Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
D.2 The rotating frame
The idea of changing to a new frame is to keep the same form than the previous
evolution equations. We start from the equation of motions for a neutrino and
an antineutrino of a given energy ω, using the polarization vector formalism:
∂tP = +ωB + λL + µ P − ¯ P × P ,
∂t ¯ P = −ωB + λL + µ P − ¯ P × ¯ P . (D.15)
where L = (0, 0, 1). We include here matter effects caused by charged leptons.
Since the case without matter has been well understood and studied, it would
be interesting to get the same equations than without matter, by working in a
new referential. Focusing on the neutrinos (the derivation is exactly the same for
antineutrinos), we impose
⎛
(
∂tP − λL × P = ⎝
˙ Px + λPy)x
( ˙
⎞ ⎛ ⎞
Puu ˙
Py − λPx)y ⎠ ≡ ⎝ Pvv ˙ ⎠. (D.16)
Pzz ˙
Pww ˙
where (u,v,w) is the new referential orthonormal basis. Since λ, the coefficient
associated with matter, does not modify the z component, the relation between
the previous referential and the new one is in its most general form:
⎛
⎝
x
y
z
⎞
⎛
⎠ = R ⎝
u
v
w
⎞
⎛
⎠ = ⎝
a b 0
c d 0
0 0 1
⎞ ⎛
⎠ ⎝
u
v
w
⎞
⎠ . (D.17)
Moreover, we know that the polarization vector is real 1 , the norm of each coordinate
has to be equal to 1 2 , and the scalar product between two coordinates has
to be zero3 . These conditions imply:
⎛
cosα(t) sin α(t) 0
R = ⎝ − sin α(t) cos α(t) 0
0 0 1
⎞
⎠ (D.18)
Thus we deduct that the basis changing matrix is a rotation matrix of angle α(t) 4
and of z axis. In the new referential the P vector writes:
⎛ ⎞ ⎛
⎞
Pxx (cosα(t) Px − sin α(t) Py)u
⎝ Pyy ⎠ ⇒ ⎝ (sin α(t) Px + cosα(t) Py)v ⎠ . (D.19)
Pzz
Pzw
1 a,b,c and d are real.
2 a 2 + b 2 = 1 and c 2 + d 2 = 1.
3 ac + bd = 0.
4 Since matter is changing with time, we take α as a function of time
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