Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
Next, we want to remove the matter term from Eq.(5.49), to do so we follow
again the procedure performed in the previous paragraph by going in a rotating
frame which give the Eq.(D.23) but with B depending on time as written in
Eq.(D.24). Here, we consider that the first two components are rotating very
rapidly and therefore can be averaged to 0, leaving just a z-component for the
magnetic field. It has been found numerically that we can consider that this
fast rotating approximation can be made but one has still to take into account
matter by reducing the vacuum mixing angle to a certain effective mixing angle
θeff similar to the in- medium mixing angle. Thus, we can make the simplification
of ignoring the ordinary matter term entirely, but using an effective mixing angle
in the B field. We rewrite the E.O.M.s in terms of an “effective Hamiltonian”
for the individual modes as
where
∂tPω = Hω × Pω
(5.52)
Hω = ωB + µD. (5.53)
The E.O.M. for D can be obtained by integrating Eq.(D.23) with sω = sign(ω):
∂tD = B × M where M ≡
+∞
−∞
dω sωωPω . (5.54)
From this equation, if we take a large µ then all Pω (−∞ < ω < ∞) remain
stuck to each other, and therefore M ∝ D. In this case, according to Eq.(5.54)
the collective vector D precesses around B with the synchronization frequency:
ωsynch = M
D =
+∞
−∞ dω sωω Pω
+∞
dω Pω
−∞
sω . (5.55)
Since we have considered that B was constant, using Eq.(5.54) it shows that
∂t(D · B) = 0 so that Dz = B · D is conserved. Since Dz represent the difference
between the flavour content of neutrinos and antineutrinos it means that
physically collective effects are only inducing pair transformations of the form
νe¯νe → νx¯νx, whereas the excess νe flux from deleptonization is conserved. This
is the first step to comprehend the spectral split phenomenon. The second step
is to consider that adiabaticity occurs.
The hypothesis of adiabaticity
We extend here the notion of adiabaticity introduced in the appendix B. In this
adiabatic limit each Hω moves slowly compared to the precession of Pω so that
the latter follows the former, in the sense that Pω move around Hω, on the surface
of a cone whose axis coincides with Hω and whose angle is constant. If one looks
at Eq.(5.53), in the case where µ is large we can assimilate the motion of Hω
106