- 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

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 ν_{1} and ν_{2}, related by a 2×2 matrix:

ν_{μ} | = | cosθ | sinθ | ν_{1} |

ν_{e} | -sinθ | cosθ | ν_{2} |

ν_{1}(t) = ν_{1}(0) exp(-iE_{1}t)

ν_{2}(t) = ν_{2}(0) exp(-iE_{2}t)

The weak eigenstate is created with a given momentum, p, and this is also the momentum of the mass eigenstates νν

E_{i} = p + m_{i}^{2}/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 ν_{1} and ν_{2} by inverting the mixing expression:

ν_{1} | = | cosθ | -sinθ | ν_{μ} |

ν_{2} | sinθ | cosθ | ν_{e} |

ν_{1}(0) = -sinθ ν_{e}(0)

ν_{2}(0) = cosθ ν_{e}(0)

and
ν

ν_{e}(t) = -sinθ ν_{1}(t) + cosθ ν_{2}(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

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

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

Copyright © Robert Harr 2003