Neutrino Oscillations

Recall from last lecture:

ν_e(t) = -sinθ ν₁(t) + cosθ ν₂(t)

If sometime later, we try to detect the electron neutrinos, we will in general detect fewer than without oscillation. The normalized amplitude for a neutrino to be in the electron neutrino state is

A_e = ν_e(t)/ν_e(0) = sin²θ exp(-iE₁t) + cos²θ exp(-iE₂t)

The intensity (probability) of the electron neutrino state normalized to the intensity at the source is given by

I_e(t)/I_e(0) = A_eA_e^* = sin⁴θ + cos⁴θ + 2sin²θ cos²θ cos ([E₂ - E₁]t)

After some massaging, the above expression reduces to

I_e(t)/I_e(0) = 1 - sin²2θ sin²([E₁ - E₂]t/2)

Using the expression for E_i, the energy difference can be written

E₁ - E₂ = (m₁² - m₂²)/2p = Δm²/2E

where we use the fact that E is approximately equal to p for small masses.

Finally, setting t=L/c, where L is the distance from the source, and reinserting the factors hbar and c we obtain:

I_e(t)/I_e(0) = 1 - sin²2θ sin²(Δm²c⁴L/4E hbar c) = 1 - sin²2θ sin²(1.27Δm²L/E)

The probability of detecting a ν_e neutrino in a detector a distance L from the source of pure ν_e is then

P(ν_e-->ν_e) = 1 - sin²2θ sin²(1.27Δm²L/E)

and the probability of detecting ν_μ is

P(ν_e-->ν_μ) = sin²2θ sin²(1.27Δm²L/E)

Experimental Evidence for Oscillations

The evidence for oscillations can be of several types:

disappearance: one flavor of neutrino disappears, other flavors not meaured
reappearance: after seeing the disappearance, the flavor reappears at a further point from the source
appearance: a neutrino flavor appears in a neutrino beam that originally (at the source) doesn't contain that flavor

Though there is no a preponderence of experimental evidence, it is all of the first type, disappearance. Experimental evidence of reappearance or appearance is being sought and will provide convincing evidence that flavor oscillations are occurring. I'll discuss the experiments in approximate chronological order.

Solar Neutrinos

Many of the fusion processes occurring in the Sun produce neutrinos. The original impetus for measuring "solar neutrinos" was to prove beyond any doubt that fusion occurs in the Sun and is the source of energy. The fundamental fusion process in the Sun is a three step reaction where by four protons are converted into a helium-4 nucleus, two positrons, two electron neutrinos, and 24.8MeV of energy. The reaction steps are:

pp --> d e⁺ ν_e + 0.42MeV
pd --> ³He γ + 5.51MeV
³He ³He --> ⁴He p p γ + 12.98MeV

Including the energy released when the two positrons annihilate with two electrons, and the energy of the gamma ray, we find that for every 25MeV of energy produced in the Sun two neutrinos are produced. From the amount of solar energy that reaches the Earth, we can conclude that the neutrino flux is 6×10¹⁰/cm² s.

The neutrinos produced in this reaction are called pp neutrinos and have a spectrum of energies with a maximum of just over 0.4MeV. Other processes occur producing neutrinos with energies up to almost 12MeV, but at several orders of magnitude less rate. Fig. 9.9 of Perkins shows the neutrino energy spectra of these processes and the flux of neutrinos produced. (I believe the dashed curves are scaled up by a few orders of magnitude to make them visible on the plot, but the figure caption doesn't indicate this.) This is important because our detectors are only sensitive above some threshold, and we need to interpret the results with that in mind.

Homestake Mine Experiment

The first apparatus to detect solar neutrinos was constructed in the Homestake mine in North Dakota by Ray Davis and collaborators. (Ray Davis was shared the 2002 Nobel Prize in Physics for detecting solar neutrinos.) His apparatus is a large tank of cleaning fluid, C₂Cl₄. Electron neutrinos can be absorbed by chlorine producing argon in the process

ν_e ³⁷Cl --> ³⁷Ar e^-

The argon atoms can be extracted from the fluid and detected with high efficiency. The threshold for the reaction is about 0.8MeV so it is insensitive to the pp neutrinos. The overall rate is low, partly explaining the long lifetime of this experiment -- it ran for more than 30 years! The other reason for its lifespan is that the observed rate was only about 30% of the expected rate. Many checks of the experiment were made to confirm this result over the years.

Kamiokande and SuperKamiokande

In the meantime, other experiments designed originally to search for proton decay found that their background events held interesting physics. These experiments are shielded from cosmic rays and natural radioactivity, but cannot be shielded from neutrinos.

SAGE and GALLEX

SNO

Summary of Solar Neutrino Measurements

The neutrino physics that has come from these measurements originally called into question the above statement, or at least questioned if the model of solar fusion was completely correct.

Atmospheric Neutrinos

Reactor Neutrinos

CHOOZ

KamLand

KamLand is another reactor experiment, located in Japan. The detector is in a location that is about 250km from several reactors. This distance is good for observing electron disappearance if we take the solar neutrino results literally.