Towards Complete Neutrino Mixing Matrix and CP-Violation
The compelling experimental evidences for oscillations of solar, atmospheric and reactor neutrinos imply the existence of 3-neutrino mixing in vacuum. We review the phenomenology of 3-neutrino mixing, and the current data on the 3-neutrino mixing parameters. The opened questions and the main goals of future research in the field of neutrino mixing and oscillations are outlined. A phenomenological approach for understanding the pattern of neutrino mixing as an interplay between the mixing, arising from the charged lepton sector, and bimaximal mixing, arising from a neutrino Majorana mass matrix, is considered with emphasis on the $CP-$violating case. We comment also on planned future steps in the experimental studies of $\nu$-mixing.
Nu04STP291204hep1.tex
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\def\nue{{\nu_e}}
\def\anue{{\bar\nu_e}}
\def\numu{{\nu_{\mu}}}
\def\anumu{{\bar\nu_{\mu}}}
\def\nutau{{\nu_{\tau}}}
\def\anutau{{\bar\nu_{\tau}}}
\newcommand{\dm}{\mbox{$\Delta m_{21}^2$~}}
% \newcommand{\dm21}{\mbox{$\Delta m_{21}^2$~}}
% \newcommand{\dm31}{\mbox{$\Delta m_{31}^2$~}}
\newcommand{\dmsol}{\mbox{$\Delta m^2_{\odot}$~}}
\newcommand{\st}{\mbox{$\sin^{2}2\theta$~}}
\newcommand{\br}{\mbox{$^{8}{B}~$}}
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\def\lsim{\:\raisebox{-0.5ex}{$\stackrel{\textstyle<}{\sim}$}\:}
\def\gsim{\:\raisebox{-0.5ex}{$\stackrel{\textstyle>}{\sim}$}\:}
\newcommand{\sss}{\sin^2 \theta_{12}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \newcommand{\dms}{\mbox{$\Delta m^2_{\odot}$}}
\newcommand{\dma}{\mbox{$\Delta m^2_{\rm A}$}}
\newcommand{\meff}{\mbox{$\left| < \! m \! > \right|$}}
\newcommand{\mefff}{\mbox{$\langle m \rangle$}}
\newcommand{\betabeta}{\mbox{$(\beta \beta)_{0 \nu}$}}
\newcommand{\hbeta}{$\mbox{}^3 {\rm H}$ $\beta$-decay }
\newcommand{\eV}{\mbox{$ \  \mathrm{eV} \ $}}
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% \def\be{\begin{equation}}
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\def\gs{\mathrel{
   \rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}}}
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   \rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}}}
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\newcommand{\ueuno}{\mbox{$ \ |U_{\mathrm{e} 1}|^2 \ $}}
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% \begin{document}

% \begin{flushright}
% SISSA 84/2004/EP\\
% hep-ph/04mmnnn\\
% \end{flushright}
% \vspace{-3cm}
\title{Towards Complete Neutrino Mixing Matrix and CP-Violation}
 
\author{S. T. Petcov
\address{Scuola Internazionale Superiore di Studi Avanzati, and
         INFN, Trieste, Italy}
        \thanks{Also at: Institute of Nuclear Research and Nuclear Energy,
            Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria}~
        \thanks{Plenary talk given at $\nu$'04 International 
Conference, June 14-19, 2004, Paris, France.}
}

\vspace{-0.8cm}
\begin{document}
\begin{abstract}
 The compelling experimental evidences for 
oscillations of solar, atmospheric and 
reactor neutrinos imply the existence of 
3-neutrino mixing in vacuum.
We review the phenomenology 
of 3-neutrino mixing, and the 
current data on the 3-neutrino mixing
parameters. The opened questions
and the main goals of future 
research in the field of neutrino 
mixing and oscillations are outlined.
A phenomenological approach 
for understanding the pattern 
of neutrino mixing 
as an interplay between
the mixing, arising from 
the charged lepton sector, and
bimaximal mixing, arising from
a neutrino Majorana mass matrix,
is considered with emphasis 
on the $CP-$violating case.
We comment also on planned future 
steps in the experimental studies of
$\nu$-mixing.


\end{abstract}

\maketitle

\vspace{-0.6cm}
\section{Introduction}

\vspace{-0.2cm}
  The hypothesis of neutrino oscillations 
was formulated in \cite{BPont57}.
In \cite{BPont67} it was suggested that 
the solar $\nu_e$ 
% neutrinos 
can take part in 
oscillations involving another active
or sterile neutrino. The evidences of 
solar neutrino ($\nu_{\odot}$-) oscillations 
obtained first in the Homestake 
experiment and strengthened by 
the results of Kamioaknde, 
SAGE and GALLEX/GNO 
experiments \cite{sol,Kam96},
were made compelling 
in the last several years by the data
of  Super-Kamiokande (SK), 
SNO and KamLAND (KL) experiments
\cite{SKsol02,SNO123,KL162}.
Under the plausible 
assumption of CPT-invariance, 
the results of the KL reactor
neutrino experiment \cite{KL162} 
established the large mixing 
angle (LMA) MSW 
oscillations/transitions \cite{MSW}
as the dominant mechanism
at the origin of the observed 
solar $\nu_e$ deficit.
The Kamiokande experiment \cite{Kam96}
provided  the first evidences
for oscillations of atmospheric 
% neutrinos,
$\nu_{\mu}$ and $\bar{\nu}_{\mu}$,
while the data of the 
Super-Kamiokande experiment
made the case of atmospheric 
neutrino oscillations convincing 
\cite{SKatmnu04}.
Evidences for oscillations 
of neutrinos were obtained also 
in the first long baseline
accelerator neutrino experiment 
K2K \cite{K2K}. Indications 
for $\nu$-oscillations
were reported by the LSND 
collaboration \cite{LSND}.

 The recent new SK data on the 
$L/E$-dependence of 
multi-GeV $\mu$-like  atmospheric 
% $\nu$ 
neutrino events \cite{SKatmnu04},
$L$ and $E$ being the distance traveled 
by neutrinos and the $\nu$ energy,
and the new 
spectrum data of KL and K2K experiments 
\cite{KL766,K2Knu04},
presented at this Conference,
are the latest significant 
contributions to the 
remarkable progress 
made in the last several
years in the studies of 
% neutrino 
$\nu$-oscillations.
For the first time the data
exhibit directly the effects of the 
oscillatory dependence on $L/E$ and $ E$ of 
the probabilities of  
$\nu$-oscillations in vacuum \cite{BP69}.
We begin to ``see''the 
oscillations of neutrinos.
As a result of these
magnificent developments, 
the oscillations of solar $\nu_e$,
atmospheric $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$, 
accelerator $\nu_{\mu}$ (at $L\sim$250 km)
and reactor $\bar{\nu}_e$ (at $L\sim$180 km), 
driven by nonzero $\nu$-masses 
and $\nu$-mixing, can be considered as 
practically established.

\vspace{-0.35cm}
\section{The Neutrino Mixing Parameters}

\vspace{-0.10cm}
   The SK atmospheric neutrino 
% ($\nu_{\rm A}$)
and K2K data are
best described in 
terms of dominant 2-neutrino
$\nu_{\mu} \rightarrow \nu_{\tau}$ 
($\bar{\nu}_{\mu} \rightarrow \bar{\nu}_{\tau}$)
vacuum oscillations.
The best fit values and the 
99.73\% C.L. allowed ranges of the 
atmospheric neutrino 
($\nu_{\rm A}$-)
oscillation parameters
% $\Delta m^2_{\rm A} \equiv \Delta m^2_{31}$
% and $\theta_{\rm A} \equiv \theta_{23}$
read \cite{SKatmnu04}: \\
$|\dma|$=2.1$\times10^{-3}~{\rm eV^2}$,~
$\sin^22\theta_{\rm A}$ = 1.0, \\ 
$|\dma|$=(1.3 -- 4.2)$\times 10^{-3}~$eV$^2$,
$\sin^22\theta_{\rm A} \geq$ 0.85.\\
\noindent The sign of $\dma$ and of 
%$\cos2\theta_{23}$ when
$\cos2\theta_{\rm A}$, if
% $\sin^22\theta_{23} \neq 1.0$. 
$\sin^22\theta_{\rm A} \neq 1.0$, 
cannot be determined using
the existing data. 
The latter implies that when, e.g., 
$\sin^22\theta_{\rm A} = 0.92$, 
one has $\sin^2\theta_{\rm A}\cong 0.64~{\rm or}~ 0.36$.

  The combined 2-neutrino oscillation 
analysis of the solar neutrino 
and the new KL 766.3 Ty spectrum  
data shows \cite{KL766,BCGPRKL2} that the 
$\nu_{\odot}$-oscillation parameters 
lie in the low-LMA region :\\  

\vspace{-0.3cm}
\noindent $\dmsol$=($7.9^{+0.6}_{-0.5}$)$\times 10^{-5}~{\rm eV^2}$, 
$\tan^2 \theta_\odot$=$(0.40^{+0.09}_{-0.07})$.\\

\vspace{-0.3cm}
\noindent The value of $\dmsol$ is 
determined with a remarkably 
high precision. 
% of 7.3\% at 1$\sigma$. 
The high-LMA solution is excluded at 
more than 3$\sigma$. 
Maximal $\nu_{\odot}$-mixing
is ruled out at $\sim 6\sigma$;
at 95\% C.L., 
$\cos 2\theta_\odot \geq 0.27$. 
One also has: $\dmsol/|\dma| \sim 0.04 \ll 1$. 
  
   The interpretation of the solar and
atmospheric neutrino, and of K2K and KL
data in terms of $\nu$-oscillations requires
the existence of 3-$\nu$ mixing
in the weak charged lepton current:\\

\vspace{-0.2cm}
%%%%%%%%%%%%%%%%%%%
% \begin{equation}
$\nu_{l \mathrm{L}}  = \sum_{j=1}^{3} U_{l j} \, \nu_{j \mathrm{L}},~~
l  = e,\mu,\tau$,\\
%\label{3numix}
%\end{equation}
%%%%%%%%%%%%%%%%%%%

\vspace{-0.2cm}
\noindent where $\nu_{j \mathrm{L}}$ is the 
%left-handed
field of neutrino $\nu_j$ having 
a mass $m_j$ and
$U$ is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) 
$\nu$-mixing matrix \cite{BPont57,MNS62}.
All existing $\nu$-oscillation data, except
the data of LSND experiment  
\footnote{In the
LSND experiment indications 
for oscillations
$\bar \nu_{\mu}\to\bar \nu_{e}$  
with $(\Delta m^{2})_{\rm{LSND}}\simeq 
1~\rm{eV}^{2}$ were obtained. 
The  LSND results are being tested 
in the MiniBooNE experiment
%at Fermilab
\cite{MiniB}.}
\cite{LSND},
can be described assuming 
3-$\nu$ mixing in vacuum
and we will consider only 
this possibility. 
The minimal 4-$\nu$ mixing scheme 
which could incorporate the 
LSND indications for 
$\nu$-oscillations
is strongly disfavored by the data \cite{Maltoni4nu}.
The $\nu$-oscillation explanation of the LSND results
is possible assuming 5-$\nu$ mixing \cite{JConrad}.

 The PMNS matrix  can be 
parametrized by 3 angles 
and, depending on whether the massive 
neutrinos $\nu_j$ are 
Dirac or Majorana particles,
by 1 or 3 CP-violation ($CPV$) phases 
\cite{BHP80,SchValle80D81}.
 In the standardly used parameterization 
(see, e.g., \cite{BPP1}),\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \bea \label{eq:Upara}
% \pmns = \left( \bad 
% c_{12} c_{13} & s_{12} c_{13} & s_{13}  \\[0.2cm] 
% -s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i \delta} 
% & c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i \delta} 
% & s_{23} c_{13} e^{i \delta} \\[0.2cm] 
% s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i \delta} & 
% - c_{12} s_{23} - s_{12} c_{23} s_{13} e^{i \delta} 
% & c_{23} c_{13} e^{i \delta} \\ 
%                \ea   \right) 
%  {\rm diag}(1, e^{i \alpha}, e^{i \beta}) \, , 
% \eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

\vspace{-0.2cm}
$\pmns = V(\theta_{12},\theta_{13},\theta_{23},\delta)
~{\rm diag}(1, e^{i \alpha}, e^{i \beta})$,\\

\vspace{-0.2cm}
\noindent where 
% we have used the usual
% notations $c_{ij} = \cos\theta_{ij}$, 
% $s_{ij} = \sin\theta_{ij}$,
$V(\theta_{12},\theta_{13},\theta_{23},\delta)$
is a CKM-like matrix, the angles
$\theta_{ij} = [0,\pi/2]$,
$\delta = [0,2\pi]$ is the 
Dirac $CPV$ phase and
$\alpha,\beta$ are two Majorana 
$CPV$ phases \cite{BHP80,SchValle80D81}. 
One can identify 
$\dmsol = \Delta m^2_{21} > 0$.
In this case 
$|\dma|$=$|\Delta m^2_{31}|\cong |\Delta m^2_{32}|$,
$\theta_{12} = \theta_{\odot}$, 
$\theta_{23} = \theta_{\rm A}$.
The angle $\theta_{13}$ is limited by 
the data from the CHOOZ and Palo Verde
experiments~\cite{CHOOZPV}. The limit
depends strongly on $|\dma|$
(see, e.g, \cite{SNO3BCGPR}).
The existing $\nu_{\rm A}$-data
is essentially insensitive 
to $\theta_{13}$ obeying the 
CHOOZ limit \cite{SKatmnu04}.
The probabilities of survival of 
reactor $\bar{\nu}_e$ and solar $\nu_e$,
relevant for the interpretation of
the KL, CHOOZ and $\nu_{\odot}$- data, depend on
$\theta_{13}$:\\

\vspace{-0.2cm}
\noindent $P^{3\nu}_{\rm KL} \cong \sin^4\theta_{13} + 
\cos^4\theta_{13}\left [1 - \sin^2 2\theta_{12}
\sin^2\frac{\Delta m^2_{21}{\rm L}}{4E} \right ]$,\\
$P^{3\nu}_{\rm CHOOZ} \cong 1 -  
\sin^2 2\theta_{13}
\sin^2\frac{\Delta m^2_{\rm 31}{\rm L}}{\rm 4E}$,\\

\vspace{-0.2cm}
$P^{3\nu}_{\odot} \cong \sin^4\theta_{13} + 
\cos^4\theta_{13}~P^{2\nu}_{\odot}
(\Delta m^2_{21},\theta_{12};\theta_{13})$,\\

\vspace{-0.2cm}
\noindent where $P_{\odot}^{2\nu}$ 
is the 2-$\nu$ mixing solar $\nu_e$
survival probability \cite{SP88,SPJRich89,PKSP88,SP97,NAOsc99} 
in the case of transitions driven by
$\Delta m^2_{21}$ and $\theta_{12}$,
in which the solar $e^-$ number density $N_e$
is replaced by $N_e \cos^2\theta_{13}$ \cite{3nuSP88},\\

\vspace{-0.3cm}
$P^{2\nu}_{\odot} = 
 \bar{P}^{2\nu}_{\odot} + P^{2\nu}_{\odot~{\rm osc}}$,
 
\vspace{-0.3cm}
\begin{equation}
\ba
\bar{P}^{2\nu}_{\odot} = \frac {1}{2} + 
 (\frac{1}{2} - P')\cos2\theta_{12}^m(t_0)\cos 2\theta_{12}, \\ [0.3cm]
P'= {{\exp(-2\pi r_{0}{\Delta m^2_{21}\over {2E}}\sin^2\theta_{12})
          - \exp(-2\pi r_{0}{\Delta m^2_{21}\over {2E}})}
         \over {1 - \exp(-2\pi r_{0}{\Delta m^2_{21}\over {2E}})}}
\ea
\label{Psolexp}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\vspace{-0.2cm}
\noindent Here $\bar{P}^{2\nu}_{\odot}$
is the average probability \cite{Parke86,SP88,SP97},
$P^{2\nu}_{\odot~{\rm osc}}$ is 
an oscillating term \cite{SP88,SPJRich89,SP97,NAOsc99},
$P'$ is the ``double exponential'' 
jump probability \cite{SP88} 
and $r_0$ is the ``running''
scale-height of the change of $N_e$ along the 
$\nu$-trajectory in the Sun
\footnote{The claims in \cite{HWLS04} that
in the LMA region
``The double exponential formula is not valid...
It requires production of neutrinos far above the 
resonance region in the density scale. 
This formula is not applicable in the range 
$\Delta m^2 \cos2\theta/(2E)\sim (1.6-8.0)\times
10^{-6}~{\rm eV^2/MeV}$ for which the density
in the production point turns out to be close
to the resonance density.'' are incorrect and/or
misleading. The analyses and the extensive numerical
studies performed in \cite{PKSP88,HaxtonExp95,NAOsc99}
show that expression (\ref{Psolexp}) for
$\bar{P}^{2\nu}_{\odot}$ provides
a high precision description 
of the average solar $\nu_e$ 
survival probability in the Sun
for any values of $\Delta m^2_{21}$ and $\theta_{12}$
(the relevant error does not exceed 
$\sim$(2-3)\%), including the values from the 
LMA region. 
Actually, it follows from the results 
in \cite{HWLS04} that 
the use of the double 
exponential expression for $P'$ 
\cite{SP88}, eq. (\ref{Psolexp}),
for description of the LMA transitions
brings in an imprecision in 
$\bar{P}^{2\nu}_{\odot}$ which does 
not exceed $\sim 10^{-6}$.
Similarly, the claim in \cite{HWLS04}
that ``... for the LMA solution, 
when the final mixing angle is large, 
one cannot use the Landau-Zener probability 
as an approximation for $P_c$. '', $P_c$ being the
jump probability, is  incorrect,
see \cite{LZ,PKSP88,HaxtonExp95}.
}
\cite{SP88,PKSP88,NAOsc99}.
In the LMA solution region 
one has \cite{SPJRich89}
$P^{2\nu}_{\odot~{\rm osc}} \cong 0$.
Using the 3$\sigma$ allowed range
of $|\dma|$=$|\Delta m^2_{31}|$ \cite{SKatmnu04}
and performing a combined analysis of
the solar neutrino, CHOOZ and KL data, 
one finds \cite{BCGPRKL2}:

\vspace{-0.3cm}
$$\sin^2\theta_{13} < 0.05,~~~~99.73\%~{\rm C.L.}$$ 

% \vspace{-0.1cm}
\noindent Similar constraint is obtained 
from a global 3-$\nu$ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\vskip -0.2cm
\includegraphics[width=7.0cm,height=6cm,clip=]{cont3g_skc_newkl_bkgd.eps}
% \includegraphics[width=7.0cm,height=6cm,clip=]{cont3g.eps}
\vspace{-0.6cm}
\caption{\label{cont3g}
The 90\%, 95\%, 99\% and 99.73\% C.L. allowed 
regions in the $\dm$-$\sss$ plane, obtained 
in a 3-$\nu$ oscillation analysis of the 
%global 
solar neutrino, KL and CHOOZ data
% for $\sin^2\theta_{13} = 0;~0.02;~0.04;~0.05$ 
\cite{BCGPRKL2}. 
}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent  oscillation analysis
of the data \cite{Maltoni4nu,3nuGlobal}.
In Fig. (\ref{cont3g})  we show 
the allowed regions in the $\dm-\sss$ plane
%obtained in a 3-$\nu$ oscillation analysis of the 
% global solar neutrino, KL and CHOOZ data
for few fixed values of $\sin^2\theta_{13}$ \cite{BCGPRKL2},
obtained using, in  particular, 
eq. (\ref{Psolexp}) for $\bar{P}^{2\nu}_{\odot}$.
% In this analysis $\Delta m^2_{31}$ 
% was allowed to vary freely within the 3$\sigma$ 
% range \cite{SLatmnu04} 
% $(1.3-4.2)\times 10^{-3}$ eV$^2$.

  Thus, the fundamental parameters 
characterizing the 3-neutrino mixing are:\\  
$\bullet$ the 3 angles $\theta_{12}$, $\theta_{23}$, $\theta_{13}$,\\
$\bullet$ depending on the nature of $\nu_j$ - 1 Dirac ($\delta$), 
or 1 Dirac + 2 Majorana ($\delta,\alpha,\beta$), 
$CPV$ phases, and\\ 
$\bullet$ the 3 neutrino masses, $m_1,~m_2,~m_3$.

  It is convenient to express the two 
larger masses in terms of the 
third mass and the measured 
$\dmsol$=$\Delta m^2_{21}>$0 
and $\dma$. In the convention 
we are using, the two possible signs of
$\dma$ correspond to two 
types of $\nu$-mass spectrum:\\
$\bullet$ with normal hierarchy,
$m_1 < m_2 < m_3$, $\dma$=$\Delta m^2_{31} >0$,
$m_{2(3)}$=$(m_1^2 + \Delta m^2_{21(31)})^{1\over{2}}$,and\\
$\bullet$ with inverted hierarchy,
$m_3 < m_1 < m_2$,  $\dma$=$\Delta m^2_{32}<$0,
$m_{2}$=$(m_3^2 - \Delta m^2_{32})^{1\over{2}}$, etc.\\
The spectrum can also be \\
$\bullet$ {\it normal hierarchical (NH)}: 
$m_1 \ll m_2 \ll m_3$,\\
\noindent $m_2 \cong (\dmsol)^{1\over{2}}\sim$0.009 eV, 
$m_3 \cong |\dma|^{1\over{2}}\sim$0.045; or\\
% $\sim$0.045eV;\\
% $m_2 \sim 9.0 \times 10^{-3}$ eV,
% $m_3 \sim 4.5 \times 10^{-2}$ eV;\\ 
$\bullet$ {\it inverted hierarchical (IH)}:
$m_3 \ll m_1 < m_2$, 
with $m_{1,2} \cong |\dma|^{1\over{2}}\sim$0.045 eV; or \\
$\bullet$ {\it quasi-degenerate (QD)}:
$m_1 \cong m_2 \cong m_3 \cong m_0$,
$m_j^2 \gg |\dma|$, $m_0 \gtap 0.20$ eV. 

 After the spectacular experimental 
progress made 
in the studies of $\nu$-oscillations, 
further understanding
of the structure of the $\nu$-masses 
and $\nu$-mixing, of their origins  
and of the status of the CP-symmetry in 
the lepton sector requires a large and
challenging program of research to be pursued 
in neutrino physics.  
The main goals of this 
research program should include:\\
$\bullet$ High precision 
measurement of
the solar and atmospheric  
neutrino oscillations parameters, 
$\Delta m^2_{21}$, $\theta_{21}$, 
and $\Delta m^2_{31}$, $\theta_{23}$.\\
$\bullet$ Measurement of, 
or improving by at least a factor 
of (5 - 10) the existing upper 
limit on, $\theta_{13}$ - 
the only small mixing angle  in 
$\pmns$.\\
$\bullet$ Determination of the
$sign(\Delta m^2_{31})$ and of
the type of $\nu$-mass spectrum 
($NH,IH,QD$, etc.).\\
$\bullet$ Determining or 
obtaining significant constraints
on the absolute scale of $\nu$-masses, 
or on $min(m_j)$.
$\bullet$ Determining 
the nature--Dirac or Majorana,
of massive neutrinos $\nu_j$.\\ 
$\bullet$ Establishing whether the CP-symmetry 
is violated in the lepton 
sector a) due to the Dirac phase $\delta$, and/or
b) due to the Majorana phases 
$\alpha$ and $\beta$ if
$\nu_j$ are Majorana particles.\\
$\bullet$ Searching with increased sensitivity
for possible manifestations, other than 
flavour neutrino oscillations,
of the non-conservation
of the individual lepton charges $L_l$, $l=e,\mu,\tau$,
such as $\mu \rightarrow e + \gamma$,
$\tau \rightarrow \mu + \gamma$, etc. decays.\\
$\bullet$ Understanding at fundamental level 
the mechanism giving rise to  
neutrino masses and mixing and to 
$L_l-$non-conservation,
i.e., finding the {\it Theory of neutrino
mixing}. This includes
understanding the origin of the 
patterns of $\nu$-mixing and $\nu$-masses 
suggested by the data.
Are the observed patterns of 
$\nu$-mixing and of $\Delta m^2_{21,31}$
related to the existence of new  
fundamental symmetry of particle interactions?
Is there any relations between quark 
mixing and neutrino mixing, e.g.,
does the relation $\theta_{12} + \theta_{c}$=$\pi/4$,
where $\theta_{c}$ is the Cabibbo angle, hold?
Is $\theta_{23} = \pi/4$, or $\theta_{23} > \pi/4$
or else $\theta_{23} < \pi/4$?
What is the physical origin 
of $CPV$ phases in $\pmns$?
Is there any relation (correlation)
between the (values of) $CPV$ phases 
and mixing angles in  $\pmns$?
Progress in the theory of $\nu$-mixing
might also lead, in particular, 
to a better understanding of the 
mechanism of generation 
of baryon asymmetry of 
the Universe \cite{LeptoG}.

  Obviously, the successful realization 
of the experimental part of this 
research program would be a formidable task
and would require many years.

The mixing angles, $\theta_{21}$,
$\theta_{23}$ and $\theta_{13}$,
Dirac $CPV$ phase $\delta$ and 
$\Delta m^2_{21}$ and $\Delta m^2_{31}$ 
can, in principle, be measured with a 
sufficiently high precision in
a variety of $\nu$-oscillation 
experiments (see further).
These experiments, however,
cannot provide information on the
absolute scale of $\nu$- masses 
and on the nature of massive neutrinos $\nu_j$.
The flavour neutrino oscillations  
are insensitive to
the Majorana $CPV$ phases 
$\alpha$ and $\beta$ \cite{BHP80,Lang86}.
Establishing whether $\nu_j$
have distinct antiparticles (Dirac fermions)
or not (Majorana fermions) is of fundamental
importance for understanding 
the underlying symmetries of 
particle interactions \cite{BiPet87} 
and the origin of $\nu$-masses. 
If $\nu_j$ are Majorana fermions,
getting experimental information about the  
Majorana $CPV $phases in $\pmns$
is a remarkably challenging problem. 
\cite{BGKP96,MajPhase1,PPR1}. 
The phases $\alpha$ and $\beta$ 
can affect significantly 
the predictions for the 
rates of the (LFV) decays
$\mu \rightarrow e + \gamma$,
$\tau \rightarrow \mu + \gamma$, etc.
in a large class of supersymmetric theories
with see-saw mechanism of neutrino mass
generation (see, e.g., \cite{PPY03}).
Majorana $CPV$ phases might be at 
the origin of the baryon asymmetry of 
the Universe \cite{LeptoG}. 

\vspace{-0.30cm}
\section{The Pattern of Neutrino Mixing}

\vspace{-0.2cm}
 The $\nu$-oscillation data suggest that 
$\theta_{12}\cong \pi/6$, $\theta_{23} \cong \pi/4$, and 
$\theta_{13} \equiv \epsilon < \pi/12$. Thus, the PMNS matrix 
is very different from the  CKM matrix:
\vspace{-0.1cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\nonumber
\pmns \cong \left(\begin{array}{ccc}
  \frac{\sqrt{3}}{2}& \frac{1}{2} & \epsilon \\
 -\frac{1}{2\sqrt{2}} & \frac{\sqrt{3}}{2\sqrt{2}} & \frac{1}{\sqrt{2}} \\
  \frac{1}{2\sqrt{2}} & -\frac{\sqrt{3}}{2\sqrt{2}} & \frac{1}{\sqrt{2}}
 \end{array} \right)
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\noindent where the $CPV$ phases 
have been suppressed.
Understanding the origin of the 
emerged patterns 
of $\nu$-mixing and of 
$\Delta m^2_{21,31}$, 
is one of the central problems 
in today's neutrino physics. 

  $\pmns$ is close in form to 
a bimaximal mixing matrix,
for which $\theta_{12}$=$\theta_{23}$=$\pi/4$ 
and $\theta_{13}$=0:
\vspace{-0.1cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\nonumber
U_{\rm bimax} =
\left(\begin{array}{ccc}
  \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\
 -\frac{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}} \\
  \frac{1}{2} & -\frac{1}{2} & \frac{1}{\sqrt{2}}
 \end{array} \right)
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
Whereas the data favor $\theta_{23} = \pi/4$
and allows for $\theta_{13} = 0$, 
$\theta_{12} = \pi/4$ is ruled out 
at $\sim$6$\sigma$ \cite{KL766,BCGPRKL2}.
The deviation of $\theta_{12}$ from $\pi/4$
can be parameterized with a 
parameter $\lambda$, which is 
very similar in value 
to the Cabibbo angle \cite{WR}:\\

\vspace{-0.3cm}
$\sin\theta_{21} \cong \frac{1}{\sqrt{2}}( 1 - \lambda),~
\lambda \cong (0.20 - 0.25) %\sin\theta_C 
\sim \theta_C$.\\

\vspace{-0.3cm}
\noindent The implied relation $\theta_\odot = \pi/4 - \theta_C$,
if confirmed experimentally, might be linked to 
GUT's \cite{marttiSMFM}.

It is natural to suppose that \cite{STP82PD,FPR}
\footnote{Extensive list of references on the subject 
is given in \cite{FPR}.} 

\vspace{-0.3cm}
$$\pmns = U^{\dagger}_{\rm L}(\lambda)~U_\nu,~~
U_\nu = U_{\rm bimax},$$

\vspace{-0.1cm}
\noindent where $U^{\dagger}_{\rm L}(\lambda)$ 
arises from diagonalization of 
the charged lepton mass matrix
and $U_{\nu}$ diagonalizes 
a neutrino Majorana mass matrix $M_{\nu}$.
The inequality $\Delta m^2_{21} \ll |\Delta m^2_{31}|$ 
and the form of $U_{\rm bimax}$ can  
be associated with an
{\it approximate} symmetry of $M_{\nu}$, 
implying (for $U_{\rm L}$=${\bf 1}$)
the conservation of the lepton 
charge \cite{STP82PD}:

\vspace{-0.3cm}
$$L' = L_e - L_{\mu} - L_{\tau}~~~~~~~~~~~(4)$$

\vspace{-0.1cm}
  If $\theta^{l}_{ij}$ are the 3 angles of
CKM-like parametrization of $U_{\rm L}(\lambda)$ and
$\sin\theta^{l}_{ij} \equiv \lambda_{ij}$,
3 generic cases are compatible with 
the $\nu$-mixing data \cite{FPR}:\\
$\bullet$ all $\lambda_{ij}$ small, $\lambda_{ij} \ltap 0.35$;\\
$\bullet$$\lambda_{23} = 1$, $\lambda_{12},\lambda_{13} \ltap 0.35$;\\
$\bullet$ all $\lambda_{ij}$ large, e.g., 
$\lambda_{ij} \geq \frac{1}{\sqrt{2}}$.\\
For $\lambda_{ij} \ltap 0.35$, the data imply:
$\lambda_{23} \ltap 0.19$,
$ \lambda_{12} \gg \lambda_{13}$ (for $\lambda_{12,13} > 0$),
$\lambda_{12} \cong $(0.21-0.25), 
$\lambda_{13} \ltap 0.03$.
With all $CPV$ phases set to 0, 
$\theta_{ij}$
are expressed in terms of $\lambda_{kl}$
and their values are correlated.
The deviation of $\theta_{12}$ from $\pi/4$ is 
determined by $s_{13}$=$\sin\theta_{13}$,
which typically implies $s^2_{13}\gtap$0.01. 
Consider two simple cases \cite{FPR}.\\
$\bullet$ Hierarchy of $\lambda_{ij}$:
$\lambda_{12}\equiv \lambda$, $\lambda_{23} \sim \lambda^{2}$, 
$\lambda_{13} \sim \lambda^{3}$.\\
Then $s_{13} \cong \lambda/\sqrt{2}$, 
$\sin^22\theta_{23}$=1-16$s^4_{13}$,
$\tan^2\theta_{12}$=1-4$s_{13}($1-2$s_{13}$+$4s^2_{13})$, and  
$\sin^22\theta_{23}\gtap$0.96; we also have
$s^2_{13}\gtap 0.01$ for $\tan^2\theta_{12}\leq 0.58$ \cite{BCGPRKL2}.\\
$\bullet$ $\lambda_{12} \equiv \lambda$,
$\lambda_{23} \sim \lambda/2$ (mild hierarchy), 
$\lambda_{13} \sim \lambda^{3}$.\\ 
Now $\sin^22\theta_{23}\gtap$0.90.\\ 
If $\lambda_{23}$=1, $\lambda_{12(13)}\ltap$0.35,
one has $\sin^22\theta_{23}\gtap$0.997, while for
$\lambda_{ij} \geq \frac{1}{\sqrt{2}}$, $\sin^22\theta_{23}\gtap$0.95
\cite{FPR}. Obviously, a sufficiently precise
measurement of $\sin^22\theta_{23}$ would allow
to distinguish between the three possibilities.

   If CP is not conserved, we have \cite{PPR3}

\vspace{-0.3cm}
$$\pmns = U_{\rm L}^\dagger U_\nu =  
\tilde{U}_{\rm lep}^\dagger P_\nu \tilde{U}_\nu Q_\nu,$$

\vspace{-0.1cm}
\noindent where, in general, 
$\tilde{U}_{\rm lep}$, $\tilde{U}_{\nu}$ are CKM-like
matrices each containing 3 angles and 1 $CPV$ phase, 
$P_{\nu}={\rm diag} (1,e^{i \phi},e^{i \omega})$ and
$Q_{\nu}={\rm diag} (1,e^{i \rho},e^{i \sigma})$.
Suppose that $\tilde{U}_\nu = U_{\rm bimax}$ and arises
from diagonalization of the simplest 
possible $M_{\nu}$ \cite{FPR},
\vspace{-0.2cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\be \label{eq:lelmlt}
M_\nu = \frac{m}{\sqrt{2}} \, 
\left( 
\bad 
0 & e^{-i\alpha'} & e^{-i\beta'} \\[0.2cm]
e^{-i\alpha'} & 0 & \epsilon~e^{-i\gamma'} \\[0.2cm]
 e^{-i \beta'} & \epsilon~e^{-i\gamma'} & 0  
\ea 
\right)~
\ee

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  
\noindent Here $\alpha',\beta',\gamma'$ are phases,
$m^2 \cong -\dma$=$\Delta m^2_{23}$ and
$\dmsol$=$\Delta m^2_{21} \cong \sqrt{2}\epsilon m^2$,
$\epsilon \sim 0.03$.
The $\nu$-mass spectrum is $IH$.
In the limit $\epsilon = 0,\tilde{U}_{\rm lep} = {\bf 1}$,
$L' = L_e - L_{\mu} - L_{\tau}$ is conserved \cite{STP82PD}.
For $\tilde{U}_{\rm lep} \neq {\bf 1}$, 
$(\alpha' - \gamma')$=$\omega$ and $(\beta' - \gamma')$=$\phi$ 
are physical $CPV$ phases, $Q_{\nu} = {\bf 1}$, 
and $U_{\rm PMNS}$=$\tilde{U}_{\rm lep}^\dagger P_\nu U_{\rm bimax}$.
The Dirac and Majorana phases in $\pmns$ 
have the same ``source'' - the $CPV$ phases in 
$\tilde{U}_{\rm lep}^\dagger$ and $P_\nu$,
$\psi$ and $\phi$, $\omega$. Even $\theta_{ij}$
depend on the latter. 

 For, e.g., ``small'' 
$\lambda_{12}$=$\lambda$, $\lambda_{23}$=$ A\lambda$,
$\lambda_{13}$=$B\lambda^3$, $A,B\sim 1$, one finds
up to terms ${\cal{O}}(\lambda^3)$ \cite{FPR}:\\ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\be \label{eq:mixcp}
%\ba 
$\tan^2 \theta_{12}\cong 1 - 2 \sqrt{2}\lambda c_\phi +  
2 (2 c^2_\phi - \sqrt{2}A c_\omega)\lambda^2$,\\
$\sqrt{2}s_{13}\cong \lambda$--$A\lambda^2c_{\omega - \phi}$,
$\sin^2 2\theta_{23}\cong$1--4$A^2\lambda^2c_{\omega - \phi}$,\\ 
% \ea 
%\ee
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
where $c_\phi$=$\cos \phi$, etc.\\
The rephasing invariants associated with $CPV$ phases
$\delta$, $\beta$ and $\alpha$-$\beta$ read, respectively: 
$J_{CP} = 
{\rm Im} \left\{ U_{e1} \, U_{\mu 2} \, U_{e 2}^\ast \, U_{\mu 1}^\ast 
\right\}$ \cite{CJ85,PKSP3nu88},\\ 
$S_1$=${\rm Im}\left\{ U_{e1}U_{e3}^\ast \right\}$,
$S_2$=${\rm Im}\left\{ U_{e2}U_{e3}^\ast \right\}$ \cite{JMaj87}.\\ 
$J_{CP}$ controls the magnitude of 
$CP$ and $T$ violating effects in 
$\nu$-oscillations \cite{PKSP3nu88}; 
$S_1$, $S_2$ appear in the effective Majorana mass 
$\meff$ in \betabeta-decay \cite{BPP1}.
In general, $J_{CP}$, $S_1$ and $S_2$ are independent.
However, in the scheme with approximate
$L_e - L_{\mu} - L_{\tau}$ symmetry we are considering,
and to leading order in $\lambda$ we find \cite{FPR}:

\vspace{-0.3cm}
$$J_{CP} 
\cong \frac{S_1}{2\sqrt{2}} \cong \frac{S_2}{2\sqrt{2}}.$$

\vspace{-0.1cm}
\noindent Thus, {\it the magnitude of CP--violating effects in 
$\nu$-oscillations is directly related 
to the magnitude of CP--violating effects associated
with the Majorana nature of neutrinos}. 
One also finds \cite{FPR}:

\vspace{-0.4cm}
$$\meff \cong  \sqrt{|\Delta m^2_{\rm A}|}
~\left |\cos2\theta_{\odot} + i~8J_{CP} \right |~,$$ 

\vspace{-0.2cm}
\noindent 
i.e., $J_{CP}$ determines the deviation 
of $\meff$ from its minimal value 
(for $IH$ spectrum) \cite{BPP1}.

  The approach to understand the pattern of 
$\nu$-mixing discussed above is by no means unique 
(see, e.g., \cite{FFerug,CAlbr}). 
It demonstrates that Dirac and Majorana
$CPV$ phases (and effects) 
can be related, and that 
$\theta_{ij}$ can depend on $CPV$ phases.

\vspace{-0.30cm}
\section{Comments on Future Progress}

\vspace{-0.20cm}
 Future progress in the studies of  
$\nu$-mixing will be crucial
for understanding at fundamental level
the mechanism generating it. 
The requisite data is forseen
to be provided by\\
$\bullet$ $\nu_{\odot}-$ and $\nu_{\rm A}-$ experiments:\\
SK, SNO, SAGE, BOREXINO \cite{BOREXINO}, LowNu 
\cite{LowNu}, and 
SK ($\nu_{\rm A}$), MINOS ($\nu_{\rm A}$) \cite{MINOS}, 
INO ($\nu_{\rm A}$) \cite{INO};\\
$\bullet$ Reactor $\bar{\nu}_e$ experiments with $L\sim$(1 - 180) km;\\
$\bullet$ Accelerator experiments:\\
K2K ($L\sim$250 km), MINOS ($L\sim$730 km),
OPERA and ICARUS \cite{OPERA,ICARUS} ($L\sim$730 km);\\ 
$\bullet$ Experiments with super beams:\\
T2K ($L\sim$295 km) \cite{T2K},
NO$\nu$A ($L\sim$800 km) \cite{NOnuA}\\
SPL+$\beta-$beams with UNO (1 megaton) detector
(CERN-Frejus, $L\sim$135 km) \cite{SPLbeta};\\
$\bullet$ $\nu$-Factory experiments ($L\sim$3000;7000 km) \cite{NuFact};\\
$\bullet$ $(\beta\beta)_{0\nu}-$
and $^{3}$H $\beta-$ decay experiments \cite{bb0nu,H3};\\
$\bullet$ Astrophysical/cosmological 
observations \cite{Astro}.\\ 

\vspace{-0.3cm}
\noindent \underline{\bf Absolute Neutrino Mass Measurements.} 
The Troitzk and Mainz \hbeta experiments 
provided information on the
%  electron (anti-)neutrino 
$\bar{\nu}_e$ mass \cite{H3}:
$m_{\bar{\nu}_e}<$2.2 eV at 95\%~C.L.
The KATRIN experiment \cite{H3}
expected to start in 2007,
is planned to reach sensitivity to 
$m_{\bar{\nu}_e}\sim$0.20 eV (95\% C.L.),
and thus to probe the region of 
QD $\nu$-mass spectrum.  
In this region $m_{1,2,3} \cong m_0 \cong m_{\bar{\nu}_e}$.

  The CMB data of the WMAP experiment 
were used to obtain an upper limit 
on $\sum_{j} m_{j}$ \cite{Astro}:\\ 
$\sum_{j} m_{j} <$ (0.7--1.8) eV   (95\%~C.L.).\\
The WMAP and future PLANCK 
experiments can be sensitive to 
$\sum_{j} m_{j} \cong 0.40$ eV.
Data on weak lensing of galaxies 
by large scale structure,
combined with WMAP and PLANCK data,
may allow one to determine 
$\sum_{j} m_{j}$ with an uncertainty 
of $\sim$(0.04--0.10) eV (see, e.g., \cite{Astro}).\\

\vspace{-0.30cm}
\noindent \underline{\bf $\betabeta-$Decay Experiments}.
The $\betabeta$-decay experiments \cite{bb0nu}
have a remarkable physics potential
\footnote{For detailed discussion and list
of references see, e.g., \cite{STPFocusNu04}}.
They can establish the Majorana nature of 
neutrinos $\nu_j$. If $\nu_j$
are Majorana particles,
they can provide unique
information \\
$\bullet$ on the type of
$\nu$-mass spectrum \cite{BGGKP99,BPP1,PPSNO2bb},\\  
$\bullet$ on the absolute scale of $\nu$-masses \cite{PPW}, and\\
$\bullet$ on the
Majorana $CPV$ phases in $\pmns$ \cite{BGKP96,PPR1}. 
The $\betabeta$-decay, 
(A,Z)$\rightarrow$(A,Z+2)+$e^-$+$ e^-$, 
is allowed if 
$\nu_j$ are Majorana particles 
(see, e.g., \cite{BiPet87}).
The nature - Dirac or Majorana, of 
neutrinos $\nu_j$ is related 
to the fundamental symmetries of 
particle interactions. Neutrinos 
$\nu_j$ will be Dirac fermions if 
particle interactions conserve 
the total lepton charge $L$. 
They can be Majorana particles
if there does not exist 
any conserved lepton charge.
Massive neutrinos are 
predicted to be of Majorana nature
by the see-saw mechanism of 
$\nu$-mass generation 
(see, e.g., \cite{seesaw}),
which also provides 
an attractive explanation of the
smallness of 
$\nu$-masses and 
- through leptogenesis theory 
(see \cite{LeptoG}), of the 
baryon asymmetry 
of the Universe.

 If $\nu_j$ are Majorana fermions,
the $\betabeta$-decay amplitude  
of interest has the form 
(see, e.g., \cite{BPP1}):\\
$A\betabeta \cong \mefff~M$, where 
$M$ is the corresponding 
nuclear matrix element (NME) and

\vspace{-0.55cm}
$$\meff=\left| m_1 |U_{\mathrm{e} 1}|^2 
+ m_2 |U_{\mathrm{e} 2}|^2e^{i\alpha}
 + m_3 |U_{\mathrm{e} 3}|^2e^{i\beta} \right|$$

\vspace{-0.1cm}
\noindent $|U_{\mathrm{e}1}|$=$c_{12}c_{13}$,
$|U_{\mathrm{e}2}|$=$s_{12}c_{13}$, 
$|U_{\mathrm{e}3}|$=$s_{13}$.
In the case of $CP$-invariance one has \cite{LW81},\\  
$\eta_{21} \equiv e^{i\alpha}$=$\pm 1$, 
$\eta_{31}\equiv e^{i\beta}$=$\pm 1$, $\eta_{21(31)}$ being the \\ 
\noindent relative 
CP-parity of Majorana neutrinos 
$\nu_{2(3)}$ and $\nu_1$. 
Thus, $\meff$ depends \cite{SPAS94}
on $\theta_{\odot}$, $\theta_{13}$,
$\dma$, $\dmsol$ as well as on 
$min(m_j)$, Majorana 
phases $\alpha$, $\beta$ and the $\nu$-mass spectrum.
The predicted value of $\meff$ for 
$\sin^2\theta_{13}$=0.04 and 
90\% C.L. allowed 
values of $\dma$ and 
$\dmsol$, $\theta_{\odot}$ (Fig. 1)
is shown as function of
$min(m_j)$ in Fig. 2.
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\vskip -0.5cm
\includegraphics[width=7cm,height=5cm,clip=]{KL2meff90SK3nus1304.epsi}
\vskip -0.5cm
\caption{The value of $\meff$ 
as function of $min(m_j)$
for $\sin^2\theta_{13}$=0.04 and 
90\% C.L. allowed ranges of values
of $\dma$, $\dmsol$ and 
$\theta_{\odot}$ \cite{PPSNO2bb,SPNobSymp04}.
}
\label{Fig2}
\end{figure}

% \vspace{-0.3cm}
  The main features of the predictions for $\meff$ 
are \cite{PPSNO2bb,PPW,STPFocusNu04}: i) for $NH$ spectrum, 
typically $\meff\ltap$0.006 eV, and
$\meff\sim$0 is possible;
ii) for $IH$ spectrum, $\meff\gtap\sqrt{|\dma|}\cos2\theta_{\odot}
\gtap$0.012 eV and $\meff\ltap\sqrt{|\dma|}\ltap$0.06 eV; 
iii) in the case $QD$ spectrum, 
$\meff\gtap m_0\cos2\theta_{\odot}\gtap$0.05 eV 
and $\meff\ltap m_0$, with $m_0\gtap$0.2 eV and
$m_0<$2.2 eV \cite{H3}, $m_0<$0.6 eV \cite{Astro}. 
Thus, for $IH$ and $QD$ spectra, $\meff$
is limited from below. 

  Many experiments have searched for
$\betabeta$-decay \cite{Morales02}. 
The best sensitivity was achieved in\\ 
Heidelberg-Moscow $^{76}$Ge experiment 
\cite{HMGe76}. A positive signal at $>$3$\sigma$,
corresponding to\\  
$\meff = (0.1 - 0.9)~{\rm eV}$  at 99.73\% C.L.,
is claimed to be observed \cite{HMGe76}. 
Two experiments, NEMO3 (with $^{100}$Mo and 
$^{82}$Se) \cite{NEMO3}
and CUORICINO (with $^{130}$Te) \cite{CUORI},
designed to reach sensitivity to 
$\meff\sim$(0.2-0.3) eV, announced 
first results:
$\meff<$ (0.7--1.2) eV \cite{NEMO3} and $\meff<$ (0.3 -- 1.6) eV
\cite{CUORI} (90\% C.L.),
where estimated uncertainties in the $NME$
\cite{bb0nuNME} are accounted for.
A number of projects aim at sensitivity to 
$\meff\sim$(0.01--0.05) eV \cite{bb0nu}:
CUORE ($^{130}$Te), GENIUS ($^{76}$Ge),
EXO ($^{136}$Xe), MAJORANA ($^{76}$Ge),
MOON ($^{100}$Mo), 
XMASS ($^{136}$Xe), etc. These experiments
will probe the region corresponding
to $IH$ and $QD$ spectra
and test the positive result 
claimed in \cite{HMGe76}. 
The knowledge of the relevant $NME$
with sufficiently small uncertainty
is crucial for obtaining 
quantitative information on
the $\nu$-mixing parameters 
from a measurement of
$\betabeta$-decay half-life.
{\it In view of their importance for
understanding the origin of
$\nu$- masses and mixing,
 performing few 
$\betabeta$-decay 
experiments with sensitivity to
$\meff\sim$(0.01--0.05) eV 
(or better)
should have highest priority in the 
future studies of $\nu$-mixing.}\\

\vspace{-0.30cm}
\noindent \underline{\bf The CHOOZ Angle $\theta_{13}$}.
The angle $\theta_{13}$ plays extremely
important role in the phenomenology of 
$\nu$-oscillations. It controls together 
with $\sin\delta$ the magnitude of
$CP$- and $T$-violating effects 
in $\nu-$oscillations.
It controls the sub-dominant 
$\nu_{\mu} \leftrightarrow\nu_e$ and
$\bar{\nu}_{\mu} \leftrightarrow \bar{\nu}_e$ 
oscillations of $\nu_{\rm A}$
and in the accelerator experiments 
MINOS, OPERA, ICARUS, T2K, NO$\nu$A, etc. 
and at $\nu-$factories.
The value of $\meff$ in $\betabeta$-decay
in the case of $NH$ spectrum
depends on $\sin^2\theta_{13}$ \cite{BPP1,PPW}.
The knowledge of $\theta_{13}$ 
is crucial for finding the correct theory 
of $\nu$-mixing as well.

  If $\sin^22\theta_{13} \ltap (0.01-0.008)$,
the $CPV$ effects in $\nu$-oscillations
would be too small to be observed in
T2K and NO$\nu$A experiments 
\cite{TMU04,MShae04}.
Thus, the future program
of searches for $CPV$ effects
in accelerator experiments
depends critically on 
the value of $\theta_{13}$.
The sensitivity of future experiments with
{\it conventional beams (MINOS+ICARUS+OPERA)},
with off-axis super beams, T2K-SK, NO$\nu$A (NuMI),
and with reactor $\bar{\nu}_e$ - Double-CHOOZ 
\cite{Reactor04},
to $\sin^22\theta_{13}$, as function of 
$|\Delta m^2_{31}|$, is illustrated in Fig. 3.
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\vskip -0.5cm
\includegraphics[width=7.5cm,height=6cm,clip=]{th13dmdep_left.eps}
% \includegraphics[width=7.5cm,height=6cm,clip=]{Manfredf6.eps}
\vskip -0.5cm
\caption{The sensitivity of 
future experiments to $\sin^22\theta_{13}$
\cite{TMU04}.
}
\label{Fig3}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{-0.35cm}
 There are several proposals for
reactor $\bar{\nu}_e$ experiments 
with baseline $L\sim$(1-2) km 
\cite{Reacth13,Reacth13J}, 
which could imporve the current limit 
$\sin^2\theta_{13}<$0.05 by a 
factor of (5-10) \cite{Reactor04}. 
The most advanced in preparation 
is the Double-CHOOZ project. 
{\it The reactor $\theta_{13}$ 
experiments, in our view,
should have highest priority
in the program 
of research in $\nu$-physics:
they can compete in sensitivity with
accelerator experiments (T2K-SK, NO$\nu$A) and 
can be done on relatively short 
(for experiments in this field) 
time scale. The planning of 
experiments to study $CPV$ effects 
in $\nu$-oscillations 
would benefit 
significantly from the results
of a high precision reactor 
$\theta_{13}$ experiment.} \\

\vspace{-0.30cm} 
\noindent \underline{\bf Measuring $\dmsol \equiv \Delta m^2_{21}$ and 
$\theta_{\odot} \equiv \theta_{12}$}.
The current solar 
neutrino and KL 766.3 Ty 
data determine $\Delta m^2_{21}$
and $\sin^2\theta_{12}$ 
with uncertainties of 12\% and 24\% at 3$\sigma$ 
\cite{BCGPTH1204}.
Accounting for possible 
reduction of 
errors in the data 
from the phase-III of 
SNO experiment \cite{SNO123}
could lead to a reduction 
only of the error in $\sin^2\theta_{12}$ 
to 21\% \cite{SKGdCP04,BCGPTH1204}. 
If instead of 766.3 Ty
one uses simulated 3 kTy
KL data in the same 
data analysis, the 3$\sigma$ errors in 
$\Delta m^2_{21}$ and $\sin^2\theta_{12}$ 
diminish to 7\% and 18\% \cite{BCGPTH1204}. 

  The most precise measurement of 
$\Delta m^2_{21}$ could be achieved 
\cite{SKGdCP04} using SK
doped with 0.1\% of Gadolinium 
(SK-Gd) for detection of reactor 
$\bar{\nu}_e$ \cite{SKGdBV04}:
SK gets the same flux of reactor
$\bar{\nu}_e$ as KamLAND and
after 3 years of 
data-taking, $\Delta m^2_{21}$
would be determined with a 
3.5\% error at 3$\sigma$ 
\cite{SKGdCP04}. 
A dedicated reactor  
$\bar{\nu}_e$ experiment with a 
baseline $L\sim$60 km tuned to 
$\bar{\nu}_e$ survival 
probability minimum, 
% (SPMIN),
could provide the most precise 
determination of $\sin^2\theta_{12}$ 
\cite{TH12,HMinth1204,BCGPTH1204}:
with statistics of $\sim$60 GWkTy 
and systematic error 
of 2\% (5\%), $\sin^2\theta_{12}$  
could be measured with uncertainty
of 6\% (9\%) at 3$\sigma$ \cite{BCGPTH1204}. 
A generic LowNu $\nu-e$ elastic
scattering experiment, 
designed to measure the $pp$ 
$\nu_{\odot}$-flux with an 
error of 3\% (1\%), would permit  
to determine $\sin^2\theta_{12}$ 
with an error of 14\% (19\%) at 3$\sigma$ 
\cite{BCGPTH1204}. 
The inclusion of the uncertainty
in $\theta_{13}$
($\sin^2\theta_{13}<$0.05)
in the analyses increases the 
quoted errors by (1-3)\% 
\cite{BCGPTH1204}.\\

\vspace{-0.30cm} 
\noindent \underline{\bf Measuring $|\dma|$, 
$\theta_{\rm A}\equiv\theta_{23}$,
$sign(\dma)$ and $\delta$}.
The expected 3$\sigma$ uncertainties in 
$|\dma|$=$|\Delta m^2_{31}|$
from studies
of $\nu_{\mu}$-oscillations in
i) MINOS + CNGS \cite{MINOS,OPERA,ICARUS},
ii) NO$\nu$A \cite{NOnuA} and
iii) T2K (SK) \cite{T2K} experiments,
if the true $|\Delta m^2_{31}|$=
2$\times 10^{-3}$ eV$^2$ and
true $\sin^2\theta_{23}$=0.5, 
read \cite{TMU04}: 
i) 26\%, ii) 25\% and iii) 12\%.
T2K (SK) and NO$\nu$A experiments
will measure also $\sin^22\theta_{23}$
with a high precision - (1-2)\% at 1$\sigma$.
However, they would not be able to
resolve the $\theta_{23}$--$(\pi/2 - \theta_{23})$
ambiguity if $\sin^22\theta_{23}<$1.
T2K and NO$\nu$A are planned to begin 
in 2009 and 2011 (or 2009 \cite{NOnuA}) and
in their first phase, both 
experiments will use $\nu_{\mu}$-beams.  
The data from this phase will not allow
to determine the $sign(\Delta m^2_{31})$.
Even if $\sin^22\theta_{13}\sim$0.10,
without the knowledge of
$sign(\Delta m^2_{31})$
it would be impossible 
to get unambiguous information
on $CP$-violation in 
$\nu$-oscillations (induced by
$\delta$) using only the phase-I 
T2K and NO$\nu$A data 
\cite{TMU04}. If $\sin^22\theta_{13}\gtap$0.05,
information on $sign(\Delta m^2_{31})$
and $\sin^2\theta_{23}\gtap$0.5
might be obtained in $\nu_{\rm A}$-experiments
by studying the Zenith angle
dependence of the multi-GeV 
$e$- and $\mu$- like, and/or
$\mu^{\pm}$, events \cite{JBSP203}.
Resolving all parameter degeneracies
\cite{PDeg,MShae04} and determining 
whether $\delta$ takes a $CPV$ value if 
$\sin^22\theta_{23}<$1 and no
information on $sign(\Delta m^2_{31})$
is available, would be a 
formidable task and would 
require high statistics 
(phase-II) data on $\nu_{\mu}$- 
and $\bar{\nu}_{\mu}$-
oscillations both from T2K and
NO$\nu$A and data on $\theta_{13}$ 
from a reactor experiment 
\cite{TMU04,PDeg,MShae04};
if $\sin^22\theta_{13}\ltap$0.01, 
data from SPL+$\beta$-beam 
experiments \cite{SPLbeta} 
or from experiments at 
$\nu$-factory \cite{NuFact}
might be required.

\vspace{-0.20cm}
\section{Instead of Conclusions}

\vspace{-0.2cm}
We are at the beginning of the road 
leading to a 
comprehensive understanding 
of the patterns of 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[htb]
\includegraphics[width=8cm,height=6.5cm,clip=]{hiroshige.eps}   
\caption{Landscape by Y. Hiroshige (Ukiyoe master, Edo epoch, Japan).}
\label{Fig4} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{-0.4cm}
\noindent neutrino masses and mixing and 
of their origin. 
The road is not easy and 
we do not quite know how long
our ``journey'' will take, 
how difficult it will be 
and what we will finally discover.  
However, I am sure the 
``view'' that will open to us
from the ``summit'' at 
the end of this ``journey'' 
will be of dazzling clarity,
perspective and beauty (Fig. 4).\\

\vspace{-0.30cm}
{\bf{Acknowledgements}}. It is a pleasure 
to thank the Organizers of the Neutrino 
2004 International Conference for 
assembling such a scientifically 
enjoyable meeting.

% \vspace{-0.40cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}




\end{document}





\newpage
















*************************************




\section{Solar Neutrino Oscillation parameters}
In this section we discuss the constraints on solar neutrino oscillation 
parameters $\Delta m^2_{\odot} \equiv
\Delta
m^2_{21}$, $\theta_{\odot} \equiv \theta_{12}$ from two flavour
$\nu_e - \nu_{\rm active}$ 
 analysis. 
\\ \\
\underline{{\bf Allowed area 
from global Solar Data  
}}:
In our analysis of  global solar data we 
include the total rates from the radiochemical experiments
Cl and Ga (Gallex, SAGE and GNO combined) \cite{sol} and
the 1496 day SK Zenith
angle spectrum data \cite{SKsolar}, the SNO spectrum data from the 
pure 
$D_2O$ phase \cite{sno1}, as well as 
the CC, NC and ES rates from the salt phase
\cite{sno2}. 
We use {BP04} fluxes \cite{bp04}
 and keep the $^{8}{B}$ flux normalisation free. 
Further details of our 
analysis can be found in  
\cite{snocc,snonc,snosaltus}.
In the left panel of Figure \ref{sol2osc}
we show the allowed region in the parameter space
including the SNO spectrum data
from the pure $D_2O$ phase while the right panel 
shows the allowed regions after including the data from the salt phase. 
The best-fit values of parameters obtained are 
\\
$\bullet$
$\dm$=6.1$\times10^{-5}$eV$^2$,
$\sss$ = 0.3,
$f_B$=0.89.\\ 
$\bullet$99\% C.L.  range {(before salt)} \\
$\dm$=(3.1-25.7) $\times 10^{-5}$eV$^2$,
$\sss$=0.21 -0.44 \\
$\bullet$99\% C.L.  range {(after salt)} \\
$\dm$=(3.2-14.8)$\times 10^{-5}$eV$^2$,
{$\sss$}=0.22 -0.37. \\
The allowed region corresponds to Large-Mixing-Angle (LMA)
MSW solution.
\begin{figure}[htb]
\includegraphics*[width=20pc]{sol_bp04_withccnclines.eps}
\caption{The 90\%, 95\%, 99\% and 99.73\% C.L.
allowed regions in the $\dm-\sin^2\theta_{\odot}$ plane from global
$\chi^2$-analysis of solar neutrino  data.
Also shown are the iso-CC/NC contours.
%ratio $R_{CC/NC}$.
}
\label{sol2osc}
\end{figure}
The impact of the salt data is to 
tighten the upper limit on {$\dm$} and {$\sss$}.   
The iso-CC/NC curves in Fig. \ref{sol2osc}
shows that for higher values of $\sss$ and
$\dm$ the 
$CC/NC$ value is higher.  Since the {CC/NC} ratio for the
salt phase data is 0.31 as opposed to 0.35 in the pure $D_2O$ phase 
the allowed regions shift to smaller $\sss$ and $\dm$ values 
following the iso-{CC/NC} contours.
\\ \\
\underline{\bf Allowed area from KamLAND spectra}:
\begin{figure}[htb]
\includegraphics[width=8cm,height=6cm,clip=]{kcomp.eps} 
\caption{Same as in figure 1 but in 
in the $\dm-\tan^2\theta_{12}$ plane from global
$\chi^2$-analysis of \kl spectral data.
}
\label{klspec}
\end{figure}
We next study the impact of the 
new results from the \kl
experiment in constraining the oscillation parameters. 
The survival probability for \kl is :
\be
P
_{\bar{e}\bar{e}}^{KL} =
1 - \sin^22\theta_{12}\sin^2\left(\frac{1.27\Delta m_{21}^2 L}
{E_{\nu}}\right)
\ee
 neglecting the small matter effect for lower values of $\Delta m^2_{21}$. 
Assuming CPT conservation i.e same oscillation parameters for
neutrinos and antineutrinos \kl is sensitive to 
$\dm$ in the  
LMA region. 

In  the right panel of 
fig \ref{klspec} we show the allowed regions obtained 
after including the latest \kl data in our analysis while the left panel 
shows the allowed regions without this data. 
The figure shows that 
higher $\Delta m^2$ regions, which correspond to an averaged survival 
probability and hence flat spectra, 
reduce in size due to the increased distortion in the  
new \kl spectrum data \cite{kl766}. 
For details of our \kl analysis we refer to \cite{kl766us,kl162us}.
\\ \\
\underline{\bf Allowed area from Solar+KamLAND}:  
\begin{figure}[htb]
\includegraphics[width=8cm,height=6cm,clip=]{sol+kl.eps}   
\caption{
Same as in figure 1
but from global
$\chi^2$-analysis of solar and \kl data.
}
\label{sol+kl}
\end{figure}
In  left(right) panel of \ref{sol+kl} we present the allowed 
regions from 
analysis of solar and  162 (766.3) ton year (Ty) \kl data.
In the left panel there are two allowed zones -- the low-LMA zone 
which is more favoured and the high-LMA zone admitted only at 3$\sigma$.
In the right panel
there is no allowed high-LMA region  
at 3$\sigma$ since 
the latest \kl data  disfavour higher $\dm$ values.  
The  allowed range of $\dm$  in low-LMA region 
also 
narrows  down considerably.
The best-fit points from combined analysis of solar 
with 162 Ty and 766.3 Ty \kl data are found 
respectively to be 
\\
$\bullet$ 
$\dm$ = 7.2 $\times 10^{-5}$eV$^2$,
$\sss$ = 0.29,
\\
$\bullet$
$\dm$ = 8.3 $\times 10^{-5}$eV$^2$,
$\sss$ = 0.27. \\
The $\theta> \pi/4$ ({Dark-Side})  solutions 
allowed by \kl spectrum data get disallowed by
the solar data as the matter effect in sun break the 
$\theta \rightarrow \pi/2 -\theta$ symmetry (cf. fig. \ref{prob}). 

In fig \ref{delchi} we plot $\Delta \chi^2 =\chi^2 -\chi^2_{min}$  
vs $\dm$ (right panel) and vs. $\sss$ (left panel).    
\begin{figure}
\vskip -0.2cm
\includegraphics[width=9cm,height=5.8cm,clip=]{del.eps}
\caption{\label{delvs12}
$\Delta \chi^2 $ vs \dm($\sss$)
marginalised over the other parameters. 
}
\label{delchi}
\end{figure} 
This figure shows that  with 
increased
statistics
\kl data constrains  
{$\Delta m^2_{21}$} more and more sharply.
However 
%further {constrained} by {KamLAND}. 
{$\sss$} is not constrained as  much. 
Maximal mixing is found to be more  disfavoured ($> 5\sigma$) 
by the new \kl data because of increased precision.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vskip 1.0cm
\begin{figure}
\includegraphics[width=16pc,clip=]{lma.eps}   
\caption{\label{prob} 
The probabilites for solar and \kl as a function of energy}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The figure \ref{prob} shows the probabilities  
vs energy 
for KL (at an average distance of 180 km)
and solar neutrinos for  \dm =$ 7\times10^{-5}$ eV$^2$
and $\sin^2\theta_{12}$ =0.3.  
For these parameters 
the high energy  $^8B$  solar neutrinos 
undergo adiabatic MSW 
transition  with $P_{ee} \approx \sin^2\theta$. 
For low energy solar neutrinos  $P_{ee}  \approx 1 - 0.5 \sin^22\theta$.
Thus with decrease in energy the survival probability increases for 
$\theta < \pi/4$ causing  an upturn in the observed energy spectrum.  
%the absence of which  may indicate presence of extra sterile neutrinos 
%\cite{smirnov}. 
Whereas for the solar probabilities the \dm dependence is completely averaged
out in KL
the probability exhibits a $L/E$ dependence which gives
it an unprecedented sensitivity  to  \dm.

The  
imprint of MSW effect \cite{msw} in Sun in the 
solar and \kl data can be quantified by parametrising  
the matter induced potential in the sun, 
{$V_{MSW} = \sqrt{2} G_F n_e$} , as
$\alpha_{MSW}.{V}$ \cite{lisimsw}.
$\alpha_{MSW}=0$ corresponds to no MSW effect
while $\alpha_{MSW}=1$ corresponds to full MSW effect.
Analysis performed in \cite{lisimsw}  with solar 
and 162 Ty \kl data shows that $\alpha_{MSW}=0$
case is rejected at 5.6$\sigma$ w.r.t the minimum which comes at 
$\alpha_{MSW}$ =1 i.e for standard MSW effect.     
However the allowed range of $\alpha_{MSW}$ 
was found to be large. 
Increased statistics from KL can put stronger constraints
on $\alpha_{MSW}$ and can provide
more precise test of {MSW} {and} "new" physics beyond MSW.
\\ \\
\underline{\bf Impact of each solar experiment}:
\begin{figure}
\includegraphics[width=8cm,height=6cm,clip=]{all-1_newkl.eps}
\caption{\label{all-1}
The allowed regions in $\dm$-$\sin^2\theta_{12}$ plane,
obtained by removing the data from
one solar neutrino experiment from the global fit.
These figures are drawn on a linear scale for $\dm$.
}
\end{figure}
In Fig. \ref{all-1} we present the allowed
regions obtained by taking out
the data from one solar neutrino experiment
from the global data set.
Taking out the SNO data allows 
smaller values of $f_B$,
and hence larger values of
of $\sin^2\theta_{12}$  ($P_{ee} \sim f_B \sin^2\theta_{12}$).
With 162 Ty \kl data excluding SNO  
allowed maximal mixing, Dark side
and higher $\Delta m^2$ solutions.
With the spectral distortion observed in the 766.3 Ty \kl data these 
regions are ruled out with increased confidence even without SNO.
\\ \\
\underline{\bf On the precision of $\nu_{\odot}$
oscillation parameters}:
\begin{figure}[htb]
\vskip -0.5cm
\includegraphics[width=6cm,height=6cm,clip=]{ppfig.ps}
\vskip -0.5cm
\caption{Allowed area  including  the rates from 
a generic pp neutrino experiment
with different errors. Figure taken from \cite{Bahcall:2003ce}}.
\label{ppfig}
\end{figure}
\begin{table*}[htb]
\caption{
3$\sigma$ allowed ranges and \% spread  of $\Delta m^2_{21}$ and
$\sin^2\theta_{12}$ obtained from 2 parameter plots. 
}
\label{spread}  
\newcommand{\m}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\tabcolsep}{2pc} % enlarge column spacing
\renewcommand{\arraystretch}{1.2} % enlarge line spacing
\begin{tabular}{@{}lllll} 
%\begin{tabular}{ccccc}
\hline
{Data set} & (3$\sigma$)Range of & (3$\sigma$)spread in  &
(3$\sigma$) Range of
&(3$\sigma$) spread in \cr
{used} & $\Delta m^2_{21}$ eV$^2$
& {$\Delta m^2_{21}$} & $\sin^2\theta_{12}$
& {$\sin^2\theta_{12}$} \cr
\hline
{only sol} & 3.0 - 17.0
&{70\%} & $0.21-0.39$ &30\%\cr
{sol+162 Ty KL}& 4.9 - 10.7
& 37\%
& $ 0.21-0.39$ & 30\%  \cr
{sol+ 766.3 Ty KL}& 7.2 - 9.5
& 14\% &
$0.21-0.37$ & 27\% \cr
\hline
\end{tabular} \\[2pt]
\end{table*}
In  Table \ref{spread} we present the  3$\sigma$
allowed ranges of $\Delta m^2_{21}$
and $\sin^2\theta_{12}$, obtained
using different data sets.
We also show the uncertainty in the value of the
parameters through a quantity ``spread'' which we define in general as
\be
{\rm spread} 
= \frac{ \Delta m^2(\sin^2\theta)_{max} - \Delta m^2(\sin^2\theta)_{min}}
{\Delta m^2(\sin^2\theta)_{max} + \Delta m^2(\sin^2\theta)_{min}}
%\times 100 ~\%
\ee
Table 1 illustrates the  
remarkable sensitivity of  \kl in reducing the uncertainty in
$\Delta m_{21}^2$.
But {$\theta_{12}$} is not constrained much better than the current
set of solar experiments. 
The reason for this 
is  the average energy and distance in \kl corresponds to 
a Survival Probability MAXimum
(SPMAX) i.e $\sin^2(\dm L/4E) \approx  0$. 
This means that the coefficient
of the $\sin^22\theta_{12}$ term
in $P_{\bar{e}{e}}^{KL}$ is relatively small,
weakening the  sensitivity of  \kl  to 
%reach
%high precision in the determination of 
$\theta_{12}$.
The precision in $\theta_{12}$ can be improved
by reducing the baseline length such that one gets
a minimum of the
$\bar{\nu}_e$ survival probability (SPMIN) 
where
$\sin^2(\dm L/4E)= 1$. 
This corresponds to a distance 
$L = 1.24  (\mathrm E/MeV)
(\mathrm eV^2/\Delta m^2_{12})$ m.
For the low-LMA solution region
and the average energy of neutrinos in
\kl experiment,
this corresponds to a
distance of approximately (50 - 70) km \cite{th12}.
For an experiment with a 70 km baseline and 24.3 GW
reactor power
%, corresponding to the Kashiwazaki
%(Daya Bay) complex in Japan (China),
$\sss$ can be determined with 
$\sim 10\%$ error at 99\% C.L. with a 3 kTy 
%(4 kTy)
statistics \cite{th12}.

Improved sensitivity in the measurement of $\theta_{12}$ 
is possible from LowNu experiments \cite{lownu}.
The {pp} flux is known with {1\%}
accuracy from Standard Solar Models as compared to 
the uncertainties of $^{8}{B} \sim$ {20\%} and $^{7}{Be} \sim$ {10\%}. 
At the low energies relevant for pp neutrinos 
$P_{ee}
\approx 1 - \frac{1}{2}\sin^22\theta$
resulting in a heightened sensitivity to $\theta_{12}$.
In \ref{ppfig} we show the allowed region  \cite{Bahcall:2003ce} in 
$\Delta m^2 -\tan^2 \theta_{12}$ plane including fictitious results 
from a generic pp scattering experiment. 
The figure shows that 
precision in $\theta_{12}$ increases with reduced error in pp 
flux measurment. 
%\begin{figure}[htb]
%\vskip -0.5cm
%\includegraphics[width=6cm,height=6cm,clip=]{ppfig.ps}
%\vskip -0.5cm
%\caption{Allowed area  including  fictitious   pp neutrino experiment
%with different errors \cite{Bahcall:2003ce}}
%\label{ppfig}
%\end{figure}

The recently proposed Gd loaded SK detector  
can measure $\dm$ with
$\sim 1\%$ and $\sss$ with $\sim 15\%$ uncertainty at 99\% C.L.
after 5 years of data taking
\cite{skgd}.

\begin{figure}[htb]
\includegraphics[width=18pc]{F-atm+k2k-3flux.eps}
\caption{The allowed regions from SK (line contours) and SK+K2K 
(shaded contours)
analysis from a
two flavour analysis.
Also shown is the  $\Delta \chi^2$ vs
$\Delta m^2_{atm}$ and $\sin^2\theta_{atm}$  marginalised 
w.r.t the undisplayed parameter\cite{maltoni}}.
\label{sk+k2k}
\end{figure}
\section{Atmospheric Neutrino Oscillation Parameters}
%\begin{figure}
%\includegraphics[width=18pc]{atm_2gen.eps} 
%\end{figure}
In this section we present the constraints  on atmospheric neutrino 
oscillation 
parameters $\Delta m^2_{32} \equiv \Delta m^2_{atm}$ and 
$\sin^2\theta_{23} \equiv \sin^2\theta_{atm}$ from
two flavour oscillation analysis 
 SK and SK+K2K 
data. 
\\ \\
\underline{\bf Allowed parameters from SK+K2K analysis}:
%\begin{figure}
%\includegraphics[width=18pc]{F-atm+k2k-3flux.eps}
%\caption{The allowed regions from SK and SK+K2K analysis from a
%two flavour analysis.
%Also shown are the behaviour of $\Delta chi^2$ vs
%$\Delta m^2_{atm}$ and $\sin^2\theta_{atm}$ \cite{maltoni}}
%\label{sk+k2k}
%\end{figure}
In 2003 the SK collaboration reanalyzed their data and reported a 
downward shift of the allowed $\Delta m^2_{atm}$ from a two-generation 
analysis
% \cite{SKatm03} 
due to incorporation of 
three-dimensional Honda atmospheric fluxes and revised cross-section and
efficiencies. 
The best-fit value of $\Delta m^2_{atm}$ from 
SK zenith angle data reported in this conference is 
 2.1$\times 10^{-3}$
eV$^2$ \cite{kearns04}.
Analysis of SuperKamiokande data
by two groups confirm this downward shift of $\Delta m^2_{atm}$
\cite{Gonzalez-Garcia:2004it,maltoni}. 

%The K2K experiment in Japan using accelerator neutrinos 
%$E_\nu \sim$ 1.3 GeV  and a source-detector distance 
%of $\sim$ 250 km 
%has also 
%reported a positive evidence of oscillation by observing disappearance   
%of muon neutrinos \cite{k2k}
The K2K  experiment which is sensitive to 
atmospheric neutrino oscillation parameters  
gives the best-fit as 
$\Delta m^2_{atm}$ = 2.8 $\times 10^{-3}$ eV$^2$ 
$\sin^22\theta_{atm}=1.0$.
% which is in the right ball park of 
%atmospheric neutrino oscillation parameters. 

In fig. \ref{sk+k2k} we show the allowed regions in 
$\Delta m^2_{atm}$ - $\sin^2\theta_{atm}$  plane from 
analysis of SK data as well as from combined analysis of 
SK+K2K data  
\cite{maltoni}.
The best-fit from the combined analysis corresponds to \\
$\bullet$
$\Delta m^2_{atm}$ = 2.3 $\times 10^{-3}$ eV$^2$, $\sin^2\theta_{atm}
= 0.5$. \\
Higher $\Delta m^2$ values are seen to be constrained
by K2K 
but $\theta_{atm}$ is not constrained any better. 
For a two generation analysis the relevant probability for 
both SK and K2K are 
vacuum oscillation probability and the allowed regions are symmetric about 
$\theta_{23}$ and $\pi/2 -\theta_{23}$. 

Recently SuperKamiokande has 
reported the observation of the first oscillation minima
in the $L/E$ distribution
\cite{ishi} .
The observation of this first dip disfavours the neutrino decay
and de-coherence solutions. 
The best-fit value of $\Delta m^2_{atm}$ from this analysis is 
2.4 $\times 10^{-3}$ eV$^2$ with 
$\sin^22\theta_{atm}$ = 1.0. 
The allowed ranges of parameters from the two different analyses are: 
\\
{$\bullet$}{90\% C.L. range}  (SK L/E) \\
{$\Delta m^2_{atm}$= {1.9} - 3.0 $\times 10^{-3}$ eV$^2$,
$\sin^22\theta_{atm} >$ 0.9 \\
$\bullet$
90\% C.L. range   (SK Zenith) \\
{$\Delta m^2_{atm}$= 1.3 - 3.0 $\times 10^{-3}$ eV$^2$,
$\sin^22\theta_{atm} >$ 0.9
\\ \\
\underline{\bf 
Precision of atmospheric neutrino oscillat-}
\underline{\bf ion parameters}:
The spread 
in $\Delta m^2_{atm}$ is 39\%    from SK zenith data 
whereas it is  22\%  from SK L/E data. 
Thus improved precision in {$\Delta m^2_{atm}$} is obtained 
with L/E data. Range of {$\sin^2\theta_{atm}$} however 
remains  unchanged: 
{$\delta(\sin^22\theta_{23}) \sim 5\%$. 
However  
$\delta(\sin^2\theta_{23}) \sim 32\%$ because   
{$\sin^2\theta_{23}$} precision is worse than
{$\sin^22\theta_{23}$} precision near maximal mixing. 
Increased statistics in
$L/E$ data can improve  
$\Delta m^2_{32}$ uncertainty to 
$\sim$ 10\%  at 90\% C.L. in  20 years of operation of SK \cite{kajitanoon}. 
This is sensitive to the {{true $\Delta m^2_{32}$}} chosen 
and the above precision is achieved  
for {$\Delta m^2_{32}$=2.5 $\times 10^{-3}$ eV$^2$}.
But precision in $\sin^2\theta_{23}$ does not improve
with increased statistics \cite{kajitanoon}. 
{Large Magnetized Iron} calorimeters 
for atmospheric neutrinos (e.g MONOLITH,INO)
\cite{monolith,ino} can improve the precision in both atmospheric 
parameters.  

\section{Three Flavour Oscillation}
The oscillation parameters for a three flavour analysis are 
two mass  squared   differences
$\Delta m^2_{21} = \Delta m^2_\odot$,~~~~
$\Delta m^2_{31} = \Delta m^2_{CHOOZ}$
$\simeq \Delta m^2_{atm} =
\Delta m^2_{32}$  
and three mixing angles which are usually denoted as  
$\theta_{12}$,$\theta_{23}$ and $\theta_{13}$  with the mixing matrix  U  
parameterized   
in the standard MNS form. 
\be
% U = 
%& = &
% R_{23}R_{13}R_{12}  \nonumber \\ &= &
\begin{pmatrix}
{c_{13}c_{12} &
s_{12}c_{13} & s_{13} \cr -s_{12}c_{23} - s_{23} s_{13} c_{12} &
c_{23} c_{12} - s_{23} s_{13} s_{12} & s_{23} c_{13} \cr s_{23}
s_{12} - s_{13} c_{23} c_{12} & -s_{23} c_{12} - s_{13} s_{12}
c_{23} & c_{23} c_{13} \cr
}
\end{pmatrix}
\nonumber
\ee
%where we have neglected the 
%CP violation phases.
In the limit 
$\Delta m^2_{21} << \Delta m^2_{32}$ as is indicated by the two 
generation analysis of atmospheric 
and solar neutrino data.
In this limit the   
atmospheric  probabilites depend on $\Delta m^2_{32}$,
$\theta_{13}$, $\theta_{23}$ and 
solar neutrino probabilites depend on
$\Delta m^2_{21}$, $\theta_{12}$, 
$\theta_{13}$. The CP violation phases  
can be neglected in this limit. 
Clearly for 
$\theta_{13} =0$ solar and atmospheric
neutrinos decouple. 
The probability for the 
CHOOZ reactor experiment depends on $\Delta m^2_{31}$ and 
$\theta_{13}$. 
\\ \\
%%%%%%%%%%%%%%%%%%
\begin{figure}
\includegraphics[width=6cm,height=6cm,clip=]{ch.eps}
\caption{Allowed area in $\Delta m^2_{31} - \tan^2\theta_{13}$ 
plane from CHOOZ data. 
The shaded regions correspond to the allowed range of 
$\Delta m^2_{31}$ using SK zenith angle (L/E) and K2K 
data \cite{Fogli:2003am,lisilbye}. }
\label{chooz} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\underline{\bf Bound on $\sin^2\theta_{13}$}:
CHOOZ/PaloVarde experiment has put stringent bounds on
$\theta_{13}$ from  non-observation of $\bar{\nu_e}$ 
disappearance \cite{choozref}. 
In fig \ref{chooz} we show the allowed area from CHOOZ experiment 
in $\Delta m^2_{31} - \tan^2\theta_{13}$ plane. 
The  area shaded by vertical blue (slanting red) lines 
correspond to the allowed range of $\Delta m^2_{31}$ 
from two generation analysis of SK L/E (zenith) + K2K
atmospheric neutrino data \cite{Fogli:2003am,lisilbye}. 
The figure shows that 
the {$\theta_{13}$} bound from CHOOZ 
depends sensitively on {$\Delta m^2_{31}$}.
Stronger bounds are obtained for  higher {$\Delta m^2_{31}$}.  
%This figure includes the effect of the $\Delta m^2_{21}$ induced 
%interference terms \cite{BNPChooz}.

\begin{figure}
\includegraphics[width=6cm, height=6cm, clip=]{del3g.eps}
\caption{\label{delchith13}
Bounds on the mixing angle $\theta_{13}$ from 
data combination. Also shown is the 3$\sigma$ $\Delta\chi^2$  limit for 
1 parameter. 
}
\end{figure}
In figure \ref{delchith13} we show the 
limits on $\theta_{13}$ obtained from   
$\Delta \chi^2$ vs 
$\theta_{13}$ plots. 
We 
marginalise over all other parameters.  
We let $\Delta m^2_{31}$ vary freely 
within the 3$\sigma$ range  obtained using the one parameter 
$\Delta \chi^2$ vs $\Delta m^2_{31}$ fit as given in the latest SK 
analysis \cite{kearns04}. 
The bounds that we get are \\
$\bullet$ $\sin^2{\theta_{13}} < 0.071 $ (CHOOZ+atm) \\
$\bullet$ $\sin^2{\theta_{13}} < 0.063 $ (sol+\kl(162)+CHOOZ+atm){\footnote{
%Note that the bound on $\sin^2\theta_{13}$ depends sensitively on the range
%of $\Delta m^2_{31}$ choosen.
Using the rage of $\Delta m^2_{31}$  from the SK+K2K analysis of
\cite{Fogli:2003am} bound comes as
{$\sin^2\theta_{13} < 0.077$}
\cite{snosaltus}}\\ 
$\bullet$ $\sin^2{\theta_{13}} < 0.05$ (sol+\kl(766.3)+CHOOZ+atm) \\
A non-zero $\theta_{13}$ prefers higher $\dm$  values which is disfavoured 
by the solar+\kl data. Because of this the combined $\chi^2$ increases 
making the bound on $\theta_{13}$ stronger than compared to 
only CHOOZ+atmospheric. Since the recent \kl data disfavours 
high $\dm$ values to a greater extent the bound on $\theta_{13}$ further 
improves by including it. 

In \ref{cont3g} we show the effect of  non-zero $\sin^2\theta_{13}$ on 
the allowed area in solar neutrino parameter space. 
The presence of a small non-zero $\theta_{13}$ can
improve the fit in the regions of
the parameter space with higher values of $\dm$  \cite{snosaltus},
i.e., in the high-LMA zone. But due to increased discord between 
the new \kl data  and high $\dm$ values 
the
high-LMA region now gets excluded at more than
 3$\sigma$ even in the presence of a
third generation in the mixing,
indicating the robustness of the low-LMA solution.
The allowed atmospheric regions in $\Delta m^2_{31} - 
\sin^2 2\theta_{23}$ plane also remains stable in presence of 
non-zero $\theta_{13}$ \cite{maltoni}. 
Including the $\Delta m^2_{21}$ terms in the analysis one can give 
bound on the 
hierarchy  parameter $\alpha = \Delta m^2_{21}/\Delta m^2_{31}$
associated with subleading oscillations  for atmospheric neutrinos 
as well as for long baseline studies. 
The best-fit for this comes as  
$\alpha$ = 0.03
\cite{maltoni}.
\begin{figure}[htb]
\vskip -0.2cm
\includegraphics[width=6cm,height=6cm,clip=]{cont3g.eps}
\caption{\label{solkl3osc}
Same as in figure 1 but from
a three flavour analysis of the
global solar and reactor neutrino data and plotted on a linear scale for $\dm$.
}
\label{cont3g}
\end{figure}

\newpage
\vspace{-0.3cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{}
%%%%%%%%%%%%%%%%%%%%%%%%%%



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% International Conference, June 14-19, 2004, Paris, France.
%
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% International Conference, June 14-19, 2004, Paris, France.
%
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% International Conference, June 14-19, 2004, Paris, France.
%
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%
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%
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% % \bibitem{sno2}
% %\bibitem{Ahmed:2003kj}
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% nucl-ex/0309004.
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%
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% International Conference, June 14-19, 2004, Paris, France.
%
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% hep-ex/0406035.
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% International Conference, June 14-19, 2004, Paris, France.


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% \bibitem{FM}
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% \bibitem{JNieves02} Private communication by J.F.~Nieves.
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% Phys.\ Rev.\ D {\bf 68}, 093009 (2003); 
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%talk given at $\nu$'04 
% International Conference, June 14-19, 2004, Paris, France.

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%talk given at $\nu$'04 
% International Conference, June 14-19, 2004, Paris, France.

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% International Conference, June 14-19, 2004, Paris, France.

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%talk given at $\nu$'04 
% International Conference, June 14-19, 2004, Paris, France.

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%talk given at $\nu$'04 
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%
% \bibitem{Morales02} A. Morales and J. Morales, hep-ph/0211332.
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%
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% C.\ Weinheimer \textit{et al.},  
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% \bibitem{PPSNO3bb} S. Pascoli and S.T. Petcov,
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% \bibitem{PPRSNO2bb} S. Pascoli, S.T. Petcov and
% W. Rodejohann, {\em Phys. Lett.} {\bf B558} (2003) 141.
%%CITATION = HEP-PH 0212113;%%
% hep-ph/0212113.

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%            S. Pascoli and  S.T. Petcov, hep-ph/0111203.
%%CITATION = HEP-PH 0110287;%%
%%CITATION = HEP-PH 0111203;%%

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%%CITATION = HEP-PH 9311204;%%

\bibitem{SPNobSymp04} S. Pascoli and S.T. Petcov, 
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% {\em Phys. Lett.} {\bf B580} (2004) 280.
%%CITATION = HEP-PH 031XXX;%%

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\bibitem{NEMO3} S. Sarazin, these Proceedings.

\bibitem{CUORI} E. Fiorini, these Proceedings.

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\bibitem{MShae04} K. McConnel,
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H. Minakata {\it et al.}, hep-ph/0211111.

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%%CITATION = HEP-PH 0404103;%%

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\bibitem{SKGdBV04} J. Beacom, M. Vagins, hep-ph/0309300.
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% hep-ph/0309300.
%%CITATION = HEP-PH 0309300;%%

\bibitem{TH12} A. Bandyopadhyay {\it et al.}, hep-ph/0302243.
% A.~Bandyopadhyay, S.~Choubey and S.~Goswami,
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%[arXiv:hep-ph/0302243].
%%CITATION = HEP-PH 0302243;%%
\bibitem{HMinth1204} H.~Minakata {\it et al.},
% H.~Nunokawa, W.~J.~C.~Teves and R.~Zukanovich Funchal,
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% Principles, accuracies and physics potentials,''
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%%CITATION = HEP-PH 0407326;%%


% \end{document}

% \newpage

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A.~Bandyopadhyay {\it et al.},
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\bibitem{snosaltus}
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%%CITATION = HEP-PH 0309174;%%

%\cite{Bandyopadhyay:2004da}
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A.~Bandyopadhyay
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%, S.~Choubey, S.~Goswami, S.~T.~Petcov and D.~P.~Roy,
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%%CITATION = HEP-PH 0406328;%%

%\cite{Bandyopadhyay:2002en}
%\bibitem{Bandyopadhyay:2002en}
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A.~Bandyopadhyay  {\it et al.},
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%%CITATION = HEP-PH 0212146;%%
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% S.~Choubey, R.~Gandhi, S.~Goswami and D.~P.~Roy,
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J.\ Phys.\ G {\bf 29}, 2465 (2003). 

%\bibitem{smirnov} 
%\cite{deHolanda:2003tx}
%\bibitem{deHolanda:2003tx}
%P.~C.~de Holanda and A.~Y.~Smirnov,
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%%CITATION = HEP-PH 0307266;%%

 
%\cite{Fogli:2003vj}
%\bibitem{Fogli:2003vj}
\bibitem{lisimsw}
G.~L.~Fogli {\it et al.}, 
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%%CITATION = HEP-PH 0309100;%%

\bibitem{lownu}See for e.g. the talk by Y. Suzuki, Neutrino 2004. 

\bibitem{th12}
%\cite{Bandyopadhyay:2003du}
%\bibitem{Bandyopadhyay:2003du
A.~Bandyopadhyay, S.~Choubey and S.~Goswami,
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%%CITATION = HEP-PH 0302243;%%

%\cite{Bahcall:2003ce}
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J.~N.~Bahcall and C.~Pena-Garay,
%``Global analyses as a road map to solar
% neutrino fluxes and oscillation  parameters,''
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%[arXiv:hep-ph/0305159].
%%CITATION = HEP-PH 0305159;%%

\bibitem{skgd}
%\bibitem{gadzooks}
%\cite{Beacom:2003nk}
%\bibitem{Beacom:2003nk}
J.~F.~Beacom and M.~R.~Vagins,
%``GADZOOKS! Antineutrino spectroscopy with large water Cherenkov detectors,''
hep-ph/0309300.
%%CITATION = HEP-PH 0309300;%%
%\cite{Choubey:2004bf}
%\bibitem{Choubey:2004bf}
S.~Choubey and S.~T.~Petcov,
%``Reactor anti-neutrino oscillations and gadolinium loaded Super-Kamiokande
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hep-ph/0404103. 
%(to appear in Phys. Lett B).
%%CITATION = HEP-PH 0404103;%%

%\bibitem{SKatm03}
%Y. Hayato \textit{et al.}, Talk given at the Int. EPS Conference
%on High Energy Physics, July 17 - 23, 2003, Aachen, Germany.

\bibitem{kearns04}E. Kearns, talk at Neutrino 2004, Paris.

%\cite{Gonzalez-Garcia:2004it}
\bibitem{Gonzalez-Garcia:2004it}
M.~C.~Gonzalez-Garcia and M.~Maltoni,
%``Status of global analysis of neutrino oscillation data,''
hep-ph/0406056.
%%CITATION = HEP-PH 0406056;%%

\bibitem{maltoni}
%\bibitem{Maltoni:2004ei}
M.~Maltoni {\it et al.},
% T.~Schwetz, M.~A.~Tortola and J.~W.~F.~Valle,
%``Status of global fits to neutrino oscillations,''
hep-ph/0405172.
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\bibitem{ishi}
Y.  Ashie et al.,
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\bibitem{kajitanoon}
T. Kajita, Talk at NOON04. 

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%\href{http://www.slac.stanford.edu/spires/find/hep/www?r=lngs-p26-2000}{SPIRES entry}
 
\bibitem{ino}
See http://www.imsc.res.in/$\sim$ino.
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% \bibitem{PaloV}
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% and
%{\em Phys. Rev.} {\bf D62} (2000) 072002.
%%CITATION = HEP-EX 9912050;%%
%%CITATION = HEP-EX 0003022;%%

%\cite{Fogli:2003am}
\bibitem{Fogli:2003am}
The $\Delta m^2_{31}$ range from SK zenith +K2K analysis is taken from  
G.~L.~Fogli {\it et al.}
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Phys.\ Rev.\ D {\bf 69}, 017301 (2004).
%%CITATION = HEP-PH 0308055;%%

\bibitem{lisilbye}The $\Delta m^2_{31}$ range from SK L/E +K2K
analysis is provided by  
G.~L.~Fogli, E.~Lisi, A.~Marrone, A.~Palazzo in a private communication. 

%\bibitem{BNPChooz} S.M. Bilenky, D. Nicolo and S.T. Petcov,
%{\em Phys. Lett.} {\bf B538} (2002) 77;
%%%CITATION = HEP-PH 0112216;%%.
%A.~Bandyopadhyay {\it et al.},
% S.~Choubey, S.~Goswami and K.~Kar,
%``Three generation neutrino oscillation parameters after SNO,''
%Phys.\ Rev.\ D {\bf 65} (2002) 073031.


\end{thebibliography}

\end{document}
 

hep-ph0412410.extracted_bib
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%

\bibitem{H3} K. Eitel, 
% these Proceedings.
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\bibitem{NEMO3} S. Sarazin, 
% these Proceedings.
Talk given at $\nu$'04 
International Conference, June 14-19, 2004, Paris, France.

\bibitem{CUORI} E. Fiorini, 
% these Proceedings.
Talk given at $\nu$'04 
International Conference, June 14-19, 2004, Paris, France.

\bibitem{bb0nuNME} J. Suhonen, 
% these Proceedings.
Talk given at $\nu$'04 
International Conference, June 14-19, 2004, Paris, France.

\bibitem{TMU04} P. Huber {\it et al.}, hep-ph/0403068.

\bibitem{MShae04} K. McConnel and
M.H. Shaevitz, hep-ex/0409028.

\bibitem{Reactor04} L. Oberauer, 
% these Proceedings. 
Talk given at $\nu$'04 
International Conference, June 14-19, 2004, Paris, France.

\bibitem{Reacth13} L.A. Mikhaelyan {\it et al.}, hep-ex/9908047
and hep-ex/0211070.

\bibitem{Reacth13J} H. Minakata {\it et al.}, hep-ph/0211111.

\bibitem{BCGPTH1204} A. Bandyopadhyay {\it et al.}, hep-ph/0410283.

\bibitem{SKGdCP04} S. Choubey and S.T. Petcov,
%``Reactor anti-neutrino oscillations 
% and gadolinium loaded Super-Kamiokande detector,''
% hep-ph/0404103. 
Phys. Lett {\bf B594} (2004) 333.
%%CITATION = HEP-PH 0404103;%%

\bibitem{SKGdBV04} J. Beacom and M. Vagins, hep-ph/0309300.
%``GADZOOKS! Antineutrino spectroscopy 
%with large water Cherenkov detectors,''
% hep-ph/0309300.
%%CITATION = HEP-PH 0309300;%%

\bibitem{TH12} A. Bandyopadhyay {\it et al.}, 
% hep-ph/0302243.
% A.~Bandyopadhyay, S.~Choubey and S.~Goswami,
%``Exploring the sensitivity of 
% current and future experiments to  Theta(odot),''
Phys.\ Rev.\ D {\bf 67}, 113011 (2003).
%[arXiv:hep-ph/0302243].
%%CITATION = HEP-PH 0302243;%%

\bibitem{HMinth1204} H. Minakata {\it et al.},
% H.~Nunokawa, W.~J.~C.~Teves and R.~Zukanovich Funchal,
%``Reactor measurement of theta(12): 
% Principles, accuracies and physics potentials,''
hep-ph/0407326.
%%CITATION = HEP-PH 0407326;%%

\bibitem{JBSP203} J. Bernabeu, 
% hep-ph/0305152;
S.~Palomares-Ruiz and S.T. Petcov,
Nucl. Phys. B {\bf 669} (2003) 255;
%%CITATION = HEP-PH 0305152;%%
S. Palomares-Ruiz and S.T. Petcov, hep-ph/0406096. 

\bibitem{PDeg} J. Burguet-Castell {\it et} {\it al.}, hep-ph/0103258;
V. Barger {\it et} {\it al.}, hep-ph/0112119;
H. Minakata {\it et} {\it al.}, hep-ph/0301210;
O. Yasuda, hep-ph/0405005.

\end{thebibliography}