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 antineutrino.
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
c_{ij} ≡ cos θ
_{ij} and
s_{ij} ≡ sin θ
_{ij}.
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.
2Flavor 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:
ν_{α} and
ν_{β}. Their
mixing with the mass eigenstates can be parameterized by
[1]
For example, imagine at
t =
0 a neutrino is produced in the
βflavor 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 
ν_{i} 〉 =
E_{i}

ν_{i}〉.
[3]
Because the neutrino masses are extremely small, for even a modestly energetic
ν_{β}, we have 
p ≈
E ≪
m_{i} and can safely apply a relativistic
approximation to the energies of
ν_{1} and
ν_{2} 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
ν_{β}) to now be in
the
ν_{α} state is the modulus square of their overlap:
[6]
where we defined
Δm^{2} ≡
m_{1}^{2}

m_{2}^{2}. 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
ν_{β} state (after switching to more conventional
units) is
[7]
Measuring Oscillation Parameters
Coming Soon...