PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

we generate the following neutrino mass matrix
L ν = 1 2
(
ν
C
Li
Σ 0 R i
) ( 0
v ′ √
2
Y T Σ ij
v ′ √
2
Y Σij
M ij
)(
νLj
Σ 0 R j
C
)
+h.c., (3.11)
and the following charged lepton mass matrix
L l = ( ) ( )( )
l Ri Σ − vY lij 0 lLj
R i v ′ Y Σij M ij Σ + C +h.c., (3.12)
R j
= ( ( )
)
l Ri Σ − lLj
Ml
R i
Σ + C +h.c., (3.13)
R j
Due to the imposed Z 2 symmetry neutrino mass matrix in Eq. 3.11 depend only on
the new Higgs VEV v ′ while in the charged lepton mass matrix both the VEV’s enter.
The value of v ′ is determined by the scale of the standard model neutrino masses and is
independent of the mass scale of all other fermions. Therefore, the neutrino masses can
be naturally light, without having to fine tune the Yukawa couplings Y Σ to unnaturally
small values.
Since we have 3 generation of triplet fermions, the mass matrix in Eq. 3.11 is 6×6.
The symmetric 6 × 6 neutrino matrix can be diagonalized by a unitary transformation
to yield 3 light and 3 heavy Majorana neutrinos. The 6 × 6 unitary matrix U which
accomplishes this satisfies the following equations,
(
0 m
U T T
D
m D M
)
U =
( )
Dm 0
, and
0 D M
where m D = v ′ Y Σ / √ 2, and
⎛ ⎞ ⎛
m 1 0 0
D m = ⎝ 0 m 2 0 ⎠, D M = ⎝
0 0 m 3
(
νL
Σ 0 C
R
) ( ν
′
= U L
Σ ′ 0C
R
)
, (3.14)
⎞
M Σ1 0 0
0 M Σ2 0 ⎠. (3.15)
0 0 M Σ3
Herem i andM Σi (i = 1,2,3)arethelowandhighenergymasseigenvaluesoftheMajorana
neutrinos respectively. In the above, the primed basis represents the fields in their mass
basis. The mixing matrix U can be parameterized as a product of two matrices
m D
U = W ν U ν (3.16)
where W ν is the matrix which brings the 6×6 neutrino matrix given by Eq. (3.11) in its
block diagonal form as
( ) ( )
0 m
Wν
T T
D mν 0
W
M ν = , (3.17)
0 ˜M
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