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Charged kaons (if±) contribution to the same decay channel but are much less abundant at low energies. At much higher energies the decay of

with a branching fraction of 38.7% is the chief contributor to the atmospheric ve and Ve flux.

Neutrino Cross Sections

Neutrinos can undergo charged current (CC), neutral current (NC) or elastic scattering (ES) reactions. The different reaction types that can occur in water and heavy water are summarized in Table 4.28, where vx stands for any kind of neutrino or antineutrino. Disregarding the large Cherenkov detector matrices in water or ice, water and heavy water are the combined target and detector materials of Super-Kamiokande (Koshiba, 1992), the largest underground neutrino experiment in operation today, and of the Sudbury Neutrino Observatory, SNO, now beginning operation (McDonald, 1996 and 1999), respectively.

Table 4.29: Cross Sections for Neutrino Elastic Scattering. (Bemporad, 1996)

Reaction

Cross Section

ve + e~ Ve + e~ »n + e~ + e"

* = °-39 •10-43 tiofb] c™2 a = 0.16 ■ 10-*3 [i] cm2

" = °"13 *10-43 tlotv] c™2

At very low energies the neutrino cross section, cr„, rises logarithmically. In the MeV region it is on the order of 10-44 cm2 and rises linearly with energy. The cross sections for elastic scattering of neutrinos of the two common flavors and their antiparticles are given in Table 4.29.

For inelastic processes above about 1 GeV where the cross section is on the order of 10-38 cm2 it rises linearly. In this energy range we can write ct„ ~ 1(T38 • Ev [cm2 GeV"1 nucleon"1] , (4.65)

with E in GeV. Above the W and Z intermediate boson thresholds the cross section saturates. For antineutrinos the cross section is approximately | of that of the neutrinos. Theoretical and experimental cross sections for charged current reactions of muon neutrinos and antineutrinos are illustrated in Figs. 4.78 and 4.79 (Hikasa et al., 1992; Frati et al., 1993).

Neutrino Flavor Ratio

For neutrino energies <1 GeV a naive calculation of the relative number of each neutrino type based on the decay channels listed above and in the absence of neutrino oscillations leads to the predictions that the ratio

where NV/1 (iV^J and N„e (NVe) are the number of (F^) and ve (Fe), respectively. Because of the excess of protons over neutrons in the primary cosmic radiation there is an excess of ve over Fe in the atmospheric neutrino flux (Gaisser et al., 1987). Thus,

The lepton-antilepton asymmetries increase with increasing energy and axe non-trivial functions of the energy and the zenith angle. They also depend weakly on the atmospheric depth of observation (Lipari, 1993).

At higher energies muon neutrinos are expected to be the dominating kind because muons are less likely to decay and thus contribute less and less to the neutrino flux in the atmosphere with increasing energy, causing the ve (Fe) fraction to diminish (see Chapter 1, Subsections 1.3.3) and 1.3.4.

Consequently, the ratio of eq. 4.66 decreases with increasing energy because muons are more and more likely to reach the ground before decaying. Comparison of the decay length to the energy-loss length of muons in the Earth leads to the conclusion that virtually all muons that reach the ground stop before decay (or capture) occurs. Therefore muons that reach the ground do not contribute to neutrinos with energy > 100 MeV, which makes their detection even for electron neutrinos via electron scattering difficult.

Up/Down Symmetry of Neutrino Flux and Geomagnetic Effects

Because of the very small but energy dependent cross sections of neutrino reactions given in eq. 4.65 and illustrated in Figs. 4.78 and 4.79 it is evident that neutrinos of energy <100 TeV remain essentially unaffected when traversing the Earth. Thus, the upward and downward fluxes should be symmetrical in the absence of neutrino oscillations, provided that the cosmic ray beam is fully isotropic.

Unfortunately the east-west effect, asymmetries and anomalies of the geomagnetic and magnetospheric fields and the highly complex and time dependent geomagnetic cutoff conditions disturb the isotropy of the cosmic radiation and therefore the up-down symmetry of the cosmic ray induced atmospheric neutrino flux (for details see Chapter 1, Section 1.8 and Chapter

6, Section 6.2). Careful consideration must therefore be given to these effects when carrying out neutrino flux calculations (Agrawal et al., 1996; Barr et al., 1989; Battistoni et al., 2000; Bugaev and Naumov, 1989; Cheung and Young, 1990; Gaisser and Stanev, 1995; Gaisser, 1999; Honda et al., 1990, 1995 and 1999; Lipari et al., 1998).

Muon Polarization and Neutrino Spectra

Muons produced in the cosmic radiation are polarized, ¡x+ have on average negative helicity and positive helicity. For accurate calculations the muon polarization must be taken into account because the neutrino spectra produced in muon decay depend on the polarization (Dermer, 1986; Barr et al., 1988; Barr et al., 1989; Volkova, 1989; Lee and Koh, 1990; Lipari, 1993; Gaisser et al., 1995).

Neutrino Oscillations

If neutrinos have mass, it is possible that the weak interaction eigenstates (j'e,^,^) are distinct from their mass eigenstates (m\, rri2,1113), and there could be vacuum oscillations of the type vM ve in analogy to the K°, K° system (Pontecorvo, 1946 and 1967; Maki et al., 1962; Bilenky and Pontecorvo, 1978). Thus, if neutrino oscillations exist then the ratio (fe + Ve)/(i/^ + V^) of eq. 4.66 could be affected.

For a two-neutrino oscillation hypothesis, i.e., v^ ve or •(-» vT, the probability for a neutrino produced in a flavor state a to be observed in a flavor state b after traveling a distance L in vacuum is given by where E„ is the neutrino energy, 0 is the mixing angle between the flavor eigenstates and the mass eigenstates, and Am2 = |m2 — m21 is the mass squared difference. (In the recent experiment of Apollonio et al. (1998) no evidence was found for neutrino oscillations in the disappearance mode ve Vx for the parameter region given approximately by Am2 > 0.9 ■ 10~3 eV2 for maximum mixing and sin2(20) > 0.18.)

For detectors near the surface of the Earth, the neutrino flight distance, and therefore the oscillation probability, is a function of the zenith angle of the neutrino trajectory. Vertically downward-going neutrinos travel about 15 km while vertically upward-going neutrinos travel roughly 13,000 km before reaching the detector.

The bulk of the downward-going muons in an underground detector are originally high energy atmospheric muons resulting from cosmic ray interactions in the atmosphere that have survived and retained sufficient energy to enter or even penetrate the detector; they are not neutrino induced muons. Only a tiny fraction of the downward-going muons are the result of neutrino interactions. If the latter take place in the detector overburden they are undistinguishable from atmospheric muons. However, if the neutrino reaction occurs inside the detector the event can be recognized as being due to a neutrino reaction. On the other hand, upward going muons that are observed in underground detectors are due exclusively to neutrino reactions in the rock below the detector or within it.

Typical events have neutrino energies of about 10 GeV if they are fully contained in a large detector such as Superkamiokande (50.000 m3 of water) and about 100 GeV for through-going muons (Kajita, 2000). The wide energy spectrum from a few hundred MeV to about 100 GeV that is available for such experiments and the range of flight distances just mentioned make measurements of atmospheric neutrinos sensitive to neutrino oscillations with a Am2 down to about 10~4 eV2.

It was pointed out by Wolfenstein (1978 and 1979) that whereas muon neutrinos interact only via neutral currents with the electrons in a medium, electron neutrinos interact both by charged and neutral currents. This fact leads to incoherence over a characteristic distance L0 (Wolfenstein length, 9.0 • 106 m in the Earth on average) between the two types of waves. Therefore oscillations are damped and destroyed and limit the sensitivity of underground experiments (Ramana Murthy, 1983; Mikheyev and Smirnov, 1986).

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