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%   Determining the oscillation parameters 
%   by Solar neutrinos and KamLAND
%   H. Nunokawa, W. J. C. Teves, R. Zukanovich Funchal
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\title{ 
\vglue -2.0cm
{\small \hfill IFUSP-DFN/02-078\\
\vglue -0.4cm
\hfill IFT-P.001/2003}\\
Determining the oscillation parameters 
by Solar neutrinos and KamLAND } 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
\author{H.~Nunokawa$^1$}\email{nunokawa@ift.unesp.br} 
\author{W.~J.~C.~Teves$^2$}\email{teves@charme.if.usp.br}  
\author{R.~Zukanovich Funchal$^2$}\email{zukanov@if.usp.br}  
 
\affiliation{\\ \\ 
$^1$ Instituto de F{\'\i}sica Te{\'o}rica,Universidade Estadual Paulista, 
     Rua Pamplona 145, 01405-900 S{\~a}o Paulo, Brazil \\ 
$^2$ Instituto de F{\'\i}sica,  Universidade de S{\~a}o Paulo  
     C.\ P.\ 66.318, 05315-970 S{\~a}o Paulo, Brazil} 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{abstract} 
 The neutrino oscillation experiment KamLAND has given the first evidence 
 for disappearance of $\bar \nu_e$ coming from nuclear reactors. 
 We have combined their data with all the solar neutrino data assuming 
 two flavor neutrino mixing and obtained allowed parameter regions
 which are compatible with the so-called large mixing angle MSW solution
 to the solar neutrino problem. 
 The allowed regions in the plane of mixing angle and 
 mass squared difference are now split into two islands at 99\% C.L. 
 We have speculated how these two islands can be distinguished in the near 
 future.   
 We have shown that a 50\% reduction of the error on SNO neutral-current 
 measurement can be important in establishing in each of these islands 
 the true values of these parameters lie.  
 We also have simulated KamLAND positron energy spectrum after 1 year of 
 data taking, assuming the current best fitted values of the oscillation 
 parameters, combined it the  with current solar neutrino 
 data and  showed how these two  split islands can be modified. 
\end{abstract} 
 
\pacs{26.65.+t,13.15.+g,14.60.Pq,91.35.-x}
 
\maketitle 
\thispagestyle{empty} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction} 
\label{sec:intro} 
\vglue -0.3cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 Many solar and atmospheric neutrino experiments have collected data 
 in the last decades which can be interpreted as evidences that neutrinos 
 produced in the Sun and in the Earth's atmosphere suffer flavor 
 conversion. While the atmospheric neutrino 
 results~\cite{atmnuobs} may be understood assuming $\nu_\mu \to \nu_\tau$ 
 conversion driven by a neutrino mass squared difference within the 
 experimental reach of the accelerator based neutrino oscillation 
 experiment K2K~\cite{k2k}, 
 the mass squared differences needed to explain the solar neutrino 
 data were, until quite recently, before the Kamioka Liquid scintillator 
 AntiNeutrino Detector (KamLAND)~\cite{kamland} has started its operation, 
 too small to be inspected by a terrestrial neutrino oscillation experiment.

 A number of different fits assuming standard neutrino 
 oscillations induced by mass and mixing~\cite{fits} as well as other 
 exotic flavor conversion mechanisms~\cite{exotics} have been performed 
 using the combined solar neutrino data from Homestake~\cite{homestake}, 
 GALLEX/GNO~\cite{gallex,gno}, SAGE~\cite{sage}, 
 Super-Kamiokande-I~\cite{superk} and SNO~\cite{sno}.  
 These analyses selected some allowed areas in the free parameter region 
 of each investigated mechanism but did not allow one to establish 
 beyond reasonable doubt which is the mechanism and what are the 
 values of the parameters that are responsible for solar $\nu_e$ flavor 
 conversion. 
 After the first result of the KamLAND (or KL hereafter) 
 experiment~\cite{kamland} 
 this picture has changed drastically.

 In the first part of this paper, we present the allowed region for the 
 oscillation parameters in two generations for the entire set of solar 
 neutrino data, for KamLAND data alone and for  KamLAND result combined 
 with all solar neutrino data, showing that this last result finally 
 establishes the so called large mixing angle (LMA)
 Mikheyev-Smirnov-Wolfenstein (MSW)~\cite{msw} solution 
 as the final answer to  the long standing 
 solar neutrino problem~\cite{bahcall}, 
 definitely discarding all the other mass 
 induced or more exotic solutions. 
 (For the first discussions on the complete ``MSW triangle'' which 
 includes the LMA region, see Ref.~\cite{mswtriangle}.)
 In the second part, we speculate on the possibility of further 
 constraining the oscillation parameters in the near future.
 For instance, we point out the importance of SNO neutral-current (NC)
 data in further constraining the LMA MSW solution. In particular, we discuss 
 the consequence of a significant reduction (50 \%) of the SNO 
 neutral-current data uncertainty.
 Finally, we simulate the expected inverse  $\beta$-decay $e^+$ energy 
 spectrum for 1 year of KamLAND data taking, based on the best fitted 
 values of 
 the oscillation parameters. We combine this with the current solar 
 neutrino data in order to show how the allowed parameter regions 
 can be modified.

\vglue -2.7cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{Determination of Oscillation Parameters} 
\label{sec:analysis} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\vglue -0.3cm
 KamLAND has observed about 40\% suppression of $\bar \nu_e$ flux 
 with respect to the theoretically expected one~\cite{kamland}, 
%KamLAND has observed a suppression of about 40\% of the expected 
%$\bar \nu_e$ flux in the absence of oscillation~\cite{kamland}, 
which is compatible with neutrino oscillations in vacuum
in two generations. 
In this case the relevant oscillation  parameters, which must be 
 determined by the fit to experimental data, are a mass squared
difference ($\Delta m^2$) and a mixing angle ($\theta$).  
 We first obtained the allowed region in the 
 ($\tan^2\theta$, $\Delta m^2$) plane for these 
 parameters compatible with all solar neutrino experimental data, 
 then with KamLAND data alone, and finally we combine  these 
 two sets of data.
\vglue -1.7cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\subsection{Solar Neutrino Experiments} 
\label{subsec:solar} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\vglue -0.3cm
 We have determined the parameter region allowed by the solar neutrino
 rates measured by Homestake~\cite{homestake}, GALLEX/GNO~\cite{gallex,gno}, 
 SAGE~\cite{sage} and SNO (elastic scattering, charged-current and 
 neutral-current reactions)~\cite{sno} (6 data points) as well as by 
 the Super-Kamiokande-I zenith spectrum data~\cite{superk} (44 data points), 
 assuming neutrino oscillations in two generations. 

 We have computed the $\nu_e \to \nu_e$  survival probability, properly taking 
 into account the neutrino production distributions in the Sun according to 
 the Standard Solar  Model~\cite{ssm}, the zenith-angle exposure of each 
 experiment, as well as the Earth matter effect as in Ref.~\cite{exotics}, 
 except that here we solved the neutrino evolution equation entirely 
 numerically. We then estimate the allowed parameter regions by 
 minimizing $\chi^2_\odot$ which is defined as
%
\begin{equation}
 \label{chi}
\chi^2_\odot = \sum_{i,j=1,...,50}
 \left[R_i^{\text{th}}-R_i^{\text{obs}} \right] \, 
 \left[\sigma_\odot^2 \right]^{-1}_{ij} \,
 \left[R_j^{\text{th}}-R_j^{\text{obs}} \right]\,,
\end{equation}
where $R_i^{\text{th}}$ and $R_i^{\text{obs}}$ denote the
theoretically expected and observed evets rates, respectively,  
which run through all the 50 data points we mentioned above, 
and $\sigma_\odot$ is the $50\times 50$ correlated error matrix, 
defined in a similar way as in Ref.~\cite{exotics}.
In this work we have treated the $^8$B
neutrino flux as a free parameter. 

 In Figure~\ref{fig1} we show in the 
 $(\tan^2 \theta,\Delta m^2)$ plane the region allowed by 
 the Super-Kamiokande-I zenith spectrum data as well as 
 the rates of all other solar neutrino experiments 
 at 90\%, 95\%, 99\% and 99.73\% C.L. In our fit we obtained a    
 $\chi^2_{\odot}(\text{min})=37.7$ for 47 d.o.f  (83 \% C.L.),  
 corresponding to the global best fit values  
 $\Delta m^2= 7.5 \times 10^{-5}$ eV$^2$ and  $\tan^2 \theta=0.42$.

%\vglue -1.0cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\subsection{KamLAND} 
\label{subsec:kam} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\vglue -0.3cm
 KamLAND~\cite{kamland} is a reactor neutrino oscillation experiment 
 searching for  $\bar \nu_e$ oscillation from over 16 power reactors 
 in Japan and South Korea which are mostly located at distances that 
 vary from 80 to 344 km from the Kamioka mine, allowing KamLAND to 
 probe the LMA MSW neutrino oscillation solution to 
 the solar neutrino problem.

 The KamLAND detector consists of about 1 kton of liquid scintillator 
 surrounded by photomultiplier tubes that registers the arrival of 
 $\bar \nu_e$ through the inverse $\beta$-decay reaction 
 $\bar \nu_e + p \to e^+ + n$, by measuring $e^+$ and the 2.2 MeV $\gamma$-ray 
 from neutron capture of a proton in delayed coincidence. The $e^+$ 
 annihilate in the detector producing the total visible energy 
 $E$  which is related to the incoming $\bar \nu_e$ energy, 
 $E_\nu$, as  $E=E_\nu-(m_n-m_p)+m_e$, where $m_n$,
 $m_p$ and $m_e$ are respectively, the neutron, proton and electron mass.

  After 145.1 days of data taking, which correspond to 162 ton yr exposure, 
  KamLAND has measured 54 inverse $\beta$-decay events where 87 were expected 
  without neutrino conversion, distributed in 13 bins of 0.425 MeV above 
  the analysis threshold of 2.6 MeV applied to contain the background under 
  about 1 event.

  We have theoretically computed the expected number of events in the 
  $i$-th bin, $N_i^{\text{theo}}$, as 
  \begin{equation}
  N_i^{\text{theo}} = \int dE_\nu \, \sigma(E_\nu) \sum_k \phi_k(E_\nu) 
  P_{\nu_e \to \nu_e} \int_i dE \,R(E,E^\prime),
  \end{equation}
  where $R(E,E^\prime)$ is the energy resolution 
  function, $E$ the observed and $E^\prime$ the 
  true $e^+$ energy, with the energy resolution $7.5\%/\sqrt{E(\text{MeV})}$. 
  Here $\sigma(E_\nu)$ is the neutrino interaction cross-section and $\phi_k$ 
  is the neutrino flux from the $k$-th power reactor, we have included all  
  reactors with baseline smaller than 350 km in the sum,
  $P_{\nu_e \to \nu_e} \equiv P_{\bar \nu_e \to \bar \nu_e}$ 
 (if CPT is conserved, which we will assume here) is the familiar 
  neutrino survival probability in vacuum (the matter effect is
  very small), which is equal to one in  case of no oscillation, 
  and explicitly depend on $\Delta m^2$ and  $\tan^2 \theta$.

  We were able to compute the regions in the $(\tan^2 \theta,\Delta m^2)$ 
  plane  allowed by KamLAND spectrum data by minimizing with respect to 
  these free parameters, the $\chi^2_{\text{KL}}$ function defined as 
  $\chi^2_{\text{KL}} = \chi^2_{\text{G}} + \chi^2_{\text{P}}$ with 
  \begin{equation}
  \chi^2_{\text{G}}= \displaystyle \sum_{i} 
  \frac{( N_i^{\text{theo}} - N_i^{\text{obs}})^2}{\sigma_i^2},
  \end{equation} 
  and 
  \begin{equation}
  \chi^2_{\text{P}}= \sum_{j} 2(N_j^{\text{theo}} - N_j^{\text{obs}}) + 
   2 \, N_j^{\text{obs}} \ln \displaystyle \frac{N_j^{\text{obs}}}
  {N_j^{\text{theo}}},
  \end{equation} 
  where  $\sigma_i = \sqrt{ N_i^{\text{obs}}+(0.0642\, N_i^{\text{obs}})^2}$ 
  is the statistical plus systematic uncertainty in the number of 
  events in the $i$-th bin and the sum in $i (j)$ is done over the bins 
  having 4 or more (less  than 4) events. 

 Using this $\chi^2_{\text{KL}}$ we have computed the allowed regions 
 at 90\%, 95\%, 99\%  and 99.73\% C.L. shown in Fig.~\ref{fig2}, 
 which are quite consistent with the ones obtained by the KamLAND group  
 in Fig. 6 of Ref.~\cite{kamland}. 
 In our fit we obtained a  $\chi^2_{\text{KL}}(\text{min})=5.4$ for 
 11 d.o.f  (91 \% C.L.), corresponding to the best fit values  
 $\Delta m^2= 7.0 \times 10^{-5}$~eV$^2$ and  $\tan^2 \theta=0.79$.

\vglue -0.5cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\subsection{Combined Results} 
\label{subsec:comb} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\vglue -0.3cm 
 Combining the results of all solar experiments with KamLAND data 
 we have obtained the allowed regions showed in Fig.~\ref{fig3}.
 The minimum value of $\chi^2_{\text{tot}}=\chi^2_\odot+\chi^2_{\text{KL}}$
 for the combined fit is      
 $\chi^2_{\text{tot}}(\text{min})=43.6$ for 60 d.o.f  (94.5 \% C.L.),  
 corresponding to the best fit values  
 $\Delta m^2= 7.1 \times 10^{-5}$~eV$^2$ and  $\tan^2 \theta=0.42$.
 We observe that there are two separated regions which are allowed  at  
 99 \% C.L.: a lower one in $\Delta m^2$ (region 1) where the global 
 best fit point is located, and an upper one (region 2) where the local best 
 fit values are $\Delta m^2= 1.5 \times 10^{-4}$~eV$^2$ 
 and  $\tan^2 \theta=0.41$, corresponding to 
 $\chi^2_{\text{loc}}(\text{min})=49.2$. We observe that 
 depending on the definition of  $\chi^2_{\text{KL}}$  used there 
 appear at 99.73\% C.L.  a third region above 
 $\Delta m^2= 2 \times 10^{-4}$~eV$^2$.

 In Fig.~\ref{fig4} we show the theoretically predicted energy spectra 
 at KamLAND for no oscillation, the best fit values of the oscillation 
 parameters for KamLAND data alone and for KamLAND combined with solar 
 data in regions 1 and 2.
 We note that the fourth energy bin, which is for the moment below the 
 analysis cut, can be quite important in determining the value of 
 the oscillation parameters in the future.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{Future Perspectives} 
\label{sec:fut} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\vglue -0.3cm  
 In this section we consider the effect of possible future experimental 
 improvements which can help in determining the oscillation parameters.
 We first consider a reduction of the error in SNO neutral-current 
 measurement then an increase of the event statistics in KamLAND. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\subsection{Effect of reducing SNO neutral-current error} 
\label{subsec:sno} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\vglue -0.3cm  
 In order to constrain the solar neutrino oscillation parameters even more, 
 in particular, to decide in which of the 99\% C.L. islands $\Delta m^2$ 
 really lie, we have investigated the effect of increasing the SNO 
 neutral-current  data precision to twice its current value.  
 We have re-calculated the region in the  $(\tan^2 \theta,\Delta m^2)$ plane 
 allowed by all current solar neutrino data artificially decreasing 
 the SNO NC measurement error but keeping the current central value, 
 as well as the other solar neutrino data, unchanged.
 The result can be seen in Fig.~\ref{fig5}. The best fit point 
 and the value of $\chi^2_\odot$(min) remain practically unchanged with 
 respect to the result obtained in Sec.~\ref{subsec:solar}, but the 
 allowed region shrinks significantly. This is because the $^8B$
 neutrino flux  normalization, which can be directly inferred from  
 SNO NC measurements, gets more constrained. 
 Combining this  with KamLAND data we obtain the allowed region 
 shown in Fig.~\ref{fig6}. We observe that this allowed region is 
 substantially smaller compared to the one shown in Fig.~\ref{fig3}. Moreover, 
 region 2 only remains at 99\%~C.L.  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\subsection{Effect of increasing KamLAND statistics} 
\label{subsec:kamfut} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\vglue -0.3cm  
  We simulate the expected KamLAND spectrum after one year of data taking 
  for three distinct assumptions. We have generated KamLAND future data 
  compatible with the best fitted values of $\Delta m^2$ and $\tan^2 \theta$ 
  obtained for : (a) KamLAND data alone,
 (b) KamLAND and current solar neutrino data in region 1 and 
 (c) KamLAND and current solar neutrino data in region 2.
  We have also included an extra bin, corresponding to the fourth bin in 
  Fig.~\ref{fig4}.  We have re-calculated the region allowed by the 
  combined fit with the current solar neutrino data in each case.
 
  The results of our calculations can be seen in Figs.~\ref{fig7}-\ref{fig9}.
  If the future KamLAND result is close to the current 
  one (see Fig.~\ref{fig7}), values 
  of $\tan^2 \theta$ larger than the ones allowed now will be possible and 
  region 2 will be excluded at 99\% C.L. For this case we have obtained 
  $\chi^2_{\text{tot}}$(min)~$=42.1$.
  On the other hand, if the future KamLAND data are more compatible with the 
  current best fit point of solar neutrino data (see Fig.~\ref{fig8}), 
  the global allowed region will diminish substantially with respect 
  to Fig.~\ref{fig3} and region 2 will only remain at 99\% C.L. 
  For this case we have obtained  $\chi^2_{\text{tot}}$(min)~$=39.1$.
  Finally, if after one year KamLAND data is more compatible with region 2 
  (see Fig.~\ref{fig9}) then one should observe an increase towards 
  larger values of $\Delta m^2$  in the combined allowed region with 
  respect to the one shown in Fig.~\ref{fig3}. In this case region 1 and 2 
  will have similar statistical significance, corresponding 
  to $\chi^2_{\text{tot}}$(min)~$ \sim 44.0$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{Discussions and Conclusion} 
\label{sec:conclusions} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\vglue -0.3cm

 We have performed a combined analysis of the complete set of solar neutrino 
 data with the recent KamLAND result in a two neutrino flavor oscillation 
 scheme. We have obtained, in agreement with  other groups~\cite{outros}, 
 two distinct islands, denominated as regions 1 and 2, in the 
 $(\tan^2 \theta, \Delta m^2)$ plane which are the most probable 
 regions where the true values of these  parameters lie. 
 Region 1, where the global best fit point was found, 
 is around $\Delta m^2=7.1 \times 10^{-5}$ eV$^2$, while region 2 
 is around $\Delta m^2=1.5 \times 10^{-4}$ eV$^2$.

 We have considered two possible improvements in the determination 
 of the neutrino oscillation parameters in the future.
 First we have investigated the effect of a decrease of 50 \% in 
 the error of the SNO NC measurement. We have shown that this will 
 substantially decrease the allowed parameter region when combined 
 with  KamLAND data. In particular, region 2 would not be allowed  at 
 95\% C.L. anymore.
 
 Second we have studied what can happen in the near future, when 
 KamLAND collects 1 year of data. We  have simulated the expected KamLAND 
 spectrum including an extra lower bin, corresponding to the fourth bin in 
 Fig.~\ref{fig3}. Three different cases were studied in combination with 
 the present solar neutrino data. In the first case we assumed that 
 the future KamLAND  spectrum will be compatible with oscillation 
 parameter values at the best fit point for the present KamLAND data 
 alone. This is the most restrictive case for region 2. 
 In the second case we have considered that future data 
 will be more compatible with the present best fit point of the solar 
 neutrino experiments. In this case the combined allowed region will be 
 much smaller than the present one and region 2 will be only allowed at 
 99\%~C.L. Finally, in the third case, we have assumed that in the future 
 KamLAND data will be compatible with the local best fit point in region 2.
 In this case the combined allowed region will suffer an increase towards 
 larger values of $\Delta m^2$  and region 1 and 2 will both remain with 
 similar statistical significance.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%  ACKNOWLEDGMENTS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{acknowledgments} 
 This work was supported by Funda{\c c}{\~a}o de Amparo 
 {\`a} Pesquisa do Estado de S{\~a}o Paulo (FAPESP) and Conselho 
 Nacional de  Ci{\^e}ncia e Tecnologia (CNPq).
\end{acknowledgments} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
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\end{thebibliography} 


%%%%%%%%%%%%%%%%%%   Fig. 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure} 
\centering\leavevmode 
\hglue -0.2cm
\includegraphics[scale=0.63]{sol.eps} 
\vglue -.8cm
\caption{Region in $(\tan^2 \theta,\Delta m^2)$ plane 
allowed by the Super-Kamiokande-I zenith spectrum 
combined with rates from Homestake, GALLEX/GNO, SAGE and SNO.  
The best fit point is marked by a cross.}
\label{fig1} 
%\vglue -0.65cm
%\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%\begin{figure} 
\centering\leavevmode 
\hglue -0.2cm
\vglue -.6cm
\includegraphics[scale=0.67]{kamland.eps} 
\vglue -0.7cm 
\caption{Regions in $(\tan^2 \theta,\Delta m^2)$ plane allowed by  
KamLAND data alone. The best fit point is marked by a cross.} 
\label{fig2} 
\vglue -0.5cm
\end{figure} 

\begin{figure} 
\centering\leavevmode 
\vglue 2.0cm 
\hglue -1.5cm
\includegraphics[scale=1.2]{comb.eps} 
\vglue -.6cm 
\caption{Region allowed by all the solar neutrino experiments combined with 
KamLAND (KL) data. The region below (above) $\Delta m^2=10^{-4}$ eV$^2$ 
is referred to as region 1 (2).  The best fit points in each region 
are also marked by cross (global best) and plus (local best).} 
\label{fig3} 
\vglue -0.5cm
\end{figure} 

 \vglue -0.5cm 
\begin{figure} 
\centering\leavevmode 
\vglue -2.2cm  
\hglue -1.0cm
\includegraphics[scale=1.0]{kamhist.eps} 
\vglue -6.cm 
\caption{Expected positron energy spectra at KamLAND (KL)
for no oscillation, the best fit values of the oscillation 
parameters for KamLAND data alone and KamLAND data combined with 
the solar neutrino data in regions 1 and 2 of Fig.~\ref{fig3}. 
The KamLAND data~\cite{kamland} is also shown as solid circles with error bars.
The energy threshold at 2.6 MeV is marked by a vertical line.} 
\label{fig4} 
\vglue 0.6cm
\end{figure} 

\begin{figure} 
\centering\leavevmode 
\hglue -0.2cm
\vglue -.5cm 
\includegraphics[scale=0.7]{sol_nc.eps} 
\vglue -.8cm 
\caption{Same as Fig.~\ref{fig1} but decreasing the SNO 
 neutral-current data error to half of its current value.}
\label{fig5} 
%\end{figure} 
%\vglue -0.3cm 
%\begin{figure} 
\centering\leavevmode 
\hglue -0.2cm
\vglue -0.2cm 
\includegraphics[scale=0.7]{comb_nc.eps} 
\vglue -0.8cm 
\caption{Same as Fig.~\ref{fig3} but decreasing the SNO 
 neutral-current data error to half of its current value.}
\label{fig6} 
\vglue -0.8cm 
\end{figure}  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure} 
\centering\leavevmode 
\hglue -0.2cm
\vglue -1.cm 
\includegraphics[scale=0.7]{sol_kam_sim1.eps} 
\vglue -0.6cm 
\caption{Same as Fig.~\ref{fig3} but for a simulated KamLAND spectrum 
after one year of data taking compatible with  the KamLAND alone best 
fit $\Delta m^2=7\times 10^{-5}$~eV$^2$ and $\tan^2\theta=0.79$.}
\label{fig7} 
%\vglue -0.3cm 
%\end{figure}
%\begin{figure} 
\centering\leavevmode 
\hglue -0.2cm
\vglue -0.4cm 
\includegraphics[scale=0.7]{sol_kam_sim2.eps}
\vglue -0.6cm 
\caption{Same as Fig.~\ref{fig7} but for the KL + Solar neutrino 
global best fit $\Delta m^2=7.1\times 10^{-5}$~eV$^2$ 
and $\tan^2\theta=0.42$ in region 1.}
\label{fig8} 
\vglue -0.3cm 
\end{figure}
\begin{figure} 
\centering\leavevmode 
\hglue -0.2cm
\vglue 0.cm 
\includegraphics[scale=0.7]{sol_kam_sim3.eps}
\vglue -0.6cm 
\caption{Same as Fig.~\ref{fig7} but for the KL + Solar neutrino 
local best fit $\Delta m^2=1.5\times 10^{-4}$~eV$^2$ and $\tan^2\theta=0.41$
in region 2.}
\label{fig9} 
\vglue -0.3cm 
\end{figure}
 
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% End Manuscript 
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\end{document} 
 

