Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
Injecting the relations (A.16) into the set of equations (A.13) we obtain for the
MNSP matrix, after some tedious but straightforward calculations,
with
⎛
U = ⎝
e iφe1 cosαcosβ e iφe2 cosαsin β e iφe3 sin α
Uµ1 Uµ2 e iφµ3 cos α sin γ
Uτ1 Uτ2 e iφτ3 cosαcosγ
⎞
⎠ (A.17)
Uµ1 = −e i(φe1+φµ3−φe3) cosβ sin γ sin α − e −i(φe2+φτ3−Φ) cos γ sin β
Uµ2 = e −i(φe1+φτ3−Φ) cos β cosγ − e i(φe2+φµ3−φe3) cosγ sin β
Uτ1 = e i(φe2+φµ3−Φ) sin β sin γ − e i(φe1+φτ3−φe3) cosβ cosγ sin α
Uτ2 = −e i(φe2+φτ3−φe3) sin β cosγ sin α − e −i(φe1+φµ3−Φ) sin β sin γ (A.18)
Our goal is to obtain the same parametrization as the PDG’s one, we first rename
the rotations angles, as:
α = θ13
β = θ12
γ = θ23
(A.19)
We can now redefine the fields by factorizing out two diagonal matrices, each side
of our U matrix containing only phases. Thus we have:
UMNSP = diag(1, e −i(φe1+φe2+φτ3−Φ) , e −i(φe1+φe2+φµ3−Φ)
× U PDG
MNSP diag(eiφe1 , e iφe2 , e −i(φe1+φe2+φµ3+φτ3−Φ) ) (A.20)
where we rename the phase that cannot be cast away by some redefinition of the
fields, namely:
φe1 + φe2 − φe3 + φµ3 + φτ3 − Φ = δ (A.21)
This is the physical phase, the CP-violating Dirac phase. That way we obtain
the same exact parametrization as the PDG’s one.
⎛
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
⎞
⎠ . (A.22)
As we can notice, there is a great deal of arbitrariness involved in the choice of the
various parameters, and many alternative choices exist for the parametrization
of the unitary matrix.
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