neutrino mass matrix in triplet higgs models with a_4 symmetry myoung chu oh miami 2008, dec. 17,...
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![Page 1: Neutrino mass matrix in triplet Higgs Models with A_4 Symmetry Myoung Chu Oh Miami 2008, Dec. 17, 2008 Based on work with Seungwon Baek](https://reader035.vdocuments.mx/reader035/viewer/2022062619/55163d02550346b2068b5187/html5/thumbnails/1.jpg)
Neutrino mass matrix in triplet Higgs Models with A_4 Symmetry
Myoung Chu Oh Miami 2008, Dec. 17, 2008
Based on work with Seungwon Baek
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Outline
1. Introduction to A_4 symmetry2. Others’ Model 1) Type I seesaw model with A_4
symmetry ( He, Keum, Volkas (2006) )
2) Triplet Higgs Model3. Our Model4. Conclusion
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Introduction to A_4 symmetry
Symmetry group of a regular tetrahedron for the proper ro-tations12 elementsEven permutations of four objects }2314{}1234){123(
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Group Multiplication table
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Representations
• 3-dim. Representation 3 :
}.,,,{)}124(),234(),143(),132{(
},,,{)}134(),142(),243(),123{(
},{)}23)(14(),24)(13(),34)(12{(
}1{}1{
331122
223311
1,23
arrarrarra
crrcrrcrrc
rrr
with
001
100
010
,
010
001
100
)1,1,1(),1,1,1(),1,1,1(
11
321
caac
diagrdiagrdiagr
He,Keum & Volkas, JHEP(2006)
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Representations
• 1-dim. Representations 1,1’,1” :• 1’(1”) :
)},({)}124(),234(),143(),132{(
)}({)}134(),142(),243(),123{(
)}1(1{)}23)(14(),24)(13(),34)(12(,1{
2
2
where3/2 ie . He, Keum & Vlokas,
JHEP(2006)
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Representations
• Tensor Products: ."',"'',"' 1111111113333 as
For
.)(
,)(
,)(
),,,()(
),,,()(
,),,(),,,(
"
'
33222
111
332
22111
3322111
1221311322323
1221311323323
321321
33
33
33
33
33
3
yxyxyx
yxyxyx
yxyxyx
yxyxyxyxyxyx
yxyxyxyxyxyx
yyyxxx
a
s
He, Keum & Volkas, JHEP(2006)
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Quark and Lepton Mixing matrices
• Quark mixing (CKM) is almost unit matrix:
1)1(2
1
)(2
1
23
22
32
AiA
A
iA
VCKM
Neutrino mixing (MNS) is approximately “tribimaximal”:
2/13/16/1
2/13/16/1
03/13/22
MNSV
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He, etal’s A_4 model
He, Keum & Volkas, JHEP (2006)Symmetry group of the Lagrangian:
4)1()2()3( AUSUSUG YLC
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:34 ZA
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: “ Tribimaximal Mixing”
with
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Triplet Higgs Model
To generate Majorana - masses, we introduce triplet Higgs T(1,3,1)
eTvfTefeTefTfTfL ccccaac 22 0
with
).(2
1
)(2
1
210
3
21
iTTT
TT
iTTT
We need small value of 0Tu to explain naturally the small
-masses. Frampton, Oh & Yoshikawa (2002)
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Our Model
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We assume .,,321 cbfedvvv
The lepton mass matrix , ,lM can be diagonalized by rotating
the left-handed lepton by the unitary matrix
The neutrino mass matrix is diagonalized with unitary matrix : U
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,121323 VVVU
.0,4/, 13233 dbam
)(UV eL
.)( UUVVV L
eLMNS
Decomposing we get
Mixing matrix in the charged lepton sector is
and
Then the becomes ( symmetric) M
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12
).,(2
2221
211212 iiT ememdiagU
dbad
dbaU
M U
.diagT MUMU
U
The remaining can be obtained by
can be diagonalized with unitary matrix :
After getting , we can compare it with the standard parametrization
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to get the ( with ): 1112 /tan UUsol ])arg[( 12121212 MM
.21
2tan
12
12
12
12
te
tei
i
sol
Now we impose the experimental data
to constrain the 5 variables and ,),/(),/(, adbydaxd b
where a, b, d are in general complex numbers: andebbeaa ba ii ,,
.diedd We can set without the loss of the generality.0d
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The analytic solutions can be obtained to be
From the condition we get ,22atmsol mΔmΔ
In principle either normal hierarchy ( ) or inverted 02 atmmΔ
hierarchy ( ) is possible. 02atmmΔ
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( ) - plane ba ,
Only , i.e. normal hierarchy is allowed. 0~b
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( ) -plane 1,mx
Lower bound of :1m
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Effective Majorana mass for neutrinoless double beta decay:
where : real positive diagonal matrix. diagM
The effective Majorana mass for : 0
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( )-plane mm ,1
There is no lower bound for . m
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The sizes of the elements of M
We do not need large hierarchy among the matrix elements in our model.
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Conclusion
• We studied a triplet Higgs model to generate Majorana neutrino masses and the mixing matrix in the frame-work of A_4 symmetry.
• The tribimaximal form of the neutrino mixing matrix can be naturally ob-tained.
• Only the normal mass hierarchy is al-lowed.
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• There is a lower bound on the light-est neutrino mass : , although it is too small to be probed in the near future experiments.
• However, there is no lower limit in the effective mass parameter of neutrinoless double beta decay.
• Our model can explain the neutrino oscillation data without fine-tuning.
m