PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

while U ν diagonalizes m ν and ˜M as
U T ν
(
mν 0
0 ˜M
)
U ν =
( )
Dm 0
. (3.18)
0 D M
The above parameterization therefore enables us to analytically estimate the mass eigenvalues
and the mixing matrix U in terms of W ν and U ν by a two step process, by first
calculating W ν and then U ν . This matrix W ν can be parameterized as [25]
(√ )
1−BB
†
B
W ν = √
−B † , (3.19)
1−B† B
where B = B 1 +B 2 +B 3 +... and B j ∼ (1/M) j , where M is the mass scale of the heavy
Majorana fermions. Using an expansion in 1/M and keeping only terms second order in
1/M, we get
( 1−
1
2
W ν ≃
m† D (M−1 ) ∗ M −1 m D m † )
D (M−1 ) ∗
−M −1 m D 1− 1 2 M−1 m D m † D (M−1 ) ∗ . (3.20)
The light and heavy neutrino mass matrices obtained at this block diagonal stage are
given by (upto second order in 1/M )
m ν = −m T D M−1 m D , (3.21)
˜M = M + 1 (
)
m D m † D
2
(M−1 ) ∗ +(M −1 ) ∗ m ∗ Dm T D . (3.22)
While Eq. (3.21) is the standard seesaw formula for the light neutrino mass matrix, Eq.
(3.22) gives the heavy neutrino mass matrix. These can be diagonalized by two 3 × 3
unitary matrices U 0 and U Σ , respectively. In our parametrization
( )
U0 0
U ν = , (3.23)
0 U Σ
where U 0 and U Σ satisfy the following equations,
U T 0 m νU 0 = D m
U T Σ ˜MU Σ = D M . (3.24)
Forthechargedleptonswefollowanidenticalmethodfordeterminingthemasseigenvalues
and the mixing matrices. However, since the charged lepton mass matrix M l given by Eq.
(3.13) is a Dirac mass matrix, one has to diagonalize it using a bi-unitary transformation
(
T † √ ml 0
2mD M
)
S =
( )
Dl 0
= M
0 D
ld , (3.25)
H
40