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.

  • $\begingroup$ 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. $\endgroup$ Commented May 3, 2019 at 15:50

2 Answers 2


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.

The one reference to go to for any particle properties is the Particle Data Group: the front page and specifically the page for neutrino oscillations in section (B) Three-neutrino mixing parameters.

  • $\begingroup$ Where are these numbers taken from? $\endgroup$
    – Omry
    Commented Oct 10, 2017 at 6:28
  • $\begingroup$ From the top of my head but I apologise, indeed. Fixed. $\endgroup$
    – user154997
    Commented Oct 10, 2017 at 9:12
  • $\begingroup$ Thanks for your answer and for the links. But don't you mean that the oscillation length is inverse proportional to $\Delta m^2$? $\endgroup$ Commented Oct 10, 2017 at 9:53
  • 1
    $\begingroup$ yes, sorry, I started the phrase with phase in mind and got interrupted! $\endgroup$
    – user154997
    Commented Oct 10, 2017 at 11:13

Oscillations play a large role, but there is another effect in play. Not all muons decay before hitting the Earth. In matter muons rapidly lose energy (they will still decay, but the resulting neutrinos carry much less energy and are thus negligible). See e.g. fig. 3 here: https://arxiv.org/abs/1502.03916. The effect of course increases with energy.


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