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\begin{document}
\vspace*{4cm}
\title{NEUTRINO DATA AND IMPLICATIONS FOR $\theta_{13}$}
\author{G.L. FOGLI$^{1,2}$, E. LISI$^{2}$, A. MARRONE$^{1}$, A. PALAZZO$^{3,\,} $\footnote{Speaker, email: antonio.palazzo@ific.uv.es} and A.M. ROTUNNO$^{1}$\\}
\address{$^1$ Dipartimento di Fisica, Universit\`a di Bari,\\
Via Amendola 173, 70126, Bari, Italy\\
$^2$ Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Bari,\\
Via Orabona 4, 70126 Bari, Italy\\
$^3$ AHEP Group, Institut de F\'isica Corpuscular, CSIC/Universitat de Val\`encia,\\
Edifici Instituts d'Investigaci\'o, Apt.\ 22085, 46071 Val\`encia, Spain}
\maketitle\abstracts{Pinning down the unknown neutrino mixing angle $\theta_{13}$ constitutes one of the most
important goals in particle physics in connection with future investigation on CP violation in the leptonic sector.
In this context, we present the results of an updated global analysis of neutrino oscillation data, focusing on
this eluding parameter. We discuss three independent and converging hints of $\theta_{13} > 0$: A first one
coming from atmospheric neutrino data; a the second one from the combination of solar and long-baseline reactor
neutrino data; and a third one from the latest MINOS measurements in the appearance
($\nu_\mu \to \nu_e$) channel. Their combination provides a preference for $\theta_{13} > 0$ at the non-negligible
statistical significance of the 2$\sigma$ (95\% C.L.). We also discuss possible refinements of the data analyses,
which might sharpen the present indication.}
\section{Introduction}
After more than a decade of data-taking, old and new neutrino experiments
continue to provide us with precious information to decipher.
The latest data not only sharpen the estimates of well known
parameters but may disclose the opportunity to probe some of
the unknown ones. This may be the case of the smallest and unknown
mixing angle $\theta_{13}$, whose determination is a fundamental target
in connection with future investigation of CP violation in the leptonic sector.
The first robust upper bound on $\theta_{13}$ was established by
the CHOOZ experiment\,\cite{Apollonio:1997xe} at the end of 1997. Since
then we have been witnessing a slow but progressive increase in the sensitivity
of the neutrino global analyses in constraining this important parameter.
Indeed, such analyses have first corroborated the CHOOZ findings and then have
progressively strengthened its upper limit. Therefore, it is not completely surprising that
neutrino data now allow us to go beyond the CHOOZ sensitivity.
What instead (pleasantly) surprise us is that they, for the first time,
point toward a non-zero value of this parameter. Remarkably, an analogous
hint has independently emerged in the recent searches of $\nu_e$
appearance in MINOS\,\cite{sanchez}, thus reinforcing the statistical significance of
the global indication for non-zero $\theta_{13}$ (now at the 95\% C.L.), together with
our hopes of pinning down this fundamental parameter.
In the following we review the current status of neutrino mass and mixing
determinations, focusing on such a weak, but nonetheless interesting, indication.
\section{The leading parameters}
Four of the fundamental parameters driving neutrino oscillations
are known with very good precision. Two couples of parameters govern
the flavor transitions in the two (almost) distinct ``atmospheric''
and ``solar'' sectors --- so called from the natural sources used
in the first searches for neutrino oscillations. In fact, precision studies
of these parameters are now complemented by ``artificial'' neutrinos
produced in reactors and accelerators. Each couple of parameters consists
of a mass squared splitting (related to the oscillation frequency) and
a mixing angle (related to the amplitude of the relevant oscillation process).
In the following we review the status of the determination of these
parameters as of May 2009.
\subsection{$\delta m^2$ and $\theta_{12}$}
Solar and KamLAND reactor neutrino oscillations are driven by two leading
parameters: the smallest mass-squared difference $\delta m^2$ and the
mixing angle $\theta_{12}$.
In our analysis~\cite{Fogli:2008jx} we have included the results
from the third phase of the Sudbury Neutrino Observatory (SNO-III)~\cite{Aharmim:2008kc},
in the form of two integral determinations
of the charged current (CC) and neutral current (NC) event rates.
Furthermore, we have included the Borexino results~\cite{Arpesella:2008mt},
and the reevaluated GALLEX datum~\cite{GALLEX}. Finally,
we have incorporated the latest KamLAND measurements~\cite{:2008ee}.
The $\delta m^2$ determination is dominated by KamLAND,
which now observes the oscillatory pattern over one entire cycle~\cite{:2008ee}.
The situation is quite different for the mixing angle $\theta_{12}$, which is
determined by an interplay of both solar and KamLAND data.
Indeed, a weak tension is present among the two independent
determinations which, as we shall see in more detail in the next section,
is at the origin of the``new" hint of $\theta_{13}>0$.
Our three flavor analysis~\cite{Fogli:2008ig}, after marginalizion over $\theta_{13}$,
provides the following determinations (at the 2$\sigma$ level),
%-------------------------------------------------------
\begin{eqnarray}
\label{limit12a}
\delta m^2 &=& 7.67\,(1^{+0.044}_{-0.047})\times 10^{-5}\mathrm{\ eV}^2\ ,\\
\label{limit12b} \sin^2\theta_{12}&=&0.312\,(1^{+0.128}_{-0.109})\ ,
\end{eqnarray}
%--------------------------------------------------------
in agreement with our previous estimates~\cite{Fogli:2006yq},
and now more precise almost by a factor of two.
\subsection{$\Delta m^2$ and $\theta_{23}$}
Atmospheric and long baseline (LBL) neutrino oscillations are
mainly driven by two leading parameters: the largest mass-squared
difference $\Delta m^2$ and the mixing angle $\theta_{23}$.
In this case, even in the limit $\theta_{13}=0$, small effects are induced by
the ``solar parameters'' ($\delta m^2,\,\theta_{12}$), which we fix at their best fit values.
The determination of the mass-squared difference $\Delta m^2$ is now dominated by
the disappearance measurements (in the $\nu_\mu \to \nu_\mu$ channel) performed by
the accelerator LBL experiment MINOS~\cite{Adamson:2008zt}, while
the mixing $\theta_{23}$ is still better constrained by atmospheric neutrino data.
Our global analysis~\cite{Fogli:2008ig} provides the allowed ranges (at the 2$\sigma$ level),
%-------------------------------------------------------
\begin{eqnarray}
\label{limit23a}
\Delta m^2 &=& 2.39\,(1^{+0.113}_{-0.084})\times 10^{-3}\mathrm{\ eV}^2\ ,\\
\label{limit23b} \sin^2\theta_{23}&=&0.466\,(1^{+0.292}_{-0.215})\ .
\end{eqnarray}
%--------------------------------------------------------
In comparison with our previous estimates~\cite{Fogli:2006yq},
the $2\sigma$ error on $\Delta m^2$ is reduced
from $\sim\! 15\%$ to $\sim\! 10\%$, due to the latest MINOS results.
It seems reasonable to expect that MINOS will appreciably
reduce the uncertainty of {\em both} parameters starting form the next data release.
\section{Hints of $\theta_{13}>0$}
The mixing angle $\theta_{13}$, if different from zero, plays a subdominant role
in each of the two sectors considered above, thus providing the main connection among them.
In the following we discuss in detail the implications of the latest neutrino data
for this parameter.
\subsection{Atmospheric data}
In~\cite{Fogli:2005cq} we pointed out a weak preference of atmospheric
neutrino data\,\footnote{Other atmospheric $\nu$ analyses as in~\cite{Schwetz:2008er,Maltoni:2008ka}
have found no or weaker hint. An indication for non-zero $\theta_{13}$ is instead
supported in~\cite{Escamilla:2008vq,Roa:2009wp}.}
%-------------
(in combination with CHOOZ) for a non-zero value of $\theta_{13}$. We traced a possible source for
this preference in subleading $3\nu$ oscillation terms driven by the
``solar parameters''~\cite{Peres:1999yi,Peres:2003wd} ,
which could partly explain the observed excess of sub-GeV atmospheric electron-like events.
We find such a hint unaltered after combination with current
(disappearance) LBL accelerator neutrino data, which are not
sufficiently sensitive to $\theta_{13}$~\cite{Fogli:2006yq}.
In particular, after inclusion of the latest MINOS (disappearance) data~\cite{Adamson:2008zt},
and marginalizing over the leading mass-mixing parameters $(\Delta m^2,\sin^2\theta_{23})$,
we still find a $\sim \! 0.9\sigma$ hint of $\theta_{13}>0$ from the combination of
atmospheric, LBL accelerator (disappearance), and CHOOZ data,
%...................................
\begin{equation}
\sin^2\theta_{13}=0.012\pm 0.013 \ \ [1\sigma,\ {\mathrm{Atm+LBL(disapp.)+CHOOZ}}]
\end{equation}
%.......................................
where the error scales almost linearly up to $\sim\!\! 3\sigma$, within the
physical range $\sin^2\theta_{13}\geq 0$ (see the magenta long-dashed curve in Fig.~3).
\subsection{Solar and KamLAND data}
In the past, the previous hint for $\theta_{13}>0$ was not corroborated by
solar and KamLAND data, which preferred
$\theta_{13}\simeq 0$ at best fit, both separately and in combination.
This trend has recently changed, however, as a consequence of the
2008 data released by KamLAND~\cite{:2008ee} which, as noted
in~\cite{fogli_NOVE}, as well as in~\cite{Balantekin:2008zm},
prefer values of the mixing angle $\theta_{12}$ somewhat higher
than those indicated by solar data (and especially by the SNO experiment).
As stressed in~\cite{fogli_NOVE,Balantekin:2008zm}, this tension
could be alleviated for $\theta_{13}>0$, as
a result of the different dependence of the survival
probability $P_{ee} = P(\nu_e \to \nu_e)$ on the mixing angles
in KamLAND and SNO, respectively.
In~\cite{fogli_NOVE} we also remarked that such a preference,
although rather small at that time ($\sim 0.5 \sigma$),
could be potentially corroborated by new solar neutrino data.
This has indeed been the case after the inclusion of the new SNO-III
data~\cite{Aharmim:2008kc}, as pointed out in~\cite{Fogli:2008jx}.
Figure~1 shows the regions allowed by our current analysis of solar (S) and
KamLAND (K) neutrino data, for both $\sin^2 \theta_{13}=0$ (left panel) and
a representative non-zero value, $\sin^2\theta_{13}= 0.03$ (right panel).
A comparison of the two panels shows that the S and K best-fit regions tend
to merge as $\sin^2\theta_{13}$ increases (up to values of few percent).
Figure~2 (left panel) shows again the regions separately allowed from S and K
data, but now in the plane spanned by the mixing parameters $(\sin^2\theta_{12},\, \sin^2\theta_{13})$.
Here the $\delta m^2$ parameter is marginalized away in the KamLAND preferred region,
which is equivalent, in practice, to set $\delta m^2$ at its best fit value.
The mixing parameters are positively and negatively correlated in the solar and KamLAND regions,
respectively, as a result of different functional forms for $P_{ee}(\sin^2\theta_{12},\sin^2\theta_{13})$
in the two cases. The S and K allowed regions, which do not overlap at $1\sigma$
for $\theta_{13}=0$, merge for $\sin^2\theta_{13}\sim\mathrm{few}\times 10^{-2}$.
The best fit (dot) and error ellipses for the solar+KamLAND combination are shown
in the middle panel of Fig.~2. A hint of $\theta_{13}>0$ emerges at $\sim\!\!1.2\sigma$ level,
%...................................
\begin{equation}
\sin^2\theta_{13}=0.021\pm 0.017 \ (1\sigma,\ {\mathrm{Solar+KamLAND}}) \ ,
\end{equation}
%.......................................
with errors scaling linearly, to a good approximation, up to $\sim\!\!3\sigma$
(see the green short-dashed curve in Fig.~3).
\subsection{Combination}
The right panel in Fig.~2 shows the $1\sigma$ and $2\sigma$ error ellipses
in the $(\sin^2\theta_{12},\, \sin^2\theta_{13})$ plane from the fit to all data.
The hint of $\theta_{13}>0$ is reinforced in the combination, with
an overall preference emerging at the level of $\sim\!\! 1.6\sigma$ ($\sim\!\! 90\%$ C.L)~\cite{Fogli:2008jx}:
%........................................
\begin{equation}
\sin^2\theta_{13}=0.016\pm 0.010 \ \ (1\sigma, \ \mathrm{all \ data~ 2008})\ ,
\end{equation}
%.......................................
with linearly scaling errors (see the black solid curve in Fig.~3).
Figure~3 summarizes the bounds we obtain on $\theta_{13}$
using different data sets. Note that the preference for nonzero $\theta_{13}$ does not
emerge significantly from solar and KamLAND data {\em taken separately},
but rather from {\em their combination}, which is sensitive to the different ($\theta_{13}$,~$\theta_{12}$)
dependence of the $\nu_e$ survival probability in matter (high-energy solar neutrinos) and
vacuum (KamLAND neutrinos). We remark that, as already stressed in~\cite{Goswami:2004cn},
a preference for nonzero $\theta_{13}$ might also emerge from future solar
neutrinos {\em alone}, by contrasting accurate data at low-energy and high-energy,
sensitive to (averaged) vacuum and matter oscillations, respectively.
In this respect, present (Borexino) and future low-energy experiments
could play an important role.
\subsection{A new hint from MINOS?}
The first MINOS data in the $\nu_\mu \to \nu_e$ appearance channel
have been presented\,\cite{sanchez} few days before this conference.
A weak preference (90\% C.L.) for $\theta_{13}>0$
emerges from the analysis. Although, very prudently, the collaboration does not
emphasize this fact, a combination of their results with ours enhances the
statistical significance of the global hint. Including the new MINOS results,
we estimate
%........................................
\begin{equation}
\sin^2\theta_{13}=0.02\pm 0.01 \ \ (1\sigma, \ \mathrm{all \ data ~2009})\ ,
\end{equation}
%.......................................
with the global hint now reaching the interesting level of 2 sigma (95\% C.L.)
\section{Conclusions}
The latest neutrino data have contributed to increase
our knowledge of neutrino properties. Notably,
they have disclosed the opportunity to probe (at an interesting C.L.\ of 95\%) the
unknown mixing angle $\theta_{13}$. This indication is especially
interesting, because a nonnegligible value of $\theta_{13}$ is required
for successful CP violation searches in the lepton sector.
Lest we be tempted to overestimate the significance of such a weak indication it is salutary
to remark that only future data will tell us if the present hints are heralding an emergent
signal or they are a mere statistical fluctuation. In this respect, our findings are even more interesting,
as they will be subject to direct verification in the near future. Indeed,
further accelerator measurements in the $\nu_\mu\to\nu_e$ appearance
channel, as well as dedicated reactor experiments in the $\nu_e\to\nu_e$
disappearance channel, are expected to provide new relevant information.
\section {Acknowledgments}
G.L.F., E.L., A.M., and A.M.R.\ acknowledge
support by the Italian MIUR and INFN through the ``Astroparticle Physics''
research project, and by the EU ILIAS through the ENTApP project.
A.P.\ acknowledges support by MEC under the I3P program, by Spanish grants
FPA2008-00319/FPA and FPA2008-01935-E/FPA
and ILIAS/N6 Contract RII3-CT-2004-506222.
%\vspace*{-0.5cm}
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\end{thebibliography}
%\verb+\cite+ to refer to the entries in the bibliography so that your
%accumulated list corresponds to the citations made in the text body.
\newpage
%---------------------------------------------------------------------------------------------------------------
%\begin{figure}[t]
%\begin{center}
%\vspace*{10cm}
%\hspace*{-.2cm}
%\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
%\includegraphics[width=25.0pc,height=23.0pc]{fig1.pdf}
%\vspace*{-45pt}
%\caption{\label{fig1} Regions allowed by Solar (S) and KamLAND (K) data,
%and by their combination (S+K), in the $2\nu$ limit ($\theta_{13} = 0$).}
%\vspace*{-0.3cm}
%\end{center}
%\end{figure}
%---------------------------------------------------------------------------------------------------------------
%------------------------------------------------------------------------------------------------
\begin{figure}[t]
\vspace*{-70pt}
\hspace*{4cm}
%\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\includegraphics[width=25.0pc,height=23.0pc]{fig1.pdf}
\vspace*{-50pt} \caption{\label{fig1} Comparison of the regions allowed by solar and KamLAND data
for two fixed values of $\theta_{13}$.}
\vspace*{-0.0cm}
\end{figure}
%-------------------------------------------------------------------------------------------------
%------------------------------------------------------------------------------------------------
\begin{figure}[t]
%begin{center}
%*\vspace*{-70pt}
%\hspace*{-7pt}
%\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\includegraphics[width=40.0pc,height=13.0pc]{fig2.pdf}
%\end{center}
\vspace{-0.5cm}
\caption{\label{fig2}
Allowed regions in the plane $(\sin^2\theta_{12},\,\sin^2\theta_{13})$:
contours at $1\sigma$ (dotted) and $2\sigma$ (solid). Left and middle
panels: solar (S) and KamLAND (K) data, separately (left)
and in combination (middle). In the left panel, the S contours are obtained
by marginalizing the $\delta m^2$ parameter as constrained by KamLAND.
Right panel: All data.}
%\vspace*{+1.3cm}
\end{figure}
%-------------------------------------------------------------------------------------------------
%------------------------------------------------------------------------------------------------
\begin{figure}[t]
\vspace*{-0.92cm}
\hspace*{4.0cm}
%\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\includegraphics[width=23.0pc,height=20.0pc]{fig3.pdf}
\vspace*{-0.5cm} \caption{\label{fig3} Bounds on $\theta_{13}$ obtained using different data sets.}
%\vspace*{-0.5cm}
\end{figure}
%-------------------------------------------------------------------------------------------------
\end{document}
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