Stars as Laboratories for Fundamental Physics - MPP Theory Group

Oscillations of Trapped Neutrinos 315
leads to P ˙ /P = −D (P T /P ) 2 with P = |P| and P T = |P T |. For a
small collision rate relative to the oscillation frequency one may use
the precession-averaged P T which is found by taking the transverse
part of the projection of P on V. Elementary geometry in Fig. 8.2
yields ⟨P T ⟩/P = cos 2θ sin 2θ so that ⟨ P ˙ /P ⟩ = −D cos 2 2θ sin 2 2θ. If
one neutrino interacts while the other is sterile (for example ν µ with
regard to charged-current absorption) D is half the collision rate Γ of
the active flavor (Stodolsky 1987). For small mixing angles one recovers
the previous intuitive relaxation rate 1 2 sin2 2θ Γ.
Equation (9.1) is based on a single-particle wave function picture
of neutrino oscillations and thus it is applicable if effects nonlinear in
the neutrino density matrices can be ignored. It does not allow one to
include the effect of Pauli blocking of neutrino phase space, which is
undoubtedly important in a SN core where the ν e Fermi sea is highly
degenerate. In this case it is rather unclear what one is supposed to
use for the damping parameter D. If ν e and ν µ scatter with equal
amplitudes, there is no damping at all because the collisions do not
distinguish between flavors, preserving the coherence between them.
Does this remain true if ν e collisions are Pauli blocked by their high
Fermi sea while those of ν µ are not Such and other related questions
can be answered if one abandons a single-particle approach to neutrino
oscillations, i.e. if one moves to a field-theoretic framework which
includes many-body effects from the start.
9.2.2 Matrix of Densities
Which quantity is supposed to replace the previous single-particle density
matrix as a means to describe a possibly degenerate neutrino
ensemble For unmixed neutrinos the relevant observables are timedependent
occupation numbers f p for a given mode p of the neutrino
field. They are given as expectation values of number operators
n p = a † pa p where a p is a destruction operator for a neutrino in mode p
and a † p the corresponding creation operator. The expectation value is
with regard to the state | ⟩ of the entire ensemble. For several flavors
it is natural to generalize the f p ’s to matrices ρ p = ρ(p) of the form 49
49 Strictly speaking ρ(p) is defined by ⟨a † j (p)a i(p ′ )⟩ = (2π) 3 δ (3) (p−p ′ )ρ ij (p) and
similar for ρ(p). Therefore, the expectation values in Eq. (9.3) diverge because they
involve an infinite factor (2π) 3 δ (3) (0) which is related to the infinite quantization
volume necessary for continuous momentum variables. In practice, this factor always
drops out of final results so that one may effectively set (2π) 3 δ (3) (0) equal to
unity.