Fermion Generations from

1
Fermion Generations from Apple Shaped Extra
Dimensions

• Douglas Singleton
• CSU Fresno
• QUARKS 2008
• Sergiev Posad, Russia May 23-29, 2008
• Work in collaboration with M. Gogberashvili, P.
  Midodashvili, and S. Aguilar JHEP 08(2007) 033
  PRD 73, 085007 (2006)

2
The family/generation puzzle

• Why are there three (and only three?)
  families/generations of fermions?
• Why do they have the masses that they do?
• Why do they have the CKM mixings that they do?

3
Brane world model for the family puzzle

• With a 6D brane world theory we construct a toy
  model to explain the family puzzle.
• One 6D fermion gives rise to three zero modes
  which become the three effective 4D fermions.
• Masses and mixings are given by coupling to a 6D
  scalar field.
• Unlike the cartoon there is only one brane.

4
6D metric and sources

• The 6D field equations M.Gogberashvili and
  D.Singleton, (Phys. Rev. D69, 026004 (2004) and
  ibid. (Phys. Lett., B582, 95 (2004) .
• 6D Metric ? 4D warp factor plus 2D cylindrical
  geometry. For blt1 we have rugby ball geometry
  for b1 soccer ball geometry for bgt1 we have
  apple shaped geometry.
• The 4D and 2D sources

5
Warp factors and matter sources

• The warp factor is given by
• The determinant goes to zero at the north and
  south poles. The radius of the 2D space is e.
• The solutions for the 4D and 2D matter sources
  are

6
6D Fermions

• We now place a 6D fermion into this background.
• Action
• Spinor
• Gamma Matrices
• 6D Dirac equation

7
Effective 4D Fermions

• The 6D fermion has solutions of the form
• The quantum number l is integer valued and acts
  at the family number.
• We are looking for zero mode solutions
• The 2D part of the spinors are

8
Three zero-modes

• We require the 6D fermions be normalizable
  separately in the 4D and 2D parts i.e.
• For this last integral to be convergent one needs
  ?
• In order to have only three zero modes (l-1, 0,
  1) we need 2ltb4 i.e. we need the 2D space to
  be apple shaped. For concreteness we take b4

9
6D scalar field

• These three zero mode fermions have zero mass and
  are orthogonal (i.e. do not mix) with one
  another.
• To generate masses and mixings we introduce a 6D
  scalar field.
• The solutions of the scalar field equations in
  the background are
• By introducing a coupling between scalar and
  fermion fields and integrating over the 2D space
  we find
• ?

10
Masses

• The Ull are mass (mixing) terms if ll (l?l).
  For Ull to be non-zero one needs the condition
  p-ll0.
• Explicitly these terms are
• For the mass case (ll) we can choose the
  constants so as to reproduce the down family
• The 2D radius has been set as 1/e1 TeV to push
  the non-zero modes to higher masses.

11
Mixings

• When l?l one has mixing terms between the
  different families.
• To get mixings for the first four terms above we
  need either p1. For last two terms we need
  p2. Carrying out the integrations gives
• It is possible to pick the constants such that
  one gets something like the CKM hierarchy.

12
Summary and Conclusions

• Examining a Dirac field placed in the 6D,
  nonsingular brane background one finds 3, m0
  modes for a steep enough f(r)
• The role of the family number is played by l
• Masses and Mixings are both generated by a scalar
  fermion coupling ? H??
• Can fit masses and mixings.

13
Acknowledgments

• D.S. Acknowledges a CSU Fresno CSM summer
  professional development grant.