Stars as Laboratories for Fundamental Physics - MPP Theory Group

270 Chapter 7
amplitude will receive radiative corrections from a variety of diagrams,
including photon exchange, which must be considered simultaneously
so that it is not at all obvious that one can extract a finite, gaugeinvariant,
observable quantity that can be physically interpreted as a
charge radius. 44
The charge radius, even if properly defined, represents only a correction
to the tree-level weak interaction and as such it is best studied
in precision accelerator experiments. In the astrophysical context, weak
interaction rates involving standard l.h. neutrinos cannot be measured
with the level of precision required to test for small deviations from
the standard model. Indeed, a recent compilation (Salati 1994) reveals
that experimental bounds on ⟨r 2 ⟩ are more sensitive than astrophysical
limits, except perhaps for ν τ for which experimental data are scarce and
so its standard-model neutral-current interactions are not well tested.
The matrix element for the anapole interaction in the Lorentz gauge
is proportional to Q 2 . Therefore, it vanishes in the limit Q 2 → 0, i.e. for
real photons coupled to the neutrino current. The role of the anapole
form factor G 1 is thus very similar to a charge radius: it represents a
correction to the standard tree-level weak interaction and as such does
not seem to be of astrophysical interest.
The form factors F 2 and G 2 are of much greater importance because
they may obtain nonvanishing values even in the Q 2 → 0 limit.
Henceforth I shall refer to µ ≡ F 2 (0) as a magnetic dipole moment,
to ϵ ≡ iG 2 (0) as an electric dipole moment, respectively. This identification
is understood if one derives the Dirac equation of motion
i∂ t ψ = Hψ for a neutrino field ψ (mass m) in the presence of an external,
weak, slowly varying electromagnetic field F µν . From Eq. (7.19)
one finds for the Hamiltonian
H = −iα · ∇ + β [ m − (µ + iϵγ 5 )(iα · E + Σ · B) ] , (7.21)
where 1σ 2 µνF µν = iα · E + Σ · B was used. In the Dirac representation
one has
( )
( )
( )
0 σ
I 0
σ 0
α = , β = , Σ = , (7.22)
σ 0
0 −I
0 σ
where σ is a vector of Pauli matrices while I is the 2 × 2 unit matrix.
For a neutrino at rest the Dirac spinor is characterized by its large
44 For recent discussions of these matters see Lucio, Rosado, and Zepeda (1985),
Auriemma, Srivastava, and Widom (1987), Degrassi, Sirlin, and Marciano (1989),
Musolf and Holstein (1991), and Góngora-T. and Stuart (1992).