Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
frame (see appendix D), we obtain the same equations than before:
∂tPω = +ωB(t) + µ Pω − ¯ ∂t
Pω × Pω ,
¯ Pω = −ωB(t) + µ Pω − ¯
Pω × Pω
¯ . (5.46)
except that the magnetic field B(t) is now time dependent as:
⎛
⎞
sin(2θV ) cos(−λt)
B(t) = ⎝ sin(2θV ) sin(−λt) ⎠ . (5.47)
− cos(2θV )
We name this new referential, the rotating frame. Using an analog derivation
as previously, we solve the differential equation in the approximation of fast
oscillation frequency λ, we obtain for the time scale for flavour conversion:
τbipolar ≈ −κ −1
˜θV κ
ln
1 +
π
ω
µQ
. (5.48)
(κ 2 + λ 2 ) 1/2
Consequently the presence of matter has little influence on the overall behaviour
of the bipolar system.
A more realistic model
Now we consider a more realistic model where the neutrino density is varying,
and where a neutrino/anti-neutrino asymmetry is present. When the density of
neutrinos is decreasing with time, such as in Fig.(5.3), the oscillations shown on
Fig.(5.6) are following this decrease. Such a decrease is the cause for an almost
complete flavour conversion occuring in the inverted hierarchy for neutrinos and
anti-neutrinos as well. Actually we observe not only a global decrease following
the µ curve but also a diminution for the amplitudes of the oscillations. Considering
next an asymmetry for the νe and ¯νe fluxes, we observe that the initial flavour
lepton asymmetry is conserved so that the net νe flux set initially remains. In
addition, we verify that when µ is very high we are in the region of synchronized
oscillations, and when µ reaches an intermediate value, bipolar oscillations take
place. Ordinary vacuum oscillation will arise at low µ. Note that the matter has
not impact on the bipolar oscillation region.
If we want to interpret figure (5.7), one can think within the polarization
vector formalism. On the left figure of Fig.(5.8), one can see the system initially
where D = P − ¯P has a component on the z-axis. Since µ is very important, the
Hamiltonian of the system H ≃ µD is initially mainly on the z-axis. Oscillations
of P and ¯P around the z-axis start because of the vector B which has a non zero
x-component 9 (O( ˜ θV )). Indeed, it allows P and ¯P to begin a rotation around
9 Without an x-component for B , the vectors P and ¯P will remain at their initial value on
the z-axis.
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