Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
The single angle approximation
The single angle approximation assumes that the flavor evolution history of a
neutrino is trajectory independent. Another way to see it is to consider that
all neutrinos interacting at radius r have been emitted with the same angle.
Mathematically, the single angle approximation sums up in the formula :
ρν(q, ϑ) = ρν(q) (5.13)
Neutrinos on any trajectory transform in the same way as neutrinos propagating
in the radial direction that we chose here to be the z–axis 3 . Consequently, we
can factorized out of the integral the terms proportional to ρνα(q ′ ) and ρ ∗ ¯να (q′ ) in
Eq.(D.15) and it reduces for the spatial dependence to the calculations of :
(1 − cos ϑ ′ )d(cos ϑ ′ 1
) = (1 − x)dx
cos ϑmax
= 1
2
[1 − cosϑmax]
2
= 1
⎡
Rν
⎣1 − 1 −
2 r
2
⎤
⎦
2
, (5.14)
recalling that ϑmax ≡ arcsin
Rν . Finally, the single-angle self-interaction Hamil-
r
tonian is given by:
√
2GF
Hνν = D(r/Rν)
[ρνα(q ′ )Lνα(q ′ ) − ρ ∗ ¯να (q′ )L¯να(q ′ )]dq ′
(5.15)
2πR 2 ν
with the geometrical factor
α
D(r/Rν) = 1
2
⎡
⎣1 −
1 −
Rν
r
2
⎤
⎦
2
. (5.16)
In the approximation of large r/Rν, the geometrical factor varies like:
D(r/Rν) ∼ 1
r 4
(5.17)
In our supernova model, the matter density is proportional to ∼ 1
r 3. Consequently,
the strength of the neutrino-neutrino interaction decreases faster than the matter
interaction strength. On Fig.(5.3), one can see the dependence of the matter
and the neutrino-neutrino interactions as a function of the distance inside the
star. It also shows the approximative ranges where self-interaction effects are
expected to produce mainly synchronization, bipolar oscillations and a spectral
split (considering that θ13 is not zero). We now explain to which physical situation
correspond each of these three stages.
3 This means choosing θ = 0
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