\section{Atmospheric Neutrinos}
\label{sec:Phys-Atm-neut}
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%\REDBLA{Creation by JEC 27/4/06 waiting for M. Maltoni Draft $\sim$22May}
%\REDBLA{Update by JEC 22/6/06}
%\REDBLA{Update by JEC 16/10/06: this is a section now}
%\REDBLA{Update by AB + JEC 3/11/06 : subsectioning + tau-neutrinos}
\subsection{Introduction}
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%use \refTab{} and \refFig{} commands to reference Tables and Figures. 
%JEC 22/6/06 START: contribution from Michele Maltoni
Atmospheric neutrinos originates from the decay chain initiated by the
collision of cosmic rays with the upper layers of the Earth's
atmosphere.
%
\begin{figure}
    \includegraphics[width=\columnwidth]{./figures/fig.octant.eps}
    \caption{ \label{fig:octant} %
      Discrimination of the wrong octant solution as a function of
      $\sin^2\theta_{23}^\mathrm{true}$, for
      $\theta_{13}^\mathrm{true} = 0$. We have assumed 10 years of
      data taking with a 440-kton detector.}
\end{figure}
The hadronic interaction between primary cosmic rays (mainly protons
and helium nuclei) and the light atmosphere nuclei produces secondary
$\pi$ and $K$ mesons, which then decay giving electron and muon
neutrinos and antineutrinos.
%
At lower energies the main contribution comes from $\pi$ mesons, and
the decay chain $\pi \to \mu + \nu_\mu$ followed by $\mu \to e + \nu_e
+ \nu_\mu$ produces essentially two $\nu_\mu$ for each $\nu_e$.  As
the energy increases, more and more muons reach the ground before
decays, and therefore the $\nu_\mu / \nu_e$ ratio increases.
%
For $E_\nu \gtrsim 1$~GeV the dependence of the total neutrino flux on
the neutrino energy is well described by a power law, $d\Phi / d_E
\propto E^{-\gamma}$ with $\gamma = 3$ for $\nu_\mu$ and $\gamma=3.5$
for $\nu_e$, whereas at sub-GeV energies the dependence becomes more
complicated because of the effects of the solar wind and of the
Earth's magnetic field~\cite{Gonzalez-Garcia:2002dz}. As for the
zenith dependence, for energies larger than a few GeV the neutrino
flux is enhanced in the horizontal direction since pions and muons can
travel a longer distance before reaching the ground, and therefore
have more chances to decay producing neutrinos.

Historically, the atmospheric neutrino problem originated in the
1980's as a discrepancy between the atmospheric neutrino flux measured
with different experimental techniques. In the previous years, a
number of detectors had been built, which could detect neutrinos
through the observation of the charged lepton produced in
charged-current neutrino-nucleon interactions inside the detector
itself.
%
These detectors could be divided into two classes: \emph{iron
calorimeters}, which reconstructed the track or electromagnetic shower
produced by the lepton, and \emph{water \v{C}erenkov}, which measured
instead the Cerenkov light emitted by the lepton as it moved faster
than light in water.
%
The oldest iron calorimeters, Frejus \cite{Daum:1994bf} and
NUSEX \cite{Aglietta:1988be}, found no discrepancy between the
observed flux and the theoretical predictions, whereas the two \WC\ detectors, IMB \cite{Becker-Szendy:1992hq} and
Kamiokande \cite{Hirata:1992ku}, observed a clear deficit in the
predicted $\nu_\mu / \nu_e$ ratio.
%
The problem was finally solved in 1998, when the water Cerenkov
detector SuperKamiokande \cite{Fukuda:1998mi} established with high
statistical accuracy that there was indeed a zenith- and
energy-dependent deficit in the muon neutrino flux with respect to the
theoretical predictions, and that this deficit was compatible with the
hypothesis of mass-induced $\nu_\mu \to \nu_\tau$ oscillations. Also,
the independent confirmation of this effect from the iron calorimeter
experiments Soudan-II \cite{Allison:1999ms} and
MACRO \cite{Ambrosio:2001je} eliminated the discrepancy between the
two experimental techniques.

Despite providing the first solid evidence for neutrino oscillations,
atmospheric neutrino experiments have received only minor
consideration during the last years. This is mainly due to two
important limitations:
%
\begin{itemize}
  \item the sensitivity of an atmospheric neutrino experiments is
    strongly limited by the large uncertainties in the knowledge of
    neutrino fluxes and neutrino-nucleon cross-section. Such
    uncertainties can be as large as 20\%.
    
  \item in general, water Cerenkov detectors do not allow an accurate
    reconstruction of the neutrino energy and direction if none of the
    two is known ``a priori''. This strongly limits the sensitivity to
    $\Delta m^2$, which is very sensitive to the resolution on $L/E$.
\end{itemize}
%
During its phase-I, Super-Kamiokande has collected 4099 electron-like
and 5436 muon-like contained neutrino events \cite{Ashie:2005ik}. With
only about a hundred events each, K2K \cite{Ahn:2006zz} and
MINOS \cite{Tagg:2006sx} already provide a stronger bound on the
atmospheric mass-squared difference $\Delta m_{31}^2$. The present
value of the mixing angle $\theta_{23}$ is still dominated by
Super-Kamiokande data, being statistics the most important factor for
such a measurement, but strong improvements are expected from the next
generation of long-baseline experiments T2K \cite{Itow:2001ee} and
NO$\nu$A \cite{Ayres:2004js}.


\begin{figure}
    \includegraphics[width=\columnwidth]{./figures/SPLBBMEMPHYS-fig16.eps}
    \caption{ \label{fig:hierarchy} %
      Sensitivity to the mass hierarchy at $2\sigma$ ($\Delta\chi^2 =
      4$) as a function of $\sin^22\theta_{13}^\mathrm{true}$ and
      $\delta_\mathrm{CP}^\mathrm{true}$ (left), and the fraction of
      true values of $\delta_\mathrm{CP}^\mathrm{true}$ (right). The
      solid curves are the sensitivities from the combination of
      long-baseline and atmospheric neutrino data, the dashed curves
      correspond to long-baseline data only. We have assumed 10 years
      of data taking with a 440-kton detector.}
\end{figure}
%
\subsection{Oscillation physics}
%
Despite these drawbacks, atmospheric detectors can still play a
leading role in the future of neutrino physics due to the huge range
in energy (from 100~MeV to 10~TeV and above) and distance (from 20~km
\begin{figure}
    \includegraphics[width=\columnwidth]{./figures/fig.theta13.eps}
    \caption{ \label{fig:theta13} %
      Sensitivity to $\sin^22\theta_{13}$ as a function of
      $\sin^2\theta_{23}^\mathrm{true}$ for LBL data only (dashed),
      and the combination LBL+ATM (solid). In the left and central
      panels we restrict the fit of $\theta_{23}$ to the octant
      corresponding to $\theta_{23}^\mathrm{true}$ and $\pi/2 -
      \theta_{23}^\mathrm{true}$, respectively, whereas the right
      panel shows the overall sensitivity taking into account both
      octants. We have assumed 8 years of LBL and 9 years of ATM data
      taking with the T2HK beam and a 1~Mton detector.}
\end{figure}
to more than 12000~Km) covered by the data. This unique feature, as
well as the very large statistics expected for a detector such as
MEMPHYS ($20\div 30$ times the present SK event rate), will allow a
very accurate study of \emph{subdominant modifications} to the leading
oscillation pattern, thus providing complementary information to
accelerator-based experiments. More concretely, atmospheric neutrino
data will be extremely valuable for:
%
\begin{itemize}
  \item resolving the octant ambiguity: although future LBL
    experiments are expected to considerably improve the measurement
    of the absolute value of the small quantity $D_{23} \equiv
    \sin^2\theta_{23} - 1/2$, they will have practically no
    sensitivity on its sign.  On the other hands, it has been pointed
    out \cite{Kim:1998bv,Peres:1999yi} that the $\nu_\mu \to \nu_e$ conversion
    signal induced by the small but finite value of $\Delta m_{21}^2$
    can resolve this degeneracy. However, observing such a conversion
    requires a very long baseline and low energy neutrinos, and
    atmospheric sub-GeV electron-like events are particularly suitable
    for this purpose. In \refFig{fig:octant} we show the potential
    of different ATM+LBL experiments to exclude the octant degenerate
    solution.

  \item resolving the hierarchy degeneracy: if $\theta_{13}$ is not
    too small, matter effect will produce resonant conversion in the
    $\nu_\mu \leftrightarrow \nu_e$ channel for neutrinos
    (antineutrinos) if the mass hierarchy is normal (inverted). The
    observation of this enhanced conversion would allow the
    determination of the mass hierarchy. Although a magnetized
    detector would be the best solution for this task, it is possible
    to extract useful information also with a conventional detector
    since the event rates expected for atmospheric neutrinos and
    antineutrinos are quite different. This is clearly visible from
    \refFig{fig:hierarchy}, where we show how the sensitivity to the
    mass hierarchy of different LBL experiments is drastically
    increased when the ATM data collected by the same detector are
    also included in the fit.

  \item measuring or improving the bound on $\theta_{13}$: although
    atmospheric data alone are not expected to be competitive with the
    next generation of long-baseline experiments in the sensitivity to
    $\theta_{13}$, they will contribute indirectly by eliminating the
    octant degeneracy, which is an important source of uncertainty for
    LBL. In particular, if $\theta_{23}^\mathrm{true}$ is larger than
    $45^\circ$ then the inclusion of atmospheric data will
    considerably improve the LBL sensitivity to $\theta_{13}$, as can
    be seen from the right panel of \refFig{fig:theta13} \cite{huber-2005-71}.

%JEC 3/11/06 START place it at the end of the section
%  \item searching for physics beyond the Standard Model: the appearance
%    of subleading features in the main oscillation pattern can also be
%    a hint for New Physics. The huge range of energies probed by
%    atmospheric data will allow to put very strong bounds on
%    mechanisms which predict deviation from the $1/E$ behavior. For
%    example, the bound on non-standard neutrino-matter interactions
%    and on other types of New Physics (such as violation of the
%    equivalence principle, or violation of the Lorentz invariance)
%    which can be derived from \emph{present} data is already the
%    strongest which can be put on these
%    mechanisms \cite{Gonzalez-Garcia:2004wg}. The increased statistics
%    expected for MEMPHYS will further improve these constraints.
%JEC 3/11/06 END 
\end{itemize}
%
%A Bueno 3/11/06 START new subsection
\subsection{Direct detection of $\nu_\tau$ in the atmospheric neutrino flux}
%
At energies above a GeV, 
we expect unoscillated events to be upward-downward going symmetric.
In contrast, we know that $\nu_\tau, \ \bar{\nu}_\tau$ induced events come from 
below the horizon (upward going events). Therefore 
the presence of $\nu_\tau$, $\bar{\nu}_\tau$ events can be revealed by a
measured excess of upward going events.
Hereafter we assume that the {$\nu_\mu$} and 
the {$\mathbf \nu_\tau$} are maximally mixed and their mass 
squared difference
is {$ \Delta m^2 = 3. \times 10^{-3}$} eV{$^2$}.
We use the Fluka 3D atmospheric neutrino fluxes.

In GLACIER, the search for $\nu_\tau$ appearance is based on the
information provided by the event kinematics and takes advantage
of the special characteristics of $\nu_\tau$ CC and the subsequent
decay of the produced $\tau$ lepton when compared to CC and NC interactions 
of $\nu_\mu$ and $\nu_e$, i.e. by making use of $\vec{P}_{candidate}$ 
and $\vec{P}_{hadron}$. Due to the large background induced by the natural 
abundance of the atmospheric neutrino flux in $\nu_e$ and
$\bar{\nu}_e$, we note that the measurement of a statistically
significant excess of 
$\nu_\tau$ events is very unlikely for the $\tau \to e$ decay mode, 
therefore we conclude that a search 
based on this channel is hopeless. Same conclusions apply to 
the muonic decay channel.

The situation is much more advantageous for the hadronic channels: 
we consider tau decays to one prong (single pion, rho) and to three
prongs ($\pi^\pm \pi^0 \pi^0 $ and three charged pions). 
After a careful evaluation of the performance of different
combinations of kinematic variables, we decided to use: $E_{visible}$, 
$y_{bj}$ (the ratio between the total hadronic energy and
$E_{visible}$) and $Q_T$ (defined as the transverse momentum of the $\tau$
candidate with respect to the total measured momentum). The chosen
variables are not independent one from another but show
correlations between them. These correlations can be exploited to reduce the
background. In order to maximize the separation between signal 
and background, we use three dimensional likelihood functions 
${\cal L}(Q_T,E_{visible}, y_{bj})$ where
correlations are taken into account. For every channel, we build three
dimensional likelihood functions 
for both signal (${\cal L}^S_\pi, \ {\cal L}^S_\rho, \ 
{\cal L}^S_{3\pi}$) and background (${\cal L}^B_\pi, \ {\cal L}^B_\rho, \ 
{\cal L}^B_{3\pi}$). To enhance the separation of $\nu_\tau$ induced 
events from $\nu_\mu, \ \nu_e$ interactions, we take a ratio of 
likelihoods as the sole discriminant variable: 
\begin{equation}
\ln \lambda_i \equiv \ln({\cal L}^S_i / {\cal L}^B_i)
\end{equation}
where $i=\pi,\ \rho, \ 3\pi$.

To further improve the sensitivity of the $\nu_\tau$
appearance search, we combine
the three independent hadronic analyses into a single one.
Events that are common to at least 
two analyses are counted only once and a survey of all possible
combinations, for a restricted set of  values of the likelihood 
ratios, is performed. Table \ref{tab:combi} illustrates the 
statistical significance achieved by several selected combinations of the 
likelihood ratios for an exposure equivalent to 100 kton$\times$year. 

\begin{table}
\caption{\label{tab:combi}Expected background and signal events for different
combinations of the $\pi$, $\rho$ and $3\pi$ analyses. The considered
statistical sample corresponds to an exposure of 100
kton$\times$year. The best
combination found is indicated in bold characters.}
\begin{center}
\begin{tabular}{cccclc}\hline\hline
$\ln \lambda_\pi$ & $\ln \lambda_\rho$ & $\ln \lambda_{3\pi}$ & 
Top & Bottom & $P_\beta$ ($\%$) \\
Cut & Cut & Cut & Events & Events &  \\ \hline
0. & 0.5 & 0. & 223 & $223 + 43 = 266$ & $2 \times 10^{-1}$ 
($3.1\sigma$)\\
1.5. & 1.5 & 0 & 92 & $92 + 35= 127$ & $2 \times 10^{-2}$ ($3.7\sigma$)\\
3. & -1 & 0. & 87 & $87 + 33 = 120 $ & $3 \times 10^{-2}$ 
($3.6\sigma$)\\
3. & 0.5 & 0. & 25 & {$25 + 22= 47$}
& {$2 \times 10^{-3}$ $(4.3\sigma)$} \\ 
3. & 1.5 & 0 & 20 & $20 + 19 = 39$ & $4 \times 10^{-3}$ ($4.1\sigma$)\\
3. & 0.5 & -1. & 59 & $59 + 30 = 89$ & $9 \times 10^{-3}$ ($3.9\sigma$)\\
3. & 0.5 & 1. & 18 & $18 + 17 = 35$ & $1 \times 10^{-2}$ ($3.8\sigma$)\\ \hline\hline
\end{tabular}
\end{center}
\end{table}
The best combination, for a 100 kton$\times$year exposure, 
is achieved for the 
following set of cuts: {$\ln \lambda_\pi > 3$, 
$\ln \lambda_\rho > 0.5$} and {$\ln \lambda_{3\pi} > 0$}. 
The expected number of NC background events amounts to 25 (top) 
while 25+22 = 47 (bottom) are expected. $P_\beta$ is the Poisson probability 
for the measured excess of upward going events to be due to a
statistical fluctuation as a function of the exposure. We have 
an effect larger than $4\sigma$ for an 
exposure of 100 kton$\times$year (one year of data taking with GLACIER).
%A Bueno 3/11/06 START
%
% JEC 3/11/06 START new section
\subsection{New phenomena beyond the "Standard Model"}
%
It is worth remembering that atmospheric neutrino fluxes are
themselves an important subject of investigation, and at the light of
the precise determination of the oscillation parameters provided by
long-baseline experiments the atmospheric neutrino data accumulated by
the proposed detectors can be used as a \emph{direct measurement} of the incoming
neutrino flux, and therefore as an indirect measurement of the primary
cosmic rays flux. 

The appearance
    of subleading features in the main oscillation pattern can also be
    a hint for New Physics. The huge range of energies probed by
    atmospheric data will allow to put very strong bounds on
    mechanisms which predict deviation from the $1/E$ behavior. For
    example, the bound on non-standard neutrino-matter interactions
    and on other types of New Physics (such as violation of the
    equivalence principle, or violation of the Lorentz invariance)
    which can be derived from \emph{present} data is already the
    strongest which can be put on these
    mechanisms \cite{Gonzalez-Garcia:2004wg}. So, the increased statistics
    expected for the proposed detectors will further improve these constraints.
% JEC 3/11/06 END 
%JEC 22/6/06 END 
