Stars as Laboratories for Fundamental Physics - MPP Theory Group

326 Chapter 9
of ν e ’s because initially they have a large chemical potential and thus
are far away from chemical equilibrium with the other flavors. Their
high degree of degeneracy implies that ν e ’s may be ignored and with
them all pair processes. Therefore, the evolution of ν e ’s mixed with one
other flavor (standard or sterile) is given by
∫
˙ρ p = i[ρ p , Ω p ] + 1 dp [ ′ W
2
P ′ P Gρ p ′G(1 − ρ p )
−W P P ′ ρ p G(1 − ρ p ′)G + h.c. ] . (9.32)
The matrix of oscillation frequencies includes vacuum and first-order
medium contributions. With the momentum-dependent oscillation period
t osc it is
Here,
Ω p = (2π/t osc ) 1 2 v p · σ with v p ≡ (s p , 0, c p ). (9.33)
s p ≡ sin 2θ p and c p ≡ cos 2θ p (9.34)
with the momentum-dependent mixing angle in the medium θ p .
The second approximation for the conditions of a SN core is the
weak-damping limit or limit of fast oscillations. It is easy to show that
for the relevant physical conditions 2π/t osc is typically much faster than
the scattering rate. Therefore, it is justified to consider density matrices
˜ρ p averaged over a period of oscillation. While the ρ p ’s are given by
four real parameters which are functions of time, the ˜ρ p ’s require only
two, for example the occupation numbers of the two mixed flavors. It
is straightforward to show that in the weak interaction basis
( f
e )
( )
˜ρ p = p 0
0 1
0 fp
x + 1 t 2 p (fp e − fp)
x , (9.35)
1 0
where t p ≡ tan 2θ p = s p /c p , f e p is the occupation number of ν e , not of
electrons, and f x p refers to a standard or sterile flavor ν x .
One way of looking at the weak-damping limit is that between collisions
neutrinos are best described by “propagation eigenstates,” i.e.
in a basis where the ˜ρ p are diagonal. Then the matrix of coupling constants
G is no longer diagonal and so flavor conversion is understood as
the result of “flavor-changing neutral currents” where “flavor” refers to
the propagation eigenstates. However, because in general the effective
mixing angle is a function of the neutrino momentum one would have
to use a different basis for each momentum, an approach that complicates
rather than simplifies the equations. Therefore, it is easiest and