Stars as Laboratories for Fundamental Physics - MPP Theory Group

308 Chapter 8
system (ν L e , ν L µ , ν R e , ν R µ ) and (ν L e , ν L µ, ν R e , ν R µ ). Moreover, one may have
both diagonal and transition magnetic moments so that for neutrinos
the r.h.s. of the equation of motion is
⎛
⎞ ⎛
∆ m c 2θ + V νe ∆ m s 2θ µ ee B T µ µe B T ν L ⎞
e
∆ m s 2θ −∆ m c 2θ + V νµ µ eµ B T µ µµ B T
ν L µ
⎜
⎟ ⎜
⎝ µ ee B T µ µe B T ∆ m c 2θ ∆ m s 2θ ⎠ ⎝ ν R ⎟
e ⎠ , (8.59)
µ eµ B T µ µµ B T ∆ m s 2θ −∆ m c 2θ νµ
R
where the right-handed neutrinos do not experience a medium-induced
energy shift.
For certain values of the parameters the oscillations between two
states may dominate which are then described by a certain 2 × 2 submatrix.
Then the general treatment is much like the flavor mixing
problems studied earlier. In general, a much richer collection of solutions
obtains—for a review see Pulido (1992).
Electric transition moments can also play a role, and precessions
in electric fields may be important. Majorana neutrinos have either
electric or magnetic transition moments, but not both as long as CP
remains conserved. Still, even electric transition moments would lead
to spin-flavor oscillations in macroscopic magnetic fields.
8.4.4 Twisting Magnetic Fields
The problem of spin oscillations is further complicated by the possibility
that the magnetic field changes its direction along the neutrino trajectory,
i.e. that it may be “twisting” (Aneziris and Schechter 1991). In
this case the equation of motion of a two-level spin-precession problem
is based on the Hamiltonian
( BL B
H = µ
T e −iφ )
B T e iφ , (8.60)
−B L
where the three parameters B L , B T , and φ vary along the neutrino
trajectory. The transverse field strength B T is a physical magnetic
field while the longitudinal one B L is an effective magnetic field which
represents the neutrino mass difference (spin-flavor oscillations!) and
refractive medium effects. As long as B T maintains its direction along
the trajectory, the phase φ can be globally chosen to be zero.
If φ varies along the trajectory so that the neutrino experiences it to
be a function of time, it is useful to transform the equation of motion to
a coordinate system which corotates with B T so that in the new frame
one is back to the old situation of a fixed direction for B T (Smirnov