PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

d
b
2
1
0
-1
-2
0.5
0
-0.5
2
1
0
-1
-2
-0.5 0 0.5
c
-0.5 0 0.5
c
-0.5 0 0.5
d
b
d
b
2
1
0
-1
-2
2
1
0
-1
-2
2
1
0
-1
-2
-2 -1 0 1 2
c
-2 -1 0 1 2
c
-2 -1 0 1 2
d
Figure 5.4: In the left pannel, scatter plot showing the 3σ [20] allowed regions for the
b−c−d parameters for the case where ξ ′ and ξ ′′ acquire VEVs. The top, middle and lower
panels show the allowed points projected on the c−b, c−d and d−b plane, respectively.
Theparameterm 0 wasallowedtotakeanyvalue. Herewehaveassumed normalhierarchy.
In the right pannel, the scatter plot for inverted hierarchy.
Therefore, the case a ≠ c = d gives us the TBM matrix and the mass eigenvalues are
shown in Table 5.4.
Since a ≠ c = d is the only allowed case for the three one dimensional Higgs case,
we find the eigenvalues and the mixing matrix for the case where c and d are not equal,
but differ by ǫ. We take c = d+ǫ and for small values of ǫ give the results in the last row
of Table 5.4, keeping just the first order terms in ǫ. The deviation from TBM is given as
follows
D 12 ≃ 0, D 23 ≃
The mass squared differences are
ǫ
4(a−d) , U e3 ≃
ǫ
2 √ 2(a−d) . (5.18)
∆m 2 21 ≃ m 2 0(2a+b+d+ ǫ 3ǫ
)(3d−b+
2 2 ), ∆m2 31 ≃ 2m 2 0b(2d−2a+ǫ) . (5.19)
From the expression of the mass eigenvalues given in the Table 5.4, one can calculate the
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