Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
Because this flux can also be expressed as
Nνα(q) = Lνα
〈Eνα〉 fνα(q), (D.8)
where Lνα, 〈Eνα〉 and fνα(q) are the energy luminosity, average energy and normalized
energy distribution function of να, respectively. Therefore one has
jνα(q) =
Lνα
4π 2 R 2 ν〈Eνα〉 fνα(q). (D.9)
Note that this formula applies both for neutrinos and antineutrinos. Finally, the
neutrino-neutrino forward scattering Hamiltonian can be written as
Hνν = √
2GF ρνα(q ′ )(1 − ˆq · ˆq ′ )dnνα(q ′ ) − ρνα(q ′ )(1 − ˆq · ˆq ′ )dn¯να(q ′
) dq ′
α
(D.10)
where q and q ′ are the momentum of the neutrino of interest and that of the background
neutrino, respectively. As mentioned above, neutrinos of the same initial
flavor, energy and emission angle have identical flavor evolution. Consequently
one must have
̺ν(q) = ̺ν(q, ϑ). (D.11)
Writing the vectors in the spherical coordinates and integrating it over the solid
angle dˆq ′ we obtain:
ˆq · ˆq ′ dˆq ′ =
[sin ϑ sin ϑ ′ (sin φ sin φ ′ + cosφcosφ ′ ) + cosϑcosϑ ′ ] d(cosϑ ′ )dφ ′
(D.12)
Because as said before the problem has a cylindrical symmetry around the z–axis,
the terms proportional to sin φ ′ and cosφ ′ will average to zero when integrating
on the azimuthal angle φ ′ :
ˆq · ˆq ′ dˆq ′
= 2π cosϑcos ϑ ′ d(cosϑ ′ ) (D.13)
Using Eqs.(D.4b) ,(D.9) and (D.12), we finally obtain:
√
2GF
Hνν =
(1 − cosϑcosϑ ′ ) (D.14)
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.
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