PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

5.3 Number of One Dimensional Higgs Representations
and their VEVs
In this section we work under the assumption that the triplet Higgs φ S has VEVs along
the direction
〈φ S 〉 = (v S ,v S ,v S ) . (5.8)
This produces the neutrino mass matrix
⎛
⎞
a+2b/3 c−b/3 d−b/3
m ν = m 0
⎝ c−b/3 d+2b/3 a−b/3 ⎠ , (5.9)
d−b/3 a−b/3 c+2b/3
where b = 2x b
v SΛ
. In the following we discuss the phenomenology of the different forms
of m ν possible as we change the number of one dimensional Higgs or put their VEVs to
zero. We assume that m ν is real.
5.3.1 No One Dimensional A 4 Higgs
If there were no Higgs which transforms as one dimensional irreducible representation
under A 4 , or if the VEV of all three of them (ξ,ξ ′ , ξ ′′ ) were zero, one would get the
neutrino mass matrix
⎛ ⎞
2b/3 −b/3 −b/3
m ν = m 0
⎝−b/3 2b/3 −b/3⎠ . (5.10)
−b/3 −b/3 2b/3
On diagonalizing this one obtains the eigenvalues
and the mixing matrix
m 1 = m 0 b, m 2 = 0, m 3 = m 0 b (5.11)
⎛
√
2
3
√
U = ⎜−
⎝ √
−
1
6
1
6
√
1
√
1
√
1
3
3
0
3
−
√
1
2
√
1
2
⎞
⎟
⎠ . (5.12)
Therefore, we can see that the tribimaximal pattern of the mixing matrix is coming directly
from the term containing the triplet Higgs φ S and does not depend on the terms
containing the one Higgs scalars ξ, ξ ′ , ξ ′′ 1 , which are charged under different one dimensional
Higgs representations of the group A 4 . However, in the absence of the one
1 The mixing pattern does not depend explicitly even on the magnitude of the VEV of φ S .
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