Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
conserved. Considering the most general case where the mass matrix of charged
leptons ml and the neutrino mass matrix mD are complex matrices, we can use
bi-unitary transformations to diagonalize them.
Let us write
l f
L = VL l m L , lf
R = VR l m R
ν f
L = UL ν m L
, νf
R = UR ν m R , (A.3)
and choose the unitary matrices VL, VR, UL and UR so that they diagonalize the
mass matrices of charged leptons and neutrinos:
V †
L ml VR = (ml)diag = diag(me, mµ, mτ) (A.4)
U †
L mD UR = (mD)diag = diag(mν1, mν2, mν3).
The fields lm iL , lm iR , νm iL and νm iR with the superscripts m are then the components
of the Dirac mass eigenstate fields em i = emiL + emiR and νm i = νm iL + νm iR . The
Lagrangian in eq. (A.2) can be written in the mass eigenstate basis as
−LW+m+D = g √ ei
2 m γ µ (V †
LUL)ij ν m Lj W − µ + mlie m LiemRi + mDi ν m Liνm Ri + h.c. , (A.5)
where mli are the charged lepton masses, namely me, mµ and mτ. and mDi are
the neutrino masses. The matrix U = V †
LUL is called the lepton mixing matrix,
or Maki-Nakagawa-Sakata-Pontecorvo (MNSP) matrix [? ]. It is the leptonic
analog of the CKM mixing matrix. It relates a neutrino flavour eigenstate |νf a 〉
produced or absorbed alongside the corresponding charged lepton, to the mass
eigenstates |νm i 〉:
|ν f a 〉 = U ∗ ai |νm i 〉 , (A.6)
A.2 The parametrization of the MNSP matrix
In general a unitary N × N matrix depends on N2 independent real parameters
that can be divided into: = N(N − 1)/2 mixing angles and N(N + 1)/2 phases.
Hence the leptonic mixing matrix with N = 3 can be written in terms of three
mixing angle and six phases.
In the Dirac case, 2N − 1 phases can be removed by a proper rephasing of
the left handed fields: the lepton mass terms ¯νRνL + h.c. remains unchanged
since the phases can be absorbed into the corresponding rephasing of the righthanded
fields. Only N(N + 1)/2 − (2N − 1) = (N − 1)(N − 2)/2 physical phases
remain. Thus, in the Dirac case CP violation is only possible in the case of
N ≥ 3 generations. Since we have three generations, the MNSP matrix is made
of 3(3 − 1)/2 = 3 mixing angles and (3 − 1)(3 − 2)/2 = 1 physical phase. This
physical phase is the CP-violating phase. The neutrino flavour eigenstate and
mass eigenstate fields are related through
⎛ ⎞ ⎛ ⎞ ⎛
⎞ ⎛ ⎞
⎝
νeL
νµL
ντL
⎠ = UMNSP ⎝
ν1L
ν2L
ν3L
⎠ = ⎝
154
Ue1 Ue2 Ue3
Uµ1 Uµ2 Uµ3
Uτ1 Uτ2 Uτ3
⎠ ⎝
ν1L
ν2L
ν3L
⎠ . (A.7)