Stars as Laboratories for Fundamental Physics - MPP Theory Group

334 Chapter 9
range where resonance is important is shown in Fig. 9.3 as diagonal
shaded band.
For conditions near resonance there is no need for a detailed calculation
because then the mixing angle is large. Therefore, one may
focus on the limiting cases where ∆m 2 is either so small or so large that
θ 2 E ≪ 1,
|θ E | = θ 0 ×
{ 1 if Eρ /E ≫ 1 (large ∆m 2 ),
E ρ /E if E ρ /E ≪ 1 (small ∆m 2 ).
(9.58)
Thus, for large ∆m 2 the ν x production rate is
ṅ νx = θ 2 0
∫ (
µν e P
e
dE E E 2 ∫ E
+ dE ′ ∑
µ νx 2π 2 µ νx a
WEE a (gx a − ge) a 2 E 2 E ′2 )
, (9.59)
′
4π 4
while for small ∆m 2 it is
(
θ0 ∆m 2 ) 2 ∫
ṅ νx =
2 √ µνe
dE
2G F n e µ νx
( P
e
E
2π 2 + ∫ E
µ νx
dE ′ ∑ a
W a EE ′ (g a xE ′ − g a e E) 2
4π 4 )
.
(9.60)
9.5.2 Neutrino Interaction Rates
In order to evaluate these integrals one first needs the production rate
PE e of ν e ’s with energy E due to the CC reaction p + e → n + ν e .
The leptons are taken to be completely degenerate, the nucleons to
be completely nondegenerate. Because they are also nonrelativistic
the absorption of an e − produces a ν e of the same energy. Therefore,
PE e = σ E n p where n p is the proton density and σ E = (CV 2 +3CA)G 2 2 FE 2 /π
is the CC scattering cross section for electrons of energy E. Here,
C V = 1 and C A = 1.26 are the usual vector and axial-vector weak
couplings. Altogether one finds
P e E = C2 V + 3C 2 A
π
G 2 FE 2 n p . (9.61)
In practice this rate is reduced by various factors. First, Pauli blocking
of nucleons cannot be neglected entirely. Second, the degeneracy of
electrons is not complete. Third, the axial-vector scattering rate may
be suppressed in a medium at nuclear densities (Sect. 4.6.7).
For NC scattering, nucleons may be neglected entirely. In Eq. (9.59)
their contribution vanishes identically because g e = g x . In Eq. (9.60) it
is suppressed because they are relatively heavy so that E ′ ≈ E.