# Atmospheric neutrino oscillations

Cosmic rays primarily consist of protons, which primarily decay into pions upon impact with the atmosphere of the Earth. The pions decay according to $$\pi^+ \to \mu^++\nu_\mu$$ and $$\pi^- \to \mu^- +\bar{\nu}_\mu$$ The muons subsequently decay according to $$\mu^+ \to e^+ + \nu_e + \bar{\nu}_\mu$$ and $$\mu^- \to e^- + \bar{\nu}_e + \nu_\mu$$ which implies that the ratio of muon neutrino flux to electron neutrino flux from the atmosphere is 2:1.

As is well known, the Super Kamiokande observatory observed a much lower value of this ratio, indicating and being consistent with neutrino oscillations $\nu_\mu \to \nu_\tau$.

However, in discussions of these results it never seems like oscillations of the electron neutrinos could be relevant. I understand they could not explain the result, since they would only make it worse, but is it not necessary to take them into account?

Electron neutrino oscillations have of course been confirmed in other experiments, so I reckon it is necessary to consider them in the interpretation of the Super Kamiokande results.

• As a passing comment "which primarily decay into pions upon impact with the atmosphere of the Earth" is pretty wrong. Cosmic proton don't decay. they are stable. Instead, then collide with atmospheric particles and those collisions generate a spray of products (exactly like what happens in an accelerator experiment) which include muons and pions and other stuff. None of which effect the rest of the question. – dmckee --- ex-moderator kitten May 3 '19 at 15:50

The oscillation length $L$ is inversely proportional to $\Delta m^2$, between the two considered flavours. But $\Delta m^2_{21}$ (for $\nu_e-\nu_\mu$) is about 3% of $\Delta m^2_{32}$ (for $\nu_\mu-\nu_\tau$).
Actually, $L \propto E/\Delta m^2$ where $E$ is the energy of the neutrino, but for the case under consideration, the $\nu_e$ and $\nu_\mu$ won't have energies two orders of magnitude apart to overturn the ratio of the $\Delta m^2$'s. On the contrary, $E$ comes into play if you compare different neutrino sources.
• Thanks for your answer and for the links. But don't you mean that the oscillation length is inverse proportional to $\Delta m^2$? – Étienne Bézout Oct 10 '17 at 9:53