Stars as Laboratories for Fundamental Physics - MPP Theory Group

Neutrino Oscillations 299
If the neutrino crosses a density region such that a resonance occurs,
this part of the trajectory yields the most restrictive adiabaticity
requirement. On resonance ξ = cos 2θ 0 and sin 2θ = 1. In this case one
defines an adiabaticity parameter
γ ≡ m2 2 − m 2 1
2ω
sin 2θ 0 tan 2θ 0
|∇ ln n e | res
, (8.39)
where the denominator is to be evaluated at the resonance point. The
adiabatic condition is γ ≫ 1. It establishes a relationship between vacuum
mixing angles and neutrino masses for which resonant oscillations
occur.
8.3.4 Inhomogeneous Medium: Analytic Results
For practical problems, notably the oscillation of solar neutrinos on
their way out of the Sun, one may easily solve the equation i∂ z Ψ =
−KΨ numerically for prescribed profiles of the electron density and
neutrino production rates. The main features of such calculations, however,
can be understood analytically because for certain simple density
profiles one can find analytic representations of Eq. (8.36) independently
of the adiabatic approximation.
Consider the situation where a ν e is produced in a medium and
subsequently escapes into vacuum through a monotonically decreasing
density profile. Because in the adiabatic limit the production and detection
points are separated by many oscillation lengths, the oscillation
pattern will be entirely washed out and one may use average probabilities
for the flavor content. Initially, the projection of the polarization
vector on the “magnetic field” direction is cos 2θ where θ is the medium
mixing angle at the production point. Because the component of P in
the B direction is conserved when B changes adiabatically, the final average
projection of P on the 3-axis is cos 2θ 0 cos 2θ. Then the survival
probability is prob(ν e →ν e ) = 1 2 (1 + cos 2θ 0 cos 2θ). This result applies
in the adiabatic limit whether or not a resonance occurs.
Next, drop the adiabatic condition but assume that the production
and detection points are many oscillation lengths away on opposite
sides of a resonance so that the oscillation pattern remains washed out;
then it remains sufficient to consider average probabilities. In this case
it is useful to write
prob(ν e →ν e ) = 1 2 + ( 1 2 − p) cos 2θ 0 cos 2θ, (8.40)
where the correction p to the adiabatic approximation is the probability
that the neutrino jumps from one branch of the dispersion relation to