Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
where the radius of the neutrino sphere is taken equal to Rν = 10 km, and Lνα are
the fluxes emitted with flavour α at the neutrino sphere as in Eq.(3.22). We work
here in the single angle approximation but the following derivation is identical
for the more general multi-angle case. The matter Hamiltonian here does not
yet take into account one-loop radiative corrections, and is made of the tree level
interaction with matter : Hm =diag( √ 2GFNe, 0, 0), where Ne is the electron
density.
The factorization
The goal here is to investigate if the S matrix containing δ can be factorized out
of the total Hamiltonian
HT = UHvacU † + Hm + Hνν(δ). (6.4)
Since we have seen previously in Eq.(4.9) that such a factorization was possible,
at the tree level, for the ”MSW” Hamiltonian made of the vacuum term and the
matter term, namely HMSW = UHvacU † + Hm, the relation
˜HMSW(δ) = S † ˜ HMSW(δ = 0) S
= S
T 0 13T12HvacT †
12T 0 †
13 + Hm
S †
(6.5)
is verified. The ˜ over the Hamiltonian means it is written in the T23 basis.
Consequently, we follow the same derivation done in the matter-only case (see
section 4.1.1). Starting from Eq.(6.1), one has to rotate in the T23 basis, since
the S matrix contained in T13 (which can be rewritten as T13 = S † T 0 13
S ) does
not commute with T23. We also multiply by S and S † to put explicitly the δ
dependence with the density matrices, such as S˜ρνα(δ)S † . We then obtain :
where
† dS˜ρνα(δ)S
i
dt
˜ρνα =
⎛
⎜
⎝
= [ ˜ HN(δ = 0) + S ˜ Hνν(δ)S † , S˜ρνα(δ)S † ], (6.6)
P(να → νe) ψνe ˜ ψ ∗ νµ ψνe ˜ ψ ∗ ντ
ψ∗ ˜ψνµ P(να → ˜νµ) ˜ ψνµ νe
˜ ψ∗ ντ
ψ∗ ˜ψντ
˜ψ νe
∗ ˜ψντ νµ P(να → ˜ντ)
⎞
⎟
⎠ (6.7)
and Eq.(4.9) is used. The idea of the derivation is to prove that the total Hamiltonian
does not depend on δ at all times, which requires to prove that
S ˜ Hνν(δ)S † = ˜ Hνν(δ = 0) (6.8)
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