Stars as Laboratories for Fundamental Physics - MPP Theory Group

Oscillations of Trapped Neutrinos 339
Because the ν x are assumed to escape freely one may set fp
x = 0 on
the r.h.s. of these equations. Moreover, for the ν x coupling constants
one may use g x = 0 because it is sterile. For an isotropic medium
and taking an angular average of the scattering rate as in the previous
section one finds
(
f˙
p x = 1 4 s2 p PE e + ∑ ∫
)
(ge) a 2 dp ′ WE a ′ E fp e . (9.72)
′
a
A summation over different target species for effective NC scattering
was restored. Taking the ν e ’s to be completely degenerate one has
fp e = 1 for E = |p| < µ νe and fp e = 0 otherwise so that
(
f˙
p x = 1 4 s2 p PE e + ∑ a
0
(g a e) 2 ∫
dE ′ E′2 WE a )
′ E
. (9.73)
2π 2
With this result and
∫ µν e
ṅ L = − dE f ˙ p x and
∫ µν e
˙Q = − dE f ˙ p x E (9.74)
the volume loss rates for lepton number and energy can be easily determined.
Turn first to the case ∆m 2 ∼ > (100 keV) 2 so that one may use the
vacuum mixing angle in a SN core. One must now include both the CC
process ep → nν x as well as the NC process ν e N → Nν x which is not
suppressed because for a sterile ν x there is no destructive interference
effect. For nondegenerate nucleons and using C 2 V + 3C 2 A ≈ 4 for both
CC and NC processes one easily finds from the results of Sect. 9.5.2
ṅ L = − 1 4 sin2 2θ 0
2G 2 F
5π 3 n B (Y p + 1 4 ) µ5 ν e
, (9.75)
and the same for ˙Q with 1 5 µ5 ν e
→ 1 6 µ6 ν e
. Electrons as NC scattering
targets may be neglected because of their degeneracy. Lepton number
is thus lost at a rate
where
Ẏ L = − sin 2 2θ 0 τ −1 (Y e + 1 4 ) Y 5/3
ν e
, (9.76)
τ −1 = 3 5 (36π)1/3 G 2 Fn 5/3
B = 7.7×10 10 s −1 ρ 5/3
15 . (9.77)
Here, Y p = Y e , Y L ≡ n L /n B , and Y ν n B = n ν = µ 3 ν/6π 2 was used, and
ρ 15 is ρ in units of 10 15 g cm −3 .
0