\subsection{Neutrino oscillation update} 

A reasonable understanding of the parameters describing three--flavour
neutrino oscillations has been developed from the results of a variety
of experiments involving solar and atmospheric neutrinos, as well as
man-made neutrinos from nuclear power plants and accelerators.
Table~\ref{tab:summary} summarises the results of two recent global
fits to world neutrino data from~\cite{GonzalezGarcia:2010er}
and~\cite{Schwetz:2008er}, both updated as of November 2010.
Details of another recent analysis can be found 
in~\cite{Fogli:2008ig,Fogli:2009ce}.
\begin{table}
  \caption{
    Determination of three--neutrino
    oscillation parameters from 2010 global data~\cite{Schwetz:2008er,
      GonzalezGarcia:2010er}. For $\Delta m^2_{31}$ and
    $\sin^2\theta_{13}$ the upper (lower) row corresponds to inverted
    (normal) neutrino mass hierarchy.
  }
  \label{tab:summary} 
  \centering
  \begin{tabular}{|@{\quad}l@{\quad}|@{\quad}c@{\quad}@{\quad}c@{\quad}|@{\quad}c@{\quad}@{\quad}c@{\quad}|}
    \hline
    & \multicolumn{2}{c|@{\quad}}{Ref.~\cite{GonzalezGarcia:2010er} (updated)}
    & \multicolumn{2}{c|}{Ref.~\cite{Schwetz:2008er} (updated)}
    \\
    parameter
    & best fit$\pm 1\sigma$ & 3$\sigma$ interval
    & best fit$\pm 1\sigma$ & 3$\sigma$ interval
    \\
    \hline
    $\Dmq_{21} ~[10^{-5}~\eVq]$
    & $7.60 \pm 0.20$ & 6.98--8.18
    & $7.59_{-0.17}^{+0.20}$  & 7.09--8.19
    \\[1mm]
    $\Dmq_{31} ~[10^{-3}~\eVq]$ &
    \begin{tabular}{c}
      $-2.35 \pm 0.09$ \\[-1mm]
      $+2.44 \pm 0.09$
    \end{tabular}
    &
    \begin{tabular}{c}
      $-$(2.07--2.65) \\[-1mm]
      $+$(2.17--2.74)
    \end{tabular}
    &
    \begin{tabular}{c}
      $-(2.34_{-0.09}^{+0.10})$ \\[-1mm]
      $+2.45\pm0.09$
    \end{tabular}
    &
    \begin{tabular}{c}
      $-(2.08-2.64)$\\[-1mm]
      $+(2.18-2.73)$
    \end{tabular}
    \\
    $\sin^2\theta_{12}$
    & $0.317_{-0.016}^{+0.017}$ & 0.27--0.37
    & $0.318_{-0.016}^{+0.019}$ & 0.27--0.38
    \\[1mm]
    $\sin^2\theta_{23}$
    & $0.45_{-0.05}^{+0.09}$ & 0.34--0.65
    & $0.51\pm{0.06}$ & 0.39--0.64
    \\[1mm]
    $\sin^2\theta_{13}$
    & $0.009_{-0.008}^{+0.015}$ & $\leq$ 0.048
    &
    \begin{tabular}{c}
      $0.017\pm0.010$ \\[-1mm]
      $0.012^{+0.010}_{-0.007}$
    \end{tabular}
    &
    \begin{tabular}{c}
      $\leq$ 0.048 \\[-1mm]
      $\leq$ 0.042
    \end{tabular}
    \\
    \hline
  \end{tabular}
\end{table}

Spectral information from the KamLAND reactor
experiment~\cite{KamLAND:2008ee, Gando:2010aa} leads to an accurate
determination of $\Dmq_{21}$ with the remarkable precision of 7\% at
$3\sigma$.
KamLAND data also start to
contribute to the lower bound on $\sin^2\theta_{12}$, whereas the upper
bound is dominated by solar data.
From the Sudbury Neutrino Observatory (SNO) we use the data of its
final phase, where the neutrons produced in the neutrino neutral
current (NC) interaction with deuterium are detected mainly by an
array of $^3$He NC detectors (NCD)~\cite{Aharmim:2008kc}, as well as
the recent joint re-analysis of data from Phase I and Phase II
(the pure D$_2$O and salt phases)~\cite{Aharmim:2009gd}. 
In this
analysis, an effective electron kinetic energy threshold of 3.5~MeV
has been used (Low Energy Threshold Analysis, LETA), and the
determination of the total neutrino flux has been improved by about a
factor two.  
These improvements have been possible thanks mainly to
the increased statistics, in particular the NC event sample in the
LETA is increased by about 70\% as a result of the decrease of the
thresholds of 5~MeV and 5.5~MeV used in the Phase~I in Phase~II
analyses respectively. 
Furthermore, energy resolution, background suppression, and
systematic uncertainties have been improved.
Data from SNO are combined with the global data on solar neutrinos
\cite{Cleveland:1998nv,Altmann:2005ix,Hosaka:2005um} including updates
on Gallium experiments \cite{Abdurashitov:2009tn, Kaether:2010ag} as
well as results from Borexino \cite{Arpesella:2008mt} on $^7$Be
neutrinos yielding an improved determination of the atmospheric
parameters.

The MINOS experiment~\cite{Adamson:2008zt} is searching for $\nu_\mu$
disappearance with a baseline of 735~km. 
The latest data were presented at the Neutrino 2010
Conference~\cite{vahle} for the neutrino (7.2$\times10^{20}$ p.o.t.)
and the anti-neutrino (1.71$\times10^{20}$ p.o.t.) running modes. 
The data confirm the energy-dependent disappearance of $\nu_\mu$,
showing significantly fewer events than expected in the case of no
oscillations in the energy range $\lesssim 6$~GeV, whereas above $\sim
6$\,GeV the spectrum is consistent with the no-oscillation
expectation. 
\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{01-PPEG/Figures/fig-minos}
  \caption{
    Allowed regions from recent MINOS data, using neutrinos-only,
    anti-neutrinos-only, and in combination.
  }
  \label{fig:minos} 
\end{figure}

Figure~\ref{fig:minos} shows a re-analysis of MINOS neutrino and
anti-neutrino data. One can see that there is a slight tension between the
neutrino and anti-neutrino results: there is no overlap of the allowed
regions at less than 90\%~CL. However, at $3\sigma$ the results are fully
consistent. We find the following $\chi^2$ minima and goodness-of-fit (GOF)
values:
\begin{equation}
  \begin{array}{rcc}
    \nu:     & \chi^2_{\text{min},\nu} = 24.4/(27-2)     & \text{GOF} = 49.6\%; \\
    \bar\nu: & \chi^2_{\text{min},\bar\nu} = 15.0/(13-2) & \text{GOF} = 18.4\%; \\
    \nu+\bar\nu: & \chi^2_\text{min,tot} = 46.1/(40-2)   & \text{GOF} = 17.3\%.
  \end{array}
\end{equation}
Hence the combined neutrino and anti-neutrino fit still provides a
very good GOF. 
Using the consistency test from reference~\cite{Maltoni:2003cu}
yields $\chi^2_\text{PG} = \chi^2_\text{min,tot} -
\chi^2_{\text{min},\nu} - \chi^2_{\text{min},\bar\nu} = 6.6$. 
The
value of $\chi^2_\text{PG}$ has to be evaluated for 2 degrees of
freedom, which implies that neutrino and anti-neutrino data are
consistent with a probability of 3.7\%. 
This number indicates a slight
tension between the sets, at the level of about $2.1\sigma$. 
In the
global analysis~\cite{Schwetz:2008er} only neutrino data are used from
MINOS, whereas in the analysis in \cite{GonzalezGarcia:2010er} both
neutrino and anti-neutrino data are included. 
It is clear from figure \ref{fig:minos} that the improvement in the
quality of the fit when the anti-neutrino data are added is
negligible.

We combine the long-baseline accelerator data from MINOS as well as
from K2K~\cite{Aliu:2004sq} with atmospheric neutrino measurements
from Super-Kamiokande I+II+III~\cite{Wendell:2010md}.
The determination of $|\Dmq_{31}|$ is dominated by spectral data from
the MINOS experiment, where the sign of $\Dmq_{31}$ (\textit{i.e.},
the neutrino mass hierarchy) is undetermined by present data. 
The
measurement of the mixing angle $\theta_{23}$ is still largely
dominated by atmospheric neutrino data from Super-Kamiokande with a
best-fit point close to maximal mixing. 
Small deviations from maximal
mixing due to sub-leading, three-flavour effects (not included in the
analysis of~\cite{Schwetz:2008er}) have been found in
references~\cite{Fogli:2005cq,GonzalezGarcia:2007ib,GonzalezGarcia:2010er}, 
see table~\ref{tab:summary}. 
A comparison
of these subtle effects can be found in reference~\cite{snow}. 
While an excess of sub-GeV $e$-like data provides an explanation for
the deviations from maximality obtained
in~\cite{Fogli:2005cq,GonzalezGarcia:2010er}, these results are not
statistically significant, and are not confirmed by a recent analysis
including $\Delta m^2_{12}$ effects from the
Super-Kamiokande collaboration \cite{Wendell:2010md}. 
\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{01-PPEG/Figures/fig-the13}
  \caption{
    Dependence of $\Delta\chi^2$ on $\sin^2\theta_{13}$ from various
    data sets, shown for NH (solid) and IH (dashed). 
    The curves labelled ``MINOS'' include disappearance and appearance
    data, and the curves labelled ``atm + LBL'' include Super-K
    atmospheric data + MINOS (disappearance and appearance) + K2K +
    CHOOZ.
  } 
  \label{fig:the13} 
\end{figure}

The third mixing angle $\theta_{13}$ is of crucial importance for
future oscillation experiments.
Figure~\ref{fig:the13} summarises the information on $\theta_{13}$ from
present data, which emerges from an interplay of different data
sets. 
An important contribution to the bound comes from
the CHOOZ reactor experiment~\cite{Apollonio:2002gd} combined with the
determination of $|\Dmq_{31}|$ from atmospheric and long-baseline
experiments.
Using this set of data, a possible hint for a non-zero $\theta_{13}$
from atmospheric data has been found in
references~\cite{Fogli:2005cq,Fogli:2008jx}. 
The origin of such a hint has been investigated in
some detail in reference~\cite{Maltoni:2008ka}, and more recently in
\cite{Fogli:2009ce, GonzalezGarcia:2010er}.
From these results one may conclude that the statistical relevance of
the hint for non-zero $\theta_{13}$ from atmospheric data depends
strongly on the details of the rate calculations and of the $\chi^2$
analysis. 
Furthermore, the origin of this effect might be traced back
to a small excess (at the $1\sigma$ level) in the multi-GeV
electron-like ($e$-like) data sample in SK-I, which however is no
longer present in the combined $\text{SK-I} + \text{SK-II}$ data and
is extremely weak in the combined 
$\text{SK-I} + \text{SK-II} + \text{SK-III}$ data set.

A recent analysis (neglecting sub-leading $\Dmq_{21}$ effects) from the
Super-Kamiokande collaboration finds no evidence of such a
hint~\cite{Wendell:2010md}. 
The results of this analysis have been
used also in the global fit of~\cite{Schwetz:2008er}. 
However, in the
combination of these data with MINOS disappearance and appearance data
a slight preference for $\theta_{13} > 0$, with $\Delta\chi^2 = 1.6
(1.9)$ at $\theta_{13}=0$ for normal hierarchy (inverted hierarchy). 
This happens due to a small
mismatch of the best fit values for $|\Delta m^3_{31}|$ at
$\theta_{13}=0$, which can be resolved by allowing for non-zero values
of $\theta_{13}$.

The MINOS Collaboration has recently reported new data from the search
of $\nu_\mu \to \nu_e$ transitions in the Fermilab NuMI
beam~\cite{Adamson:2010uj}. 
The new data are based on a total exposure
of 7$\times$10$^{20}$ protons-on-target, more than twice the size of
the previous data release~\cite{Adamson:2009yc}. 
The new MINOS far
detector data consists of 54 electron-neutrino events, while,
according to the measurements in the MINOS Near Detector, $49.1 \pm
7.0 (stat) \pm 2.7 (syst)$ background events were expected. 
Hence the
observed number of events is in agreement with background expectations
within 0.7$\sigma$ and the hint for a non-zero value of $\theta_{13}$
present in previous data has largely disappeared.

An important piece of information on $\theta_{13}$ comes from solar
and KamLAND data. 
The relevant survival probabilities are given by:
\begin{equation}\label{eq:Pee}
  P_{ee} \approx
  \begin{cases}
    \cos^4\theta_{13} \left(1- \sin^22\theta_{12}\right \langle \sin^2\phi\rangle)
    & \text{solar, low energies / KamLAND}
    \\
    \cos^4\theta_{13} \, \sin^2\theta_{12}
    & \text{solar, high energies}
  \end{cases}
  \, ;
\end{equation}
where $\phi = \Dmq_{21} L / 4E$ and $\langle \sin^2\phi\rangle \approx
1/2$ for solar neutrinos.  
Equation~(\ref{eq:Pee}) implies an
anti-correlation of $\sin^2\theta_{13}$ and $\sin^2\theta_{12}$ for
KamLAND and low-energy solar neutrinos. 
In contrast, for the high-energy part of the spectrum, which undergoes
the adiabatic MSW conversion inside the sun and which is constrained
by the SNO CC/NC measurement, a positive correlation of
$\sin^2\theta_{13}$ and $\sin^2\theta_{12}$ emerges. 
As discussed
in~\cite{Maltoni:2004ei, Goswami:2004cn}, this complementarity leads
to a non-trivial constraint on $\theta_{13}$ and allows the hint for a
non-zero value of $\theta_{13}$ to be understood, which helps 
to reconcile the slightly different best fit points for $\theta_{12}$
as well as for $\Dmq_{21}$ for solar data and data from KamLAND
separately~\cite{Balantekin:2008zm,Goswami:2004cn,Fogli:2008jx,Maltoni:2003da,GonzalezGarcia:2007ib}.  
We found that the inclusion
of the new solar data, and in particular of the SNO-LETA results, tends
to lower the statistical significance of $\theta_{13}\neq 0$ from this
data set. 
There is a minor dependence of the hint
for $\theta_{13}$ on the solar model, as discussed for example in
\cite{GonzalezGarcia:2010er}. 

From the global data a significance for $\theta_{13} > 0$ of
$1.8\sigma$ for the inverted hierarchy and $1.7\sigma$ for normal the
hierarchy (NH) is obtained in~\cite{Schwetz:2008er} and $1.3\sigma$
in~\cite{GonzalezGarcia:2010er}. 
We find that the inverted hierarchy (IH)
gives a slightly better fit, however, with only $\Delta\chi^2 = 0.7$
\cite{Schwetz:2008er} with respect to the best fit in normal hierarchy
and even less ($\Delta\chi^2 = 0.1$) in \cite{GonzalezGarcia:2010er}.

Recently the MiniBooNE collaboration announced updated results of their
search for $\bar\nu_\mu\to\bar\nu_e$
transitions~\cite{AguilarArevalo:2010wv}. 
In the full energy range from 200~MeV to 3~GeV they find an excess of
$43.2\pm22.5$ events over the expected background (the error includes
statistical and systematical uncertainties). 
In the oscillation-sensitive region of 475~MeV to 1250~MeV the
background-only hypothesis has a probability of only
0.5\%~\cite{AguilarArevalo:2010wv}. 
This result is consistent with the
evidence for $\bar\nu_\mu \to \bar\nu_e$ transitions reported by
LSND~\cite{Aguilar:2001ty} if interpreted in terms of neutrino oscillations,
see figure~\ref{fig:mboone}. 
Any explanation of these hints for
$\bar\nu_\mu \to \bar\nu_e$ transitions at the scale of $E/L \sim 1$~eV$^2$
has to satisfy strong constraints from various experiments. 
First, no
evidence for transitions has been found in MiniBooNE neutrino data above
475~MeV~\cite{AguilarArevalo:2007it}. 
This suggests that CP (or even CPT)
violation has to be invoked to reconcile neutrino and anti-neutrino data.
Second, severe constraints exist for $\bar\nu_e$ \cite{Declais:1994su,
Apollonio:2002gd} and $\nu_\mu, \bar\nu_\mu$ \cite{Dydak:1983zq,
AguilarArevalo:2009yj, Adamson:2010wi, Bilenky:1999ny} disappearance at this
scale, which have to be respected by any explanation of the
$\bar\nu_\mu\to\bar\nu_e$ excesses.
\begin{figure}
  \includegraphics[width=0.8\textwidth]{01-PPEG/Figures/fig-mboone}
  \caption{\label{fig:mboone} Constraint from no-evidence data (NEV)
  compared to the combined allowed regions from LSND and MiniBooNE $\bar\nu$
  data (shaded) at 90\% and 99\% CL for (3+1) oscillations. We also show the
  individual regions from LSND and MiniBooNE $\bar\nu$ data.}
\end{figure}

The standard approach to the LSND problem is to introduce one or more
sterile neutrinos at the eV scale. 
Adding one sterile neutrino one obtains
the so-called (3+1) mass scheme. 
In this framework there is no CP violation
at short baselines, and disappearance experiments strongly disfavour an
explanation of the appearance signals, see for
example~\cite{Maltoni:2002xd}. 
The latest situation is summarised in
figure~\ref{fig:mboone} (taken from \cite{Akhmedov:2010vy}), which shows
the well known tension in the (3+1) oscillation fit: the region indicated
from LSND combined with MiniBooNE anti-neutrino data (shaded region) and the
constraint from all other experiments seeing no positive signal (blue
curves) touch each other at $\Delta\chi^2 = 12.7$, which corresponds to
99.8\%~CL for 2 degrees of freedom.
If two neutrino mass states at the eV scale are present \cite{Peres:2000ic,
Sorel:2003hf} ((3+2) scheme), the possibility of CP violation opens
up~\cite{Karagiorgi:2006jf}, which allows the LSND and MiniBooNE
neutrino data to be reconciled \cite{Maltoni:2007zf}. 
However, constraints from disappearance
data still impose a challenge to the fit, and the overall improvement with
respect to the (3+1) case is not significant~\cite{Maltoni:2007zf,
Karagiorgi:2009nb}, see \cite{Akhmedov:2010vy} for updated (3+2) results.

Apart from sterile-neutrino oscillations, various more exotic explanations
of the LSND signal have been proposed, among them, sterile-neutrino
decay~\cite{Ma:1999im, PalomaresRuiz:2005vf, Gninenko:2010pr}, CPT
violation~\cite{Murayama:2000hm, Barenboim:2002ah, GonzalezGarcia:2003jq,
Barger:2003xm}, violation of Lorentz symmetry~\cite{Kostelecky:2004hg,
deGouvea:2006qd, Katori:2006mz}, quantum decoherence~\cite{Barenboim:2004wu,
Farzan:2008zv}, mass-varying neutrinos~\cite{Kaplan:2004dq, Barger:2005mh},
shortcuts of sterile neutrinos in extra dimensions~\cite{Pas:2005rb},
sterile neutrino oscillations with a non-standard energy
dependence~\cite{Schwetz:2007cd}, sterile neutrinos plus non-standard
interactions~\cite{Akhmedov:2010vy,Nelson:2010hz}. 
Some of these proposals
involve very speculative physics, and many of them fail to describe all data
consistently.

Unfortunately the LSND puzzle is far from solved. 
The statistical
significance of MiniBooNE data is not high enough to settle this issue
unambiguously, and recent anti-neutrino data added more confusion on the
topic. 
Moreover, the event excess in the low energy region observed at about
$3\sigma$ in MiniBooNE neutrino data \cite{AguilarArevalo:2007it} provides
another puzzle, which might or might not be related to the LSND one.
Currently only global analyses relying on various different data sets can
address the LSND issue, with disappearance experiments playing a crucial
role. 
In view of future high precision oscillation experiments it would be
highly desirable to understand better whether there is interesting neutrino
physics happening at the scale $E_\nu/L \sim 1$~eV$^2$.



