1. Family Symmetry Solution to the SUSY Flavour and CP Problems Plan of talk: I.Family Symmetry II.Solving SUSY Flavour and CP Problems Work with and Michal Malinsky

2. Universal form for mass matrices, with Georgi-Jarlskog factors Texture zero in 11 position Fermion mass spectrum well described by Symmetric Yukawa textures Introduction to Family Symmetry G.Ross et al

3. To account for the fermion mass hierarchies we introduce a spontaneously broken family symmetry It must be spontaneously broken since we do not observe massless gauge bosons which mediate family transitions The Higgs which break family symmetry are called flavons  The flavon VEVs introduce an expansion parameter  = /M where M is a high energy mass scale The idea is to use the expansion parameter  to derive fermion textures by the Froggatt-Nielsen mechanism (see later) In SM the largest family symmetry possible is the symmetry of the kinetic terms In SO(10),  = 16, so the family largest symmetry is U(3) Candidate continuous symmetries are U(1), SU(2), SU(3) etc If these are gauged and broken at high energies then no direct low energy signatures

4. Nothing Candidate Family Symmetries

5. Simplest example is U(1) family symmetry spontaneously broken by a flavon vev For D-flatness we use a pair of flavons with opposite U(1) charges Example: U(1) charges as Q (  3 )=0, Q (  2 )=1, Q (  1 )=3, Q(H)=0, Q(  )=-1,Q(  )=1 Then at tree level the only allowed Yukawa coupling is H  3  3 ! The other Yukawa couplings are generated from higher order operators which respect U(1) family symmetry due to flavon  insertions: When the flavon gets its VEV it generates small effective Yukawa couplings in terms of the expansion parameter Froggatt-Nielsen Mechanism

6. What is the origin of the higher order operators? To answer this Froggat and Nielsen took their inspiration from the see-saw mechanism Where  are heavy fermion messengers c.f. heavy RH neutrinos

7. There may be Higgs messengers or fermion messengers Fermion messengers may be SU(2) L doublets or singlets

8. 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:

9. 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.

10. 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)

11. The motivation for  123 from tri-bimaximal neutrino mixing For tri-bimaximal neutrino mixing we need

12. A Realistic SU(3)£ SO(10) Model Yukawa Operators Majorana Operators Varzielas,SFK,Ross

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

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

15. In SUSY we want to understand not only the origin of Yukawa couplings But also the soft masses The SUSY Flavour Problem  See-saw parts

16. The Super CKM Basis Squark superfields Quark mass eigenstates Quark mass eigenvalues

17. Super CKM basis of the squarks (Rule: do unto squarks as we do unto quarks)

18. Squark mass matrices in the SCKM basis Flavour changing is contained in off-diagonal elements of Define  parameters as ratios of off-diagonal elements to diagonal elements in the SCKM basis  ij = m 2 ij /m 2 diag

19. Typical upper bounds on  Clearly off-diagonal elements 12 must be very small Quarks Leptons

20. 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

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

22. Recall Yukawa matrices, ignoring phases: Where Under the same assumptions we predict:

23. In the SCKM basis we find: Yielding small  parameters

24. The SUSY CP Problem Neutron EDM d n <4.3x10 -27 e cm Electron EDM d e <6.3x10 -26 e cm Abel, Khalil,Lebedev Why are SUSY phases so small? In the universal case

25. Postulate CP conservation (e.g. real) with CP is spontaneously broken by flavon vevs This is natural since in the SU(3) limit the Yukawas and trilinears are zero in any case So to study CP violation we must consider SU(3) breaking effects in the trilinear soft masses as we did for the scalar soft masses Ross,Vives Solving the SUSY CP Problem with SU(3) Family Symmetry

26. Soft trilinear operators in SU(3) Using flavon VEVs previously N.B parameters c i f and  i f are real

27. Compare the trilinears to the Yukawas They only differ in the O(1) real dimensionless coefficients

28. Since we are interested in the (1,1) element we focus on the upper 2x2 blocks The essential point is that , , ,  are real parameters and phases only appear in the (2,2,) element (due to SU(3) flavons) Thus the imaginary part of A d 11 in the SCKM basis will be doubly Cabibbo suppressed

29. To go to SCKM we first diagonalise Y d Then perform the same transformation on A d c.f. universal case Extra suppression factor of

30. Conclusions SU(3) gauged family symmetry, when spontaneously broken by particular flavon vevs, provides an explanation of tri-bimaximal neutrino mixing When combined with SUSY it gives approx. universal squark and slepton masses, suppressing SUSY FCNCs It also suppresses SUSY contributions to EDMs by an extra order of magnitude compared to mSUGRA or CMSSM  remaining phase must be <0.1 Maybe SUGRA can help with this remaining 10% tuning problem – work in progress