Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
(like for other angles, it is when the emission is along the z–axis) and π/2 which
corresponds to a neutrino emitted in a tangent way at point B on Fig. 5.2. Moreover,
because of assumptions 1 and 2 in the neutrino bulb model, all the neutrino
beams with the same emission angle ϑ0 and the same initial physical properties
must be equivalent. In simulating the flavor transformations of neutrinos in the
neutrino bulb model, it is only necessary to follow a group of neutrinos which are
uniquely indexed by their initial flavors, energies and emission angles. At any
given radius r, all the geometric properties of a neutrino beam may be calculated
using r and ϑ0. For example, ϑ and Θ are related ϑ0 through the following
identity:
where
and
sin ϑ0
r
= sin Θ
l − l0
= sin ϑ
Rν
, (5.9)
l ≡ AP ≡ r cosϑ, (5.10)
l0 ≡ AE ≡ Rν cosϑ0. (5.11)
The length l − l0 in Eq.(5.9) is also the total propagation distance along the
neutrino beam.
The differential number density of neutrinos
The neutrino-neutrino interaction Hamiltonian depends on the probability for
a neutrino emitted initially at the neutrino sphere with flavour α to be in a
certain flavour β at a certain distance, this probability can be expressed via the
element ρνβνβ of the density matrix ρνα. Since we are interested in the interaction
between a certain neutrino and the neutrino background with which it interacts,
one also has to take into account the set of neutrinos which have the same flavour
evolution history describe by ρνβνβ . Thus, let us now express this set of neutrinos
by a differential number density of neutrinos dnνα(q) at radius r which has the
contribution from all να with energy q which propagate in directions within the
range between ˆq and ˆq + dˆq. The calculation of this differential number is made
in appendix D, it leads to:
√
2GF
Hνν =
(1 − cosϑcosϑ ′ ) (5.12)
2πR 2 ν
α
ρνα(q ′ , ϑ ′ )fνα(q ′ ) Lνα
〈Eνα〉 − ρ∗¯να (q′ , ϑ ′ )f¯να(q ′ ) L¯να
d(cos ϑ
〈E¯να〉
′ )dq ′ .
This is the multi-angle neutrino-neutrino interaction Hamiltonian where, in addition
to the momentum, we also integrate over the emission angle ϑ ′ when we
consider the direction of interaction given by ϑ. Such a computation is numerically
very demanding, that is why an approximation is often made to obtain
more easily numerical results taking into account the neutrino self-interaction.
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