Stars as Laboratories for Fundamental Physics - MPP Theory Group

284 Chapter 8
equation of motion for these quantities which is most compact in terms
of a density matrix
ρ ab = Ψ ∗ bΨ a . (8.8)
Then i∂ t ρ k = [Ω k , ρ k ] or i∂ z ρ ω = −[K ω , ρ ω ] where [A, B] = AB − BA
is a commutator of matrices. Therefore,
i∂ t ρ = (2k) −1 [M 2 , ρ] or i∂ z ρ = (2ω) −1 [M 2 , ρ] , (8.9)
where the indices ω or k have been dropped.
A beam evolves from the z = 0 state or density matrix as
Ψ ω (z) = e iKz Ψ ω (0) or ρ ω (z) = e −iKz ρ ω (0) e iKz . (8.10)
An analogous result applies to the case of temporal rather than spatial
oscillations. In the weak-interaction basis e iKz will be denoted by W ,
W (z) ≡ (e iKz ) weak = U(e iKz ) mass U † . (8.11)
If the neutrino is known to be a ν e at the source (z = 0) its probability
for being measured as a ν e at a distance z (“survival probability”) is
|W ee (z)| 2 .
One may be worried that the simple-minded derivation and interpretation
of these results is problematic because the neutrino wave function
is never directly observed. What is observed are the charged leptons
absorbed or emitted in conjunction with the neutrino production and
detection. However, if one performs a fully quantum-mechanical calculation
of the probability (or cross section) for the compound process of
neutrino production, propagation, and absorption, the naive oscillation
probability described by the elements of the W matrix factors out for
all situations of practical interest, and notably in the relativistic limit
(Giunti et al. 1993; Rich 1993).
8.2.2 Two-Flavor Oscillations
The neutrino mixing matrix U can be parametrized exactly as the
Cabbibo-Kobayashi-Maskawa (CKM) matrix in the quark sector in
Eq. (7.6). If one of the three two-family mixing angles is much larger
than the others (as for the quarks) one may study oscillations between
the dominantly coupled families as a two-flavor mixing problem. Moreover,
because so far all experiments—with the possible exception of
solar and certain atmospheric neutrino observations—yield only upper
limits on oscillation parameters one usually restricts the analysis to a