1. Minimal SO(10)×A4 SUSY GUT ABDELHAMID ALBAID In Collaboration with K. S. BABU Oklahoma State University

2. Outline 1- Motivation 2- Minimal SO(10) and A4 3- Hierarchy & Mass Relations 4-Doubly Lopsided Structure 5- Analytical Fit 6- Numerical Fitting at the Gut Scale 7-Predictions at the Low Scale 8-Conclusion

3. Motivation To construct SO(10)×A4 model such that SO(10) is broken in the minimum way to SM in order to preserve the gauge coupling unification and make the unified gauge coupling perturbative below the Planck scale To explain most of the quark and lepton features, such as, quark and neutrino mixing angles, mass hierarchies and relations, CP violation,……..

4. Minimal SO(10) and A4 No Mixing matrix for quark sector Good for 3rd generation, bad for 1st and 2nd generations It is the finite group of the even permutation of four objects and contains 12 elements. It has four irreducible representation 1, 1', 1'', 3. A4 is the smallest discrete group that has a three dimensional irreducible representation [E. Ma, G. Rajasekaran, 2001] A4 flavor symmetry very easily gives tri-bi-maximal mixing matrix [P.F Harrison et al, 2002] SUSY A4 solves FCNC problem Accommodates all the SM multiplets in three 16-dimensional spinor representations. Considered as first approximation to CKM matrix Why SO(10)? What is A4? Why A4?

5. Hierarchy & Mass Relations The Matter Fields SO(10) A4 3 1 1 1 3 3 3 3 The Higgs Fields SO(10) A4 1 1 1 1 3 3 3

6. Hierarchy & Mass Relations Consider the following Feynman diagram The effective operator after integrating out the heavy states

7. Hierarchy & Mass Relations So the light fermions' masses WhereFrom the above matrix Bad Relations Hierarchy between the second and third generation

8. Doubly Lopsided Structure Doubly lopsided structure leads to [K.S. Babu, S. Barr, 2001] Consider the combination of the following two diagrams, the first (second) gives flavor symmetric (anti-symmetric) contribution, Small CKM matrix Large Neutrino mixing matrix

9. Doubly Lopsided Structure And rewrite the in terms of we get After combining the above three diagrams (i.e. ) and redefining our parameters such that :

10. Doubly Lopsided Structure SU(5) limit Consider the following limit Having bothof order one leads to doubly lopsided structure

11. Analytical Fit By using this approximation The following expressions are obtained

12. Numerical fitting at Gut scale The following are the numerical fit at GUT scale for The inputs The outputs

13. Predictions at the Low Scale We do the running of the above fitting from the Gut Scale to Msusy =500 GeV then from Msusy to low scale using RGE with

14. Conclusion From the invariant SO(10) × A4 superpotential we obtain the doubly lopsided structure which explains the largeness of the neutrino mixing angles and smallness of the quark mixing angles simultaneously. We fit numerically at low scale, the fermion masses, quark and lepton mixing angles (except atmospheric angle), and the CP violation parameter The atmospheric neutrino mixing angle is still needed to be corrected by including the appropriate right handed neutrino structure to our model The constraint of preserving the gauge coupling unification and making the unified gauge coupling perturbative all the way to the Planck scale guides us to consider invariant SO(10) × A4 model with the minimum Higgs representation breaking scheme.