Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
This matrix is a (complex) unitary matrix, therefore we have:
det(U) = e iΦ
(A.8)
= Ue1(Uµ2Uτ3 − Uτ2Uµ3) − Uµ1(Ue2Uτ3 − Uτ2Uµ3) + Uτ1(Ue2Uµ3 − Uµ2Ue3)
We choose to express Uµ1, Uτ1, Uµ2, Uτ2 as a function of the other terms of the
U matrix. To do so, we multiply the relation (A.8) by each of those terms, take
the conjugate of the equation and use the unitarity conditions. Let us make an
example with Uµ1. From Eq.(A.8), we obtain
Knowing that
we have :
e −iΦ U ∗ µ1 = U ∗ µ1 (det(U))∗
= U ∗ µ1 Ue1(Uµ2Uτ3 − Uτ2Uµ3)
− | Uµ1 | 2 (Ue2Uτ3 − Uτ2Uµ3)
+ U ∗ µ1 Uτ1(Ue2Uµ3 − Uµ2Ue3) (A.9)
U ∗ µ1Ue1 = −Ue2U ∗ µ2 − Ue3U ∗ µ3 (A.10)
and
U ∗ µ1 Uτ1 = −Uτ2U ∗ µ2 − Uτ3U ∗ µ3 (A.11)
Uµ1 = e iΦ (Uτ2Ue3 − Ue2Uτ3) ∗
With similar derivations, we obtain for Uτ1, Uµ2, Uτ2
Uτ1 = e iΦ (Ue2Uµ3 − Uµ2Ue3) ∗
Uµ2 = e iΦ (Ue1Uτ3 − Uτ1Ue3) ∗
Uτ2 = e iΦ (Uµ1Ue3 − Ue1Uµ3) ∗
(A.12)
(A.13)
For the remaining terms we have the following two relations due to the unitarity:
and
| Ue1 | 2 + | Ue2 | 2 + | Ue3 | 2 = 1 (A.14)
| Uτ3 | 2 + | Uµ3 | 2 + | Ue3 | 2 = 1. (A.15)
These relations are the same as those for spherical coordinates in a 3D Euclidian
space. We choose:
Ue1 = e iφe1 cos α cosβ
Ue2 = e iφe2 cos α sin β
Ue3 = e iφe3 sin α
Uµ3 = e iφµ3 cos α sin γ
Uτ3 = e iφτ3 cosαcos γ (A.16)
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