Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
where S(δ) = diag(1, 1, e iδ ) 1 . Since S does not commute with the rotation matrix
T23, we have to work with rotated wave functions written in another basis that
we call the T23 basis. We therefore introduce the combinations
˜Ψµ = cosθ23Ψµ − sin θ23Ψτ, (4.6)
˜Ψτ = sin θ23Ψµ + cosθ23Ψτ. (4.7)
This corresponds to multiplying the neutrino column vector in Eq. (4.1) with T †
23
from the left. Eq. (4.1) then becomes
i ∂
⎛ ⎞ ⎡
Ψe
⎝ ˜Ψµ ⎠ = ⎣ST
∂t ˜Ψτ
0 13T12 ⎛ ⎞
E1 0 0
⎝ 0 E2 0 ⎠ T
0 0 E3
†
12T 0 ⎛ ⎞⎤
⎛ ⎞
Vc 0 0 Ψe
† †
13 S + ⎝ 0 0 0 ⎠⎦
⎝ ˜Ψµ ⎠,
0 0 0 ˜Ψτ
(4.8)
since S commute with T12 and of course diag(E1, E2, E3). It is easy to notice
that the S matrix commutes with the matter term of the total Hamiltonian that
we call ˜ H in this basis. The Hamiltonian ˜ H depends on the CP-violating phase,
δ via the formula
˜H(δ) = S ˜ H(δ = 0) S † . (4.9)
We now know that the explicit dependence of δ can be factorized out of the
Hamiltonian in that particular basis. To see the consequence of such a factorization,
it is more interesting to work with the transition amplitudes that are
present in the evolution operator formalism (See appendix C for details.). We
are interested in solving the evolution equation corresponding to Eq.(4.8):
i dŨ(δ)
dt = ˜ H(δ) Ũ(δ). (4.10)
It is important to recall that we need to solve this equation with the initial
condition
Ũ(δ)(t = 0) = 1. (4.11)
Defining
and using the relation in Eq.(4.9) we get
U0 = S † Ũ(δ), (4.12)
i dU0
dt = ˜ H(δ = 0) U0, (4.13)
i.e. U0 provides the evolution when the CP-violating phase is set to zero. Using
Eq.(4.11) we see that the correct initial condition on U0 is U0(t = 0) = S † .
However, Eq.(4.13) is nothing but the neutrino evolution equation with the CPviolating
phase set equal to zero. If we call the solution of this equation with the
1 †
From this formula we have the straightforward but useful relation T 13 = S T 0 † †
13 S .
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