Fermion Families From Two Layer Warped Extra Dimensions

1
Fermion Families From Two Layer WarpedExtra
Dimensions

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

2

• Plan of this talk
• 1.Motivations
• 2.Simple introduction to the method we used
• and an examplefermion 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 fermions 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,Midoda
  shvili,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

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

Thank you!