Quark and Lepton Masses in SO(10) x A4

1. Quark and Lepton Masses in SO(10) x A4 Abdelhamid Albaid In Collaboration with K. S. Babu Oklahoma State University

2. Outline 1- Motivation 2- Minimal SO(10) 3- What is A4, and why A4? 4- The Model 5-Mass hierarchies and relations, and quark mixings 6-Conclusion

3. Motivation Primary Motivation • To find a model with the smallest possible Higgs representation such that gauge coupling unification is not broken. • To explain features of the quark and lepton masses. • This talk is devoted to explain the hierarchy among the three generations, mass relations of down quarks and charged leptons, and quark mixing predictions.

4. Minimal SO(10) • Accommodates all the SM multiplets in three 16-dimensional spinor representations • Shortcomings of the minimal SO(10) : Good for 3rd generation, bad for 1st and 2nd generations. No Mixing matrix for quark sector In addition there is no way to couple 45 Higgs with only the ordinary family 16-plets.

5. What is A4 and why A4? The A4 properties • The group A4 is the finite group of the even permutation of four objects and contains 12 elements • There are four irreducible representation 1, 1', 1'', 3 If and are triplets then their tensorial product decomposes in irreducible representation as

6. A4(cont.) Why A4? • A4 is the smallest discrete group that has three dimensional irreducible representation • A4 flavor symmetry very easily gives tri-bi-maximal mixing matrix [ arXiev:hep-ph/0202074] • SUSY A4 solves FCNC problem sfermions have degenerate masses

7. The Model The Matter Fields The Higgs Fields

8. The Model (cont.) Let us consider the following Feynman diagram: Based on the above diagram, the Superpotental of the Yukawa terms is:

9. The Model (cont.)

10. The Model (cont.)

11. The Model (cont.) The three light matter states can be obtained by integrating out the heavy states

12. The Model (cont.) • The quantity Q is a linear combination of SO(10) generators. There are two generators that commute with the SM. So we can write Where is the third generators of SU(2)R For example:

13. Mass Hierarchies and Relations, and Quark Mixings Hierarchy of Fermion Masses Note that the hierarchy between the second and third generation masses can be explained In the limit , it is remarkable that a relation among generations is related to the vacuum alignment of A4 Higgs

14. Mass Hierarchies and Relations, and Quark Mixings (cont.) Since both D and L get mass from the same Higgs, we get Therefore Let us look at mass relations of the down quark and charged lepton for the second generations

15. Mass Hierarchies and Relations, and Quark Mixings (cont.) Mass Relation for 1st family In order to fix mass ratio for 1st generation, we need to add two 10-plet to our model. Consider the following diagram

16. Mass Hierarchies and Relations, and Quark Mixings (cont.) Above figuregives a flavor-symmetric contribution to the down quark and charged lepton mass matrix:

17. Mass Hierarchies and Relations, and Quark Mixings (cont.) • Mass matrix for up quarks is still rank 2. This corresponds to the fact that

18. Mass Hierarchies and Relations, and Quark Mixings (cont.) Quark Mixings The symmetry of the contributions to the first row and column of down quark leads to the following form After diagonalizing the (23) block

19. Conclusion • We constructed a model with a Lagrangian invariant under SO(10)xA4 with minimum possible Higgs representation in order to preserve gauge coupling unification. • SUSY SO(10)xA4 gives degenerate sfermion masses and solves the problem of FCNC. • This Model explains the mass ratio of down quarks and charged leptons, and predicts quark mixing. • The hierarchy of the three families of fermions can be understood from A4 symmetry breaking. • We are currently extending our work to include neutrino sector