# 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.

# What Are Atmospheric Neutrinos?

Returning to the subject of the Nobel Prize, you might be wondering what the physics prize this year was actually about.

What matters for these experiments is (1) neutrino flavor, (2) neutrino baseline (distance traveled from being in a more or less known flavor configuration) and (3) neutrino energy. Atmospheric neutrinos are one of the several different ways that we can study oscillations.

High energy particles coming in from space (i.e. cosmic rays) are generally nuclei (most often just protons). Upon reaching the atmosphere, these particles often have enough energy to undergo inelastic collisions with the particles in the atmosphere. The most common types of events would be hadronic showers, where the strong force allows for more hadrons to be created. The most common hadrons created in showers are pions, which are the lightest mesons (bound quark-antiquark states). Charged pions decay through the weak force into neutrinos and charged leptons. It turns out that pions decay to muons and muon neutrinos well over 99% of the time. Thus, cosmic rays create muon neutrinos (and antineutrinos) within the atmosphere. Typical energies are also much higher than is found in nuclear processes, so atmospheric neutrinos can be distinguished from solar and beta decay neutrinos.

Because neutrinos don’t really interact much with anything else, cosmic ray neutrinos can also pass straight through Earth. If a neutrino detector can reconstruct the direction of the neutrino, it can then look for short baseline neutrinos (from above) and long baseline neutrinos (from below). A pretty broad energy spectrum is also found. It also happens that neutrino oscillations (in a simplified two-neutrino case) evolve like:

$P(\nu_\ell\rightarrow\nu_{\ell'})\propto \sin^2 \frac{\Delta m^2 L}{4E}$

Theshape of the oscillations depends on the mass splitting and on the ratio of baseline to neutrino energy. If the mass splitting is too small, then not enough oscillations will happen to measure them. If the mass splitting is too large, the oscillations will be happening so fast that any realistic detector will just measure the unoscillated spectrum with an extra scaling constant of 1/2.

Fortunately, it just so happens that one of the mass splittings relevant to a sample of nearly pure muon neutrinos leads to oscillations that are readily measurable to the GeV-scale atmospheric neutrinos. So, the oscillations should be apparent if one looks at the ratio of the upward-going (long baseline/oscillated) neutrinos to downward-going (short baseline/unoscillated) neutrinos. A characteristic sinusoidal shape should be seen if this ratio is plotted as a function of L/E. This allows for precision fits of oscillation parameters using atmospheric neutrino data.

# Measurement of the Ultra High Energy Cosmic Ray Spectrum

The Pierre Auger Observatory has uploaded a new measurement of the energy spectrum of ultra high energy cosmic rays. They looks at events with energies greater than 1018 eV. Since most cosmic rays are protons, this is actually an incredibly large amount of energy for a single subatomic particle. In fact, these are particles and nuclei with macroscopic amounts of kinetic energy. The huge amount of energy means that these particles create huge air showers high in the atmosphere. Auger measures the light signal emitted as the shower progresses as well as signals from muons hitting a massive array of water Cherenkov tanks on the ground at the detector site in Argentina.

The spectrum of these cosmic rays is actually quite important in astrophysics. The cosmic ray spectrum generally follows a power law spectrum, but the shape is expected to change around 5×1019 eV. A knee-like feature should be seen around that energy as the spectrum changes from one power law to another.

The reason for this expected ship is the so-called GZK cutoff. The cosmic microwave background consists of a thermal distribution of microwaves (photons) with a temperature of 2.7 K, or an energy of around 225 μeV. It turns out that a proton can interact with a photon to create a short-lived Δ that then decays to a nucleon (proton or neutron) and a pion. The Δs decays via the strong force, so while they have a mass of around 1232 MeV, they also have widths of over 100 MeV. This means that the lifetime is very short and so we generally refer to a Δ “resonance” rather than a Δ “particle.” The GZK cutoff represents the approximate energy at which significant numbers of protons will interact with the CMB to create Δs.

The pion in the final state after the Δ decays carries a sizable fraction of the initial proton energy. So, the proton will continue to lose energy until the energy is too low for this process to continue. As a result, the proton spectrum at energies higher than the GZK cutoff should be suppressed compared to the spectrum at energies lower than the cutoff. There is a mean free path for this process that is still quite large, so some protons might be expected at energies above the cutoff, but they should originate in some region in the vicinity (on cosmological scales) of Earth.

# IceCube Search for Anisotropy in Neutrino Data

Also out this week is a paper by IceCube in which they search for an anisotropy in the spatial distribution of very high energy cosmic ray neutrinos. This is basically just a search for point sources of these neutrinos on top of a more or less uniform background. Point sources would be very exciting because then we might be able to figure out what kinds of objects create these neutrinos.

IceCube is an enormous detector located at the South Pole. It consists of photodetectors dropped into holes bored in the Antarctic ice and looks for Cherenkov radiation from high energy particles created in neutrino interactions in the ice. The whole detector has a volume of around a cubic kilometer: far larger than any detector we could feasibly construct instead of using a natural target like ice in Antarctica.

In this search, IceCube searches for an anisotropy in two ways: a multipole expansion (decomposition of the spatial distribution to spherical harmonics) and an autocorrelation-based statistical test. In both methods they find results consistent with and actually slightly smoother than expected backgrounds.