Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
where ∆ ˜m 2 is the mass squared difference between νH and νL in matter:
∆ ˜m 2 = ∆m 2 − 2EV (x) 2
(7.4)
in the small angle approximation.
The term sin 2
x2 ∆ ˜m
x1
2
4E dx
in Eq.(7.3) is the interference term between the
matter eigenstates. It oscillates with the energy and the resonance locations.
This is why one can see fast oscillations as a function of the energy but also for a
given energy as a function of time. Indeed, the shock wave moving, it will change
a little the locations of the resonances, modifying the phase in the interference
term, so that a neutrino of a given energy will also feel rapid oscillations. The
interference term represents the phase effects.
Discussion
We discuss here the condition for the phase effects to be present in the oscillating
probabilities (see Fig.(7.3)). From Eq.(7.3), one can see that the two resonances
encountered by the neutrino must be semi-adiabatic. Indeed, if one of them is
completely adiabatic then cosχi = 1 which yields sin 2χi = 0 and the interference
term vanishes. If one of the resonances is completely non-adiabatic then
sin χi = 1 gives sin 2χi = 0 and the interference term is zero again. Therefore,
only multiple semi-adiabatic resonances will not cancel the oscillating term. The
semi-adiabaticity depends on the derivative of the matter density but also on
the relevant mixing angle. Such a condition is typically satisfied for the range
10 −5 sin 2 θ13 10 −3 .
The second condition for the phase effects to exist is that the coherence between
the two mass eigenstates must be conserved at the resonances, i.e no decoherence
occurs. To estimate this coherence conservation over the distance,
one can write the coherence length defined as the distance over which the wave
packets separate [93]:
Lcoh ∼ 4√ 2σE 2
∆m 2
, (7.5)
where σ is the width of the wavepacket at source. Taking σ ∼ 10 −9 cm near the
neutrino sphere [10] in the relevant energy range of 5–80 MeV, the coherence
length for SN neutrinos is Lcoh ∼ 10 8 –10 10 cm. Resonances separated by distances
well larger than Lcoh may be taken to be incoherent. Since the distances involved
are O(10 8 − 10 9 cm) (see figure 7.2), coherence length may be conserved and
phase effects can occur.
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