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Fermion Masses and Unification

Fermion Masses and Unification. Steve King University of Southampton. Lecture III Family Symmetry and Unification. 1.Introduction to family symmetry 2.Froggatt-Nielsen mechanism 3.Gauged U(1) family symmetry and its shortcomings

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Fermion Masses and Unification

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  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.A4 and vacuum alignment 6. A4 Pati-Salam Theory Appendix A. A4 Appendix B. Finite Groups

  3. Recall Symmetric Yukawa textures • Universal form for mass matrices, with Georgi-Jarlskog factors • Texture zero in 11 position 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

  5. Nothing

  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 2.Froggatt-Nielsen Mechanism 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

  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)C2U(1), SU(2)L2U(1) and U(1)Y2U(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:

  10. Suppose we restrict the sums of charges to satisfy Then A1, A2, A3 anomalies are cancelled a’ la GS for any values of x,y,z,u,v But we still need to satisfy the A1’=0 anomaly cancellation condition.

  11. The simplest example is for u=0 and v=0 which is automatic in SU(5)GUT since10=(Q,Uc,Ec) and 5*=(L,Dc)  qi=ui=ei and di=li so only two independent ei, li. In this case it turns out that A1’=0 so all anomalies are cancelled. Assuming for a large top Yukawa we then have: SO(10) further implies qi=ui=ei=di=li

  12. F=(Q,L) and Fc=(Uc,Dc,Ec,Nc)  In this case it turns out that A1’=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=(hu, hd)  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

  13. Shortcomings of U(1) Family Symmetry 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 li and ei that can give the desired texture e.g. previous example l1=e1=3, l2=e2=1, l3=e3=hf=0 gave: The desired texture can be achieved with non-Abelian family symmetry. Another motivation for non-Abelian family symmetry comes from neutrino physics.

  14. Diagonal RH nu basis columns See-saw Sequential dominance Dominant m3 Subdominant m2 Decoupled m1 Tri-bimaximal Constrained SD Sequential dominance can account for large neutrino mixing SFK

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

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

  17. 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

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

  19. First set up an orthonormal basis: FA=0 flatness  <1>=1 FD=0 flatness 1 .2 =0 FB=0 flatness  <2>=2 FE=0 flatness 1 .3 =0 FC=0 flatness  <3>=3 FF=0 flatness 2 .3 =0 SFK ‘05 Vacuum Alignment in SO(3)

  20. FR=0  <23> gets vevs in the (2,3) directions FT=0  <123> 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 Then align 23 and 123 relative to 1 , 2 , 3 using additional terms:

  21. A4 is similar to the semi-direct product Same invariants as A4 2=12+22+32 , 3 =123 5. A4 and Vacuum Alignment We can replace SO(3) by a discrete A4 subgroup: De M.Varzielas, SFK, Ross The main advantage of using discrete family symmetry groups is that vacuum alignment is simplified…

  22. The Diamond (A4) Crystal Structure

  23. A nice feature of MSSM is radiative EWSB Ibanez-Ross (s)top loops drive negative Varzielas, SFK, Ross, Malinsky Radiative Vacuum Alignment Similar mechanism can be used to drive flavon vevs using D-terms Leads to desired vacuum alignment with discrete family symmetry A4 negative for negative  for positive  for positive 123

  24. Ma; Altarelli, Feruglio; Varzeilas, Ross, SFK, Malinsky Symmetry group of the tetrahedron Comparison of SO(3) and A4 Discrete set of possible vacua

  25. Dirac Operators: SFK, Malinsky 6. A4 Pati-Salam Theory

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

  27. Majorana Operators Majorana Neutrino matrix: Dirac Neutrino matrix: . . . . • CSD in neutrino sector due to vacuum alignment of flavons • m3» m2 »1/ and m1» 1 is much smaller since ¿ 1 • See-saw mechanism naturally gives m2» m3 since the  cancel

  28. Dirac: Majorana: The Messenger Sector

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

  30. Appendix A. A4 SFK, Malinsky hep-ph/0610250

  31. Appendix B. Finite Groups Ma 0705.0327

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