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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