Stars as Laboratories for Fundamental Physics - MPP Theory Group

288 Chapter 8
Fig. 8.3. Oscillation pattern for a Gaussian source distribution with s =
1
5 l osc according to Eq. (8.21).
Fig. 8.4. Oscillation pattern for a mixture of neutrino energies with δ =
1
10 π/l 0 according to Eq. (8.23).
These phenomena are well described in the picture of a precessing
polarization vector which represents a density matrix and thus is designed
to deal with incoherent or partially coherent beams. Notably, for
a distribution of energies the polarization vector is P = ∫ dω g(ω) P ω .
Because the components P ω precess with different frequencies about a
common “magnetic field” direction the component of P transverse to B
disappears as the P ω approach a uniform distribution on the precession
cone in Fig. 8.2. The projection of P on B, however, is conserved so
that P → (P · ˆB) ˆB for t → ∞. If originally P = (0, 0, 1) for initial ν e ’s
the geometry of Fig. 8.2 indicates that P 3 → cos 2 2θ for t → ∞ and so
prob(ν e → ν µ ) = 1 2 sin2 2θ.