%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[prl,twocolumn,showpacs,byrevtex]{revtex4}
%\documentclass[12pt]{article}
\input epsf
\usepackage{graphicx}
\usepackage{dcolumn}
\usepackage{bm}
\def\UU{Y_U^\dagger Y_U}
\def\DD{Y_D^\dagger Y_D}
\def\EE{Y_E^\dagger Y_E}
\def\dthodt{d\theta_{23}\over dt}
\def\dthtdt{d\theta_{13}\over dt}
\def\dththdt{d\theta_{12}\over dt}
\def\dsodt{ds_{23}\over dt}
\def\dstdt{ds_{13}\over dt}
\def\dsthdt{ds_{12}\over dt}
\def\uoo{c_{13}c_{12}}
\def\uot{c_{13}s_{12}}
\def\uoth{s_{13}}
\def\uto{-c_{23}s_{12}-c_{12}s_{13}s_{23}}
\def\utt{c_{12}c_{23}-s_{12}s_{13}s_{23}}
\def\utth{c_{13}s_{23}}
\def\utho{s_{12}s_{23}-c_{12}s_{13}c_{23}}
\def\utht{-c_{12}s_{23}-c_{23}s_{13}s_{12}}
\def\uthth{c_{13}c_{23}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\ba{\begin{eqnarray}}
\def\ea{\end{eqnarray}}
\def\br{\begin{array}}
\def\er{\end{array}}
\def\dmdt{16\pi^2{dm\over dt}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\ba{\begin{eqnarray}}
\def\ea{\end{eqnarray}}
\def\br{\begin{array}}
\def\er{\end{array}}
\def\Dmott{\Delta m^2_{12}}
\def\Dmttht{\Delta m^2_{23}}
\def\Dmotht{\Delta m^2_{13}}
\begin{document}
%\preprint{}
\title{\Large High Scale Mixing Unification and Large Neutrino Mixing
Angles}
\author{R. N. Mohapatra}
\email{rmohapat@physics.umd.edu}
\affiliation{Department of Physics, University of Maryland,
College Park, MD 20742, USA.}
\author{M. K. Parida}
\email{mparida@sancharnet.in}
\affiliation{Department of Physics, North Eastern Hill University,
Shillong 793022, India.}
\author{G. Rajasekaran}
\email{graj@imsc.ernet.in}
\affiliation{Institute of Mathematical Sciences, Chennai 600 113, India.}
\begin{abstract}
Starting with the hypothesis that quark and lepton mixings are
identical at or near the GUT scale, we show that the large solar and
atmospheric neutrino mixing angles together with the small reactor angle
$U_{e3}$
can be understood purely as a result of renormalization group evolution.
The only requirements are that the three neutrinos must be quasi
degenerate in mass and have same CP parity. It predicts that the common
Majorana mass for the neutrinos must be larger than $0.1$ eV making the
idea testable in the currently planned or ongoing 
experiments searching for neutrinoless-double-beta decay.
\end{abstract}
\date{\today}
\pacs{14.60.Pq, 11.30.Hv, 12.15.Lk}
%\input epsf
%\def\lsim{\raise0.3ex\hbox{$\;<$\kern-0.75em\raise-1.1ex\hbox{$\sim\;$}}}
%\def\gsim{\raise0.3ex\hbox{$\;>$\kern-0.75em\raise-1.1ex\hbox{$\sim\;$}}}
\rightline{}\rightline{UMD-PP-03-038;~~NEHU-PHY-MP-02-03;~~
IMSc/2003/01/02}
%
\maketitle
%\section{INTRODUCTION}
%\label{sec1}
The idea that disparate physical parameters describing forces and matter
at low
energies may unify at very short distances (or high mass scales) has been
a very helpful tool in seeking a unified understanding of apparently
unrelated phenomena \cite{ref0}. In the context of supersymmetric grand
unifed theories, such an approach explains the weak mixing angle
$\sin^2\theta_W$ and thereby the different strengths of the weak,
electromagnetic and strong forces. One of the key ingredients of the grand
unified theories is the unification between quarks and leptons. One may,
therefore, hope that in a quark-lepton unified theory, the weak interaction
properties of quarks and leptons parameterized by means of the flavor
mixing matrices will become identical at high energies. 

On the
experimental side, recent measurements on atmospheric and solar neutrino
fluxes and those at K2K and KamLAND which are a manifestation of the
phenomena of
neutrino oscillations suggest that two of the neutrino mixings i.e.
the mixings between
$\nu_e- \nu_{\mu}$ and  $\nu_{\mu}-\nu_{\tau}$
(to be denoted by $\theta_{12}$ and $\theta_{23}$, respectively)
are large \cite{ref1,ref2,ref3,ref4,ref5} while the third mixing
between the $\nu_e-\nu_{\tau}$ is bounded to be very small by the
CHOOZ-Palo Verde reactor
experiments i.e.  $\sin^2{2\theta_{13}}< 0.15$
\cite{ref6}. On the other hand, it is now quite well established that
all observed quark mixing angles are very small. One may therefore ask
whether there is any trace of quark lepton unification in the mixing
angles as we move to higher scales.

The first question in this connection is whether high scales have anything
to do with neutrino masses or it is purely a weak scale phenomenon. One of
the simplest ways to understand small neutrino masses is via the
seesaw mechanism \cite{seesaw} according to which the neutrino mixing
is indeed a
high scale phenomenon, the new high scale being that of
 the right handed neutrino masses ($M_R$) in an appropriate extension of
the standard model. Present data put the seesaw
scale $M_R$ very close to the conventional GUT scales. It is therefore
tempting to speculate whether quark and lepton mixing angles are indeed
unified at the GUT-seesaw scale. This would of course imply that 
all neutrino mixing angles at the high scale $M_R$ are very small whereas
at the weak scale two of them are known to be large. In this letter we
show that simple radiative correction effects embodied in the
renormalization group evolution of parameters from seesaw scale to the
weak scale can indeed provide a complete understanding of all neutrino
mixings at the weak scale, starting with very small mixings at the
GUT-seesaw scale. 

\par
The fact that renormalization group evolution from the seesaw
scale to the weak scale \cite{ref10,ref10a} can lead to drastic changes
in the magnitudes of the mixing angles was
pointed out in several papers \cite{ref10,ref11,ref12,ref13,ref14,ref14a}.
In particular, it was shown in \cite{ref11} that this dependence on
renormalization group
evolution can be exploited in simple seesaw extensions of the minimal
supersymmetric standard model (MSSM) to explain the large value of the
atmospheric mixing angle starting with a small mixing at the seesaw scale,
provided two conditions are satisfied: (i) the two neutrino-mass eigen
states have same CP and (ii) they are very nearly degenerate in mass. In
general, in gauge models that attempt to explain the large neutrino
mixings \cite{ref7}, one needs to make many assumptions to constrain the
parameters. In contrast, in this class of ``radiative magnification''
models \cite{ref11},~there is no need to invoke
special constraints on the parameters at high scales  beyond those needed
to guarantee the quasi-degeneracy. In fact  the main content of radiative 
magnification models is the quasi-degeneracy assumption and since the  
value of common
Majorana mass $m_0$ for all neutrinos
is required 
to be in sub-eV range ($\geq 0.1$ eV,), this assumption is 
experimentally testable in the ongoing neutrinoless double beta decay
searches \cite{ref12}. 

In this letter, we show that under the same
conditions for radiative magnification as just outlined, if we start with
the hypothesis that at the
seesaw scale the quark and neutrino mixings are unified to a common set of
values, i.e. the known extrapolated values of the well known CKM angles, 
after renormalization group evolution to the weak scale, we can obtain the
solar and the atmospheric mixing angles that are in agreement with
observations  without contradicting the CHOOZ-Palo Verde
bound on $\theta_{13}$.
 
This result has two important implications: (i) it would provide a very
simple and testable way to understand the observed large neutrino mixings
and (ii) if confirmed by the neutrinoless double beta decay experiments,
it would provide a strong hint of quark lepton unification at high scales. 
\par
%\section{RENORMALIZATION GROUP EQUATIONS FOR MASSES AND  MIXINGS}
%\label{sec2}
Our basic assumption will be a seesaw type model which will lead to equal
quark and lepton mixing angles at the seesaw scale as well as to a
quasi-degenerate neutrino spectrum (see later). We will
then follow the ``diagonalize and run'' procedure for the neutrino
parameters and use the RGEs directly for the physical observables,
namely, the mass eigenvalues $m_i$ and the mixing angles $\theta_{ij}$
($i,j=1,2,3$).  We also assume  the
neutrino mass eigenstates to possess the same CP and ignore CP
violating phases in the mixing matrix. Also for
 simplicity, we adopt the mass ordering among the
quasi-degenerate eigenstates to be of type $m_3\agt m_2\agt
 m_1$. The real $3\times 3$ mixing matrix is
parametrized as,
\par
\noindent
\be U=\left[\br{ccc}
\uoo&\uot&\uoth\\
\uto&\utt&\utth\\
\utho&\utht&\uthth
\er\right],\label{eq1}\ee
\par
\noindent
where $c_{ij}=\cos\theta_{ij}$ and $s_{ij}=\sin\theta_{ij} (i,j=1, 2,
3$). $U$
diagonalizes the mass matrix
$M$ in the flavor basis with $U^TMU={\rm diag}(m_1, m_2, m_3)$.
The RGEs for the mass eigen values  can be written as \cite{ref13, ref14}
\par
\noindent
\be{dm_i\over dt}=-2F_{\tau}m_iU_{\tau
i}^2-m_iF_u,\,\left(i=1,2,3\right).\label{eq2}\ee
\par
\noindent
For every $\sin\theta_{ij}=s_{ij}$, the corresponding RGEs are,
\par
\noindent
\ba\dsodt&=&-F_{\tau}{c_{23}}^2\left( 
-s_{12}U_{\tau1}D_{31}+c_{12}U_{\tau2}D_{32}
\right),\label{eq3}\\
\dstdt&=&-F_{\tau}c_{23}{c_{13}}^2\left( 
c_{12}U_{\tau1}D_{31}+s_{12}U_{\tau2}D_{32}
\right),\label{eq4}\\
\dsthdt&=&-F_{\tau}c_{12}\left(c_{23}s_{13}s_{12}U_{\tau1}
D_{31}-c_{23}s_{13}c_{12}U_{\tau2}D_{32}\right.\nonumber \\
&&\left.+U_{\tau1}U_{\tau2}D_{21}\right).\label{eq5}\ea
\par
\noindent
where $D_{ij}={\left(m_i+m_j)\right)/\left(m_i-m_j\right)}$
and,~for MSSM,
\par
\noindent
\ba F_{\tau}&=&{-h_\tau^2}/{\left(16\pi^2\cos^2\beta\right)},\nonumber\\
F_u&=&\left(1\over{16\pi^2}\right)\left({6\over5}g_1^2+6g_2^2-
6{h_t^2\over\sin^2\beta}\right),\label{eq6}\ea
\par
\noindent
%but, for SM,
%\par
%\noindent
%\ba F_{\tau}&=&{3h_\tau^2}/\left( 32\pi^2\right),\nonumber \\
%F_u&=&\left(3g_2^2-2\lambda-6h_t^2-2h_\tau^2\right)/\left(16\pi^2\right).
%\label{eq7}\ea
%\par
\noindent
When the mass difference between $m_i$ and $m_j$ tends to vanish,~$D_{ij}\to
\infty$, and the corresponding term in the RHS of (\ref{eq3})-(\ref{eq5})
predominantly drives the RG evolution for the mixing
angle which might become large or even approach its quasii-fixed point maximal
value anywhere between
$\mu=M_{\rm SUSY}-M_R$. Since $m_i$ and $m_j$ are scale dependent, the
initial difference existing between them at $\mu=M_R$ is narrowed down
during
the course of RG evolution as we approach $\mu=M_{\rm SUSY}$. This causes
$D_{ij}\to\infty$ and hence large magnification to the mixing angle due to
radiative effects. Also $F_\tau$ is enhanced by a factor $\sim 10^3$ in the
large $\tan\beta(\simeq 50$) region in the case of MSSM
as compared to the SM where such effects do not exist. Thus if the SUSY scale
is significantly larger  with $M_Z < M_{\rm SUSY} < 1$TeV, radiative
magnification to large mixing may occur through RG evolution from the see-saw
scale,~$M_R=10^{13}$~GeV, down to $M_{\rm SUSY}$. Then the standard model
evolution  below $M_{\rm SUSY}$
causes negligible contribution to the masses and the magnified mixings
because of two reasons:
(i)absence of $\tan^2\beta$ effects, and (ii)small range of RG evolution
from $M_{\rm SUSY}$ to $M_Z$. Thus the predicted mixings remains almost flat
and very close to $\sin\theta_{ij}(M_{\rm SUSY}$) for all energies below
$M_{\rm SUSY}$.
\par
The mixing unification hypothesis implies that we choose all neutrino
mixings at the seesaw scale equal to the corresponding quark mixings,
which in the Wolfenstein parameterization are dictated by the parameter 
$\lambda_0=.2$. We then have, at the seesaw scale,
 $s_{12}\simeq\lambda_0$, $s_{23}\simeq O(\lambda_0^2)$ and
$s_{13}\simeq O(\lambda_0^3)$. These values get substantially magnified
in the region around $M_{\rm SUSY}$. Using 
$|D_{31}|\simeq |D_{32}|\ll |D_{21}|$, we see from 
(\ref{eq3})-(\ref{eq5}), that the dominant contribution to RG evolution of
$s_{23}(\mu)$ is due
to the term $\sim \lambda_0^2F_{\tau}D_{32}$. Similarly the terms 
contributing
to the evolution of $s_{13}(\mu)$ are $\sim \lambda_0^3F_{\tau}D_{32}$ or
$\sim\lambda_0^3F_{\tau}D_{31}$. On the other hand 
the evolution of $s_{12}$ is dominated 
by the term $\sim \lambda_0^5F_{\tau}D_{21}$ where the large enhancement
likely to be caused by the
largeness in $|D_{21}|$ is damped out due to higher 
power of $\lambda_0^5$.
Since the mixing angles change substantially 
only around $M_{\rm SUSY}$, such
dominance to RG evolutions holds approximately at 
all other lower scales below $M_R$. 

\par
If the neutrino mixing angles are to be compatible with experimental
observations at low energies, we need at most the  magnification
factors:
${(\sin\theta_{23}/\sin\theta_{23}^0)}\simeq 20$,
${(\sin\theta_{13}/\sin\theta_{13}^0)}
\le 60$, and
 ${(\sin\theta_{12}/\sin\theta_{12}^0)} \simeq 4$,
where we have used the experimental neutrino mixings for $\theta_{ij}$
\cite{ref1, ref2, ref3, ref4, ref5, ref6} and quark mixings for
$\theta_{ij}^0$ \cite{ref15}.
~That the CHOOZ-Palo Verde bound can tolerate  a magnification factor as large
as 60 is crucial to achieve bilarge mixings by radiative magnification while
keeping the magnified angle $\theta_{13}$ at low energies well below the upper
bound.
%\section{BILARGE NEUTRINO MIXINGS BY RG EVOLUTION}
%\label{sec3}

Starting from known values of gauge couplings,  masses of quarks and charged
leptons, and CKM mixings in the quark sector at low energies, at first we
use the bottom-up approach and all the relevant RGEs to obtain the
corresponding quantities at higher scales,
$10^{13}$ GeV-$10^{16}$ GeV. Assuming the neutrino mixing at
$\mu=M_R\simeq 10^{13}$ GeV
to be small and similar to quark mixings, we then expect the initial
conditions at $\mu=M_R$ to be $\sin\theta^0_{23}\simeq 0.035$, 
$\sin\theta^0_{13}\simeq 0.0025$
and $\sin\theta^0_{12}\simeq 0.20$ \cite{ref15}. Using these as input and the
mass eigenvalues $m^0_i$ as unknown parameters at the high scale, we then
follow  the top-down approach though
(\ref{eq2})-(\ref{eq5}) and other standard RGEs. The unknown
parameters $m^0_i$
are determined in such a way that the solutions
obtained at low energies
agree with mass squared differences and the mixing angles given by the
experimental data within $90\%$ C.L.~\cite{ref1, ref2, ref3, ref4, ref5, 
ref6}
\par
\noindent
\ba \Dmott&=&\left(2-50\right)\times 10^{-5}{\rm eV}^2,\nonumber \\
    \Dmttht&=&\left(1.2-5\right )\times10^{-3}{\rm eV}^2,\nonumber \\
    \sin\theta_{23}&=& 0.54-0.83, \sin\theta_{12}= 0.40-0.70, \nonumber \\
    \sin\theta_{13}&\le& 0.16~.\label{eq8}\ea
\par
In Table I we prersent input mass eigenvalues at the see-saw scale and
our solutions to RGEs at  $M_{\rm SUSY}$ in the
large
$\tan\beta$ region. Our solutions clearly exhibit radiative magnification of
both the mixing angles, $\theta_{23}$ and $\theta_{12}$
for a wide range of input values of $m_i^0$. We find that
although
enhancement due to RG evolution occurs in the $\nu_e-\nu_{\tau}$
sector also,  $\sin\theta_{13}$ remains
well within the CHOOZ-Palo Verde bound \cite{ref6}.
\par
In Fig. \ref{fig1} we present the RG evolutions of the two mixing angles
 starting from $M_R=10^{13}$ GeV down to $M_Z$ for one  set of
input masses:
$m^0_1=0.2983$ eV, $m^0_2=0.2997$ eV, and $m^0_3=0.3383$ eV.
The flatness of the curves below $M_{SUSY}$ is due to negligible
renormalization effect from SM.
 The corresponding low-energy solutions are
$m_1=0.2410$ eV, $m_2=0.2411$ eV,  $m_3=0.2435$ eV,
   $\Dmott=4.8\times10^{-5}$ eV$^2$, $\Dmttht=1.1\times 10^{-3}$ eV$^2$,
~$\sin\theta_{23}=0.680$, $\sin\theta_{13}=0.080$, and
$\sin\theta_{12}=0.568$.
\par
We have also noted that the radiative magnification mechanism leading to bilarge
mixings works more easily if we take all other initial values same  
as mentioned above but 
$\sin\theta^0_{13}=0.0$ which could be relevant to certain neutrino 
mass textures. In
this case the CHOOZ-Palo Verde bound is  always protected.
\par
%\section{PREDICTIONS FOR BETA-DECAY, DOUBLE BETA DECAY AND $U_{e3}$}
%\label{sec4}

Let us now discuss the experimental tests of the model.
The quasi-degenerate neutrino mass spectrum needed for radiative magnification
in this analysis has several interesting predictions as shown below.
 
\par\noindent {\large \bf Double Beta and Tritium Beta Decays:}
Experiments on $(\beta\beta)_{0\nu}$-decay  measure the effective mass
$|<M_{ee}>|=|\sum m_iU_{ei}^2|$. Recent searches have obtained stringent limits
at $90\%$ C.L. with $|<M_{ee}>|<(0.33-1.35)$ eV ~\cite{db}. Also
through a repetition of analysis of previous
experimental data, evidences of $(\beta\beta)_{0\nu}$-decay have been reported
with $|<M_{ee}>|=(0.05-0.84)$ eV ~\cite{ref16}. The current searches by
various double beta experiments such as GENIUS, EXO, MAJORANA, CUORE
and MOON ~\cite{ref17a} are expected to cover the range of interest in
this paper i.e. $|<M_{ee}>|<(0.1-.35)$ eV.
It is quite interesting to note that our RG solutions permit radiative
magnification
consistent with the available experimental data on $\Dmott$, $\Dmttht$, and
mixing angles within $90\%$  confidence level given in (\ref{eq8}),
if the input mass eigenvalues at
$M_R=10^{13}$ GeV are within the range: $0.20$ eV $\leq m^0_i(M_R)\leq
0.9$ eV. This corresponds to the  low energy limit
$0.15$ eV $\leq m_i(M_Z) \leq 0.65$ eV.
Then, apart from uncertainties due to nuclear-physics  effects,
our choice of phases
leads to the   prediction for the effective mass
\be |<M_{ee}>|=0.15\,{\rm eV}-0.65\,{\rm eV}. \nonumber \ee
This range overlaps the one reported in \cite{ref16}, or under investigation
in \cite{ref17a}. Thus, a clear and testable prediction of the
bilarge radiative magnification mechanism is that neutrinoless double-beta
decay should be observed in the next round of experiments.
\par
Further, the quasi-degenerate neutrino mass spectum can be directly measured
in Tritium $\beta$-decay searches which are sensitive to
$m_{\bar\nu_e} \simeq m_i(i=1,2,3)$.
Although the present bound is  $m_{\bar\nu_e} < 2.2$ eV, mass values as
low as $m_{1,2,3} \simeq 0.35$ eV can be reached by
KATRIN experiment \cite{ref18}.


\par\noindent {\large \bf Prediction for $U_{e3}$:}
Starting from the  allowed range of high-scale input values of  the CKM
mixing angle with  $V_{ub}\simeq U^0_{e3}\simeq 0.0025-0.004$, the
RG-evolutions predict enhancement of $\sin\theta_{13}$ at low
energies
\be U_{e3}=\sin\theta_{13}=0.08-0.10. \nonumber \ee
Although this prediction is well below the present experimental upper
bound \cite{ref6}, it is accessible to several planned long-baseline
neutrino experiments in future such as NUMI-Off-Axis or JHF proposals.
%\section{DEGENERATE NEUTRINOS FROM TYPE II SEESAW}

Finally let us address the question of how a quasi-degenerate
neutrino spectrum can arise within the seesaw framework. As has been
emphasized in literature, in models with asymptotic parity symmetry,
the conventional seesaw formula (to be called type I) is replaced by
a modified formula (type II) 
\begin{eqnarray}
M~=~fv_L-M_D(fv_R)^{-1}M^T_D
\end{eqnarray}
where $f$ is the Yukawa coupling of the right handed neutrinos that leads
to the RH neutrino mass matrix after symmetry breaking and $M_D$ is the
Dirac mass for the neutrinos. In quark-lepton unification schemes, one
expects  $M_D$ to be related to the up-quark mass matrix. Therefore, it is
reasonable to expect its eigenvalues to be hierarchical. On the other
hand, the Yukawa coupling matrix $f$ carries information corresponding to
physics beyond the standard model and is therefore apriori unconstrained
by standard model physics. We could therefore choose $f$ to be close to
the unit matrix. In this case, quark-lepton unification requires that the
lepton mixing angles be very close to the quark mixing angles but the
neutrino mass spectrum dominated by the first term in Eq. (10) in
combination with second term can easily
lead to a quasi-degenerate spectrum of Majorana neutrinos. This can, for
example, be achieved by using flavor symmetries in the context of 
$SU(2)_L\times SU(2)_R\times SU(4)_{PS}$ gauge symmetric models. We will 
discuss detailed construction of such models in a subsequent
paper\cite{deg}. An
important point is that the renormalization group equations hold for this
type II seesaw formula as long as we assume that the $SU(2)_L$ triplet
Higgs whose vev responsible for the first term in Eq. (10) is heavier than
the seesaw scale. This is true in models realizing the type II seesaw.

%\section{CONCLUSION}
%\label{sec5}
 In summary, we have shown that in the MSSM, the hypothesis of quark-
lepton mixing unification at the seesaw scale seems to
generate the correct observed mixing pattern for neutrinos i.e. two large
mixings needed for $\nu_e-\nu_\mu$
and  $\nu_\mu-\nu_\tau$ and small mixing for $U_{e3}$ at low energies.
Quasi-degenerate neutrino spectrum with a common mass for neutrinos $\geq
0.1$ eV is a testable prediction of the model. Important new result of our
analysis is that although magnification occurs for the $U_{e3}$ parameter,
it remains small due to the fact that $V_{ub}$ is very small. The
prediction for $U_{e3}$ also provides another test of the model. 
\begin{figure}
\epsfxsize=8.5cm
\epsfbox{bi1.ps}
\caption{Evolution of small quark-like mixings at the see-saw scale to bilarge
neutrino mixings at low energies .The solid, dashed and dotted lines represent
$\sin\theta_{23}$, $\sin\theta_{13}$, and $\sin\theta_{12}$, respectively, 
as defined in the text.}  
\label{fig1}
\end{figure}
\begin{table*}
\caption{Radiative magnification to bilarge
mixings
at low energies for input values of $\sin\theta_{23}^0
=0.035$,~$\sin\theta_{13}^0=0.0025$, ~and $\sin\theta_{12}^0=0.2$ 
at the see-saw scale $M_R= 10^{13}$ GeV. }
\begin{ruledtabular}
\begin{tabular}{lcccc}\hline
$m_1^0$(eV)&0.2579&0.2983&0.4064&0.6050\\
$m_2^0$(eV)&0.2590&0.2997&0.4088&0.6087\\
$m_3^0$(eV)&0.2935&0.3383&0.4621&0.6887\\
$m_1$(eV)&0.2066&0.2410&0.3244&0.4789\\
$m_2$(eV)&0.2067&0.2411&0.3247&0.4794\\
$m_3$(eV)&0.2088&0.2435&0.3283&0.4847\\
$\Dmott$(eV$^2$)&$5.0\times 10^{-5}$&$4.8\times 10^{-5}$&
$2.0\times 10^{-4}$&$4.2\times 10^{-4}$\\
$\Dmttht$(eV$^2$)&$0.85\times 10^{-3}$&$1.1\times 10^{-3}$&
$2.3\times 10^{-3}$&$5.1\times 10^{-3}$\\
$\sin\theta_{23}$&0.547&0.680&0.610&0.644\\
$\sin\theta_{13}$&0.080&0.080&0.080&0.085\\
$\sin\theta_{12}$&0.563&0.568&0.519&0.731\\\hline
\end{tabular}
\end{ruledtabular}
\label{tab1}
\end{table*}
\begin{acknowledgments}
M.K.P. thanks the Institute of Mathematical Sciences
for Senior Associateship.
The work of R.N.M is supported by the NSF grant No.~. The work
of M.K.P is supported by the DST project No.~SP/S2/K-30/98 of the
Govt.~of India.  
\end{acknowledgments}
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