Stars as Laboratories for Fundamental Physics - MPP Theory Group

286 Chapter 8
Fig. 8.1. Oscillation pattern for two-flavor oscillations (neutrino energy ω).
The flavor oscillations described by Eq. (8.16) are fully analogous to
the rotation of the plane of polarization in an optically active medium
or to the spin precession in a magnetic field. This analogy is brought
out more directly if one starts with the equation of motion for the
density matrix Eq. (8.9). Suppressing the index ω the matrices can be
expressed as
ρ = 1 (1 + P · σ) and K = ω − b 2 0 + 1 B · σ, (8.18)
2
and a similar representation for Ω where B is expressed as a function
of k by virtue of ω → k to lowest order for relativistic neutrinos.
The vector P is a flavor polarization vector. In the weak-interaction
basis |Ψ e | 2 = 1 2 (1 + P 3) and |Ψ µ | 2 = 1 2 (1 − P 3) give the probability for
the neutrino to be measured as ν e or ν µ , respectively. P 1 and P 2 contain
phase information and thus reveal the degree of coherence between the
flavor states. For a pure state |P| = 1 while in general |P| < 1. For
P = 0 one has a completely incoherent equal mixture of both flavors.
In optics, P describes the degree of polarization of a light beam in
the Poincaré sphere representation of the Stokes parameters (Poincaré
1892; Born and Wolf 1959).
The equation of motion for the polarization vector in any flavor
basis is found to be (Stodolsky 1987; Kim, Kim, and Sze 1988)
∂ z P = B × P or ∂ t P = B × P. (8.19)
Here, B plays the role of a “magnetic field” and P that of a “spin
vector.” The precession of P for an initial ν e where P(0) = (0, 0, 1) is
shown in Fig. 8.2.