Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
standard initial condition Ũ(δ = 0)(t = 0) = 1 to be Ũ(δ = 0), we see that we
should set U0 = Ũ(δ = 0)S† , which yields
Ũ(δ) = S Ũ(δ = 0) S† . (4.14)
This relation can also be obtained by a simpler but equivalent derivation.
If we consider Eq.(4.10) respectively in the case where δ = 0 and the case where
δ = 0:
dŨ(δ = 0)
i =
dt
˜ H(δ = 0) Ũ(δ = 0). (4.15)
and
i dŨ(δ)
dt = ˜ H(δ) Ũ(δ). (4.16)
We use the relation Eq.(4.9) and multiplied each side of the equation by the S
matrix to obtain:
i d S† Ũ(δ) S
=
dt
˜ H(δ) S † Ũ(δ) S. (4.17)
We then take the difference between Eq.(4.16) and Eq.(4.15) multiplied by S, we
have:
i d
S
dt
†
Ũ(δ) S − Ũ(δ = 0) = ˜
H(δ = 0) S †
Ũ(δ) S − Ũ(δ = 0) . (4.18)
that we can rewrite:
i d
dt (∆(δ)) = ˜ H(δ = 0) (∆(δ)). (4.19)
Therefore, realizing that S † Ũ(δ) S has the same initial condition that Ũ(δ = 0),
the variable ∆ which is a priori a function of δ is zero initially. By recurrence
the variable ∆ will remain zero and therefore we find of Eq.(4.14). This equation
illustrates how the effects of the CP-violating phase separate in describing the
neutrino evolution. It is valid both in vacuum and in matter for any density
profile. It is easy to verify that this result does not depend on the choice of the
parametrization for the neutrino mixing matrix.
The transition probabilities and the CP-violating phase
The solution of Eq. (4.16) can also be written in the form:
⎛
Ũ(δ) = ⎝
Aee A˜µe A˜τe
Ae˜µ A˜µ˜µ A˜τ ˜µ
Ae˜τ A˜µ˜τ A˜τ˜τ
70
⎞
⎠. (4.20)