Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
with the motion of D. Since a large µ means also that all Pω are aligned, using
Eq.(5.54), one realize that the motion of D is essentially a precession around B
and one can define a frame with the two vectors B and D that moves around B.
According to Eq. (5.53) all Hω obviously lie in the plane spanned by B and D
which we call the “co-rotating plane.” If we assume that µ varies slowly enough
with time, then we are in the adiabatic limit: each of Pω follows Hω and stays
mainly in this plane which seems the right frame to see an adiabatic evolution.
Consequently M, also evolves in that plane and therefore we can decompose:
and rewrite the EOM of Eq.(5.54) as
M = bB + ωcD (5.56)
∂tD = ωc B × D. (5.57)
Therefore D and the co-rotating plane precess around B with the common or
“co-rotation frequency” ωc. Projecting Eq.(5.56) on the transverse plane one
has:
ωc =
+∞
−∞ dω sω ω Pω⊥
+∞
dω sωPω⊥
−∞
=
+∞
−∞ dω sω ω Pω⊥
D⊥
. (5.58)
ωc is clearly a function of µ. When µ → ∞ and all Pω are aligned, this is identical
with the synchronization frequency Eq. (5.55).
The corotating frame
To study the feature of the spectral split, it is easier to work in a corotating
frame. To do so, we apply the same method used in the previous subsection
(see appendix D) for studying the neutrino evolution in a new frame. Therefore
writing the E.O.M. given in Eq.(5.52) in the corotating frame yields:
∂tPω = Hω × Pω − ωcB × Pω
= ((ω − ωc)B + µD) × Pω
(5.59)
Initially when µ is very large, all individual Hamiltonians are essentially aligned
with D. In turn, D is aligned with the weak-interaction direction if initially
all polarization vectors Pω are aligned with that direction. In other words, all
neutrinos are prepared in interaction eigenstates and initially Pω ∝ Hω. The
adiabatic evolution would imply that if µ changes slowly enough, Pω follows
Hω(µ) and therefore remains aligned with Hω(µ) at later times as well. So the
adiabatic solution of the EOMs for our initial condition is given by
Pω(µ) = ˆ Hω(µ) Pω , (5.60)
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