Stars as Laboratories for Fundamental Physics - MPP Theory Group

Nonstandard Neutrinos 269
dipole moment for F 2 , and an electric dipole moment for G 2 . In the
matrix element derived from this Lagrangian these couplings should
be viewed as form factors which are functions of Q 2 where Q is the
energy-momentum transfer to the fermion, i.e. the energy momentum
of the photon line attached to the fermion current. The interpretation
of a charge etc. then pertains to the Q 2 → 0 limit.
It is usually assumed that neutrinos are electrically neutral, i.e. that
F 1 (0) = 0 because electric charge quantization implies that elementary
particles carry only charges in multiples of 1 e where e is the electron
3
charge. In recent discussions of electric charge quantization 43 it was
stressed, however, that the standard model of electroweak interactions
without grand unification requirements does allow neutrinos to carry
small electric charges. Their possible magnitude is thus an experimental
issue; existing limits are reviewed in Sect. 15.8. Because these limits
are very restrictive, i.e. because neutrino electric charges must be very
small, it appears likely that electric charge is quantized after all so that
neutrino electric charges vanish identically.
Even if neutrinos are electrically neutral, as shall be assumed henceforth,
they can virtually dissociate into charged particles and so they
will have a form factor F 1 (Q 2 ) which does not vanish for Q 2 ≠ 0. One
may visualize the neutral object as a superposition of two charge distributions
of opposite sign with different spatial extensions. In terms
of a power series expansion of F 1 (Q 2 ) one usually defines the charge
radius by virtue of
⟨r 2 ⟩ = 6 ∂F 1(Q 2 )
e ∂Q 2 ∣ ∣∣∣∣Q 2 =0
(7.20)
where ⟨r 2 ⟩ may be both positive or negative.
For neutrinos, the interpretation of the charge radius as an observable
quantity is a rather subtle issue as it is probed by “off-shell”
photons (Q 2 ≠ 0), i.e. by intermediate photons in processes such as
scattering by photon exchange. Because the form factor is proportional
to Q 2 such scattering processes do not exhibit a Coulomb divergence.
The charge radius induces a short-range or contact interaction similar
to processes involving Z ◦ exchange. Therefore, the charge radius
represents a correction to the standard tree-level electroweak scattering
amplitude between neutrinos and charged particles. This tree-level
43 Babu and Mohapatra 1990; Babu and Volkas 1992; Takasugi and Tanaka 1992;
Foot, Lew, and Volkas 1993; Foot 1994. References to earlier works are given in
these papers.