Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 243
medium particles 40 which are left-handed and thus polarized. Therefore,
(1 − γ 5 )ψ ν = 2ψ ν and C V ′ = 2C V .
Because we are dealing with forward scattering where recoil effects
do not occur, the contributions from free or bound nucleons are the
same. Therefore, Eq. (6.106) allows one to determine the refractive
index of any normal medium. Because electric neutrality implies an
excess density of electrons over positrons which balances against the
protons, their neutral-current contributions cancel. (A possible exception
is a π − condensate that may exist in neutron stars.) An excess
of ν e over ν e appears to occur only in a young supernova core where
neutrinos have a large chemical potential for the first few seconds after
collapse.
All told, the dispersion relation for unmixed neutrinos, valid even in
the nonrelativistic limit (Chang and Zia 1988), can be written in terms
of a potential energy as
(ω − V ) 2 = k 2 + m 2 , (6.107)
where V = −(n refr − 1) ω. For all practical cases
V = ± √ ⎧
⎨ (− 1
2G F n B ×
Y 2 n + Y e + 2Y νe ) for ν e ,
⎩ (− 1Y 2 n + Y νe ) for ν µ,τ ,
(6.108)
(upper sign ν, lower sign ν). Here, n B is the baryon density and
Y f ≡ n f − n f
n B
(6.109)
are the particle number fractions commonly used in astrophysics. Numerically,
√
2 GF n B = 0.762×10 −13 eV
ρ
(6.110)
g cm −3
with the mass density ρ.
A remark concerning the absolute sign of V is in order. The relative
signs between the different C V ’s can be worked out easily from the
weak interaction structure of the standard model. Also, the relative
sign of the effective neutral-current amplitudes which follow from Z ◦
and W exchange follows directly, for example, from the FTD approach
(Nötzold and Raffelt 1988). Thus to fix the overall sign it is enough
40 In a supernova core or in the early universe it is not possible to distinguish
between a “test neutrino” and a “medium neutrino.” There, one has to study the
nonlinear evolution of the entire ensemble self-consistently (Sect. 9.3.2).