[416] | 1 | \section{Atmospheric Neutrinos} |
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| 2 | \label{sec:Phys-Atm-neut} |
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| 3 | % |
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| 4 | %\REDBLA{Creation by JEC 27/4/06 waiting for M. Maltoni Draft $\sim$22May} |
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| 5 | %\REDBLA{Update by JEC 22/6/06} |
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| 6 | %\REDBLA{Update by JEC 16/10/06: this is a section now} |
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| 7 | %\REDBLA{Update by AB + JEC 3/11/06 : subsectioning + tau-neutrinos} |
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| 8 | \subsection{Introduction} |
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| 9 | %% |
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| 10 | %use \refTab{} and \refFig{} commands to reference Tables and Figures. |
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| 11 | %JEC 22/6/06 START: contribution from Michele Maltoni |
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| 12 | Atmospheric neutrinos originates from the decay chain initiated by the |
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| 13 | collision of cosmic rays with the upper layers of the Earth's |
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| 14 | atmosphere. |
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| 15 | % |
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| 16 | \begin{figure} |
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| 17 | \includegraphics[width=\columnwidth]{./figures/fig.octant.eps} |
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| 18 | \caption{ \label{fig:octant} % |
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| 19 | Discrimination of the wrong octant solution as a function of |
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| 20 | $\sin^2\theta_{23}^\mathrm{true}$, for |
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| 21 | $\theta_{13}^\mathrm{true} = 0$. We have assumed 10 years of |
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| 22 | data taking with a 440-kton detector.} |
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| 23 | \end{figure} |
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| 24 | The hadronic interaction between primary cosmic rays (mainly protons |
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| 25 | and helium nuclei) and the light atmosphere nuclei produces secondary |
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| 26 | $\pi$ and $K$ mesons, which then decay giving electron and muon |
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| 27 | neutrinos and antineutrinos. |
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| 28 | % |
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| 29 | At lower energies the main contribution comes from $\pi$ mesons, and |
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| 30 | the decay chain $\pi \to \mu + \nu_\mu$ followed by $\mu \to e + \nu_e |
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| 31 | + \nu_\mu$ produces essentially two $\nu_\mu$ for each $\nu_e$. As |
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| 32 | the energy increases, more and more muons reach the ground before |
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| 33 | decays, and therefore the $\nu_\mu / \nu_e$ ratio increases. |
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| 34 | % |
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| 35 | For $E_\nu \gtrsim 1$~GeV the dependence of the total neutrino flux on |
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| 36 | the neutrino energy is well described by a power law, $d\Phi / d_E |
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| 37 | \propto E^{-\gamma}$ with $\gamma = 3$ for $\nu_\mu$ and $\gamma=3.5$ |
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| 38 | for $\nu_e$, whereas at sub-GeV energies the dependence becomes more |
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| 39 | complicated because of the effects of the solar wind and of the |
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| 40 | Earth's magnetic field~\cite{Gonzalez-Garcia:2002dz}. As for the |
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| 41 | zenith dependence, for energies larger than a few GeV the neutrino |
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| 42 | flux is enhanced in the horizontal direction since pions and muons can |
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| 43 | travel a longer distance before reaching the ground, and therefore |
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| 44 | have more chances to decay producing neutrinos. |
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| 45 | |
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| 46 | Historically, the atmospheric neutrino problem originated in the |
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| 47 | 1980's as a discrepancy between the atmospheric neutrino flux measured |
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| 48 | with different experimental techniques. In the previous years, a |
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| 49 | number of detectors had been built, which could detect neutrinos |
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| 50 | through the observation of the charged lepton produced in |
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| 51 | charged-current neutrino-nucleon interactions inside the detector |
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| 52 | itself. |
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| 53 | % |
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| 54 | These detectors could be divided into two classes: \emph{iron |
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| 55 | calorimeters}, which reconstructed the track or electromagnetic shower |
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| 56 | produced by the lepton, and \emph{water \v{C}erenkov}, which measured |
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| 57 | instead the Cerenkov light emitted by the lepton as it moved faster |
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| 58 | than light in water. |
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| 59 | % |
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| 60 | The oldest iron calorimeters, Frejus \cite{Daum:1994bf} and |
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| 61 | NUSEX \cite{Aglietta:1988be}, found no discrepancy between the |
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| 62 | observed flux and the theoretical predictions, whereas the two \WC\ detectors, IMB \cite{Becker-Szendy:1992hq} and |
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| 63 | Kamiokande \cite{Hirata:1992ku}, observed a clear deficit in the |
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| 64 | predicted $\nu_\mu / \nu_e$ ratio. |
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| 65 | % |
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| 66 | The problem was finally solved in 1998, when the water Cerenkov |
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| 67 | detector SuperKamiokande \cite{Fukuda:1998mi} established with high |
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| 68 | statistical accuracy that there was indeed a zenith- and |
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| 69 | energy-dependent deficit in the muon neutrino flux with respect to the |
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| 70 | theoretical predictions, and that this deficit was compatible with the |
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| 71 | hypothesis of mass-induced $\nu_\mu \to \nu_\tau$ oscillations. Also, |
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| 72 | the independent confirmation of this effect from the iron calorimeter |
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| 73 | experiments Soudan-II \cite{Allison:1999ms} and |
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| 74 | MACRO \cite{Ambrosio:2001je} eliminated the discrepancy between the |
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| 75 | two experimental techniques. |
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| 76 | |
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| 77 | Despite providing the first solid evidence for neutrino oscillations, |
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| 78 | atmospheric neutrino experiments have received only minor |
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| 79 | consideration during the last years. This is mainly due to two |
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| 80 | important limitations: |
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| 81 | % |
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| 82 | \begin{itemize} |
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| 83 | \item the sensitivity of an atmospheric neutrino experiments is |
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| 84 | strongly limited by the large uncertainties in the knowledge of |
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| 85 | neutrino fluxes and neutrino-nucleon cross-section. Such |
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| 86 | uncertainties can be as large as 20\%. |
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| 87 | |
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| 88 | \item in general, water Cerenkov detectors do not allow an accurate |
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| 89 | reconstruction of the neutrino energy and direction if none of the |
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| 90 | two is known ``a priori''. This strongly limits the sensitivity to |
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| 91 | $\Delta m^2$, which is very sensitive to the resolution on $L/E$. |
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| 92 | \end{itemize} |
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| 93 | % |
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| 94 | During its phase-I, Super-Kamiokande has collected 4099 electron-like |
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| 95 | and 5436 muon-like contained neutrino events \cite{Ashie:2005ik}. With |
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| 96 | only about a hundred events each, K2K \cite{Ahn:2006zz} and |
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| 97 | MINOS \cite{Tagg:2006sx} already provide a stronger bound on the |
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| 98 | atmospheric mass-squared difference $\Delta m_{31}^2$. The present |
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| 99 | value of the mixing angle $\theta_{23}$ is still dominated by |
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| 100 | Super-Kamiokande data, being statistics the most important factor for |
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| 101 | such a measurement, but strong improvements are expected from the next |
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| 102 | generation of long-baseline experiments T2K \cite{Itow:2001ee} and |
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| 103 | NO$\nu$A \cite{Ayres:2004js}. |
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| 104 | |
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| 105 | |
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| 106 | \begin{figure} |
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| 107 | \includegraphics[width=\columnwidth]{./figures/SPLBBMEMPHYS-fig16.eps} |
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| 108 | \caption{ \label{fig:hierarchy} % |
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| 109 | Sensitivity to the mass hierarchy at $2\sigma$ ($\Delta\chi^2 = |
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| 110 | 4$) as a function of $\sin^22\theta_{13}^\mathrm{true}$ and |
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| 111 | $\delta_\mathrm{CP}^\mathrm{true}$ (left), and the fraction of |
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| 112 | true values of $\delta_\mathrm{CP}^\mathrm{true}$ (right). The |
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| 113 | solid curves are the sensitivities from the combination of |
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| 114 | long-baseline and atmospheric neutrino data, the dashed curves |
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| 115 | correspond to long-baseline data only. We have assumed 10 years |
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| 116 | of data taking with a 440-kton detector.} |
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| 117 | \end{figure} |
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| 118 | % |
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| 119 | \subsection{Oscillation physics} |
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| 120 | % |
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| 121 | Despite these drawbacks, atmospheric detectors can still play a |
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| 122 | leading role in the future of neutrino physics due to the huge range |
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| 123 | in energy (from 100~MeV to 10~TeV and above) and distance (from 20~km |
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| 124 | \begin{figure} |
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| 125 | \includegraphics[width=\columnwidth]{./figures/fig.theta13.eps} |
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| 126 | \caption{ \label{fig:theta13} % |
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| 127 | Sensitivity to $\sin^22\theta_{13}$ as a function of |
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| 128 | $\sin^2\theta_{23}^\mathrm{true}$ for LBL data only (dashed), |
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| 129 | and the combination LBL+ATM (solid). In the left and central |
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| 130 | panels we restrict the fit of $\theta_{23}$ to the octant |
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| 131 | corresponding to $\theta_{23}^\mathrm{true}$ and $\pi/2 - |
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| 132 | \theta_{23}^\mathrm{true}$, respectively, whereas the right |
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| 133 | panel shows the overall sensitivity taking into account both |
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| 134 | octants. We have assumed 8 years of LBL and 9 years of ATM data |
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| 135 | taking with the T2HK beam and a 1~Mton detector.} |
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| 136 | \end{figure} |
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| 137 | to more than 12000~Km) covered by the data. This unique feature, as |
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| 138 | well as the very large statistics expected for a detector such as |
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| 139 | MEMPHYS ($20\div 30$ times the present SK event rate), will allow a |
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| 140 | very accurate study of \emph{subdominant modifications} to the leading |
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| 141 | oscillation pattern, thus providing complementary information to |
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| 142 | accelerator-based experiments. More concretely, atmospheric neutrino |
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| 143 | data will be extremely valuable for: |
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| 144 | % |
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| 145 | \begin{itemize} |
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| 146 | \item resolving the octant ambiguity: although future LBL |
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| 147 | experiments are expected to considerably improve the measurement |
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| 148 | of the absolute value of the small quantity $D_{23} \equiv |
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| 149 | \sin^2\theta_{23} - 1/2$, they will have practically no |
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| 150 | sensitivity on its sign. On the other hands, it has been pointed |
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| 151 | out \cite{Kim:1998bv,Peres:1999yi} that the $\nu_\mu \to \nu_e$ conversion |
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| 152 | signal induced by the small but finite value of $\Delta m_{21}^2$ |
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| 153 | can resolve this degeneracy. However, observing such a conversion |
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| 154 | requires a very long baseline and low energy neutrinos, and |
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| 155 | atmospheric sub-GeV electron-like events are particularly suitable |
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| 156 | for this purpose. In \refFig{fig:octant} we show the potential |
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| 157 | of different ATM+LBL experiments to exclude the octant degenerate |
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| 158 | solution. |
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| 159 | |
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| 160 | \item resolving the hierarchy degeneracy: if $\theta_{13}$ is not |
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| 161 | too small, matter effect will produce resonant conversion in the |
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| 162 | $\nu_\mu \leftrightarrow \nu_e$ channel for neutrinos |
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| 163 | (antineutrinos) if the mass hierarchy is normal (inverted). The |
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| 164 | observation of this enhanced conversion would allow the |
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| 165 | determination of the mass hierarchy. Although a magnetized |
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| 166 | detector would be the best solution for this task, it is possible |
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| 167 | to extract useful information also with a conventional detector |
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| 168 | since the event rates expected for atmospheric neutrinos and |
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| 169 | antineutrinos are quite different. This is clearly visible from |
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| 170 | \refFig{fig:hierarchy}, where we show how the sensitivity to the |
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| 171 | mass hierarchy of different LBL experiments is drastically |
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| 172 | increased when the ATM data collected by the same detector are |
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| 173 | also included in the fit. |
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| 174 | |
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| 175 | \item measuring or improving the bound on $\theta_{13}$: although |
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| 176 | atmospheric data alone are not expected to be competitive with the |
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| 177 | next generation of long-baseline experiments in the sensitivity to |
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| 178 | $\theta_{13}$, they will contribute indirectly by eliminating the |
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| 179 | octant degeneracy, which is an important source of uncertainty for |
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| 180 | LBL. In particular, if $\theta_{23}^\mathrm{true}$ is larger than |
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| 181 | $45^\circ$ then the inclusion of atmospheric data will |
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| 182 | considerably improve the LBL sensitivity to $\theta_{13}$, as can |
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| 183 | be seen from the right panel of \refFig{fig:theta13} \cite{huber-2005-71}. |
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| 184 | |
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| 185 | %JEC 3/11/06 START place it at the end of the section |
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| 186 | % \item searching for physics beyond the Standard Model: the appearance |
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| 187 | % of subleading features in the main oscillation pattern can also be |
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| 188 | % a hint for New Physics. The huge range of energies probed by |
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| 189 | % atmospheric data will allow to put very strong bounds on |
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| 190 | % mechanisms which predict deviation from the $1/E$ behavior. For |
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| 191 | % example, the bound on non-standard neutrino-matter interactions |
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| 192 | % and on other types of New Physics (such as violation of the |
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| 193 | % equivalence principle, or violation of the Lorentz invariance) |
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| 194 | % which can be derived from \emph{present} data is already the |
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| 195 | % strongest which can be put on these |
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| 196 | % mechanisms \cite{Gonzalez-Garcia:2004wg}. The increased statistics |
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| 197 | % expected for MEMPHYS will further improve these constraints. |
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| 198 | %JEC 3/11/06 END |
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| 199 | \end{itemize} |
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| 200 | % |
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| 201 | %A Bueno 3/11/06 START new subsection |
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| 202 | \subsection{Direct detection of $\nu_\tau$ in the atmospheric neutrino flux} |
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| 203 | % |
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| 204 | At energies above a GeV, |
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| 205 | we expect unoscillated events to be upward-downward going symmetric. |
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| 206 | In contrast, we know that $\nu_\tau, \ \bar{\nu}_\tau$ induced events come from |
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| 207 | below the horizon (upward going events). Therefore |
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| 208 | the presence of $\nu_\tau$, $\bar{\nu}_\tau$ events can be revealed by a |
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| 209 | measured excess of upward going events. |
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| 210 | Hereafter we assume that the {$\nu_\mu$} and |
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| 211 | the {$\mathbf \nu_\tau$} are maximally mixed and their mass |
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| 212 | squared difference |
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| 213 | is {$ \Delta m^2 = 3. \times 10^{-3}$} eV{$^2$}. |
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| 214 | We use the Fluka 3D atmospheric neutrino fluxes. |
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| 215 | |
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| 216 | In GLACIER, the search for $\nu_\tau$ appearance is based on the |
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| 217 | information provided by the event kinematics and takes advantage |
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| 218 | of the special characteristics of $\nu_\tau$ CC and the subsequent |
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| 219 | decay of the produced $\tau$ lepton when compared to CC and NC interactions |
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| 220 | of $\nu_\mu$ and $\nu_e$, i.e. by making use of $\vec{P}_{candidate}$ |
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| 221 | and $\vec{P}_{hadron}$. Due to the large background induced by the natural |
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| 222 | abundance of the atmospheric neutrino flux in $\nu_e$ and |
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| 223 | $\bar{\nu}_e$, we note that the measurement of a statistically |
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| 224 | significant excess of |
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| 225 | $\nu_\tau$ events is very unlikely for the $\tau \to e$ decay mode, |
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| 226 | therefore we conclude that a search |
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| 227 | based on this channel is hopeless. Same conclusions apply to |
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| 228 | the muonic decay channel. |
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| 229 | |
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| 230 | The situation is much more advantageous for the hadronic channels: |
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| 231 | we consider tau decays to one prong (single pion, rho) and to three |
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| 232 | prongs ($\pi^\pm \pi^0 \pi^0 $ and three charged pions). |
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| 233 | After a careful evaluation of the performance of different |
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| 234 | combinations of kinematic variables, we decided to use: $E_{visible}$, |
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| 235 | $y_{bj}$ (the ratio between the total hadronic energy and |
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| 236 | $E_{visible}$) and $Q_T$ (defined as the transverse momentum of the $\tau$ |
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| 237 | candidate with respect to the total measured momentum). The chosen |
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| 238 | variables are not independent one from another but show |
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| 239 | correlations between them. These correlations can be exploited to reduce the |
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| 240 | background. In order to maximize the separation between signal |
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| 241 | and background, we use three dimensional likelihood functions |
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| 242 | ${\cal L}(Q_T,E_{visible}, y_{bj})$ where |
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| 243 | correlations are taken into account. For every channel, we build three |
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| 244 | dimensional likelihood functions |
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| 245 | for both signal (${\cal L}^S_\pi, \ {\cal L}^S_\rho, \ |
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| 246 | {\cal L}^S_{3\pi}$) and background (${\cal L}^B_\pi, \ {\cal L}^B_\rho, \ |
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| 247 | {\cal L}^B_{3\pi}$). To enhance the separation of $\nu_\tau$ induced |
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| 248 | events from $\nu_\mu, \ \nu_e$ interactions, we take a ratio of |
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| 249 | likelihoods as the sole discriminant variable: |
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| 250 | \begin{equation} |
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| 251 | \ln \lambda_i \equiv \ln({\cal L}^S_i / {\cal L}^B_i) |
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| 252 | \end{equation} |
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| 253 | where $i=\pi,\ \rho, \ 3\pi$. |
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| 254 | |
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| 255 | To further improve the sensitivity of the $\nu_\tau$ |
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| 256 | appearance search, we combine |
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| 257 | the three independent hadronic analyses into a single one. |
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| 258 | Events that are common to at least |
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| 259 | two analyses are counted only once and a survey of all possible |
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| 260 | combinations, for a restricted set of values of the likelihood |
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| 261 | ratios, is performed. Table \ref{tab:combi} illustrates the |
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| 262 | statistical significance achieved by several selected combinations of the |
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| 263 | likelihood ratios for an exposure equivalent to 100 kton$\times$year. |
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| 264 | |
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| 265 | \begin{table} |
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| 266 | \caption{\label{tab:combi}Expected background and signal events for different |
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| 267 | combinations of the $\pi$, $\rho$ and $3\pi$ analyses. The considered |
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| 268 | statistical sample corresponds to an exposure of 100 |
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| 269 | kton$\times$year. The best |
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| 270 | combination found is indicated in bold characters.} |
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| 271 | \begin{center} |
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| 272 | \begin{tabular}{cccclc}\hline\hline |
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| 273 | $\ln \lambda_\pi$ & $\ln \lambda_\rho$ & $\ln \lambda_{3\pi}$ & |
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| 274 | Top & Bottom & $P_\beta$ ($\%$) \\ |
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| 275 | Cut & Cut & Cut & Events & Events & \\ \hline |
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| 276 | 0. & 0.5 & 0. & 223 & $223 + 43 = 266$ & $2 \times 10^{-1}$ |
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| 277 | ($3.1\sigma$)\\ |
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| 278 | 1.5. & 1.5 & 0 & 92 & $92 + 35= 127$ & $2 \times 10^{-2}$ ($3.7\sigma$)\\ |
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| 279 | 3. & -1 & 0. & 87 & $87 + 33 = 120 $ & $3 \times 10^{-2}$ |
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| 280 | ($3.6\sigma$)\\ |
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| 281 | 3. & 0.5 & 0. & 25 & {$25 + 22= 47$} |
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| 282 | & {$2 \times 10^{-3}$ $(4.3\sigma)$} \\ |
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| 283 | 3. & 1.5 & 0 & 20 & $20 + 19 = 39$ & $4 \times 10^{-3}$ ($4.1\sigma$)\\ |
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| 284 | 3. & 0.5 & -1. & 59 & $59 + 30 = 89$ & $9 \times 10^{-3}$ ($3.9\sigma$)\\ |
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| 285 | 3. & 0.5 & 1. & 18 & $18 + 17 = 35$ & $1 \times 10^{-2}$ ($3.8\sigma$)\\ \hline\hline |
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| 286 | \end{tabular} |
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| 287 | \end{center} |
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| 288 | \end{table} |
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| 289 | The best combination, for a 100 kton$\times$year exposure, |
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| 290 | is achieved for the |
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| 291 | following set of cuts: {$\ln \lambda_\pi > 3$, |
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| 292 | $\ln \lambda_\rho > 0.5$} and {$\ln \lambda_{3\pi} > 0$}. |
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| 293 | The expected number of NC background events amounts to 25 (top) |
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| 294 | while 25+22 = 47 (bottom) are expected. $P_\beta$ is the Poisson probability |
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| 295 | for the measured excess of upward going events to be due to a |
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| 296 | statistical fluctuation as a function of the exposure. We have |
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| 297 | an effect larger than $4\sigma$ for an |
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| 298 | exposure of 100 kton$\times$year (one year of data taking with GLACIER). |
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| 299 | %A Bueno 3/11/06 START |
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| 300 | % |
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| 301 | % JEC 3/11/06 START new section |
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| 302 | \subsection{New phenomena beyond the "Standard Model"} |
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| 303 | % |
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| 304 | It is worth remembering that atmospheric neutrino fluxes are |
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| 305 | themselves an important subject of investigation, and at the light of |
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| 306 | the precise determination of the oscillation parameters provided by |
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| 307 | long-baseline experiments the atmospheric neutrino data accumulated by |
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| 308 | the proposed detectors can be used as a \emph{direct measurement} of the incoming |
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| 309 | neutrino flux, and therefore as an indirect measurement of the primary |
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| 310 | cosmic rays flux. |
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| 311 | |
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| 312 | The appearance |
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| 313 | of subleading features in the main oscillation pattern can also be |
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| 314 | a hint for New Physics. The huge range of energies probed by |
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| 315 | atmospheric data will allow to put very strong bounds on |
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| 316 | mechanisms which predict deviation from the $1/E$ behavior. For |
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| 317 | example, the bound on non-standard neutrino-matter interactions |
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| 318 | and on other types of New Physics (such as violation of the |
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| 319 | equivalence principle, or violation of the Lorentz invariance) |
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| 320 | which can be derived from \emph{present} data is already the |
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| 321 | strongest which can be put on these |
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| 322 | mechanisms \cite{Gonzalez-Garcia:2004wg}. So, the increased statistics |
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| 323 | expected for the proposed detectors will further improve these constraints. |
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| 324 | % JEC 3/11/06 END |
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| 325 | %JEC 22/6/06 END |
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