PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

where m l = vY l , while D l and D H are diagonal matrices containing the light and heavy
charged lepton mass eigenvalues. With the above definition for the diagonalization, the
right-handedandleft-handedweak andmass eigenbasis forthecharged leptonsarerelated
respectively as,
( ) ( )
lL
lR
Σ + C
R
( l
′
= S L
Σ ′ + C
R
)
, and
Σ − R
= T
(
l
′
R
Σ ′ −
R
)
. (3.26)
We denote the four component mass eigenstates of the standard model leptons and the
fermion triplet by l m ± = l′± R +l′± L , ν m = ν L ′ +ν′ C
L , Σ0 m = Σ′ 0
R +Σ′ 0 C
R and Σ
±
m = Σ ′± R +Σ′ ∓
RC
.
Instead of using Eq. (3.25) directly forthe diagonalization, we will work with thematrices
M † l M l = SM † l d
M ld S † , and M l M † l
= T M ld M † l d
T † , (3.27)
to obtain S and T respectively. As for the neutrinos, we parameterize
S = W L U L , and T = W R U R , (3.28)
where W L and W R are the unitary matrices which bring M † l M l and M l M † l
to their block
diagonal forms, respectively,
( )
( )
W † L M† l M ˜ml† ˜m
lW L = l 0
0 ˜M† ˜M , and W † R M lM † †
l W ˜ml ˜m
R = l 0
(3.29) .
H H 0 ˜MH ˜M†
H
Using arguments similar to that used for the neutrino sector, and keeping terms up to
second order in 1/M, we obtain
( √
1−m
†
W L =
D (M−1 ) ∗ M −1 m D 2m
†
)
D (M−1 ) ∗
− √ 2M −1 m D 1−M −1 m D m † D (M−1 ) ∗ , (3.30)
W R =
(
√
1 2ml m † D (M−1 ) ∗ M −1
− √ 2(M −1 ) ∗ M −1 m D m † l
1
)
, (3.31)
The square of the mass matrices for the light and heavy charged leptons in the flavor
basis obtained after block diagonalization by W R and W L are given by
and
˜m l˜m † l
= m l m † l −2m lm † D (M−1 ) ∗ M −1 m D m † l , (3.32)
˜M H ˜M†
H = MM† +2m D m † D +(M−1 ) ∗ M −1 m D m † l m lm † D
+ m D m † l m lm † D (M−1 ) ∗ M −1 , (3.33)
˜m † l ˜m l = m † l m l −[m † D M∗−1 M −1 m D m † l m l +h.c] (3.34)
˜M † H ˜M H
= M † M +M −1 m D m † D M +M† m D m † D (M−1 ) ∗ +M −1 (m D m † D )2 (M −1 ) ∗
+ [M −1 (M −1 ) ∗ M −1 m D (m † l m l)m † D M−1 2 M−1 m D m † D M∗−1 M −1 m D m † D M +
h.c] (3.35)
41