Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
onances on the other hand. As discussed above and in section (1.2.2), to study the
neutrino conversion that occurs in a supernova or in a dense environment, one can
calculate the conversion probability between the neutrino eigenstates in matter
at each resonance region 3 , and combine the result of different resonances, assuming
them to be independent. Such a derivation includes the implicit hypothesis
that though the neutrino mass eigenstates develop a relative phase between each
other, the coherence disappears between successive resonances. This is indeed
what is done in [63, 113]
While separating the H- and L-resonances is legitimate, one cannot put apart
the multiple resonances that can encounter neutrinos in a supernova where a
shock wave develops. Indeed, if such resonances are close to each other, the
coherence is kept only if one considers the amplitudes of neutrino flavour conversions
at the resonances. This gives rise to what is called phase effects which can
be seen as rapid oscillations in the neutrino probabilities (see figure (7.3)). We
use in the following the notation and framework described in [22, 41] to explain
the pahse effects.
Analytical derivation
We perform here a similar derivation than the one made in the appendix B,
concerning the evolution of the matter eigenstates crossing a resonance but for
multiple resonances. We consider the simple case of two resonances due to a dip
in the SN density profile, as an example. A neutrino with energy E encounters
two resonances R1 and R2 at x1 and x2 respectively. We write νH and νL, the
heavier mass and the lighter mass eigenstates respectively. At x ≪ x1, the density
ρ(x) ≫ ρR, so that:
νH(x ≪ x1) ≈ νe . (7.1)
We consider the evolution to be adiabatic till it reaches the resonance region.
There, the resonance mixes the matter eigenstates before the crossing (x1−) to
yield new matter eigenstates at (x1+) such as:
νH(x1+)
=
νL(x1+)
cosχ1 sin χ1eiϕ − sin χ1e−iϕ cosχ1
νH(x1−)
. (7.2)
νL(x1−)
where P1 ≡ sin 2 χ1 is the “jump probability” if it were an isolated resonance.
The matter eigenstates propagate to the other resonance gaining a relative phase.
After the second crossing one can write, the νe survival probability far from the
second resonance as :
Pee = cos 2 (χ1 − χ2) − sin 2χ1 sin 2χ2 sin 2
x2
3 Such a calculation is performed in appendix B for a single resonance.
124
x1
∆ ˜m 2
4E dx
. (7.3)