Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
where the quantity Axy is the amplitude of the oscillation from νx to νy when δ
is non-zero. Similarly we define the amplitude for the process νx → νy to be Bxy
when δ = 0 so that
P(νx → νy, δ = 0) = |Axy| 2 . (4.21)
and
P(νx → νy, δ = 0) = |Bxy| 2 . (4.22)
Using Eq. (4.14) it is possible to relate survival probabilities for the two cases
with δ = 0 and δ = 0. These solutions will be related as
⎛
Aee
⎝ Ae˜µ
A˜µe
A˜µ˜µ
⎞ ⎛
A˜τe 1 0
A˜τ ⎠ ˜µ = ⎝ 0 1
0
0
Ae˜τ A˜µ˜τ A˜τ˜τ 0 0 eiδ ⎞ ⎛
Bee
⎠ ⎝ Be˜µ
B˜µe
B˜µ˜µ
⎞ ⎛
B˜τe 1 0
B˜τ ⎠ ⎝ ˜µ 0 1
0
0
Be˜τ B˜µ˜τ B˜τ˜τ 0 0 e−iδ ⎞
⎠
(4.23)
One can immediately see that the electron neutrino survival probability does not
depend on the CP-violating phase, as in vacuum,
which implies:
One can further write
By solving these equations one gets
and
Aee = Bee
(4.24)
P(νe → νe, δ = 0) = P(νe → νe, δ = 0). (4.25)
c23Aµe − s23Aτe = c23Bµe − s23Bτe
s23Aµe + c23Aτe = e −iδ [s23Bµe + c23Bτe]
Aµe = (c 2 23 + s2 23 e−iδ )Bµe + c23s23(e −iδ − 1)Bτe, (4.26)
Aτe = c23s23(e −iδ − 1)Bµe + (s 2 23 + c 2 23e −iδ )Bτe. (4.27)
Clearly the individual amplitudes in Eqs.(4.26) and (4.27) depend on the CPviolating
phase. However, taking absolute value squares of Eqs.(4.26) and (4.27),
after some algebra, one obtains:
or equivalently
|Aµe| 2 + |Aτe| 2 = |Bµe| 2 + |Bτe| 2 , (4.28)
P(νµ → νe, δ = 0) + P(ντ → νe, δ = 0) = P(νµ → νe, δ = 0) + P(ντ → νe, δ = 0).
(4.29)
This relation could also be immediately obtained by the relation of conservation
of probability
P(νe → νe) + P(νµ → νe) + P(ντ → νe) = 1 (4.30)
which comes from the unitarity of the evolution operator.
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