Astron. Astrophys. 363, 825-836 (2000)

2. Oscillations of degenerated neutrinos

The neutrino oscillations take place due to the fact that neutrino mass eigenstate components propagate differently because they have different energies, momenta and masses. In the standard assumption of the mixing of massive neutrinos, the neutrino flavor eigenstates are described by a superposition of the mass eigenstate components (Particle Data Group 1998). For oscillations occurring in vacuum between two neutrino flavors and the mixing can be written as:

where and are the mass eigenstate components and is the vacuum mixing angle. The mixing of antineutrinos can be obtained from Eq. (1) by performing the transformations and . It is usual to consider that each neutrino/antineutrino of a definite flavor is dominantly one mass eigenstate (Particle Data Group 1998). In this circumstance we refer to the dominant mass eigenstate component of / as , that of / as and to their difference of the squared masses as .

The mixing of the mass eigenstate components are modified in the presence of the asymmetric background. The background mixing angle and the vacuum mixing angle are related through (see e.g. Foot et al. 1996):

Here (), is the averaged neutrino momentum and is the neutrino effective potential due to the interaction with the asymmetric background (Enqvist et al. 1992; Foot et al. 1996):

where is a numerical factor (), is the neutrino temperature, is the Fermi constant () is the mass of Z boson, (GeV), is the photon number density (), and are the neutrino and antineutrino number densities.

The functions are defined as (Foot et al. 1996; Foot & Volkas 1997):

Here is approximately equal to the baryon to photon ratio, and s are the lepton asymmetries defined as:

where s are the number densities.

For the purpose of this work we consider large neutrino asymmetries (). Also, we consider that is non-degenerated and consequently . In this case the functions may be written as:

The effective potentials of antineutrinos can be obtained by replacing with in Eq. (3).

The neutrino flavor eigenstates oscillate via weak interaction processes. The average oscillation probability of a flavor eigenstate () is defined as (see e.g.: Raffelt 1996; Foot et al.1996; Enqvist et al. 1992):

where is the neutrino averaged elastic collision rate, is the mean distance between interactions (Foot et al. 1996), and the neutrino oscillation length is given by (see e.g. Raffelt 1996):

where is the neutrino energy. We will consider the and then (Foot et al. 1996). The averaged collision rate is given by (Enqvist et al. 1992):

where is the number density of massless non-degenerated neutrino species.

Eq. (7) together with Eqs. (2), (8), (9) gives the averaged oscillation probability used in the next section to evolve in time the neutrino distribution functions. The oscillation probabilities of antineutrinos can be obtained by performing the transformations and in Eqs. (7)-(9).

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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