PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

m 2 . The eigenvectors are given as
⎛ ⎞
U i =
⎝ −y 1+y 2 −y 4 + √ a
2y 3 b
1
b
1
b
⎠ , U j =
⎛
⎝ −y 1+y 2 −y 4 − √ a
2y 3 c
1
c
1
c
⎞
⎛
⎠ , U 3 = ⎝
⎞
0
−√ 1
2
1√
2
⎠ , (6.19)
where U i corresponds to the eigenvalue given in Eq. (6.16) and U j to that in Eq. (6.17).
WhetherU 1 ≡ U i orU j dependsonwhetherm i issmallerorlargerthanm j . Thequantities
b and c are the normalization constants given by
and
and a is given as
b 2 = 2+ (y 1 +y 2 −y 4 + √ a) 2
(2y 3 ) 2 , (6.20)
c 2 = 2+ (y 1 +y 2 −y 4 − √ a) 2
(2y 3 ) 2 , (6.21)
a = y 2 1 +y 2 2 +y 2 4 +8y 2 3 +2y 1 y 2 −2y 1 y 4 −2y 2 y 4 . (6.22)
From Eqs. (6.16), (6.17) and (6.18) we obtain
∆m 2 21 = 4v 2 1 (y 1 +y 2 +y 4 ) √ a ,
∆m 2 31 = v 2 1(3y 1 −y 2 +y 4 − √ a)(y 1 −3y 2 −y 4 + √ a) . (6.23)
The mixing angles can be seen from Eq. (6.19) to be
θ ν 13 = 0 ,
tanθ ν 23 = 1 ,
tanθ ν 12 = (y 1 +y 2 −y 4 − √ a)b
(y 1 +y 2 −y 4 + √ a)c . (6.24)
In the above, neither the ratio of the two mass squared differences ∆m 2 21/∆m 2 31, nor the
mixinganglesdependonthevalueofthetripletVEVv 1 . TheyonlydependontheYukawa
couplings. Onlytheabsolutemass square differences ∆m 2 21 and∆m2 31 individually depend
on the triplet VEV. The effective neutrino mass predicted for neutrino-less double beta
decay is given as
|m νee | = |2v 1 y 4 | , (6.25)
while the effective mass squared observable in beta decay m 2 β and the total neutrino mass
crucial for cosmology m t are given as
m 2 β = ∑ i
|m i | 2 |U ei | 2 , and m t = ∑ i
|m i | , (6.26)
respectively.
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