Stars as Laboratories for Fundamental Physics - MPP Theory Group

Neutrino Oscillations 307
say, between a helicity-minus ν e to a helicity-plus ν µ which, because of
its assumed Majorana nature, is identical to a ν µ . Because typically the
ν e and ν µ masses will be different they must be included in the phase
evolution of the neutrinos. Thus, for relativistic states (momentum p)
one arrives at a two-level equation of motion of the form
i∂ t
( νe
ν µ
)
=
( m
2
νe
/2p µB T
µB T m 2 ν µ
/2p
) ( νe
ν µ
)
. (8.56)
The transition magnetic moment µ thus leads to simultaneous spin and
flavor oscillations (Schechter and Valle 1981). Moreover, neutrinos will
be converted to antineutrinos (and the reverse). If such spin-flavor
oscillations occurred, say, in the Sun one might actually be able to
measure a flux of antineutrinos from that source.
A transition magnetic moment implies that the individual flavor
lepton numbers are not conserved. Thus most likely there is also standard
flavor mixing which allows for, say, ν e ↔ ν µ oscillations. If the
mass eigenstates are m 1 and m 2 , respectively, and if the vacuum mixing
angle is θ, the four-level equation of motion in vacuum is
⎛ ⎞ ⎛
⎞ ⎛ ⎞
ν e ∆ m c 2θ ∆ m s 2θ 0 µB T ν e
ν i∂ µ
t ⎜ ⎟
⎝ ν e ⎠ = ∆ m s 2θ −∆ m c 2θ µB T 0
ν µ
⎜
⎟ ⎜ ⎟
⎝ 0 µB T ∆ m c 2θ ∆ m s 2θ ⎠ ⎝ ν e ⎠ , (8.57)
ν µ µB T 0 ∆ m s 2θ −∆ m c 2θ ν µ
where ∆ m ≡ (m 2 2 − m 2 1)/4p, c 2θ ≡ cos 2θ, and s 2θ ≡ sin 2θ. Any of ν e ,
ν µ , ν e , and ν µ can oscillate into any of the others.
Including medium effects further complicates this matrix because
the energies of ν e and ν µ are shifted by different amounts, and each
is shifted in the opposite directions from its antiparticle. With V νe =
√
2GF (n e − 1 2 n n) and V νµ = √ 2G F (− 1 2 n n) the matrix becomes
⎛
⎞
∆ m c 2θ + V νe ∆ m s 2θ 0 µB T
∆ m s 2θ −∆ m c 2θ + V νµ µB T 0
⎜
⎟
⎝ 0 µB T ∆ m c 2θ − V νe ∆ m s 2θ ⎠ . (8.58)
µB T 0 ∆ m s 2θ −∆ m c 2θ − V νµ
If there are density gradients (solar neutrinos!) one may have resonant
magnetic conversions between, say, ν e and ν µ where the barrier to spinprecessions
caused by the mass difference is compensated by matter
effects, i.e. one may have resonant spin-flavor oscillations (Akhmedov
1988a,b; Barbieri and Fiorentini 1988; Lim and Marciano 1988).
For Dirac neutrinos, transitions to antineutrinos are not possible
and so one needs to consider oscillations separately in the four-level