Stars as Laboratories for Fundamental Physics - MPP Theory Group

240 Chapter 6
Λ αβ
A does not contribute so that only the Λ 00
V component remains of
interest. In the static limit Π 00 is simply given by π L (0, k) which in
turn can be identified with the square of the screening scale kS 2 in a
medium (Sect. 6.4.1). This implies that the neutrino interacts with the
external electric field as if it had a charge
e ν = −(C V /e) √ 2 G F k 2 S. (6.102)
In a classical (nondegenerate, nonrelativistic) hydrogen plasma the
screening scale is given by the Debye scale through kS 2 = 2kD 2 = 2e 2 n e /T
with the electron density n e so that the induced neutrino charge is e ν =
C V e 2 √ 2 G F n e /T . This induced charge is explained by the medium
polarization caused by the weak force exerted by the presence of the
neutrino.
While this induced charge is conceptually very interesting it does
not seem to have any immediate practical consequences. Notably, it
is not the relevant quantity for the interaction with a static magnetic
field, i.e. one may not infer that neutrinos move on curved paths in
magnetic fields. The presence of a “neutrino charge” was derived for
the interaction with a static electric field! The relevant form factor
for the interaction with a static magnetic field is identified by noting
that now only the spatial components of A µ are nonzero. In the static
limit only the component Π 00 of the polarization tensor survives when
for the
neutrino interaction with a magnetic field. The contribution from Λ αβ
A
can be interpreted as a “normal” or “Dirac magnetic moment” induced
by the medium (Semikoz 1987a; D’Olivo, Nieves, and Pal 1989)
contracted with A µ . Then there is no contribution from Λ αβ
S
µ ν = −eC A
√
2GF 4π
∫ ∞
0
dp [ f e −(p) − f e +(p) ] . (6.103)
In the limit of a classical plasma this is µ ν = (e ν /2m e )(2C A /C V ) where
e ν is the induced electric charge of Eq. (6.102).
This induced Dirac magnetic moment is to be compared with the
electron’s Dirac moment e/2m e , not with an anomalous moment. The
former arises from the eψ e γ µ ψ e A µ coupling, the latter is described by
1
µ 2 eψ e σ µν ψ e F µν . This means that the induced dipole moment does
not lead to neutrino spin precession—it only couples to left-handed
states. It entails an energy difference between neutrinos moving in
opposite directions along a magnetic field. The transverse part of the
field has no impact on the neutrino—there is no spin precession, and no
curvature of the trajectory. (These conclusions pertain to the limit of
weak magnetic fields. For strong fields the modification of the electron