Stars as Laboratories for Fundamental Physics - MPP Theory Group

298 Chapter 8
This case is best understood for two-flavor mixing if one studies
the (temporal) evolution of the neutrino flavor polarization vector P.
Recall that it evolves according to the spin-precession formula Ṗ =
B × P where the “magnetic field” is now a function of time. If B
varies slowly relative to the precession frequency the “spin” follows the
magnetic field in the sense that it moves on a precession cone which is
“attached” to B. If the spin is oriented essentially along the magnetic
field direction it stays pinned to that direction. Therefore, it can be
entirely reoriented by slowly turning the external magnet.
For the case of neutrino oscillations this means that in the adiabatic
limit a state can be entirely reoriented in flavor space, i.e. an
initial ν e can be turned almost completely into a ν µ even though the
vacuum mixing angle may be small. Consider θ 0 ≪ 1, an initial density
so large that the medium effects dominate, begin with a ν e , and
let m 1 < m 2 . This means that in Fig. 8.8 (lowest panel) begin on
the upper branch of the dispersion relation far to the right of the
crossover. Then let the neutrino propagate toward vacuum through
an adiabatic density gradient. This implies that it stays on the upper
branch and ends up at n e = 0 (vacuum) as the mass eigenstate
of the upper eigenvalue m 2 which corresponds approximately to a ν µ .
This behavior is known as “resonant neutrino oscillations” or MSW effect
after Mikheyev, Smirnov, and Wolfenstein. Mikheyev and Smirnov
(1985) first discovered this effect when they studied the oscillation of
solar neutrinos while Wolfenstein (1978) first emphasized the importance
of refraction for neutrino oscillations. A simple interpretation of
resonant oscillations in terms of an adiabatic “level crossing” was first
given by Bethe (1986).
In order to quantify the adiabatic condition return to the picture
of a spin precessing around a magnetic field. For flavor oscillations
the precession frequency is 2π/l osc . The “magnetic field” is tilted with
an angle 2θ (medium mixing angle) against the 3-direction (Fig. 8.2)
and so its speed of angular motion is 2 dθ/dt. For spatial rather than
temporal oscillations the adiabatic condition is
|∇θ| ≪ π/l osc . (8.37)
This translates into ∇θ = 1 2 ξ (sin2 2θ/ sin 2θ 0 ) ∇ ln n e while l osc is given
by Eq. (8.32) and ξ by Eq. (8.31). Thus, the adiabatic condition is
ξ sin 3 2θ |∇ ln n e | ≪ sin 2 2θ 0 |m 2 2 − m 2 1|/2ω. (8.38)
This condition must be satisfied along the entire trajectory.