1. Fermion Masses and Unification Steve King University of Southampton

2. Lecture 4 Family Symmetry and Unification II 1.SO(3) or A 4 family symmetry and unification 2.SU(3) or  27 family Symmetry and unification 3.Quark-lepton connections 4.SUSY flavour and SU(3) 5.Do we need a family symmetry? Appendix Finite Groups

3. SFK, Malinsky Realistic SO(3)£ Pati-Salam Model Flavon vacuum alignment Model also works with SO(3) replaced by its A  subgroup

4. Symmetry group of the tetrahedron Discrete set of possible vacua Ma; Altarelli, Feruglio; Varzeilas, Ross, SFK, Malinsky Comparison of SO(3) and A 4

5. . Majorana Operators Dirac Neutrino matrix:Majorana Neutrino matrix: Dirac Operators: CSD in neutrino sector due to vacuum alignment of flavons m 3 » m 2 » 1/  and m 1 » 1 is much smaller since  ¿ 1 See-saw mechanism naturally gives m 2 » m 3 since the  cancel

6. Gauged SU(3) family symmetry Now suppose that the fermions are triplets of SU(3)  i = 3 i.e. each SM multiplet transforms as a triplet under a gauged SU(3) with the Higgs being singlets H» 1 This “explains” why there are three families c.f. three quark colours in SU(3) c The family symmetry is spontanously broken by antitriplet flavons Unlike the U(1) case, the flavon VEVs can have non-trivial vacuum alignments. We shall need flavons with vacuum alignments:  3 >/ (0,0,1) and / (0,1,1) in family space (up to phases) so that we generate the desired Yukawa textures from Froggatt-Nielsen:

7. Frogatt-Nielsen in SU(3) family symmetry In SU(3) with  i =3 and H=1 all tree-level Yukawa couplings H  i  j are forbidden. In SU(3) with flavons  the lowest order Yukawa operators allowed are: For example suppose we consider a flavon with VEV then this generates a (3,3) Yukawa coupling Note that we label the flavon with a subscript 3 which denotes the direction of its VEV in the i=3 direction.

8. Next suppose we consider a flavon with VEV then this generates (2,3) block Yukawa couplings Writing and these flavons generate Yukawa couplings If we have  3 ¼ 1 and we write  23 =  then this resembles the desired texture To complete the texture there are good motivations from neutrino physics for introducing another flavon / (1,1,1)

9. Realistic SU(3)£ SO(10) Model Yukawa Operators Majorana Operators Varzielas,SFK,Ross Model also works with SU(3) replaced by its  27 subgroup

10. Inserting flavon VEVs gives Yukawa couplings After vacuum alignment the flavon VEVs are Writing Yukawa matrices become:

11. Assume messenger mass scales M f satisfy Then write Yukawa matrices become, ignoring phases: Where

12. ..... From above we see that Quark-Lepton Connections

13. Assume II: all 13 angles are very small Assume I: charged lepton mixing angles are small Charged Lepton Corrections and sum rule SFK,Antusch; Masina,…. Note the sum rule In a given model we can predict and.

14. The Neutrino Sum Rule Measured by experiment – how well can this combination be determined? Predicted by theory e.g.1. bi-maximal predicts 45 o e.g.2. tri-bimaximal predicts 35.26 o

15. Tri-bimaximal sum rule.. Antusch, Huber, SFK, Schwetz Bands show 3  error for a neutrino factory determination of  13 cos  Current 3  A Prediction

16. An old observation: SU(3) family symmetry predicts universal soft mass matrices in the symmetry limit However Yukawa matrices and trilinear soft masses vanish in the SU(3) symmetry limit So we must consider the real world where SU(3) is broken by flavons Solving the SUSY Flavour Problem with SU(3) Family Symmetry This was discussed in SUGRA by Ross, Velasco-Sevilla and Vives – here we re-examine this from a bottom-up point of view using only symmetry properties of SU(3) or  27

17. Soft scalar mass operators in SU(3) Using flavon VEVs previously

18. Recall Yukawa matrices, ignoring phases: Where This predicts almost universal squark and slepton masses:

19. Do we need a family symmetry? Ferretti, SFK, Romanino; Barr One family of “messengers” dominatesThree families of quarks and leptons Suppose then in a particular basis Not bad! But… Accidental sym Need broken Pati-Salam… Conclusion: partial success, but little predictive power esp. in neutrino sector

20. Appendix Finite Groups Ma 0705.0327