Stars as Laboratories for Fundamental Physics - MPP Theory Group

282 Chapter 8
oscillations, except that here the two helicity components rather than
the flavor components get transformed into each other. Again, this is
a standard effect familiar from the behavior of electrons in magnetic
fields. In astrophysical bodies large magnetic fields exist, especially in
supernovae, so that magnetic helicity oscillations are potentially interesting.
However, much larger magnetic dipole moments are required
than are predicted for standard massive neutrinos. Thus, flavor oscillations
have rightly received far more attention.
Presently I will develop the theoretical tools for neutrino oscillations,
and summarize the current experimental situation. In Chapter 9
I will discuss the more complicated phenomena that obtain when oscillating
neutrinos are trapped in a supernova core. The story of solar
neutrinos (Chapter 10) is inextricably intertwined with that of neutrino
oscillations, especially of the MSW variety. Finally, oscillations
may also play a prominent role for supernova neutrinos and the interpretation
of the SN 1987A signal (Chapter 11).
8.2 Vacuum Oscillations
8.2.1 Equation of Motion for Mixed Neutrinos
In order to derive a formal equation for the oscillation of mixed neutrinos
I begin with the equation of motion of a Dirac spinor ν i which
describes the neutrino mass eigenstate i. It obeys the Dirac and thus
the Klein-Gordon equation (∂ 2 t − ∇ 2 + m 2 i ) ν i = 0. One may readily
combine all mass eigenstates in a single equation
where
(∂ 2 t − ∇ 2 + M 2 ) Ψ = 0, (8.1)
M 2 ≡
⎛
m 2 1 0 0
⎜
⎝ 0 m 2 2 0
0 0 m 2 3
⎞
⎟
⎠ and Ψ ≡
⎛ ⎞
ν 1
⎜ ⎟
⎝ ν 2 ⎠ . (8.2)
ν 3
Eq. (8.1) may be written in any desired flavor basis, notably in the
basis of weak-interaction eigenstates to which one may transform by
virtue of Eq. (7.8),
⎛ ⎞ ⎛ ⎞
ν e ν 1
⎜ ⎟ ⎜ ⎟
⎝ ν µ ⎠ = U ⎝ ν 2 ⎠ . (8.3)
ν τ ν 3
The mass matrix transforms according to M 2 → UM 2 U † and is no
longer diagonal. (Note that U −1 = U † because it is a unitary matrix.)