Stars as Laboratories for Fundamental Physics - MPP Theory Group

306 Chapter 8
substantial reduction of the left-handed solar neutrino flux then requires
magnetic dipole moments of order 10 −10 µ B . Typical galactic
magnetic fields are of order 10 −6 G with coherence scales of order kpc.
Therefore, neutrinos from a galactic source such as a supernova could
be “flipped” before reaching Earth if their dipole moment is of order
10 −12 µ B . These values are in the neighborhood of what is experimentally
and observationally allowed so that there remains interest in the
possibility that magnetic spin oscillations could have a significant impact
on the neutrino fluxes from the Sun or from supernovae.
8.4.2 Spin Precession in a Medium
The picture of magnetic spin oscillations thus developed is incomplete.
If neutrinos propagate in a medium as in the Sun they are subject to
medium-induced refractive effects which shift the energy of left-handed
states relative to right-handed ones which propagate as in vacuum because
they are sterile. Therefore, including medium effects the effective
Hamiltonian for the evolution of the two helicity states of ν e is
( )
0 µBT
H = √
µB T 2GF (n e − 1n , (8.54)
2 n)
where n e and n n are the electron and neutron densities, respectively.
Here, “spin up” refers to the direction of motion, i.e. to right-handed
states, “spin down” to left-handed ones. The two helicity states are
no longer degenerate, and the spin precession will no longer lead to
a complete reversal of the spin. One may write H = b + µB · σ so
that there is effectively a longitudinal magnetic field, i.e. parallel to the
direction of motion or z-direction
B L = G F(n e − 1 2 n n)
√
2 µ
= 66.6 kG 10−10 µ B
µ
ρ (Y e − 1 2 Y n)
g cm −3 . (8.55)
Here, ρ is the mass density and Y e,n the electron and neutron number
per baryon. If B L ≫ B T the precession is around a direction close to
the direction of motion, and so the spin reversal will always be far from
complete. Hence, the presence of the medium suppresses magnetic spin
oscillations (Voloshin, Vysotskiĭ, and Okun 1986).
8.4.3 Spin-Flavor Precession
If neutrinos were Majorana particles they could not have magnetic
dipole moments, but they could still possess transition moments between
different flavors. In this case the magnetic oscillation would be,