# Atmospheric Neutrinos, Redux

To go into a bit more detail about atmospheric neutrinos than in my previous post, let’s start with a detector at Earth’s surface:

Now consider a downward neutrino of energy E at a zenith angle θ. We’re considering long (on geological scales) baseline oscillations, so this neutrino has an oscillation baseline that is effectively 0. If we consider an upward neutrino at the equivalent zenith angle, the neutrino now has to go through a significant chunk of the planet. It’s baseline is basically 2Rcosθ, where R is the radius of Earth. We can also see that the upward going neutrino still has the same zenith angle with respect to the surface when it first leaves the atmosphere and enters the planet.

If we assume a perfectly spherical Earth and an isotropic cosmic ray flux, the fact that both neutrinos hit the surface at the same angle is very important. It means that the two sources of neutrinos have the same flux as seen by our detector. So, if no oscillations happen, the ratio of upward to downward neutrinos is unity, even if we look at them as functions of energy and direction.

If there are neutrino oscillations, we will instead see the ratio evolve as a sinusoid in (cosθ/E):

$\frac{R_{\rm up}(E,\theta)}{R_{\rm dn}(E,\theta)} = \sin^2(2\Theta)\sin^2\left(\frac{R\Delta m^2 \cos\theta}{2E}\right)$

If we add in units, this becomes

$\frac{R_{\rm up}(E,\theta)}{R_{\rm dn}(E,\theta)} = \sin^2(2\Theta)\sin^2\left(1.62\times10^4\frac{\Delta m^2 \cos\theta}{E}\right)$

where E is in GeV and Δm2 is in eV2. We can then readily see that, for example, for a pretty typical angle of cosθ=0.5 and E=10 GeV, we should be sensitive to a mass splitting of order 10-3 eV2 or so with a pretty modest dataset. The actual splittings are all reasonably close to this, so atmospheric neutrinos are useful in studying some types of oscillations.

In reality, things are a bit more complicated than this. There are in fact three types of neutrinos that we know about, so we have to consider that there are multiple neutrino mixing angles and neutrino mass splittings if we want to be more rigorous.

The initial neutrino flux also isn’t just a single type of neutrino. Muon neutrinos as well as muon antineutrinos will be the predominant types of neutrino created in cosmic ray showers, but there will still be electron neutrinos generated by muon decays in flight, decays of heavier mesons and hadrons (you can also get tau neutrinos from some of these), and even from a small fraction (about 0.01%) of pion decays.

Furthermore, Earth isn’t completely spherical and the cosmic ray flux is probably not completely isotropic, so there will be slight differences due to that. The effect of oscillations is pretty large, so probably these would only affect precision measurements but would have basically no bearing on an attempt just to see if there are oscillations. (Actually, I don’t know if these things even have any real effect on current experiments).

Finally, matter does affect how neutrinos oscillate. In particular, electron neutrinos have an effective mass change due to their extra interactions with electrons (readily found in matter) that other neutrinos don’t have. This means that the mixing parameters are all somewhat different depending on the density of the matter.

In the end, if you really want to make a precise measurement with atmospheric neutrinos, it’s probably best to come up with some simulation to account for any effects that might be large enough to be seen in your experiment.