K. Wood

# Neutrino States

Neutrinos are very light, fundamental particles that only feel the weak force. Like the electron (e-) and it's heavier cousins the muon (μ-) and tau (τ-), neutrinos are classified as leptons within the Standard Model. Unlike the charged leptons, neutrinos are electrically neutral which makes them difficult to detect. For example, the nuclear processes that produce the Sun's light (photons) also produce a swarm of neutrinos that our bodies don't even notice. At this moment you are being bombarded by ~1,000,000,000 neutrinos every second!

Neutrinos can be produced in conjunction with a charged lepton (i.e. e± , μ±, or τ±) when W± bosons, the electrically charge mediators of the weak force, decay. In the language of quantum mechanics, the neutrino is said to be in a weak eigenstate defined by the charge and flavor of its decay parter. The convention is set to conserve lepton number and electric charge at the interaction vertex. For example, when a W+ decays into a neutrino and a μ+, the resulting neutrino is said to be a muon neutrino. If it had been a W- decaying into a neutrino and an e-, the result would be an electron anti-neutrino.

Quantum mechanics tells us that the time evolution of a free particle is determined by the mass eigenstates. There is no reason why these need to be the same states as the weak eigenstates defined by their production. Generally speaking, the mass eigenstates are some superposition of the weak eigenstates:

The matrix above describes the flavor content of each mass state (and vice versa) and is called the PMNS matrix. The following parameterization of the PMNS matrix automatically encodes unitarity and contains the exact number of free parameters (3 angles and 1 phase ) necessary to account for all allowed possibilities:

where cij ≡ cos θij and sij ≡ sin θij. The fact that weak eigenstates are not eigenstates of the free Hamiltonian is responsible for the notion of neutrino oscillations. To see this without getting distracted by the cumbersome math introduced by the 3x3 PMNS matrix, consider a world where only 2 neutrino flavors exist.

# 2-Flavor Mixing

In this section, we will apply basic quantum mechanics to a simplified neutrino model to derive the oscillation probablities in some detail. Suppose there are only two flavors of neutrinos: να and νβ. Their mixing with the mass eigenstates can be parameterized by

[1]

For example, imagine at t = 0 a neutrino is produced in the β-flavor state. We can write the same state as a linear combination of mass eigenstates

[2]

We can time evolve this state using the fact that the mass states are eigenstates of the Hamiltonian:
Ĥ |νi 〉 = Ei |νi〉.

[3]

Because the neutrino masses are extremely small, for even a modestly energetic νβ, we have |p| ≈ Emi and can safely apply a relativistic approximation to the energies of ν1 and ν2 in the expression above:

[4]

Putting this and the inverse of Equation [1] into Equation [3] gives the time evolved state in terms of the weak eigenstates:

[5]

The probability of the state (that began as a νβ) to now be in the να state is the modulus square of their overlap:

[6]

where we defined Δm2m12 - m22. Neutrinos travel close to the speed of light due to their tiny mass, so (in natural units) we can replace t in Equation [6] with L, the distance the neutrino travels since its production at t = 0. At any given time the neutrino must be in one of the two weak eigenstates, so the probability to remain in the νβ state (after switching to more conventional units) is

[7]