1. 1 Unification of Quarks and Leptons or Quark-Lepton Complementarities Bo-Qiang Ma Peking University (PKU) Peking University (PKU) in collaboration with Nan Li in collaboration with Nan Li November 16, 2007, Talk at NCTS @ NTHU November 16, 2007, Talk at NCTS @ NTHU ?

2. 2 Basic structure of matter Basic interactions

3. 3 Properties of Fermions

4. 4 The unification of quarks and leptons The unification of quarks and leptons Why there are three generations ？ The origin of masses and their relations ？ The mixing between different generations ： why mixing ？ Is it possible for a unification of quarks and leptons ？

5. Parametrization of Lepton Mixing Matrix Are there connections between the parametrizations of quark and lepton mixing matrices ？

6. 6 Cabibbo, Kobayashi, Maskawa (CKM) Matrix Unitarity (or lack thereof) of CKM matrix tests existence of further quark generations and possible new physics (eg. Supersymmetry) u,c,t quarks couple to superposition of other quarks Weak eigenstatesMass eigenstates w/ Particle Data Group ‘01 Central Values

7. .. Neutrino Oscillations This is only possible if neutrinos have mass  new physics beyond the Standard Model

8. 8 Two flavor case Assuming are flavor eigenstates, are mass eigenstates. Their mixing are Assuming at, there is only electron neutrino, i.e.,

9. 9 at t, Express in terms of ， one obtains Then the probablity of finding neutrino at t is using and

10. 10 one obtains The oscillation period is And the oscillation probability is Generating to three generations ， i.e., the lepton mixing matrix ， is called PMNS matrix. B. Pontecorvo, JETP 7 (1958) 172; Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. 28 (1962) 870.

11. 11 Mixing matrices of quarks and leptons

12. 12 Data for quark and lepton mixing angles 夸克与轻子混合角的实验数据 quark mixing angles lepton mixing angles

13. 13 Data for quark mixing matrix 夸克混合矩阵的实验数据 Data for lepton mixing matrix 轻子混合矩阵的实验数据

14. 14 Parametrization of quark mixing matrix 夸克混合矩阵的参数化 L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945.

15. 15 Parametrization of lepton mixing matrix 轻子混合矩阵的参数化 Base matrix ： （ 1 ） Bimaximal Matrix （ 2 ） Tribimaximal Matrix

16. 16 1.Parametrization Based on Bimaximal Matrix Introduce parameters a, b, λ : W. Rodejohann, Phys. Rev. D 69 (2004) 033005; N. Li and B.-Q. Ma, Phys. Lett. B 600 (2004) 248.

17. 17 We get trigonometric functions ：

18. 18 Expansion of Lepton Mixing （ PMNS ） Matrix ：

19. 19 Ranges of Data ： Ranges of Parameters ： The expansion is reasonable ， and converges fast 。 To take phase angle into account ， set U e3 = bλe -iλ.

20. 20 The Jarlskog parameter to describe CP violation In this parametrization ， The value of J is C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039; C. Jarlskog, Z. Phys. C 29 (1985) 491.

21. 21 2. Parametrization Based on Tribimaximal Matrix N. Li and B.-Q. Ma, Phys. Rev. D 71 (2005) 017302.

22. 22 Expansion of Lepton Mixing （ PMNS ） Matrix ： The Tribimaximal expansion converges more fast than the Bimaximal expansion, but is of less symmetry 。

23. 23 Quark-Lepton Complementarity Lepton mixings Quark mixings Present experimental data allows for relations like the bimaximal complementarity relation: Is such quark-lepton complementarity a hint of an underlying quark-lepton unification? Two possibilities: 1. Bimaximal complementarity 2. Tri-bimaximal complementarity Raidal ('04) Smirnov, Minakata ('04) SFK (’05)

24. 24 3. Parametrization based on QLC Quark mixing angles Lepton mixing angles

25. 25 夸克轻子互补性 (QLC) Quark-Lepton Complementarity A. Yu. Smirnov, hep-ph/0402264; M. Raidal, Phys. Rev. Lett. 93 (2004) 161801.

26. 26 For quark mixing ： The trigonometric functions are ：

27. 27 Assuming As there are uncertainties in data ， we assume (1) (2) therefore obtain unified parametrization of quark and lepton mixing matrix. N. Li and B.-Q. Ma, Phys. Rev. D 71 (2005) 097301.

28. 28 (1) From above assumptions, we obtain ：

29. 29 Expansion of Lepton Mixing （ PMNS ） Matrix ：

30. 30 Significance of the expansion 1.The Wolfenstein parameter λcan measure both the deviation of quark mixing matrix from the unit matrix, and the deviation of lepton mixing matrix from Bimaximal mixing pattern 。 2. The Bimaximal mixing pattern is derived as the leading-order term naturally 。

31. 31 3. The Bimaximal expansion at first-order can be naturnal obtained: Bimaximal expansion Expansion base on QLC By re-scaling the Bimaximal expansion can be naturally obtained 。

32. 32 4. The values of λ ： after re-scaling in Bimaximal expansion, the value is in the right range of the Wolfenstein parameter. 5. The value of, at best value i.e., Jarlskog parameter can be fixed by the CP violation of the lepton sector ， thus one obtain the range of. Therefore all of the 4 parameters can be fixed

33. 33 （ 2 ） Under the assumption The expansion of lepton mixing （ PMNS ） matrix

34. 34 Relations between quark and lepton mixing matrices Possible relations between quark and lepton mixing matrices ： （ 1 ） （ 2 ） And their connections with QLC 。 N. Li and B.-Q. Ma, Euro.Phys.J.C 42, 17 (2005).

35. 35 A remark 1.The quark and lepton mixing matrices can be parametrized with the same set of Wolfenstein parameters A and λ, therefore the two parameters can describe both the derivation of the quark mixing matrix from the unit matrix and the lepton mixing matrix from the Bimaximal mixing pattern. 2.If λand A are different for quarks and leptons, we can consider the expansion of the lepton mixing matrix as a general parametrization form in similar to the Wolfenstein parametrization of quark mixing matrix.

36. 36 Relations between masses of quarks and leptons 夸克轻子质量的关系 : Relations between masses of quarks and leptons 夸克轻子质量的关系 : 质量的起源和关系？ Quark and lepton masses are 12 free parameters in the standard model. Are there relations between masses of different generations? Are there relations between masses of quarks and leptons ？

37. 37 Koide mass relation Y. Koide, Lett. Nuovo Cimento 34, 201 (1982); Phys. Lett. B 120, 161 (1983).

38. 38 QLC of the masses The extended Koide’s relations Y. Koide, Lett. Nuovo Cimento 34, 201 (1982); Phys. Lett. B 120, 161 (1983).

39. 39 Some conjectures The neutrino masses

40. 40 夸克轻子质量的关系 夸克轻子质量的关系 N. Li and B.-Q. Ma, Phys. Lett. B 609, 309 (2005).

41. 41 Energy scale insensitivity of Koide's relation N. Li and B.-Q. Ma, Phys. Rev. D 73, 013009 (2006). The relation is insensitive of energy scale in a huge energy range from 1 GeV to 2x10 16 GeV. the quark-lepton complementarity of masses is also insensitive of energy scale.

42. 42 1.QLC of the mixing angles 2. QLC of the mixing matrices 3. QLC of the masses Unification or Complementarity of Quarks and Leptons 夸克轻子的统一性或互补性 (QLC) ？

43. 43 Conclusions 1.Provide possible relations connecting quark and lepton mixing and masses. 2.Provide useful hints for model construction toward a unification of quarks and leptons. 2. Provide a general form of parametrization of lepton mixing matrix, in similar to the Wolfenstien parametrization of quark mixing matrix.

44. 44 Publications 1. N. Li and B.-Q. Ma, Phys. Lett. B 600, 248 (2004). 2. N. Li and B.-Q. Ma, Phys. Rev. D 71, 017302 (2005). 3. N. Li and B.-Q. Ma, Phys. Rev. D 71, 097301 (2005). 4. N. Li and B.-Q. Ma, Euro.Phys.J.C 42, 17 (2005). 5. N. Li and B.-Q. Ma, Phys. Lett. B 609, 309 (2005). 6. N. Li and B.-Q. Ma, Phys. Rev. D 73, 013009 (2006).