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\def\be{\begin{eqnarray}}
\def\ee{\end{eqnarray}}

\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{$\Delta{m}^{2}$~}}
\newcommand{\st}{\mbox{$\sin^{2}2\theta$~}}
\def\lsim{\:\raisebox{-0.5ex}{$\stackrel{\textstyle<}{\sim}$}\:}
\def\gsim{\:\raisebox{-0.5ex}{$\stackrel{\textstyle>}{\sim}$}\:}
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\begin{document}

\title{Global analysis of solar neutrinos}


\author{Sandhya Choubey
\footnote[3]{sandhya@hep.phys.soton.ac.uk}
}

\address{Department of Physics and Astronomy,\\ University of Southampton,
Highfield, Southampton S017 1BJ, UK}


\begin{abstract}
The recent data from SNO is analysed with the global data 
from the other solar neutrino experiments, first in a model 
independent way and then assuming two-flavor oscillations.
Neutrino oscillations to active flavors 
can best explain the global data with 
the LMA solution emerging as the best-fit. 
However transitions to ``mixed'' states with upto 30\% sterile mixture 
are still allowed at $1\sigma$.

\end{abstract}

%Uncomment for PACS numbers title message
%\pacs{00.00, 20.00, 42.10}

% Uncomment for Submitted to journal title message
%\submitto{\JPA}

% Comment out if separate title page not required
%\maketitle

%\section{Introduction}

{\bf Introduction:}
For over four decades solar neutrino experiments at Homestake (Cl), 
Sage, Gallex-GNO (Ga) \cite{globalsolar} and Super-Kamiokande (SK)
\cite{Fukuda:2002pe} have observed a deficit of the observed 
solar neutrino flux from that predicted by the Standard Solar Model
(SSM) \cite{Bahcall:2000nu}. 
The Sudbury Neutrino Observatory (SNO) has provided
the direct evidence for oscillations of solar $\nue$ 
to another non-electron type active neutrino
by simultaneously observing the charged
current (CC) and neutral current (NC) interactions of neutrinos on 
deuteron \cite{Ahmad:2002jz,Ahmad:2002ka}. 
This single experiment gives evidence for the presence 
of another active neutrino flavor in the solar neutrino beam at the 
$5.3\sigma$ level. When combined with the electron scattering (ES)
data from SK, oscillations to active neutrinos is confirmed at the 
$5.5\sigma$ level. 

%\section{Model Independent Analysis}
{\bf Model Independent Analysis:}
We first use just the data on total rates from SK and SNO 
and obtain model independent bounds on the total $^8B$ flux 
normalization factor $f_B$ and the solar neutrino suppression rate 
$P_{ee}$ \cite{Bandyopadhyay:2002xj,Bandyopadhyay:2002bu}. 
Since both SK and SNO are consistent with no spectral distortion 
we use it as an emperical 
justification to assume that $P_{ee}$ is effectively energy independent and 
express the SK, CC and NC rates as
\be
R^{el}_{SK} = f_B P_{ee} + f_B r P_{ea},~~~
R^{CC}_{SNO} = f_B P_{ee}, ~~~
R^{NC}_{SNO} = f_B (P_{ee} + P_{ea})
\label{one}
\ee
where $P_{ea}$ is the transition probability to active neutrino and 
$r = \sigma_{\nu_\mu,\tau}/\sigma_{\nu_e} \simeq 0.157$ 
is the ratio of $\numu$ to $\nue$ scattering cross-section 
folded with the $^8B$ neutrino spectrum and averaged over energy. 
%Note that $r$ depends on the detector characteristics. We have computed 
%$r$ for SK above 5 MeV. 
%All the rates
%are defined with respect to the BPB2000
%SSM \cite{Bahcall:2000nu}.
%If we assume that the solar $\nue$ are converted entirely to another 
%active flavor then $R^{el}_{SK}$ and $R^{CC}_{SNO}$ can be used to 
%predict the total observed $^8B$ in SNO
%\cite{Barger:2001zs,Bandyopadhyay:2002bu},
%$
%R^{NC}_{SNO} = R^{CC}_{SNO} + (R^{el}_{SK} - R^{CC}_{SNO})/r.
%$
%The NC rate observed in SNO is $R^{NC}_{SNO} = 1.01 \pm 0.12$.
%Thus
%the SNO NC measurement has achieved a precision in $f_B$, which is already
%better than that obtained from the SK and SNO CC combination.

If we assume that the solar neutrinos are completely transformed to active 
neutrinos then 
$P_{ea} = 1 - P_{ee}$. We use eq.(\ref{one}) to show 
in figure
1 the 1$\sigma$ and 2$\sigma$ contours in the $f_B-P_{ee}$
plane from the combinations SK + CC (the outer lines) 
and SK + CC + NC (the inner lines). The best-fit $f_B$ comes at 
1, consistent with the SSM.
The inclusion of the NC data narrows down the ranges of $f_B$
and P$_{ee}$. The error in $f_B$ after inclusion of the NC data
is almost half the size of the corresponding error from SSM.

\begin{figure}[t]
\begin{center}
\vskip 2.5cm
\begin{tabular}{cc}
\epsfysize=7.5cm \epsfxsize=7.5cm \hfil \epsfbox{1.epsi}\hfil
& \hskip 0cm 
\epsfysize=7.5cm \epsfxsize=7.5cm \hfil \epsfbox{2.epsi}
\hfil
\end{tabular}
%\vskip 0.4cm
%\caption{ 
%}
\vskip -2cm
\end{center}
\end{figure}
%\vskip -2cm

We next take up a general case where the $\nue$ 
oscillate into a state $\nu^\prime$, where 
$
\nu^\prime =  \nu_{\rm active} \sin\alpha + \nu_{\rm sterile} \cos\alpha
$,
$\sin^2\alpha(\cos^2\alpha)$ being the fraction of active(sterile) 
component in the resultant beam on Earth.
In this case $P_{ea} = \sin^2\alpha (1 - P_{ee})$ and 
%one gets the following sets of equations for $f_B$ and 
%$\sin^2\alpha$
%\be
%\sin^2\alpha (f_B - R^{CC}_{SNO}) &=& (R^{el}_{SK} - R^{CC}_{SNO})/r
%\label{fbsk}
%\\[2mm]
%\sin^2\alpha (f_B - R^{CC}_{SNO}) &=& R^{NC}_{SNO} - R^{CC}_{SNO}
%\label{fbnc}
%\ee
for each value of $\sin^2\alpha$ we 
show in figure 2 the central and
the 1$\sigma$ and 2$\sigma$ ranges of $f_B$.
We note that transition to pure 
active components ($\sin^2\alpha=1$) is completely consistent 
while transitions to pure sterile states ($\sin^2\alpha=0$) 
are completely forbidden.
Combining the $1\sigma(2\sigma)$ 
lower limit of $f_B$ from this fit with the $1\sigma(2\sigma)$ 
upper limit from SSM (vertical lines are the $2\sigma$ limit)
gives a lower limit of $\sin^2\alpha >0.68(0.45)$.
This means that at $1\sigma(2\sigma)$ we can still have upto
32\%(65\%) admixture of sterile component in the solar neutrino beam. 
While we have obtained the bound on $\cos^2\alpha$ in 
a (quasi)model independent way, the same can be obtained in the framework of 
neutrino oscillations \cite{Bahcall:2002zh,Maltoni:2002ni}. 
Our bound
is in excellent agreement with the bounds 
obtained in \cite{Bahcall:2002zh,Maltoni:2002ni}.

%\section{Model Dependent Analysis}

%\begin{figure}
%\centerline{\epsfxsize=0.2\textwidth\epsfbox{all_spec.eps}}
%\caption{The regions in the parameter space allowed from the analysis
%of global solar neutrino data.}
%\label{fig4}
%\end{figure}

{\bf Model Dependent Analysis:}
We now look at the implications of the 
global solar neutrino data in the framework of two-generation 
oscillations\footnote{A complete list of references on global analyses 
including the recent SNO data can be found in \cite{Choubey:2002nc}.}. 
Since SNO disfavors the sterile option at $5.5\sigma$ 
we consider transitions to active flavors only. For the 
global data we consider the total rates observed in Cl and 
Ga (SAGE, GALLEX and GNO combined), the 1496 day SK 
zenith angle energy spectrum data and the recent data from SNO.
Since it is not yet possible to identify the ES, 
CC and NC events
separately in SNO,
they have made available their results as a combined
CC+ES+NC data in 17 day and 17 night energy bins.
We employ this full day-night SNO spectrum in our statistical analysis.
For the details of our data analysis procedure and the definition of 
the $\chi^2$ function we refer to \cite{Choubey:2002nc,Bandyopadhyay:2001aa}.
The results of the global $\chi^2$ analysis is shown in Table 1 
\cite{Choubey:2002nc}.
The best-fit comes in the LMA region as before \cite{Bandyopadhyay:2001aa} but 
LOW is still allowed with a very high probability. However SMA is 
seen to be virtually ruled out by the data.

\begin{figure}[t]
\begin{center}
\vskip 2.0cm
\begin{tabular}{cc}
\hskip -1cm
\epsfysize=7.5cm 
\epsfxsize=10.5cm \hfil 
\epsfbox{t1.epsi}\hfil
& \hskip -1.5cm 
\epsfysize=7.5cm 
\epsfxsize=7.5cm \hfil 
\epsfbox{all1.epsi}\hfil
\end{tabular}
%\vskip 0.4cm
%\caption{ 
%}
\vskip -3cm
\end{center}
\end{figure}

Figure 3 shows the allowed areas in the $\Delta m^2-\tan^2\theta$ 
plane from the analysis of the global solar neutrino data in the 
framework of two-generation $\nue-\nu_{\rm active}$ oscillations. 
We note that the
(i) NC rate disfavors the LOW solution which 
appears only at $3\sigma$, 
(ii) values of $\Delta m^2$ above $3.5\times 10^{-4}$ eV$^2$ are
disfavored, (iii) maximal Mixing is disfavored at $3\sigma$,
(iv) {\it dark Side} solutions are gone, which implies that 
$\Delta m^2_{\rm solar} > 0$.

{\bf Conclusions:}
We perform a detail analysis of the solar neutrino 
data. We use the SK and SNO rates to 
obtain bounds on $f_B$, $P_{ee}$ and $\cos^2\alpha$ 
in a (quasi)model independent way. We find that transitions 
to states with about 30\% sterile mixture is still allowed at 
$1\sigma$. We analyse the global data in the framework of oscillations 
and present the best-fit solutions and the allowed 
areas in the two-flavor parameter space. The LMA 
solution emerges as the only undisputed solution to the solar neutrino 
problem.

%\section*{Acknowledgments}
%I thank the organizers of NuFact'02 for their hospitality 
%during the conference.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newpage
\section*{References}

\begin{thebibliography}{99}

\bibitem{globalsolar}
%\cite{Cleveland:nv}
%\bibitem{Cleveland:nv}
B.~T.~Cleveland {\it et al.},
%``Measurement Of The Solar Electron Neutrino Flux With The Homestake  Chlorine Detector,''
Astrophys.\ J.\  {\bf 496}, 505 (1998),
%%CITATION = ASJOA,496,505;%%
;
%\cite{Abdurashitov:2002nt}
%\bibitem{Abdurashitov:2002nt}
J.~N.~Abdurashitov {\it et al.}  [SAGE Collaboration],
%``Measurement of the solar neutrino capture rate by the Russian-American  gallium solar neutrino experiment during one half of the 22-year cycle of solar activity,''
arXiv:astro-ph/0204245
%%CITATION = ASTRO-PH 0204245;%%
;
%\cite{Hampel:1998xg}
%\bibitem{Hampel:1998xg}
W.~Hampel {\it et al.}  [GALLEX Collaboration],
%``GALLEX solar neutrino observations: Results for GALLEX IV,''
Phys.\ Lett.\ B {\bf 447}, 127 (1999)
%%CITATION = PHLTA,B447,127;%%
;
E. Bellotti, Talk at Gran Sasso National Laboratories, Italy, May 17, 2002
;
T. Kirsten, Talk at Neutrino 2002, Munich, Germany, May 2002.

%\cite{Fukuda:2002pe}
\bibitem{Fukuda:2002pe}
S.~Fukuda {\it et al.}  [Super-Kamiokande Collaboration],
%``Determination of solar neutrino oscillation parameters using 1496 days  of Super-Kamiokande-I data,''
Phys.\ Lett.\ B {\bf 539}, 179 (2002).
%[arXiv:hep-ex/0205075].
%%CITATION = HEP-EX 0205075;%%

%\cite{Bahcall:2000nu}
\bibitem{Bahcall:2000nu}
J.~N.~Bahcall, M.~H.~Pinsonneault and S.~Basu,
%``Solar models: Current epoch and time dependences, neutrinos, and  helioseismological properties,''
Astrophys.\ J.\  {\bf 555}, 990 (2001).
%[arXiv:astro-ph/0010346].
%%CITATION = ASTRO-PH 0010346;%%

%\cite{Ahmad:2002jz}
\bibitem{Ahmad:2002jz}
Q.~R.~Ahmad {\it et al.}  [SNO Collaboration],
%``Direct evidence for neutrino flavor transformation from neutral-current  interactions in the Sudbury Neutrino Observatory,''
Phys.\ Rev.\ Lett.\  {\bf 89}, 011301 (2002).
%[arXiv:nucl-ex/0204008].
%%CITATION = NUCL-EX 0204008;%%

%\cite{Ahmad:2002ka}
\bibitem{Ahmad:2002ka}
Q.~R.~Ahmad {\it et al.}  [SNO Collaboration],
%``Measurement of day and night neutrino energy spectra at SNO and  constraints on neutrino mixing parameters,''
Phys.\ Rev.\ Lett.\  {\bf 89}, 011302 (2002).
%[arXiv:nucl-ex/0204009].
%%CITATION = NUCL-EX 0204009;%%

%\cite{Bandyopadhyay:2002xj}
\bibitem{Bandyopadhyay:2002xj}
A.~Bandyopadhyay, S.~Choubey, S.~Goswami and D.~P.~Roy,
%``Implications of the first neutral current data from SNO for solar  neutrino oscillation,''
Phys.\ Lett.\ B {\bf 540}, 14 (2002).
%[arXiv:hep-ph/0204286].
%%CITATION = HEP-PH 0204286;%%

%%\cite{Barger:2001zs}
%\bibitem{Barger:2001zs}
%V.~D.~Barger, D.~Marfatia and K.~Whisnant,
%%``Unknowns after the SNO charged-current measurement,''
%Phys.\ Rev.\ Lett.\  {\bf 88}, 011302 (2002).
%%[arXiv:hep-ph/0106207].
%%%CITATION = HEP-PH 0106207;%%

%\cite{Bandyopadhyay:2002bu}
\bibitem{Bandyopadhyay:2002bu}
A.~Bandyopadhyay, S.~Choubey, S.~Goswami and D.~P.~Roy,
%``What can the SNO neutral current rate teach us about the solar neutrino  anomaly,''
Mod.\ Phys.\ Lett.\ A {\bf 17}, 1455 (2002).
%[arXiv:hep-ph/0203169].
%%CITATION = HEP-PH 0203169;%%

%\cite{Bahcall:2002zh}
\bibitem{Bahcall:2002zh}
J.~N.~Bahcall, M.~C.~Gonzalez-Garcia and C.~Pena-Garay,
%``If sterile neutrinos exist, how can one determine the total B-8 and  Be-7 solar neutrino fluxes?,''
arXiv:hep-ph/0204194.
%%CITATION = HEP-PH 0204194;%%  

%\cite{Maltoni:2002ni}
\bibitem{Maltoni:2002ni}
M.~Maltoni, T.~Schwetz, M.~A.~Tortola and J.~W.~Valle,
%``Constraining neutrino oscillation parameters with current solar and  atmospheric data,''
arXiv:hep-ph/0207227
%%CITATION = HEP-PH 0207227;%%
;
%\cite{Maltoni:2002xd}
%\bibitem{Maltoni:2002xd}
M.~Maltoni, T.~Schwetz, M.~A.~Tortola and J.~W.~Valle,
%``Ruling out four-neutrino oscillation interpretations of the LSND  anomaly?,''
arXiv:hep-ph/0207157.
%%CITATION = HEP-PH 0207157;%%

%\cite{Choubey:2002nc}
\bibitem{Choubey:2002nc}
S.~Choubey, A.~Bandyopadhyay, S.~Goswami and D.~P.~Roy,
%``SNO and the solar neutrino problem,''
arXiv:hep-ph/0209222.
%%CITATION = HEP-PH 0209222;%%

%\cite{Bandyopadhyay:2001aa}
\bibitem{Bandyopadhyay:2001aa}
A.~Bandyopadhyay, S.~Choubey, S.~Goswami and K.~Kar,
%``Impact of the first SNO results on neutrino mass and mixing,''
Phys.\ Lett.\ B {\bf 519}, 83 (2001).
%[arXiv:hep-ph/0106264]
%%CITATION = HEP-PH 0106264;%%

\end{thebibliography}

\end{document}




