1. Twisted Flavors and Tri-bimaximal Neutrino Mixing Atsushi Watanabe (Kyushu U. ) with Koichi Yoshioka (Kyushu U.), Naoyuki Haba (Tokushima U. & Munich, Tech. U.) 28/09/2006 @DESY Phys.Rev.Lett.97,041601,2006 arXiv:hep-ph/0603116

2. Contents  Introduction  Scherk-Schwarz flavor twisting  Tri-bimaximal neutrino mixing  Summary Flavor symmetry breaking by the boundary conditions on the extra dimension realistic neutrino mixings

3. Introduction  Lepton flavor mixing (3 generations) [M.C.Gonzalez-Garcia, C.Pena-Garay, ’ 03;A.Bandyopadhyay, S.Choubey, S.Goswami, S.T. Petcov, D.P. Roy, ’ 05] flavor symmetry breaking observables

4. Symmetry breaking  Vacuum expectation values of scalar fields  Structure of extra-dimensional space

5. Compactification Scherk-Schwarz compactification translation: identification of points: [Scherk and Schwarz,’79]

6. Orbifolding Boundary conditions are reflection:

7. Neutrino flavor twisting  5-dim model (for simplicity)  5-dim Dirac fermion (gauge singlet)  Other fields are confined on 4-dim [K.Dienes, E.Dudas, T.Gherghetta, ’ 99] 4-dim

8. Flavor symmetry  permutation group A simple non-abelian discrete group [S.Pakvasa, H.Sugawara, ’ 78; H.Harari, H.Haut, J.Weyers, ’ 78] 1 2 3

10. Lagrangian the mode expansion of integrating over integrating out the infinite tower of bulk neutrinos

11. Mass matrix of left-handed neutrino eigenvalueseigenvectors

12. Mass spectrum Inverted hierarchy or degenerate Typical mass scale

13. Comment on the mass eigenvalues ordinary seesaw bulk Majorana mass boundary Dirac mass conventional behavior

14. Mixing angles for neutrinos [P.F. Harrison, D.H. Perkins, W.G. Scott, ’ 02] “ tri-bimaximal mixing ”

15. symmetry breaking In general, in order to produce １． Large symmetry breaking term ２． A special type of symmetry breaking the last term is not general form of,

16. Summary  We utilize Scherk-Schwarz twist for handling flavor symmetry breaking.  Assuming as a flavor symmetry for neutrino sector, light neutrino phenomenology has rich and robust predictions such as tri-bimaximal mixing form of generation mixing. quark sector, GUT, phenomenology, …

17. and This boundary condition corresponds to two parities on

18. small

19. Charged-lepton sector triplet small mixing for left-handed direction

20. と定義すると この操作はに関する折り返しになっている 先ほどの は 両方で non-trivial にひねってみると..

21. Consistency conditions is a parity Furthermore, must satisfy and

22. Possible boundary conditions : e.g. from

23. Possible boundary conditions