Etudes des proprietes des neutrinos dans les contextes ...
Etudes des proprietes des neutrinos dans les contextes ...
Etudes des proprietes des neutrinos dans les contextes ...
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tel-00450051, version 1 - 25 Jan 2010<br />
4.1 CP effects with neutrino interactions at tree<br />
level<br />
4.1.1 Exact analytical formulas<br />
In this section, our aim is simple: try to obtain relations on the oscillating<br />
probabilities in matter at tree level, showing explicit dependence of the Dirac<br />
CP-violating phase δ. All along this section we work with <strong>neutrinos</strong> but the<br />
derivation and the implications would be exactly the same for anti-<strong>neutrinos</strong>.<br />
The factorization<br />
To do so, we start naturally with the evolution equation of <strong>neutrinos</strong> in matter<br />
within the wave functions formalism. In three flavours, the MSW equation (1.44)<br />
is<br />
i ∂<br />
∂t<br />
⎛<br />
Ψe<br />
⎞<br />
⎡<br />
⎛<br />
E1 0 0<br />
0 E2 0<br />
0 0 E3<br />
⎞<br />
⎛<br />
Vc + Vn 0 0<br />
0 Vn 0<br />
⎞⎤<br />
⎛<br />
⎝ Ψµ<br />
Ψτ<br />
⎠ = ⎣T23T13T12 ⎝ ⎠ T †<br />
12T †<br />
13T †<br />
23 + ⎝<br />
0 0<br />
⎠⎦<br />
⎝<br />
Vn<br />
(4.1)<br />
where T23T13T12 accounts for the MNSP matrix U, and their definition is<br />
⎛<br />
T12 = ⎝<br />
c12 s12 0<br />
−s12 c12 0<br />
0 0 1<br />
We remind that<br />
⎞<br />
for the charged-current and<br />
⎛<br />
⎠ , T13 = ⎝<br />
c13 0 s13 e −iδ<br />
0 1 0<br />
−s13 e iδ 0 c13<br />
⎞<br />
⎛<br />
⎠ , T23 = ⎝<br />
Ψe<br />
Ψµ<br />
Ψτ<br />
1 0 0<br />
0 c23 s23<br />
0 −s23 c23<br />
(4.2)<br />
Vc(x) = √ 2GFNe(x) (4.3)<br />
Vn(x) = − 1<br />
√ 2 GFNn(x). (4.4)<br />
for the neutral current. Since Vn only contributes an overall phase to the neutrino<br />
evolution we ignore it. For now we ignore any corrections to the interactions<br />
between <strong>neutrinos</strong> and matter beyond tree-level. We also do not consider here<br />
the neutrino-neutrino interactions. Such additional interactions will be discussed<br />
in section (4.2) and chapter 5 and 6 respectively. Our goal is to make the phase<br />
δ explicitly appear. This phase is contained in the T13 rotation matrix and can<br />
be factorized easily to yield the relation:<br />
T13 = S T 0 13 S†<br />
68<br />
(4.5)<br />
⎞<br />
⎠ ,<br />
⎞<br />
⎠ ,
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