Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
The proof
The neutrino-neutrino interaction Hamiltonian, ˜ Hνν, depends on S˜ρνα(δ)S † (see
Eqs.(6.1) and (6.2)) which is also the unknown of the problem, making the evolution
equations non-linear. The subtlety is that such terms are summed over
the different initial conditions in Hνν, since initially either a νe, a νµ, or a ντ are
produced. The terms S˜ρνα(δ)S † are calculated for one initial condition να only.
Since we wish to prove Eq.(6.8), we consider the evolution equation of the linear
combination B(q, δ) =
να LναS˜ρνα(q, δ)S † at a given momentum q:
i d
LναS˜ρνα(q, δ)S
dt
να
†
= [ ˜ HMSW(δ = 0) + S ˜ Hνν(δ)S † ,
LναS˜ρνα(q, δ)S
να
† ].
(6.9)
The second key point of this derivation is to look at the initial conditions. At
the initial time, B(q, δ) reads, in the T23 basis of Eq.(4.6), as :
LναS˜ρνα(q, δ, t = 0)S
να
† ⎛
Lνe 0 0
= ⎝ 0 c2 23Lνµ + s2 23Lντ c23s23e−iδ (Lνµ − Lντ)
0 c23s23eiδ (Lνµ − Lντ) s2 23Lνµ + c2 ⎞
⎠
23Lντ
(6.10)
One immediately sees that B(q, δ, t = 0) does not depend on δ if and only if Lνµ =
Lντ. The quantity να LναS †˜ρ
∗ να (q, δ, t = 0)S , which corresponds to the antineutrinos
does not depend on δ as well, if and only if Lνµ = Lντ. Moreover, the
total Hamiltonian HT is independent of δ at initial time since ˜ HMSW(δ) does not
depend on δ (at any time) at tree level, and
S ˜ Hνν(t = 0, δ)S † = (6.11)
√
2GF (1 − ˆq · ˆq ′ )[S˜ρνα(t = 0, q ′ )S † Lνα(q ′ ) − S˜ρ ∗ να (t = 0, q′ )S † Lνα(q ′ )]dq ′
α
is equal to ˜ Hνν(t = 0, δ = 0) initially when Lνµ = Lντ(and Lνµ = Lντ). In
that case, one can see, by recurrence, from the Liouville-Von Neumann equation
Eq.(6.9), that the evolution of the term
να LναS˜ρνα(q, δ)S † is exactly the same
as the term
να Lνα ˜ρνα(q, δ = 0), since they have the same initial conditions
(for any q) and the same evolution equations. Moreover, the exact same relation
applying in the same time for anti-neutrinos (Eq.(6.2)) (but with an opposite
sign of δ), one simultaneously obtains that at any time:
hence :
˜Hνν(δ) = S ˜ Hνν(δ = 0)S † , (6.12)
˜HT(δ) = S ˜ HT(δ = 0)S † . (6.13)
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