Stars as Laboratories for Fundamental Physics - MPP Theory Group

314 Chapter 9
Even coherent or partially coherent density matrices can be diagonalized
in some basis. The interactions with the background medium
will also be diagonal in some basis; for neutrinos, this is the weak interaction
basis. If the density matrix and the interactions are diagonal in
the same basis, there is no decoherence effect. For example, a density
matrix diagonal in the weak interaction basis implies that there is no
relative phase information between, say, a ν e and a ν µ and so collisions
which affect ν e ’s and ν µ ’s separately have no impact on the density
matrix. However, a density matrix which is diagonal in the mass basis,
assumed to be different from the weak interaction basis, will suffer a
loss of coherence in the same medium.
All told, the loss of coherence is given by a shrinking of the length
of P. More precisely, only the component P T is damped which represents
the part “transverse” to the interaction basis, i.e. which in the
interaction basis represents the off-diagonal elements of ρ. Thus, in the
presence of collisions the evolution of P is given by (Stodolsky 1987)
Ṗ = V × P − DP T . (9.1)
The first part is the previous precession formula, except that here a
temporal evolution is appropriate since one has in mind the evolution
of a spatially homogeneous ensemble rather than the spatial pattern of
a stationary beam. The “magnetic field” V is
⎛ ⎞
V = 2π sin 2θ
⎜ ⎟
⎝ 0 ⎠ , (9.2)
t osc
cos 2θ
where θ is the mixing angle in the medium and t osc the oscillation
period. They are given in terms of the neutrino masses and momentum,
the vacuum mixing angle, and the medium density by Eqs. (8.30)
and (8.32) where strictly speaking the neutrino energy is to be replaced
by its momentum as we have turned to temporal rather than spatial
oscillations. The damping parameter D is determined by the scattering
amplitudes on the background.
The evolution described by Eq. (9.1) is a precession around the
“magnetic field” V, combined with a shrinking of the length of P to
zero. This final state corresponds to ρ = 1 where both flavors are
2
equally populated, and with vanishing coherence between them. For
D = t −1
osc and sin 2θ = 1 the evolution of the flavors is shown in Fig. 9.1
2
(solid line). Ignoring the wiggles in this curve it is an exponential as
can be seen by multiplying both sides of Eq. (9.1) with P/P 2 which