# Neutrino Oscillations

### Recall from last lecture:

• neutrinos are neutral and nearly massless, invented to balance energy in β-decays
• there are three separate types of neutrinos, νe, νμ, and ντ
• left-handed neutrinos and right-handed anti-neutrinos couple to W's and Z's

## Neutrino Mixing

In the remaining discussion, we will assume that neutrinos are "standard" Dirac particles.

For many years, the Standard Model assumed that neutrinos are massless, only because all evidence was consistent with this assumption, and the resulting equations were somewhat simplified. But what happens if we drop this assumption? Then neutrinos can have small, but finite masses. Additionally, they may have non-diagonal couplings to W bosons, in the same way that the down-type quarks have non-diagonal couplings. The down-type quark couplings involve the CKM matrix, and in a similar spirit, the neutrino couplings involve a complex, 3×3 matrix known as the MNS matrix.

As with the quarks, the result is that the mass eigenstates, and the weak eigenstates are not the same. Consider an example with two neutrinos, weak eigenstates νe and νμ, and mass eigenstates ν1 and ν2, related by a 2×2 matrix:

 νμ = cosθ sinθ ν1 νe -sinθ cosθ ν2
We assume that in a reaction, the weak eigenstate is created. But ν1 and ν2 are the mass eigenstates, with masses m1 and m2. (The experimental evidence suggests that the masses are very small, both in the eV range. The mass difference is undetectable in the energy of other outgoing particles.) The propagation of the mass eigenstates in space is governed by:
ν1(t) = ν1(0) exp(-iE1t)
ν2(t) = ν2(0) exp(-iE2t)
The weak eigenstate is created with a given momentum, p, and this is also the momentum of the mass eigenstates ν1 and ν2. For neutrino masses much less than the momentum, the energies are approximately
Ei = p + mi2/2p

Assume that at t=0 we start off with a pure electron neutrino state, νe(0) = 1, and νμ(0) = 0. We can find the initial amounts of ν1 and ν2 by inverting the mixing expression:

 ν1 = cosθ -sinθ νμ ν2 sinθ cosθ νe
Therefore
ν1(0) = -sinθ νe(0)
ν2(0) = cosθ νe(0)
and
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. This can be seen by including the time dependence of the

## 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
All the experimental evidence to date is of the first type, disappearance. 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.

#### Homestake Mine Experiment

The first apparatus to detect solar neutrinos was constructed in the Homestake mine in North Dakota by Ray Davis and collaborators.

#### 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.