1. arXiv:0904.2301

4. Unitarity Tests of Mixing Matrices The quark sector

5. More general W- interaction with quarks Example: Left-Right symmetric model with more than 3 generations can induce right-handed current to V CKM and make the 3x3 first 3 generation V CKM non-unitary. Use unitary gauge if not known the full particle contents to calculate loops. He, Tandean & Valencia

28. Mixing in Quarks and Leptons Xiao-Gang He Department of Physics and Center for Theoretical Sciences NTU, Taipei NTU, Taipei 1. Mixing in Quarks and Neutrinos 2. Quark -Lepton Complementarity 3. Unitarity Tests of Mixing Matrices 4. Conclusions

29. 1. Mixing in Quarks and Neutrinos

30. Quark mixing A convenient parameterization: The Wolfenstein parameterization

32. Neutrino Mixing Three light neutrinos, Z decay, N = 2.983\pm 0.009, 3 neutrino mixing

35. Summary of mixing angles

36. Good approximation for neutrino mixing: The tri-bimaximal matrix Harrison, Perkins & Scott; Zhi-Zhong Xing; Xiao-Gang He & A. Zee Good approximation for quark mixing: The unit matrix Very different mixing patterns for quarks and neutrinos! But: Some interesting features

37. Natural Zero-th order mixing matrices

38. The natural 0-th order mixing matrix for quark

39. The natural 0-th order mixing matrix for neutrino – tri-bimaximal mixing Babu and He, He, Keum & Volkas

43. The natural 0-th order mixing matrix for neutrino – tri-bimaximal mixing- Independent of lepton masses

44. Good approximation for neutrino mixing: The tri-bimaximal matrix Harrison, Perkins & Scott; Zhi-Zhong Xing; Xiao-Gang He & A. Zee Good approximation for quark mixing: The unit matrix But: Hint some deeper reason? Q-L Complementarity He, Li & Ma 2. Quark –Lepton Complementarity

45. QLC - Start with unit matrix for quark mixing If one takes unit matrix for the 0th quark mixing N. Li & B.-Q. Ma The corresponding QLC predicted mixing for lepton is the bi-maximal mixing of the form The corresction is of order Wolfenstein parameters.

46. A better 0th order expansion for quarks? A new proposal: Tri-minimal parameterization S.-W.Li & Q.-Q. Ma

47. Much faster convergence than Wolfenstein parameterization! He Li & Ma

49. Exact Q-L complementarity With deviations A theoretical understanding of Q-L compplementarity is still lacking! Exact Q-L complementarity With deviations A theoretical understanding of Q-L compplementarity is still lacking!

50. 3. Unitarity Tests of Mixing Matrices The quark sector

51. More general W- interaction with quarks Example: Left-Right symmetric model with more than 3 generations can induce right-handed current to V CKM and make the 3x3 first 3 generation V CKM non-unitary. Use unitary gauge if not known the full particle contents. He, Tandean & Valencia, Xiao et al.,,,

53. There are rooms for violation of unitarity. Further tests are needed

54. Neutrino Mixing Three light neutrinos, Z decay, N = 2.983\pm 0.009, 3 neutrino mixing

55. The lepton sector Summary from Valle. Still has large room for non-unitarity at 10 percent level, something new may be there to make it happen. Example: Seesaw models.

56. Large light and heavy neutrino mixing in Seesaw models

59. Can Seesaw Models cause large non-unitary deviation in U PMNS ? Naively, No! … But … Constraints on elements in U \nu N

60. Kerstin & Smirnov; Xing et al.; He, Oh, Tandean &Wen

62. Possible to have large elements in U \nu N and therefore observable non-unitarity in lepton mixing!

63. andean Interesting application of large U \nu N at the LHC He, Oh, Tandean & Wen; Li & He

64. Model realization for large mixing between light and heavy neutrinos X.-G. He and E. Ma, arXiv:0907.2737[hep-ph]

67. Seesaw 3+3 or 3+2?

69. The U PMNS is not unitary in any case.

70. 5. Conclusions The CKM and PMNS mixing matrices for quark and lepton sectors describe related phenomena well. The quark mixing is approximated by a unit matrix and the lepton mixing is by the tri- bimaximal matrix. The tri-bimaximal mixing can be understood from theoretical point of view. There are interesting relations between quark and lepton mixing, the quark-lepton complementarity. Theoretical understanding these relations are poor. There are rooms of violating the unitarity of the mixing matrices both in quark and lepton sector. Seesaw models can give large mixing between light and heavy neutrinos, and therefore large violation of unitarity in lepton mixing. Theoretical models can be constructed. There are interesting LHC physics may results. The FL symmetry can link 3+3 seesaw model to a 3+2 seesaw model.