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