Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
4.2.2 Explicit CP-violation phase dependence
The factorization?
If we include the radiative corrections to the interactions with matter, the neutrino
evolution equations get modified:
i ∂
⎛
⎝
∂t
Ψe
Ψµ
Ψτ
⎞
⎠ =
⎡
⎣T23T13T12
⎛
⎝
E1 0 0
0 E2 0
0 0 E3
⎞
⎠ T †
12T †
13T †
23 +
⎛
⎝
Vc 0 0
0 0 0
0 0 Vµτ
⎞⎤
⎛
⎠⎦
⎝
(4.59)
As seen in the case of interactions with matter at tree level, to obtain explicit
relations between probabilities and the CP-violating phase, it is necessary to work
within a new basis where a rotation by T23 is performed. In this case, factorizing
the S matrix containing δ, out of the Hamiltonian, one has:
i ∂
⎛
⎝
∂t
Ψe
⎞
⎛
Ψe
˜Ψµ ⎠ =
˜Ψτ
˜ HT(δ) ⎝ ˜Ψµ
˜Ψτ
⎡ ⎛
= S
+
⎛
⎝
⎣T 0 13T12
⎝
⎞
⎠ = S ˜ H ′ T(δ) S †
E1 0 0
0 E2 0
0 0 E3
⎞
⎛
⎝
Ψe
˜Ψµ
˜Ψτ
⎠ T †
12T 0 †
13
Vc 0 0
0 s 2 23 Vµτ −c23s23e iδ Vµτ
0 −c23s23e −iδ Vµτ c 2 23 Vµτ
⎞
⎠
⎞⎤
⎛
⎠⎦S
† ⎝
Ψe
˜Ψµ
˜Ψτ
Ψe
Ψµ
Ψτ
(4.60)
Contrary to Eq.(4.9), the S matrix does not commute with the matter Hamiltonian
in the T23 basis. This fact implies that:
and therefore that
⎞
⎠ ,
˜HT(δ) = S ˜ HT(δ = 0) S † , (4.61)
Ũm(δ) = S Ũm(δ = 0)S † . (4.62)
Consequence on the electron neutrino survival probability
Nevertheless, we can factorize the S matrices but we will have an Hamiltonian
which depends on δ. We can rewrite Eq.(4.60) in the evolution operator formalism
to have:
⎛
⎞ ⎛
⎞
i ∂
∂t
⎝
Aee A˜µe A˜τee iδ
Ae˜µ A˜µ˜µ A˜τ ˜µe iδ
Ae˜τe −iδ A˜µ˜τe −iδ A˜τ˜τ
⎠ = ˜ H ′ T(δ) ⎝
85
Aee A˜µe A˜τee iδ
Ae˜µ A˜µ˜µ A˜τ ˜µe iδ
Ae˜τe −iδ A˜µ˜τe −iδ A˜τ˜τ
⎠
(4.63)
⎞
⎠ .