1. Quark and Lepton Mixing in S 4 Flavor Model September 28, 2010 Max-Planck-Institut für Kernphysik Heidelberg, Germany Morimitsu Tanimoto (Niigata University) with H. Ishimori and Y. Shimizu 1

2. Niigata ↓ 2

3. Niigata University Niigata City Population: 811,996 Niigata City is an urban center developed by its port. Even though it is located on a substantial expansion of agricultural landscapes, it has also easy accesses to major cities by airplanes, express omnibuses, and bullet trains. Also from its international airport, there are regular flights to Harbin, Shanghai, Seoul, Vladivostok, Khabarovsk, Guam. Niigata aspires to be a gateway to the East Asia.

4. 1 Tri-bi maximal mixing and Flavor Symmetry 2 S 4 Flavor Model in Quarks and Leptons 3 S 4 Flavor Model in Sleptons 4 Summary Plan of my talk 4

5. Three Flavor analysis strongly suggests Tri-bimaximal Mixing of Neutrinos 5 Harrison, Perkins, Scott (2002) 1 Tri-bimaximal mixing and Flavor symmetry Recent experiments of the neutrino oscillations go into a new phase of precise determination of mixing angles and mass squared differences. indicates Non-Abelian Flavor Symmetry ?

6. Mixing angles are independent of mass eigenvalues Different from quark mixing angles Consider the structure of Neutrino Mass Matrix, which gives Tri-bi maximal mixing 6

7. Quark Sector 7

8. Let us consider Flavor Symmetry. 8

9. Need some ideas to realize Tri-bi maximal mixing by S3 flavor symmetry × 9

10. ○ T’, S 4, Δ(54) flavor models also give Tri-bi maximal mixing ! A 4 Symmetry may be hidden. triplet ( ν e, ν μ,ν τ ) L A 4 should be broken ! 3 L ×3 L 3 L ×3 L ×3 H

11. Δ （ 27 ）, Δ （ 54 ）, Σ （ 81 ） 11

12. Stringy origin of non-Abelian discrete flavor symmetries T. Kobayashi, H. Niles, F. Ploeger, S. Raby, M. Ratz, NPB768,135(2007) hep-ph/0611020 Non-Abelian Discrete Flavor Symmetry from T 2 /Z N Orbifolds A.Adulpravitchai, A. Blum, M. Lindner, JHEP0907, 053 (2009), 0906.0468 Non-Abelian Discrete Flavor Symmetries from Magnetized/Intersecting Brane Models H. Abe, K-S. Choi, T. Kobayashi, H. Ohki, NPB820, 317 (2009) 0904.2631 H. Abe, K-S. Choi, T. Kobayashi, H. Ohki, NPB820, 317 (2009), 0904.2631 Origin of the non-Abelian Flavor symmetry ? Tri-bimaximal neutrino mixing from orbifolding, G.Altarelli, F.Feruglio, Y.Lin, NPB775, 31 (2007) hep-ph/0610165 Non-Abelian Discrete Groups from the Breaking of Continuous Flavor Symmetries A.Adulpravitchai, A. Blum, M. Lindner, JHEP0909, 018 (2009), 0907.2332 A.Adulpravitchai, A. Blum, M. Lindner, JHEP0909, 018 (2009), 0907.2332

13. Non-Abelian Discrete Symmetries in Particle Physics Hajime IshimoriHajime Ishimori, Tatsuo Kobayashi, Hiroshi Ohki,Tatsuo KobayashiHiroshi Ohki Hiroshi OkadaHiroshi Okada, Yusuke Shimizu, Morimitsu Tanimoto,Yusuke ShimizuMorimitsu Tanimoto e-Print: arXiv:1003.3552 [hep-th] Prog.Theor.Phys.Suppl.183:1-163,2010 We review pedagogically non-Abelian discrete groups and show some applications for physical aspects. This article includes a brief view on general aspects of group theory, i.e. something basic and useful theorems. Reference 13

14. ● CKM mixing in Quarks ? Cabibbo angle? We need Quark-lepton unification in a GUT. We need Quark-lepton unification in a GUT. ● Ue3=0 in Tri-bimaximal mixing! There are hints Non-zero U e3 in experiments. How can one predict U e3 ? Flavor Symmetry of Neutrinos is related with Physical Phenomena. ● SUSY Flavor Sector, SUSY FCNC, EDM We discuss the case of S 4 symmetry.

15. 11’1”311’1”3 Four irreducible representations in A 4 symmetry Before discussing S 4 model, let us understand how to get the tri-bimaximal mixing in the example of A 4 flavor model. E. Ma and G. Rajasekaran, PRD64(2001)113012 15

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17. 1’ × 1” → 1 3 L × 3 L → 1 3 L × 3 flavon → 1 3 L × 3 flavon → 1” 3 L × 3 flavon → 1’ 3 L × 3 L × 3 flavon → 1 17

18. Can one get Desired Vacuum in Spontaneous Symmetry Breaking ? Scalar Potential Analysis These mass matrices do not yet predict tri-bimaximal mixing ! 18

19. ------------------------ 19

20. As seen in this A 4 model, in order to reproduce the tri-bi maximal mixing, we need Non-Abelian Discrete Symmetry (A 4, T ’, S 4 … ) and Symmetry Breaking Vacuum Alignment of flavons. 20 Spontaneous Breaking ? ( Scalar potential ) Explicit Breaking ? (Boundary condition in extra-dim.)

21. H. Ishimori, K. Saga, Y. Shimizu, M. Tanimoto, arXiv:1004.5004 2 S 4 Flavor Model in Quarks and Leptons S 4 ×Z 4 with SUSY SU(5) GUT ⇒ Tri-bimaximal, Cabibbo angle C.Hagedorn, M.Lindner, R.N.Mohapatra, JHEP 0606, 042 (2006) SO(10) B.Dutta, Y. Mimura, R.N. Mohapatra, arXiv:0911.2242 SO(10) C.Hagedorn, S. F. King, C. Luhn, arXiv:1003.4249 SU(5) R.d.A. Toorop, F. Bazzocchi, L. Merlo, arXiv: 1003.4502 Pati-Salam

22. Up quarks MRMR Dirac Neutrinos Charged leptons Down quarks We take l =m=1, n=2. S 4 ×Z 4 ×U(1) FN with SUSY SU(5) GUT

23. S 4 invariant superpotential for leptons 23 3 L ×2 R ×3 flavon 3 L ×1 R ×3 flavon 2 R ×2 R ×2 flavon 3 L ×1 R ×3 flavon 2 R ×2 R 1 R ×1 R 3 L ×2 R ×3 flavon

24. We take VEV ’ s We get Lepton Mass Matrices Due to m-n<0 ○ ○ ○ ○ 24

25. No mixing in the left-hand ! Θ 12 =60°in the right-hand ! Vacuum alignment

26. After seesaw, we get the tri-bimaximal mixing 26

27. Deviation from the Tri-bimaximal mixing due to Higher dimensional mass operators Superpotential of next-to-leading order 27

28. The charged lepton mass matrix including the next-to-leading terms Since the lepton mixing is given as we have non-zero U e3 0.003

29. Next-to-leading in Neutrino sector 29

30. Determination of magnitudes Putting observed masses and M=10 12 GeV, we get FN charges l =m=1, n=2 Desired Vacuum Alignments

31. We can predict mixing angles. 31

32. Quark Sector is predicted. Down Quarks Left-handed mixing is given as 32

33. Including next-to-leading order, we get 33

34. Up Quark Sector We add the next-to-leading mass matrix 34 Direct Yukawa coupling

35. We take alignment, we get After rotating it by the orthogonal matrix, We obtain Up Quarks

36. We obtain CKM matrix elements In the leading order, we predict 36

37. Including next-to-leading order corrections, we get The parameter set reproduces observed values very well. Values of parameters are consistent with our mass matrices. 37 CP violation can be discussed !

38. Flavor symmetry constrains not only quark/lepton mass matrices, but also mass matrices of their superpartner, i.e. squark/slepton Specific patterns of squark/slepton mass matrices could be tested in future experiments. In this talk, we concentrate on lepton FCNC. 3. S 4 Flavor Symmetry in Sleptons Consider Soft SUSY Breaking Term in Supergravity.

39. Second order Slepton mass matrices are derived from 39

40. For the left-handed sector, higher dimensional terms are given as Left-handed Slepton mass matri ｘ is 40

41. Right-handed Slepton mass matrix is 41

42. Move to Super-CKM basis ( Diagonal Basis of Charged Lepton) in order to estimate magnitudes of FCNC. Mass Insertion Parameters F. Gabbiani, E. Gabrielli, A. Masiero and L. Silvestrini, Nucl. Phys. B477(1996) 321 Experimental Constraint from μ→eγ Numerical analyses are required. where ○ Dominant term

43. A terms are obtained as Experimental Constraint We need numerical analyses of μ→eγ. Dangerous !

44. μ→eγ Decay ○ ○ ○○

45. EDM of Electron J.Hisano, M. Nagai, P. Paradisi, Phys.Rev.D80:095014,2009. ○ ○

46. Preliminary

47. Assume the maximal phase

50. 4 Summary ☆ S 4 Flavor Symmetry in SU(5) can give realistic quark and lepton mixing matrices. Tri-bimaximal mixing, Cabibbo angle ☆ S 4 discrete symmetries work to suppress FCNC in the framework of gravity mediation in SUSY breaking. ★ Squark sectors in S 4 Symmetry ? ★ Origin of S 4 Symmetry ? 50 ★ Mass Spectrum ?

51. One of Future Problems One of Future Problems Can we predict Neutrino Masses? Symmetry cannot predict mass spectrum. Symmetry breaking gives mass spectrum. Normal mass hierarchy Inverted mass hierarchy T2K and NOν Ａ ! T2K and NOν Ａ ! 51

52. We know Koide formula ! Can we predict neutrino spectrum consistent with Tri-bimaximal Mixing ？ We need more studies of Symmetry Breakings ! Accuracy 10 -5 for neutrinos 52

53. Thank you ! 53

54. ★ Tiny Neutrino Masses ? Seesaw, Extra-Dimensions Mass Spectrum? ★ Large Flavor Mixings ? However Θ 13 ? CP? ★ Majorana Neutrinos ？ L number violation? ★ Right-handed Neutrinos ？ LHC, Leptogenesis ★ Sterile Neutrinos ？ ★ New Interaction of Neutrinos ？ ★ Neutrino Soft Mass ？ ★ Cosmic Neutrinos? What is interesting in Neutrino Physics ? 54

55. A.D. Dolgov, arXiv: 0803.3887 m ν < 0.63 eV 95% c.l K.Ichikawa, M. Fukugita, M. kawasaki, PRD71(2005)043001 M. Fukugita, K. Ichikawa, M. Kawasaki, O. Lahav, PRD74(2006)027302 55

56. 56

57. Our Multiplication Rule of S 4 57

58. S 4 invariant superpotential 58

59. Realization of Vacuum Alignment Introduce driving fields with R charge 2 59

60. We obtain Desired Vacuum Alignment Scalar potential

61. 61

62. The first neutrino was detected on 24 th Feb 2010. 62

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64. 64

65. A 4 symmetry (Tetrahedral Symmetry) 65

66. EDM of Electron J.Hisano, M. Nagai, P. Paradisi, Phys.Rev.D80:095014,2009. Anomalous Magnetic Moment of Muon ○○

67. Neutrino Parameters Global fit for 3 flavors 67