Stars as Laboratories for Fundamental Physics - MPP Theory Group

Oscillations of Trapped Neutrinos 333
r.h. “species.” The scattering with ν e and ν x is then described by the
effective NC Hamiltonians
H L,R = G F
2 √ 2 ψ eγ µ (1 ∓ γ 5 )ψ e ΨG L,R γ µ (1 − γ 5 )Ψ, (9.52)
where Ψ is again a neutrino column vector in flavor space. Further,
( 2 sin 2 )
θ
G L =
W + 1 0
0 2 sin 2 ,
θ W − 1
( 2 sin 2 )
θ
G R =
W 0
0 2 sin 2 , (9.53)
θ W
where sin 2 θ W ≈ 1 will be used. Hence the effective NC coupling constants
are different for ν e and ν x interacting with l.h. electrons while
4
they are the same for r.h. ones.
The ν e Fermi sea is very degenerate. With regard to the neutrino
distributions one may thus use the approximation T = 0 so that neutrino
occupation numbers are 1 below their Fermi surface, and 0 above.
Because the chemical potential µ νe of the ν e population exceeds µ νx , and
because neutrinos can only down-scatter in the T = 0 limit, the term
proportional to fp(1 x − fp e ′) vanishes. Moreover, the detailed-balance
requirement (1 − fp)P e E e = fpA e e E implies PE e = 0 for E > µ νe where
fp e = 0. Then altogether
∫ (
µν e θ
2
ṅ νx = dE E PEE e 2 ∫ E
+
µ ν x 2π 2 µ ν xdE ′ ∑ W a (gxθ a E − ge a θ E ′) 2 E 2 E ′2 )
EE .
′
a
4π 4 (9.54)
Because for degenerate neutrinos n ν = µ 3 ν/6π 2 Eq. (9.54) can be written
as a differential equation for µ νx .
According to Eq. (8.30) the mixing angle in a medium which is
dominated by protons, neutrons, and electrons is given by
sin 2θ 0
tan 2θ E =
, (9.55)
cos 2θ 0 − E/E ρ
where the density-dependent “resonance energy” is
E ρ ≡
∆m2
2 √ 2 G F n e
, (9.56)
with the electron density n e . For E = µ νe
E
E ρ
=
one finds
(68 keV)2
∆m 2 Y e Y 1/3
ν e
ρ 4/3
14 , (9.57)
where ρ 14 is the density in units of 10 14 g cm −3 and as usual Y j gives the
abundance of species j relative to baryons. The approximate parameter