The expressions derived for neutrino fluxes in Chapter 6 apply to atmospheric neutrinos at production. Because of their small cross section, most neutrinos pass through the Earth without absorption, so atmospheric neutrinos from the whole sky can be observed from a single detector. This makes it possible to compare neutrino fluxes over a range of path lengths from ∼ 10 to 10,000 km. If the neutrinos change in some way as they propagate after production, then the fluxes will differ from those obtained by integrating their production spectra from the top of the atmosphere to the ground.

The evidence that atmospheric neutrinos do indeed suffer an identity change during propagation is summarized in Figure 7.1 from the Super-Kamiokande experiment [59]. (See Ref. [220] for a complete review.) Crosses show the data for low energy (top row) and higher energy (bottom row) as a function of zenith angle. The dashed lines show the expected number of events in the absence of oscillation, while the solid lines show the fitted fluxes assuming oscillations. Electron neutrinos are not much affected at these energies, while muon neutrinos show a characteristic behavior in which high-energy downward muons are unaffected while muons that cross the Earth are affected, and the deviation begins already above the horizon for low-energy muon neutrinos. The description of this behavior and its implications for theory are the subject of this chapter. As we will see, the existence of oscillations requires that the neutrinos have a nonzero rest mass.

Neutrino mixing

The hypothesis of neutrino mixing was first anticipated by Pontecorvo in 1957, and a few years later developed by Maki, Nakagawa and Sakata [71] on the basis of a two-neutrino hypothesis. The first evidence for oscillations came from measurements of solar neutrinos by Davis over a 25-year period starting in 1970. The solar neutrino studies began in 1964 with back-to-back papers by Bahcall [221] and Davis [222] predicting the number of electron neutrinos expected from fusion

reactions in the Sun and proposing the experiment using chlorine as the target and looking for

A deficit of electron neutrinos (by a factor of 1/3) was found [223] as measurements continued and ever more detailed calculations were made [224].