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Neutrino oscillations

The observation of atmospheric neutrinos with a neutrino telescope provides a means to study neutrino oscillations with a base-line length up to the order of the diameter of the Earth. The focus of investigation is the muon neutrino, but there is also a very interesting possible signature for extremely high energy (>100 TeV) tau neutrinos.

Atmospheric $\nu _\mu $ oscillations

Atmospheric neutrinos are emitted in the decay of hadrons produced by the interactions of cosmic rays with atmospheric nuclei. The production of electron neutrinos and of muon neutrinos is dominated by the processes :
${\pi}^{\pm} \rightarrow {\mu}^{\pm} + {\nu}_{\mu}/{\overline{\nu}}_{\mu}$
followed by ${\mu}^{\pm} \rightarrow e^{\pm} + {\overline{\nu}}_{\mu} /
{\nu}_{\mu} + {\nu}_e/{\overline{\nu}}_e$.

In an infinite medium the ratio, r, of the flux of  $\nu _\mu $ and ${\overline{\nu}}_{\mu}$ to the flux of $\nu _e$ and  ${\overline{\nu}}_{e}$ is expected to be two. Since the atmosphere is not an infinite medium, this ratio increases with increasing neutrino energy, because not all high energy muons can decay before they are absorbed by the ground. Furthermore, the magnetic field of the Earth has some influence on low energy charged particles and this modifies the energy spectra below a few GeV.

These effects are taken into account for the calculations of the predicted neutrino fluxes. The overall normalisation uncertainty is estimated to be about 20%, which is due to systematic theoretical uncertainties in the energy spectra of the primary cosmic rays and to the uncertainties in their composition. Generally, experimental results are reported as R = rDATA/rMC in order to cancel common systematic uncertainties thereby reducing the overall uncertainty in R to about 5%.

Measurements published by underground experiments show evidence of a deficit in the number of muon neutrinos with respect to electron neutrinos. No anomaly was observed by the Fréjus and NUSEX experiments.

Neutrino oscillations have been proposed as an explanation for the low value of the ratio R. With the hypothesis of two-neutrino mixing, the oscillation probability is:

\begin{displaymath}P = \sin^2 2 \theta \sin^2 \left( 1.27 \frac{L}{E}
\Delta m^2 \right)
\end{displaymath}
where $\theta$ is the mixing angle, L is the distance travelled by the neutrino (in km), E is the neutrino energy (in GeV) and $\Delta m^2$ is the difference of the square of the masses (in eV2). 

As the neutrinos are produced in the atmosphere, the distance L ranges between 15km, for vertically downward-going neutrinos, and almost 13000km, for vertically upward-going neutrinos.

 Figure below shows the variation of the survival probability as a function of L/E.

Variation of the neutrino survival probability as a function of L/Efor the squared mass difference favoured by the Superkamikokande experiment (0.035 eV2 and maximal mixing).

Several points remain to be clarified. First of all, there are three different regions of $\Delta m^2$ which have been explored with solar neutrinos, atmospheric neutrinos and short baseline beam neutrinos respectively, all of which indicate evidence of non-zero $\Delta m^2$. This cannot be supported in a three-flavour neutrino scheme. 

An independent measurement of the region 60<L/E<1250km/GeV could resolve the uncertainties in the interpretation of these different experiments. This region contains the principal oscillation (first dip) for the $\Delta m^2$ values in question: $1~\times~10^{-3}~<~{\Delta}m^2~< 2~\times~10^{-2}$ eV2.

 ANTARES is perfectly suited to this task. The peak sensitivity of the ANTARES detector is very near the most probable value reported by Super-Kamiokande. Accurate measurement of the position of the principal oscillation would provide a precise measurement of $\Delta m^2$.

Determination of the neutrino survival probability requires a measurement of the energy of the incident muon neutrino. For the isotropic $\nu q$ and $\overline{\nu}\overline{q}$charged-current interactions, half of the neutrino energy goes to the hadron shower. For the $\overline{\nu}q$ and $\nu\overline{q}$ interactions, a much larger fraction of the energy goes to the muon, but these interactions are three times less frequent than the isotropic interactions. The energy of the hadron showers is difficult to estimate accurately, so the measurement of the oscillation parameters in ANTARES depends principally on the measurement of the muon momentum. 

The energy of the muon is determined by its range and the precision depends on the vertical spacing of the PMTs, 4 GeV for a spacing of 16 m. Reconstruction inefficiencies degrade the energy resolution in an energy-dependent way, depending on the reconstruction algorithms employed.

Tau neutrinos

Although the contribution of tau neutrinos to the atmospheric neutrino flux is negligible, their interactions must be taken into account when studying neutrino oscillations. If oscillations of the type $\nu_\mu \rightarrow \nu_\tau$ occur, the charged current interactions of $\nu_\tau$ will produce charged $\tau$ leptons that can contribute to the signal observed. 

Most of the hadronic and electronic $\tau$ decays will escape detection, but the muonic decays $\tau^- \rightarrow \mu^-\bar\nu_\mu\nu_\tau$ can be seen in the ANTARES detector, and these events might be mis-identified as $\nu _\mu $ interactions.

The muonic branching ratio of the $\tau$ is 17%. Furthermore, the charged-current interactions are considerably suppressed by the limited phase space due to the mass of the $\tau$ (1.78 GeV). For very large $\Delta m^2$ (1 eV2), the number of $\nu_\tau$ would be equal to the number of $\nu _\mu $ over the entire atmospheric flux (< 500 GeV), but the number of $\tau^- (\tau^+)$ produced with energies above 10 GeV would be only 53% (64%) of the number of $\mu^- (\mu^+)$ because of the limited phase space, and the contamination of the muon sample due to tau decays would be about 9% (11%). For smaller values of $\Delta m^2$, the contamination in the region of the main oscillation dip could be larger.

An attractive possibility for detecting very high energy neutrinos exists :

 The Earth is nearly transparent to low-energy neutrinos, but opaque to neutrinos above 100 TeV. Nonetheless, tau neutrinos well above 100 TeV can produce a signal in ANTARES because, unlike the $e^\pm$ and $\mu^\pm$ produced in $\nu _e$ and $\nu _\mu $ interactions, the $\tau^\pm$ produced in $\nu_\tau$ interactions decay before they are absorbed, producing $\nu_\tau$ of lower energy which continue along the original $\nu_\tau$ flight path, but with decreasing interaction probability. Once the $\nu_\tau$ energy has been degraded to about 100 TeV, the $\nu_\tau$ can penetrate the Earth and produce an accumulation of very-high-energy events in the detector. Such an accumulation would be a signal for tau neutrinos. Moreover, the flux of $\nu_\tau$ from a given source would be constant during the Earth's rotation, whereas the flux of $\nu _\mu $ would vary with the sidereal day because of the change in elevation seen from ANTARES. The variation of the $\nu _\mu $ flux could not be observed from AMANDA, because it is located at the South Pole. A comparison of signals from the same sources observed at the different elevations corresponding to AMANDA and ANTARES could lead to very exciting results.

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Author : Thierry Stolarczyk