1. Speaker: Zhi-Qiang Guo Advisor: Bo-Qiang Ma School of Physics, Peking University 17 th, September, 2008

2. Plan of this talk: 1.Motivations 2.Simple introduction to the method we used and an example:fermion families from warped extra dimensions 3.Conclusions This talk is based on our recent paper: Z.-Q.Guo,B.-Q.Ma, JHEP,08(2008)065, hep-ph/0808.2136.

3. Two puzzles in particle physics: 3 generations: why fermions replicate themselves Hierarchy structure of fermion’s masses

4. Many papers about these puzzles Especially their solutions in Extra Dimension background Hierarchy structure of fermion masses The mass hierarchy in 4D originate from the small overlap of wave functions in high dimensions Integrating out the high dimensions, get the effective coupling in 4D

5. Small overlap of wave functions can induces hierarchy structure

6. Works in Warped spacetime: Slice of AdS spacetime, i.e. Rundall-Sundrum (RS) model Rundall,Sundrum PRL(1999) The metric Massive Dirac fermion Action

7. Equation of Motion (EOM) M is the bulk mass parameter Many papers in this approach Grossman,Neubert PLB(2000); Gherghetta,Pomarol NPB(2000) My talk will be based on a concise numerical examples given by Hosotani et.al, PRD(2006) Gauge-Higgs Unification Model in RS spacetime

8. The Hierarchy structure in 4D are reproduced by the bulk mass parameters of the same order in 5D Questions: the origin of the same order bulk mass parameters? Correlate with the family puzzle: one bulk mass parameter stands for a flavor in a family The purpose of our work is try to give a solution to this question.

9. Family problem in Extra Dimensions: Families in 4D from one family in high dimensions These approaches are adopted in several papers recently Frere,Libanov,Nugaev,Troitsky JHEP(2003); Aguilar,Singleton PRD (2006); Gogberashvili,Midodashvili,Singleton JHEP(2007) The main point is that fermion zero modes can be trapped by topology objiect-such as vortex; or special metric

10. Votex solution in 6D, topological number k Football-like geometry

11. One Dirac fermion in the above bachground EOM in 6D The 4D zero mode solution There exist n zero modes, n is limited by the topological number k, or the parameters in metric So families in 4D can be generated from one family in 6D

12. The idea that 1 family in high dimension can produce several families in lower dimensions can help us address the question Question: the origin of the same order bulk mass parameters? M is the bulk mass parameters

13. 2. The main point of our paper Consider a 6D spacetime with special metric: For convenience, we suppose extra dimensions are both intervals. A massive Dirac fermion in this spacetime with action and EOM:

14. Let Make conventional Kluza-Klein (KK)decompositions Note : we expand fermion field in 6D with modes in 5D, that is, at the first step, we reduce 6D spacetime to 5D.

15. We can get the following relations for each KK modes: If, that is, RS spacetime, the first equation above will equals to

16. The correspondence So the origin of the same order bulk mass parameters are of the same order Note: should be real numbers by Eq.(2.10). Further, are determined by the equations

17. For zero modes, these equations decoupled For massive modes, we can combine the first order equation to get the second order equations

18. They are 1D Schrodinger-like equations correspond to eigenvalues of a 1D Schrodinger-like equation. They are of the same order generally. So it gives a solution of the origin of the same order bulk mass parameters.

19. A problem arises from the following Contradiction: On one side : the number of eigenvalues of Schrodinger-like equation is infinite. There exists infinite eigenvalues that become larger and larger So they produce infinite families in 4D.

20. On another side: the example of Hosotani et.al means that larger bulk mass parameter produces lighter Fermion mass in 4D

21. So it needs a mechanics to cut off the infinite series and select only finite eigenvalues. It can be implemented by selecting special metric and imposing appropriate boundary conditions. We will give an example below. Before doing that, we discuss the normalization conditions and boundary conditions from the action side at first.

22. Rewriting the 6D action with 5D fermion modes, we get The conventional effective 5D action

23. There are two cases for the normalization conditions: Case (I): the orthogonal conditions We can convert these orthogonal conditions to the boundary conditions

24. Two simple choices: Case (II): the orthogonal conditions are not satisfied. Then K and M are both matrices it seems bad, because different 5D modes mixing not just among the mass terms, but also among the kinetic terms. However, we can also get conventional 5D action if there are finite 5D modes

25. It can be implemented by diagonalizing the matrices K and M. The condition is that K is positive-definite and hermitian.

26. The new eigenvalues in the action are We should check that whether they are of the same order.

27. In the following, we give an example that case II happens. Suppose a metric The 1D Schrodinger-like equations are

28. Solved by hypergeometrical functions Suppose the range of z to be When, the boundary conditions requires that

29. Then n is limited to be finite This boundary conditions determines the solutions up to the normalization constants

30. We have no freedom to impose boundary conditions at Then K and M must be matrices. We should diagonalize them to get the conventional 5D action. We give an example that only 3 families are permitted. The eigenvalues are determined to be They are of the same order.

31. Conclusions: M M2 M3 Fermion masses in 4D: Hierarchy structure C1 C2 C3 Correspondences in 5D: Same order M1 Families in 5D from 1 family in 6D Eigenvalues of 1D Schrodinger-like equatuion

32. The problem is that eigenvalues of a Schrodinger-like equation are infinite generally. We suggest a special metric and choose special boundary conditions to bypass this problem above. There also exists alternative choices: 1. discrete the sixth dimension: the differential equation will be finite difference equation, in which the number of eigenvalues are finite naturally; 2. construct non-commutative geometry structure in extra dimension, by appropriate choice of internal manifold, we can get finite KK particles. (see Madore, PRD(1995))

33. Only a rough model, far from a realistic one, may supply some hints for future model building.