PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

Sin 2 θ 12
Sin 2 θ 13
Sin 2 θ 23
0.35
0.345
0.34
0.335
0.33
0.05
0.04
0.03
0.02
0.01
0
0.6
0.5
0.4
-0.4 -0.2 0 0.2 0.4
ε
-0.4 -0.2 0 0.2 0.4
ε
-0.4 -0.2 0 0.2 0.4
ε
∆m 2 21
∆m 2 31
8.2e-05
8e-05
7.8e-05
7.6e-05
7.4e-05
7.2e-05
m t
0.0028
0.0026
0.0024
0.0022
0.002
0.12
0.11
0.1
-0.4 -0.2 0 0.2 0.4
ε
-0.4 -0.2 0 0.2 0.4
ε
-0.4 -0.2 0 0.2 0.4
ε
Figure 5.3: The left panels show sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 vs ǫ and the right panels
show ∆m 2 21 , ∆m2 31 and m t vs ǫ respectively. Here ξ ′ and ξ ′′ acquire VEVs. The other
parameters c, b and m 0 are allowed to vary freely.
5.3.4 Three One-Dimensional A 4 Higgs
Finally, we let all three Higgs which transform as different one dimensional irreducible
representations under the symmetry group A 4 contribute to m ν . In this case one has
to diagonalize the most general mass matrix given in Eq. (5.9). This matrix has four
independent parameters. If we assume that the VEVs of ξ, ξ ′ and ξ ′′ are such that
a = c = d, then the eigenvalues and mixing matrix are given in the first row of Table 5.4.
This gives us a mass matrix, which give us two of the mass eigenstates as degenerate.
To get the correct mass splitting in association with TBM mixing, it is essential that (i)
we should have contribution from the VEVs of the one dimensional Higgs and (ii) the
contribution from the the three one dimensional Higgs ξ, ξ ′ and ξ ′′ in m ν should not be
identical. If we assume that a = c ≠ d, then one can easily check that m ν has e − τ
exchange symmetry, and hence the resulting mass matrix is disallowed. This is because
for a = c, as discussed before we get e−τ exchange symmetry and the ξ ′ term has an inbuilt
e−τ symmetry. Similarly for a = d ≠ c, one gets e−µ symmetry in m ν and is hence
disfavored. Only when we impose the condition c = d, we have µ−τ symmetry in m ν ,
since the ξ term and the sum of the ξ ′ and ξ ′′ terms are now separately µ−τ symmetric.
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