Stars as Laboratories for Fundamental Physics - MPP Theory Group

Oscillations of Trapped Neutrinos 313
interactions with a medium, whether the neutrinos are degenerate or
not. As a free spin-off it will provide a proper formalism to deal with
the refractive effects of neutrinos propagating in a bath of neutrinos,
a problem that is of interest in the early universe where interactions
among neutrinos produce the dominant medium effect, and in the neutrino
flow from a SN core where the density of neutrinos is larger than
that of nonneutrino background particles (Sect. 11.4).
Besides neutrino flavor oscillations, magnetically induced spin or
spin-flavor oscillations are also of potential interest because in supernovae
and the early universe strong magnetic fields are believed to exist.
Spin relaxation (the process of populating the r.h. degrees of freedom by
the simultaneous action of spin oscillations and collisions) is a very similar
problem to that of achieving chemical equilibrium between different
flavors which is discussed here. Therefore, similar kinetic methods can
be applied (Enqvist, Rez, and Semikoz 1995 and references therein).
9.2 Kinetic Equation for Oscillations and
Collisions
9.2.1 Stodolsky’s Formula
The loss of coherence between mixed neutrinos in collisions cannot be
properly understood on the amplitude level because it is not the amplitudes,
but only their relative coherence that is damped—the flavor
states “decohere,” they do not disappear. This is different from the
decay of mixed particles where one of the amplitudes can be viewed
as decreasing exponentially so that the total number of particles is
not conserved. A natural description of decoherence is achieved by a
density matrix as in Eq. (8.8); for a two-flavor mixing problem it was
expressed in terms of a polarization vector P according to Eq. (8.18),
ρ = 1 (1+P·σ). In the weak interaction basis its diagonal elements are
2
the probabilities for measuring ν in, say, the ν e or ν µ state, respectively,
while the off-diagonal elements contain relative phase information.
In a two-level system, the length of the polarization vector measures
the degree of coherence: length 1 corresponds to a pure state, shorter
P ’s to some degree of incoherence, and length zero is the completely
mixed or incoherent state (Stodolsky 1987). In this latter case ρ is
proportional to the unit matrix which is invariant under a transformation
of basis. This state of “chemical” or “flavor equilibrium” has no
off-diagonal elements in any basis.