Stars as Laboratories for Fundamental Physics - MPP Theory Group

Nonstandard Neutrinos 271
component, a Pauli two-spinor ϕ, which is then found to evolve as
i∂ t ϕ = (µB 0 + ϵE 0 ) · σ ϕ, (7.23)
where the index 0 refers to quantities in the neutrino rest frame. A
neutrino polarized in or opposite to the field direction has the energy
±µB 0 or ±ϵE 0 , respectively, so that µ and ϵ are indeed magnetic and
electric dipole moments, respectively.
The electromagnetic form factors obey certain constraints for Dirac
and Majorana neutrinos; detailed discussions were provided by a number
of authors. 45 For Dirac neutrinos, all form factors must be real
relative to each other (no relative phases) if CP invariance holds. For
the diagonal case (coupling to one neutrino species) all form factors
must be real, and CP invariance implies that the electric dipole moment
must vanish. For Majorana neutrinos, a magnetic transition moment
(F 2 ) must be imaginary, an electric transition moment (G 2 ) real. If
CP invariance holds, in addition one of them must vanish, i.e. there is
either a transition electric, or a transition magnetic moment, but not
both. Majorana neutrinos cannot have diagonal electric nor magnetic
moments, nor can they have a charge or charge radius; they may have
an anapole form factor.
7.3.3 Two-Photon Coupling
Discussions of neutrino electromagnetic form factors are usually restricted
to the effective neutrino coupling to an electromagnetic wave
or static field. However, a two-photon coupling of neutrinos is also possible
and of some interest. Historically, it was thought for some time
that the process γγ → νν could be of great importance for the emission
of neutrinos from stars until it was shown by Gell-Mann (1961)
that the amplitude for this process vanishes identically if neutrinos
have only l.h. local interactions with electrons. Several authors discussed
the γγ → νν process when neutrinos are massive, or when they
have more general interaction structures (Halprin 1975; Fischbach et al.
1976, 1977; Natale, Pleitez, and Tacla 1987; Gregores et al. 1995). However,
there does not seem to be a plausible scenario where this process
would be of serious astrophysical interest.
In the standard model, there is an effective two-photon coupling
to neutrinos because the interaction is not local; rather, it is mediated
by finite-mass gauge bosons. Early calculations of the effective
45 For example Nieves (1982), Kayser (1982), Shrock (1982), and Li and Wilczek
(1982).