Stars as Laboratories for Fundamental Physics - MPP Theory Group

Oscillations of Trapped Neutrinos 335
For l.h. electrons the coupling strengths are different, and they are
relativistic so that recoil effects are not small. However, they are degenerate
so that their contribution is expected to be smaller than the ep
process. It is not entirely negligible, however, especially if the nucleon
contribution is partly suppressed by many-body effects. The transition
rates were worked out in detail by Raffelt and Sigl (1993) for entirely
degenerate leptons. They found
W R EE ′
W L EE ′
≈ G2 Fµ 2 e
3π
≈
G2 Fµ 2 e
3π
(E − E ′ )E ′
,
E 2
(E − E ′ )E
, (9.62)
E ′2
a result which is the lowest-order term of an expansion in powers of
E/µ e (electron chemical potential µ e ).
9.5.3 Time Scale for Flavor Conversion
Given enough time the ν τ ’s will reach the same number density as the
ν e ’s. Therefore, it is most practical to discuss the approach to chemical
equilibrium in terms of a time scale
τ −1 ≡ − d ( )
dt ln nνe − n νx
. (9.63)
n νe
If τ were a constant independent of n νx /n νe the difference between the
number densities would be damped exponentially.
Collecting the results of the previous section one then finds easily
for the case of a “large” ∆m 2 of Eq. (9.59)
τ −1 = θ 2 0
3G 2 F
5π n pµ 2 ν e
[
(C 2 V + 3C 2 A) F p + µ ν e
µ e
F e
]
, (9.64)
where F p and F e give the contributions of protons (CC process) and
electrons (effective NC process). They are functions of µ νx which is
parametrized by
η ≡ µ νx /µ νe . (9.65)
One finds (Fig. 9.2, left panel)
F p = (1 − η 5 )/(1 − η 3 ),
F e = ( 5 6 + 1 2 η + 1 4 η2 + 1 12 η3 ) (1 − η) 3 /(1 − η 3 ), (9.66)