Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
1.1.2 3 flavors in vacuum
Consider now the case of three neutrino flavours. Similarly to the two flavour
case, a 3 × 3 unitary mixing matrix relating the flavour eigenstates to the mass
eigenstates can be defined such as:
⎛
⎝
νeL
νµL
ντL
⎞
⎛
⎠ = ⎝
Ue1 Ue2 Ue3
Uµ1 Uµ2 Uµ3
Uτ1 Uτ2 Uτ3
⎞ ⎛
⎠ ⎝
ν1L
ν2L
ν3L
⎞
⎠ . (1.14)
In general, in the case of Dirac neutrinos, the lepton mixing matrix U, which is
made of 3 rotation matrices, depends on three mixing angles θ12, θ13 and θ23 and
one CP-violating phase δ 4 . It is convenient to use for the matrix U the standard
parametrization of the quark mixing matrix:
⎛
c12 c13 s12 c13 s13 e −iδ
U = ⎝ −s12 c23 − c12 s23 s13 eiδ c12 c23 − s12 s23 s13 eiδ s23 c13
s12 s23 − c12 c23 s13 e iδ −c12 s23 − s12 c23 s13 e iδ c23 c13
⎞
⎠ . (1.15)
One can also factorize the U matrix in three rotation matrices and obtain:
where
⎛
T12 = ⎝
c12 s12 0
−s12 c12 0
0 0 1
⎞
U = T23T13T12D ≡ TD , (1.16)
⎛
⎠ , T13 = ⎝
c13 0 s13 e −iδ
0 1 0
−s13 e iδ 0 c13
⎞
⎛
⎠ , T23 = ⎝
1 0 0
0 c23 s23
0 −s23 c23
(1.17)
and D = diag(e −iϕ1 , 1, e −iϕ2 ). The phases ϕ1 and ϕ2 are present only for neutrinos
in the Majorana case. It immediately follows that the Majorana phases have
no effect on neutrino oscillations. Therefore one can omit the factor D and write
U = T. It can be useful, to factorize T13 as follows
4 See appendix A.
T13 = ST 0 13S †
13
(1.18)
⎞
⎠ ,