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

2. Outline 1. Introduction to A_4 symmetry 2. Others Model 1) Type I seesaw model with A_4 symmetry ( He, Keum, Volkas (2006) ) 2) Triplet Higgs Model 3. Our Model 4. Conclusion

3. Introduction to A_4 symmetry Symmetry group of a regular tetrahedron for the proper rotations 12 elements Even permutations of four objects

4. Group Multiplication table

5. Representations 3-dim. Representation 3 : with He,Keum & Volkas, JHEP(2006)

6. Representations 1-dim. Representations 1,1,1 : 1(1) : where. He, Keum & Vlokas, JHEP(2006)

7. Representations Tensor Products: For He, Keum & Volkas, JHEP(2006)

9. Quark and Lepton Mixing matrices Quark mixing (CKM) is almost unit matrix: Neutrino mixing (MNS) is approximately tribimaximal:

10. He, etals A_4 model He, Keum & Volkas, JHEP (2006) Symmetry group of the Lagrangian:

11. :

12. : Tribimaximal Mixing with

14. Triplet Higgs Model To generate Majorana - masses, we introduce triplet Higgs T(1,3,1) with We need small value ofto explain naturally the small -masses. Frampton, Oh & Yoshikawa (2002)

15. Our Model

16. We assume The lepton mass matrix,can be diagonalized by rotating the left-handed lepton by the unitary matrix The neutrino mass matrix is diagonalized with unitary matrix :

17. Decomposing we get Mixing matrix in the charged lepton sector is and Then the becomes ( symmetric)

18. The remainingcan be obtained by can be diagonalized with unitary matrix : After getting, we can compare it with the standard parametrization

19. to get the ( with ): Now we impose the experimental data to constrain the 5 variables and where a, b, d are in general complex numbers: We can set without the loss of the generality.

20. The analytic solutions can be obtained to be From the condition we get In principle either normal hierarchy ( ) or inverted hierarchy ( ) is possible.

21. ( ) - plane Only, i.e. normal hierarchy is allowed.

22. ( ) -plane Lower bound of

23. Effective Majorana mass for neutrinoless double beta decay: where : real positive diagonal matrix. The effective Majorana mass for :

24. ( )-plane There is no lower bound for.

25. The sizes of the elements of We do not need large hierarchy among the matrix elements in our model.

26. Conclusion We studied a triplet Higgs model to generate Majorana neutrino masses and the mixing matrix in the framework of A_4 symmetry. The tribimaximal form of the neutrino mixing matrix can be naturally obtained. Only the normal mass hierarchy is allowed.

27. There is a lower bound on the lightest 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.