Stars as Laboratories for Fundamental Physics - MPP Theory Group

Neutrino Oscillations 283
As usual one expands the neutrino fields in plane waves of the form
Ψ(t, x) = Ψ k (t) e ik·x for which Eq. (8.1) is
(∂ 2 t + k 2 + M 2 ) Ψ k (t) = 0 . (8.4)
In general one cannot assume a temporal variation e −iωt because there
are three different branches of the dispersion relation with ωi 2 = k 2 +m 2 i .
A mixed neutrino cannot simultaneously have a fixed momentum and
a fixed energy!
In practice one has always to do with very relativistic neutrinos for
which k = |k| ≫ m i . In this limit one may linearize Eq. (8.4) by virtue
of ∂t 2 + k 2 = (i∂ t + k)(−i∂ t + k). For each mass eigenstate i∂ t → ω i ≈ k
and one needs to keep the exact expression only in the second factor
where the difference between energy and momentum appears. Thus
∂t 2 + k 2 ≈ 2k(−i∂ t + k), leading to the Schrödinger-type equation
(
i∂ t Ψ k = Ω k Ψ k where Ω k ≡ k + M 2 )
. (8.5)
2k
The vector Ψ originally consisted of neutrino Dirac spinors but it was
reinterpreted as a vector of (positive-energy) probability amplitudes.
For negative-energy states (antineutrinos) a global minus sign appears
in Eq. (8.5).
The Schrödinger equation (8.5) describes a spatially homogeneous
system with a nonstationary temporal evolution. In practice one usually
deals with the opposite situation, namely a stationary neutrino
flux such as that from a reactor or the Sun with a nontrivial spatial
variation. Then it is useful to expand Ψ(t, x) in components of fixed
frequency Ψ ω (x)e −iωt , yielding
(−ω 2 − ∇ 2 + M 2 ) Ψ ω (x) = 0. (8.6)
In the relativistic limit and restricting the spatial variation to the
z-direction one obtains in full analogy to the previous case
i∂ z Ψ ω = −K ω Ψ ω where K ω ≡
(
ω − M 2
2ω
)
. (8.7)
This equation describes the spatial variation of a neutrino beam propagating
in the positive z-direction with a fixed energy ω.
Ultimately one is not interested in amplitudes but in the observable
probabilities |Ψ l | 2 = Ψ ∗ lΨ l with l = e, µ, or τ. One may derive an