Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
know to which matter eigenstates corresponds which flavour eigenstates, we use
the evolution operator in matter:
⎛ ⎞ ⎛
⎞ ⎛ ⎞
⎝
ν m 1L
ν m 2L
ν m 3L
⎠ = ⎝
U m 1e Um 1µ Um 1τ
U m 2e U m 2µ U m 3µ
U m 3e U m 2τ U m 3τ
⎠ ⎝
νeL
νµL
ντL
⎠ . (1.59)
Since the evolution operator of Eq.(1.59) linking the matter eigenstates to the
flavour eigenstates must correspond at zero density to the MNSP matrix, we can
define three matter mixing angles11 : θm 13 , θm 23 , and θm 12 . In the previous section we
have seen that in the approximation of infinite density, the matter mixing angle
tends to the value π/2. This yields:
ν1m = νµ , ν2m = ντ , ν3m = νe . (1.60)
Probabilities of conversion in dense matter
Let us now follow these matter eigenstates until they exit the star where they will
coincide with the mass eigenstates. Similarly as the derivation in the appendix
B, we call PH and PL, the hopping probability for the H-resonance and the Lresonance
respectively. For instance, if we want to calculate the probability for
νe emitted at the neutrino sphere to exit as ν1, we start at high density on the
diagram of Fig(1.5), and follow the matter eigenstates ν3m (top curve) which
is initially a νe. Since we want to obtain P(νe → ν1), we start with ν3m being
approximatively equal to νe, at the H-resonance crossing, there is a probability PH
for ν3m to hop on the ν2m branch. Encoutering the L-resonance, the probability of
hopping is PL for ν2m to go into ν1m which finally exits in vacuum as ν1. Finally,
we can write the probability for a νe initially emitted to oscillate into a ν1 at the
star surface:
P(νe → ν1) = PHPL
(1.61)
For νµ and ντ we find with the same reasoning:
P(νµ → ν1) = (1 − PL) and P(ντ → ν1) = (1 − PH)PL (1.62)
The fluxes on Earth
In supernova, the density is high enough that the two resonances can occur. Let us
calculate what the flux of νe will be on Earth as an application of the factorization
method. Considering equal initial fluxes for νµ and ντ: F 0 µ = F 0 τ = F 0 x , we can
easily calculate the fluxes of the mass eigenstates at the star surface.
F 0 1m = F 0 x , F 0 2m = F 0 x , F 0 3m = F 0 e . (1.63)
11 For simplicity, we do not take into account the CP-violating phase δ
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