Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 241
wavefunctions, i.e. Landau levels rather than plane waves, would have
to be used for a self-consistent treatment of the photon polarization
tensor and thus, for the neutrino coupling to a magnetic field. For a
first discussion see Oraevskiĭ and Semikoz 1991.)
In summary, on the basis of the existing literature it appears that
the medium-induced electromagnetic form factors of neutrinos are of
practical importance only for the photon decay process that was discussed
in the previous section.
6.7 Neutrino Refraction
6.7.1 Neutrino Refractive Index
When neutrinos propagate in a medium they will experience a shift of
their energy, similar to photon refraction, due to their coherent interaction
with the medium constituents (Wolfenstein 1978). The neutrino
refractive index can be calculated in the same way as that for any
other particle which propagates in a medium, namely on the basis of
the forward scattering amplitudes as discussed in Sect. 6.2.1. As one
needs only forward scattering, and as the relevant medium constituents
are protons, neutrons, electrons, and possibly other neutrinos, only the
Feynman graphs of Fig. 6.15 need to be considered. 38
In most situations of practical interest the energies of the neutrinos
and of the medium particles are much smaller than the W and Z mass
(80.2 and 91.2 GeV) so that the energy and momentum transferred by
the gauge bosons is always much less than their mass. 39 This justifies
to expand their propagators (energy-momentum transfer Q) as
D µν (Q) = g µν
m 2 Z,W
+ Q2 g µν − Q µ Q ν
m 4 Z,W
+ . . . (6.104)
and keep only the first term. (The second term is needed if the contribution
of the first one cancels as in a CP symmetric medium—see
38 In the formalism of finite temperature and density (FTD) field theory the amplitudes
may be written in a more compact form so that the relevant Feynman
graphs reduce to a tadpole and a bubble graph (Nötzold and Raffelt 1988; Nieves
1989; Pal and Pham 1989). Apart from a more compact notation, however, the
FTD formalism leads to the same expressions as the “pedestrian” approach chosen
here.
39 See however Learned and Pakvasa (1995) as well as Domokos and Kovesi-
Domokos (1995) for a discussion of the oscillations of very high-energy cosmic neutrinos
for which this approximation is not adequate.