Stars as Laboratories for Fundamental Physics - MPP Theory Group

Oscillations of Trapped Neutrinos 321
9.3.2 Neutrino Refraction
As a first simple application one may recover neutrino refraction by
a medium that was previously discussed in Sect. 6.7.1. An explicit
evaluation of the second term of Eq. (9.13) with H int = H NC yields
i ˙ρ = [Ω 0 p, ρ p ] + ∑ a
n a [G a , ρ p ] , (9.17)
where n a ≡ ⟨B µ a ⟩P µ /P 0 where P is the neutrino four momentum and
thus P/P 0 their four velocity. In an isotropic medium the spatial parts
of ⟨B µ a ⟩ vanish so that n a is the number density of fermions a. If the
medium is unpolarized, axial currents do not contribute.
In the standard model with the coupling constants of Appendix B
one finds for an isotropic, unpolarized medium of protons, neutrons,
and electrons,
i ˙ρ p = [ (Ω 0 p + √ 2 G F N l ), ρ p
]
,
−i˙ρ p = [ (Ω 0 p − √ 2 G F N l ), ρ p
]
, (9.18)
where N l = diag(n e , 0, 0) in the flavor basis (electron density n e ). For
two flavors this is equivalent to the previous precession formula. Notably,
the ν and ν oscillation frequencies are shifted in opposite directions
relative to the vacuum energies.
Neutrino-neutrino interactions make an additional contribution to
the refractive energy shifts, i.e. to the first-order term in Eq. (9.13).
After the relevant contractions one finds 51 (Sigl and Raffelt 1993)
Ω S p = √ 2 G F
∫
dq { G S (ρ q − ρ q )G S + G S Tr [ (ρ q − ρ q )G S
]}
.
(9.19)
Ω S p is given by the same formula with ρ q and ρ q interchanged. The
trace expression implies the well-known result that neutrinos in a bath
of their own flavor experience twice the energy shift relative to a bath
of another flavor.
The early universe is essentially matter-antimatter symmetric so
that higher-order terms to the refractive index must be included as
discussed in Sect. 6.7.2; see also Sigl and Raffelt (1993). In stars,
51 If the neutrino ensemble is not isotropic one has to include a factor (1−cos θ pq )
under the integral, where θ pq is the angle between p and q.