# Neutrino Oscillations

### Recall from last lecture:

νe(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

Ae = νe(t)/νe(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
Ie(t)/Ie(0) = AeAe* = sin4θ + cos4θ + 2sin2θ cos2θ cos ([E2 - E1]t)
After some massaging, the above expression reduces to
Ie(t)/Ie(0) = 1 - sin22θ sin2([E1 - E2]t/2)
Using the expression for Ei, 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:

Ie(t)/Ie(0) = 1 - sin22θ sin2(Δm2c4L/4E hbar c) = 1 - sin22θ sin2(1.27Δm2L/E)
The probability of detecting a νe neutrino in a detector a distance L from the source of pure νe is then
P(νe-->νe) = 1 - sin22θ sin2(1.27Δm2L/E)
and the probability of detecting νμ is
P(νe-->νμ) = 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+ νe + 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

νe 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.

#### 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.

### Reactor Neutrinos

#### 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.