Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 237
6.6 Neutrino Form Factors in Media
In a medium, neutrinos can interact with photons using electrons or
other charged particles as go-betweens. The basic idea is to consider the
Compton process of Fig. 6.11 with the initial- and final-state electrons
in the same state, i.e. forward scattering for the electrons. Then one
may sum over all electrons of the medium. This coherent superposition
of the amplitudes from all electrons was used in the previous section to
calculate the standard-model plasmon decay rate. There, only on-shell
(propagating) photons were considered. In general one may consider
other cases, for example electromagnetic scattering by the exchange of
a space-like photon, or the behavior of neutrinos in an external electric
or magnetic field.
The neutrino electromagnetic form factors in vacuum will be studied
in Sect. 7.3.2. They can be classified as a charge radius, an anapole
moment, and an electric and a magnetic dipole moment. They are
induced by intermediate (virtual) charged particles such as charged
leptons or W bosons. In the present case the form factors are induced
by the real particles of the ambient heat bath. The effective Lagrangian
Eq. (7.19) is fundamentally Lorentz covariant, a fact which reduces
the number of possible form factors to four. While in a medium the
couplings may also be written in a nominally Lorentz covariant form,
the medium singles out an inertial frame, leading to more complicated
structures. This is analogous to dispersion which is simple in vacuum
(a mass term is the only possibility) while in a medium the dispersion
relations can be excruciatingly complicated.
Limiting the couplings to the ones mediated by electrons and protons,
the induced photon coupling to the neutrino is proportional to A µ
because these fermions couple by the usual eψγ µ ψA µ interaction. The
neutrinos couple by their standard effective neutral-current interaction
(G F / √ 2) ψ ν γ α (1 − γ 5 )ψ ν ψ e γ α (C V − C A γ 5 )ψ e with the weak coupling
constants C V and C A given in Appendix B. Therefore, after summing
over all intermediate electron states the effective neutrino-photon interaction
may be written in the form
L eff = − √ 2 G F ψ ν γ α
1
2 (1 − γ 5)ψ ν Λ αβ A β , (6.98)
where Λ αβ is a matrix which depends on the medium properties and
on the energy-momentum transfer, i.e. the energy momentum K of the
photon line. It consists of a symmetric piece Λ αβ
V which is proportional
to C V , and an antisymmetric piece Λ αβ
A which is proportional to C A .