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, ν_e, ν_μ, and ν_τ
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 ν_e and ν_μ, and mass eigenstates ν₁ and ν₂, related by a 2×2 matrix:

ν_μ	=	cosθ	sinθ	ν₁
ν_e	=	-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

E_i = p + m_i²/2p

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

ν₁	=	cosθ	-sinθ	ν_μ
ν₂	=	sinθ	cosθ	ν_e

Therefore

ν₁(0) = -sinθ ν_e(0)
ν₂(0) = cosθ ν_e(0)

and

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. 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.

Kamiokande and SuperKamiokande

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.

Neutrino Oscillations

Recall from last lecture:

Neutrino Mixing

Experimental Evidence for Oscillations

Solar Neutrinos

Homestake Mine Experiment

Kamiokande and SuperKamiokande

SAGE and GALLEX

SNO

Summary of Solar Neutrino Measurements

Atmospheric Neutrinos

Reactor Neutrinos

CHOOZ

KamLand

Neutrino Beams

LSND

MiniBoone

Minos