1. Fermion Masses and Unification Steve King University of Southampton

2. Lecture III Family Symmetry and Unification 1.Introduction to family symmetry 2.Froggatt-Nielsen mechanism 3.Gauged U(1) family symmetry and its shortcomings 4.Gauged SO(3) family symmetry and vacuum alignment 5.A 4 and vacuum alignment 6. A 4 Pati-Salam Theory Appendix A. A 4 Appendix B. Finite Groups

3. Universal form for mass matrices, with Georgi-Jarlskog factors Texture zero in 11 position Recall Symmetric Yukawa textures 1. Introduction to Family Symmetry

4. 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) or SO(3) etc If these are gauged and broken at high energies then no direct low energy signatures

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6. 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 2.Froggatt-Nielsen Mechanism

7. 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

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

9. 3. Gauged U(1) Family Symmetry Problem: anomaly cancellation of SU(3) C 2 U(1), SU(2) L 2 U(1) and U(1) Y 2 U(1) anomalies implies that U(1) is linear combination of Y and B-L (only anomaly free U(1)’s available) but these symmetries are family independent Solution: use Green-Schwartz anomaly cancellation mechanism by which anomalies cancel if they appear in the ratio:

11. Suppose we restrict the sums of charges to satisfy Then A 1, A 2, A 3 anomalies are cancelled a’ la GS for any values of x,y,z,u,v But we still need to satisfy the A 1 ’=0 anomaly cancellation condition.

12. The simplest example is for u=0 and v=0 which is automatic in SU(5) GUT since10=(Q,U c,E c ) and 5 * =(L,D c )  q i =u i =e i and d i =l i so only two independent e i, l i. In this case it turns out that A 1 ’=0 so all anomalies are cancelled. Assuming for a large top Yukawa we then have: SO(10) further implies q i =u i =e i =d i =l i

13. F=(Q,L) and F c =(U c,D c, E c,N c )  In this case it turns out that A 1 ’=0. PS implies x+u=y and x=x+2u=y+v. So all anomalies are cancelled with u=v=0, x=y. Also h=(h u, h d )  The only anomaly cancellation constraint on the charges is x=y which implies Note that Y is invariant under the transformations This means that in practice it is trivial to satisfy

14. A Problem with U(1) Models is that it is impossible to obtain For example consider Pati-Salam where there are effectively no constraints on the charges from anomaly cancellation There is no choice of l i and e i that can give the desired texture e.g. previous example l 1 =e 1 =3, l 2 =e 2 =1, l 3 =e 3 =h f =0 gave: Shortcomings of U(1) Family Symmetry The desired texture can be achieved with non-Abelian family symmetry. Another motivation for non-Abelian family symmetry comes from neutrino physics.

15. columns SFK Sequential dominance can account for large neutrino mixing See-saw Sequential dominance Dominant m 3 Subdominant m 2 Decoupled m 1 Diagonal RH nu basis Tri-bimaximal Constrained SD

16. Large lepton mixing motivates non-Abelian family symmetry Need Suitable non-Abelian family symmetries must span all three families e.g. 2$ 3 symmetry (from maximal atmospheric mixing) 1$ 2 $ 3 symmetry (from tri-maximal solar mixing) SFK, Ross; Velasco-Sevilla; Varzelias SFK, Malinsky with CSD

17. 4. Gauged SO(3) family symmetry Antusch, SFK 04 Left handed quarks and leptons are triplets under SO(3) family symmetry Right handed quarks and leptons are singlets under SO(3) family symmetry Real vacuum alignment (a,b,c,e,f,h real) Barbieri, Hall, Kane, Ross To break the family symmetry introduce three flavons  3,  23,  123

18. But this is not sufficient to account for tri-bimaximal neutrino mixing If each flavon is associated with a particular right-handed neutrino then the following Yukawa matrix results

19. For tri-bimaximal neutrino mixing we need The motivation for  123 is to give the second column required by tri-bimaximal neutrino mixing How do we achieve such a vacuum alignment of the flavon vevs?

20. Vacuum Alignment in SO(3) First set up an orthonormal basis: F A =0 flatness  =  1 F B =0 flatness  =  2 F C =0 flatness  =  3 F D =0 flatness   1.  2 =0 F E =0 flatness   1.  3 =0 F F =0 flatness   2.  3 =0 SFK ‘05

21. Then align  23 and  123 relative to  1,  2,  3 using additional terms: F R =0  gets vevs in the (2,3) directions F T =0  gets vevs in the (1,2,3) directions (vevs of equal magnitude are required to minimize soft mass terms) Finally  23 is orthogonal to  123 due to  123.  23 =0

22. De M.Varzielas, SFK, Ross We can replace SO(3) by a discrete A 4 subgroup: A 4 is similar to the semi-direct product Same invariants as A 4  2 =  1 2 +  2 2 +  3 2,  3 =  1  2  3 The main advantage of using discrete family symmetry groups is that vacuum alignment is simplified… 5. A 4 and Vacuum Alignment

23. The Diamond (A4) Crystal Structure

24. Radiative Vacuum Alignment Varzielas, SFK, Ross, Malinsky (s)top loops drive negative A nice feature of MSSM is radiative EWSB Ibanez-Ross Similar mechanism can be used to drive flavon vevs using D-terms Leads to desired vacuum alignment with discrete family symmetry A 4 negative for negative for positive for positive 123

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

26. SFK, Malinsky 6. A 4 Pati-Salam Theory Dirac Operators:

27. Further Dirac Operators required for quarks: Dirac Operators: Dirac Neutrino matrix:

28. . Majorana 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 Dirac Neutrino matrix: Majorana Neutrino matrix:

29. The Messenger Sector Majorana: Dirac:

30. Including details of the messenger sector: Messenger masses:

31. Appendix A. A 4 SFK, Malinsky hep-ph/0610250

35. Appendix B. Finite Groups Ma 0705.0327