Stars as Laboratories for Fundamental Physics - MPP Theory Group

Neutrino Oscillations 297
when the quantity ξ of Eq. (8.31) is much smaller than unity. In the
opposite limit one finds for the mixing angle in the medium
sin 2θ = m2 2 − m 2 1
√
2 GF n e 2ω sin 2θ 0. (8.34)
The oscillation length becomes
l osc =
2π
√
2 GF n e
= 1.63×10 4 km g cm−3
Y e ρ , (8.35)
independent of the neutrino masses or energy.
Typically the effect of the medium is, therefore, to suppress the
mixing angle and thus the possibility to observe oscillations. Of course,
for normal materials with a density of a few g cm −3 and neutrino masses
in the eV range one needs TeV neutrino energies for the medium to be
relevant at all. For a review of the impact of oscillations in a medium
on neutrino experiments or the observation of atmospheric, solar, or
supernova neutrinos see, for example, Kuo and Pantaleone (1989).
8.3.3 Inhomogeneous Medium: Adiabatic Limit
In an inhomogeneous medium the oscillation problem is much more
complicated. Recall that the spatial variation of a stationary neutrino
beam in the z-direction is given by i∂ z Ψ = −KΨ according to Eq. (8.7)
if one drops the index ω. The matrix K = ω−Meff/2ω 2 is now a function
of z. Formally, the solution is Ψ(z) = W Ψ(0) with
( ∫ z )
W = S exp i K(z ′ ) dz ′ , (8.36)
0
where S is the space-ordering operator. An explicit solution is not
available because the matrices K(z) generally do not commute for different
z. Of course, for a constant K one recovers the previous result
W = e iKz .
In certain limits one may still find simple solutions. The most interesting
case of neutrinos moving through an inhomogeneous medium
is the emission from stars, notably the Sun, where they are produced
in a relatively high-density region and then escape into vacuum. The
density at the center of the Sun is about 150 g cm −3 , the solar radius
6.96×10 10 cm, yielding an extremely shallow density variation by terrestrial
standards! Therefore, consider the adiabatic limit where the
density of the medium varies slowly over a distance l osc which is the
characteristic length scale for the oscillation problem.