
\begin{equation}
 \quad  m_{\n 2}
\equiv
\sqrt{\Delta m^2_{\rm atm}} \  , \quad
 m_{\n 1}
=
 m_{\n 2}-\frac{\Delta m_{\odot}^2}{2 m_{\n 2}} \ , \quad  
 m_{\n 3}\simeq 0\ ,
\end{equation}
the hierarchical type ($m_{\n 1}\ll m_{\n 2}\ll m_{\n 3}$) gives
\begin{eqnarray}
 m_{\n 1}\sim 0 \  ,\qquad m_{\n 2}=\sqrt{\Delta m^2_{\odot}} \  ,
  \qquad m_{\n 3}=\sqrt{\Delta m^2_{\rm atm}}\ .
\end{eqnarray}
Taking the three cases of the neutrino mass spectra,
one obtains the following forms, repectively,
\begin{eqnarray}
({\bf Y}_{\n}^{\dag}{\bf Y}_{\n})_{21}
&\simeq&
\frac{M_{\rm R}}{v_{u}^{2}}
\left[U_{\m2}U_{e2}^{*}(m_{\n 2}-m_{\n 1})+
 U_{\m3}U_{e3}^{*}(m_{\nu 3}-m_{\n 1})
\right] 
\label{muemass3}\\
\nonumber \\
&\simeq&
\frac{M_{R}}{\sqrt{2}v_{u}^{2}}\frac{\Delta m^2_{\rm atm}}{2m_\n}
\left[
\frac{1}{\sqrt{2}}U_{e2}^* \frac{\Delta m_{\odot}^2}{\Delta m^2_{\rm atm}}
+ U_{e3}^*
\right] \ ,          
\quad ({\rm Degenerate}) 
\nonumber\\
\nonumber \\
&\simeq& 
\frac{M_{R}}{\sqrt{2}v_{u}^{2}}\sqrt{\Delta m_{\rm atm}^2}
\left[ 
\frac{1}{2\sqrt{2}} U_{e2}^*\frac{\Delta m_{\odot}^2}{\Delta m^2_{\rm atm}} 
- U_{e3}^*
\right] \ , 
\quad ({\rm Inverted }) 
\nonumber\\
\nonumber\\
&\simeq&
\frac{M_{R}}{\sqrt{2}v_{u}^{2}}\sqrt{\Delta m_{\rm atm}^2}
\left[
\frac{1}{\sqrt{2}}U_{e2}^*
\sqrt{\frac{\Delta m_{\odot}^2}{\Delta m^2_{\rm atm}}}
+ U_{e3}^*
\right] \ , 
\quad
({\rm Hierar. }) 
\nonumber
\end{eqnarray}
where we take the maximal   mixing for the atmospheric neutrinos. 
Since  $U_{e2}\simeq 1/\sqrt{2}$ for the bi-large mixing matrix, 
the first terms in the square brackets of the right hand sides of  
eqs.(\ref{muemass3}) can be estimated by putting the experimental data.
For the case of the degenerate neutrino masses,
$({\bf Y}_{\n}^{\dag}{\bf Y}_{\n})_{21}$ depends on 
the unknown neutrino mass scale $m_{\n}$.  
As one takes the smaller $m_\n$, one predicts the larger branching ratio. 
In our calculation, we take $m_{\n}=0.3 \eV$, which is close to the upper 
bound from the neutrinoless double beta decay experiment,
and also leads to the smallest branching ratio. 
It is easy to see the fact that the second terms in eqs.(\ref{muemass3}) 
are dominant as far as  
$U_{e3} ~\>~ 0.01({\rm degenerate}), ~ 0.01({\rm inverted})$ 
and $0.07({\rm hierarchical})$, respectively.
The comparison between the cases of  the inverted masses and 
the normal hierarchical masses depends on the magnitude 
and the phase of $U_{e3}$.

We  present numerical results in the case of the degenerate 
neutrino masses assuming ${\bf M_R}=M_R {\bf I}$.
In Fig.\ref{D1202}, the branching ratio of  
$\m \ra e+\r$ is given versus the left-handed 
selectron mass $m_{\t e_L}$ for each $\tan\b=3,\  10,\  30$
and a fixed wino mass $M_2$ at the electroweak scale.
Our predictions depend on $M_R$
strongly, because the magnitude of
the neutrino Yukawa coupling is determined by  $M_R$.
If  $M_R$ increases to $10^{14} $GeV,
the branching ratio becomes $10^{4}$ times larger since it
is proportional to $M_R^2$ approximately.
In general, 
we expect the relation BR(degenerate)$\ll$BR(inverted)$\cong$
BR(hierarchical)\cite{tanimoto}. 
 
The branching ratio of $\m\ra e+\r$ 
will be improved to the lebel $10^{-14}$ in the PSI.
Therefore, future experiments can probe the framework for 
the neutrino masses.
%%%%%%%%%%%%%%  Figure 1  %%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\epsfxsize=6.0 cm
\epsfbox{D1202.ai}
\end{center}
%\caption{\label{label}here}
\caption{Predicted branching ratio BR$(\m \ra e+\r)$
versus the left-handed selectron mass for $\tan\b=3,\ 10,\ 30$
in the case of the degenerate neutrino masses.
Here $M_R=10^{12}$GeV and $U_{e3}=0.2$ are taken.
The solid curves correspond to  $M_2=150 $GeV and the dashed ones to  
$M_{2}=300 $GeV.}
\label{D1202}
\end{figure}

\section*{References}
\begin{thebibliography}{1}
\bibitem{Hisano}
J. Hisano, T. Moroi, K. Tobe, M. Yamaguchi
and T. Yanagida, 1995 {\it Phys. Lett.}{\bf B357} 579;
J. Hisano, D. Nomura and T. Yanagida, 1998 {\it Phys. Lett.}{\bf B437} 351;
S. Baek, T. Goto, Y. Okada and K. Okumura, 2001 {\it Phys. Rev.}{\bf D63} 
051701;
J. Sato, K. Tobe, and T. Yanagida, 2001 {\it Phys. Lett.}{\bf B498}189;
J. R. Ellis, J. Hisano, M. Raidal, Y. Shimizu {\it hep-ph/0206110}.

\bibitem{tanimoto}
A. Kageyama, S. Kaneko, N. Shimoyama and M. Tanimoto, 
2002 {\it Phys. Lett.}{\bf B527} 206,
A. Kageyama, S. Kaneko, N. Shimoyama and M. Tanimoto, 
2002 {\it Phys. Rev.}{\bf D65} 096010.
\end{thebibliography}
\end{document}



















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