Stars as Laboratories for Fundamental Physics - MPP Theory Group

Neutrino Oscillations 305
system, however, involves a time-dilation factor γ −1 so that the Hamiltonian
for the spin evolution in the laboratory system is H = µγ −1 B·σ
or explicitly
H = µ [ (m/ω)(B · ˆv)ˆv + ˆv × (B × ˆv) + E × v ] · σ. (8.50)
If the spin is quantized along the direction of motion (helicity states)
and if there is only a magnetic field in the laboratory (E = 0), the
Hamiltonian is
( γ
H = µB
−1 cos θ e −iφ )
sin θ
e +iφ sin θ −γ −1 , (8.51)
cos θ
with θ the magnetic field direction relative to the neutrino direction of
motion and φ its azimuthal direction which may be chosen as φ = 0.
The bottom line is that very relativistic neutrinos (γ → ∞) spinprecess
like nonrelativistic ones, except that the relevant magnetic field
is the transverse component B T ; the effective laboratory Hamiltonian is
( ) 0 1
H = µB T . (8.52)
1 0
Therefore, if one begins with left-handed neutrinos a complete precession
into right-handed ones will always occur (assuming a sufficient
path length), independently of the direction between the laboratory
magnetic field and the neutrino direction of motion (Fujikawa and
Shrock 1980).
If there is only an electric field in the laboratory frame, magnetic
oscillations will nevertheless occur because Eq. (8.50) implies that the
neutrino sees an effective magnetic field B T = E×v (Okun 1986). The
precession thus takes place in the plane of E and v.
Contrary to flavor oscillations, the oscillation length does not depend
on the energy. Therefore, a broad energy spectrum would not
cause a depolarization of the neutrino flux from an astrophysical source
(Sun, supernovae). Rather, neutrinos of all energies would spin-precess
in step with each other.
Because the precession frequency is 2µB T the neutrinos will have
reversed their spin after a distance
1
l 2 osc = π = 5.36×10 13 cm 10−10 µ B
2µB T µ
1 G
B T
, (8.53)
with µ B = e/2m e the Bohr magneton. With a solar B T in the kG regime
the oscillation length is of order a solar radius R ⊙ = 6.96×10 10 cm. A