Stars as Laboratories for Fundamental Physics - MPP Theory Group

304 Chapter 8
8.4 Spin and Spin-Flavor Oscillations
8.4.1 Vacuum Spin Precession
According to Sect. 7.2 neutrinos may have magnetic dipole moments.
In particular, if neutrinos have masses they inevitably have small but
nontrivial electromagnetic form factors. Novel interactions can induce
large dipole moments even for massless neutrinos. In the presence of
magnetic fields such a dipole moment leads to the familiar spin precession
which causes neutrinos to oscillate into opposite helicity states.
If a state started out as a helicity-minus neutrino, the outgoing particle
is a certain superposition of both helicities. The “wrong-helicity”
(right-handed) component does not interact by the standard weak interactions,
diminishing the detectable neutrino flux. As the Sun has
relatively strong magnetic fields it is conceivable that magnetic spin
oscillations could be responsible for the measured solar neutrino deficit
(Werntz 1970; Cisneros 1971)—see Sect. 10.7. Magnetic spin precessions
could also affect supernova neutrinos, and in the early universe it
could help to bring the “wrong-helicity” Dirac neutrino degrees of freedom
into thermal equilibrium. Presently I will focus on some general
aspects of neutrino magnetic spin oscillations rather than discussing
specific astrophysical scenarios.
In the rest frame of a neutrino the evolution of its spin operator S =
1
2 σ (Pauli matrices σ) is governed by the Hamiltonian H 0 = 2µB 0 · S;
this follows directly from Eq. (7.23). Here, B 0 refers to the magnetic
field in the neutrino’s rest frame as opposed to B in the laboratory
frame. The equation of motion of the spin operator in the Heisenberg
picture is then given by i∂ t S = [S, H 0 ] or
Ṡ = 2µB 0 × S, (8.48)
which represents the usual precession with frequency 2µB 0 around the
magnetic field direction. Equivalently, one may consider Eq. (7.23) for
the neutrino spinor which has the form of a Schrödinger equation.
Usually one will be concerned with very relativistic neutrinos so
that the magnetic field B 0 in the rest frame must be obtained from the
electric and magnetic fields in the laboratory frame E and B by virtue
of the usual Lorentz transformation
B 0 = (B · ˆv)ˆv + γ[ˆv × (B × ˆv) + E × v], (8.49)
where v is the velocity of the neutrino, ˆv the velocity unit vector, and
γ = (1 − v 2 ) −1/2 = E ν /m the Lorentz factor with E ν the neutrino energy
and m its mass. The spin precession as viewed from the laboratory