Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
becomes:
−i ∂
⎛
⎝
∂t
Ψe
Ψµ
Ψτ
⎞
⎠ =
⎡
⎣T23T13T12
⎛
⎝
E1 0 0
0 E2 0
0 0 E3
⎞
⎠ T † † †
12T 13T 23
⎤⎛
⎦⎝
Ψe
Ψµ
Ψτ
⎞
⎠ , (C.2)
If we take the complex conjugate of Eq.(C.2) and redefine the antineutrino wave
functions as Ψ ∗
α = ψ α then we’ll obtain:
i ∂
⎛
⎝
∂t
ψ e
ψ µ
ψ τ
⎞
⎠ =
⎡
⎣T23T ∗ 13T12
⎛
⎝
E1 0 0
0 E2 0
0 0 E3
⎞
⎠T †
12T † ∗
13 T †
23
⎤⎛
⎦⎝
ψ e
ψ µ
ψ τ
⎞
⎠ . (C.3)
The evolution equation is the same for neutrinos and antineutrinos except that
the c.c. of T13 is taken for antineutrinos, this is the reason why the sign of δ, the
CP-violating phase, has to be changed for antineutrinos.
C.2 The Schrödinger equation for evolution operators
This is a equivalent way to express the previous equation on a matrix form
containing all different initial conditions. The evolution operator U(t, 0) links the
neutrino wave functions at a given time t to the initial neutrino wave functions
created at time t = 0.
where
⎛
⎝
ψe(t)
ψµ(t)
ψτ(t)
⎞
U(t, 0) = ⎝
⎛
⎠ = U(t, 0) ⎝
⎛
ψe(0)
ψµ(0)
ψτ(0)
Uee Ueµ Ueτ
Uµe Uµµ Uµτ
Uτe Uτµ Uττ
⎞
⎞
⎠ , (C.4)
⎠ . (C.5)
To understand what’s representing the Uij coefficient, let’s take a physical case
where a electron neutrino is initially created:
⎛ ⎞ ⎛ ⎞
ψe(0) 1
⎝ ψµ(0) ⎠ = ⎝ 0 ⎠. (C.6)
ψτ(0) 0
Replacing Eq.(C.6) in Eq.(C.4), one obtains:
⎛ ⎞ ⎛
ψe,e(t)
⎝ ψµ,e(t) ⎠ = ⎝
ψτ,e(t)
168
Uee
Uµe
Uτe
⎞
⎠ , (C.7)