neutrino mass matrix in triplet higgs models with a_4 symmetry myoung chu oh miami 2008, dec. 17,...
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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
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
Introduction to A_4 symmetry
Symmetry group of a regular tetrahedron for the proper ro-tations12 elementsEven permutations of four objects }2314{}1234){123(
Group Multiplication table
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)
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)
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)
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
He, etal’s A_4 model
He, Keum & Volkas, JHEP (2006)Symmetry group of the Lagrangian:
4)1()2()3( AUSUSUG YLC
:34 ZA
: “ Tribimaximal Mixing”
with
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)
Our Model
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
,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
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
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
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
( ) - plane ba ,
Only , i.e. normal hierarchy is allowed. 0~b
( ) -plane 1,mx
Lower bound of :1m
Effective Majorana mass for neutrinoless double beta decay:
where : real positive diagonal matrix. diagM
The effective Majorana mass for : 0
( )-plane mm ,1
There is no lower bound for . m
The sizes of the elements of M
We do not need large hierarchy among the matrix elements in our model.
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.
• 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
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