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)]
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?
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.
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 b<1 we have
rugby ball geometry; for b=1 soccer ball geometry; for b>1 we have apple
shaped geometry.
The 4D and 2D sources
Warp factors and matter sources
r
The warp factor is given by:
1.0
0.8
0.6
The determinant goes to zero at the “north “ and
“south poles”. The radius of the 2D space is ε.
0.4
0.2
0.5
1.0
1.5
2.0
2.5
3.0
0.5
1.0
1.5
2.0
2.5
3.0
r
The solutions for the 4D and 2D matter sources are:
1
r
30
25
20
15
10
5
r
6D Fermions
We now place a 6D fermion into this background.
Action:
Spinor:
Gamma Matrices:
6D Dirac equation:
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:
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 2<b≤4 i.e. we need the 2D
space to be apple shaped. For concreteness we take b=4
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

Masses
The Ull’ are mass (mixing) terms if l=l’ (l≠l’). For Ull’ to be non-zero one needs the condition
p-l+l’=0.
Explicitly these terms are
For the mass case (l=l’) we can choose the constants so as to reproduce the “down” family
The 2D radius has been set as 1/ε=1 TeV to push the non-zero modes to higher masses.
Mixings
When l≠l’ one has mixing terms between the different families.
To get mixings for the first four terms above we need either p=±1. For last two terms we
need p=±2. Carrying out the integrations gives:
It is possible to pick the constants such that one gets something like the CKM hierarchy.
Summary and Conclusions
Examining a Dirac field placed in the 6D, nonsingular brane
background one finds 3, m=0 modes for a steep enough φ(r)
The role of the family number is played by l
Masses and Mixings are both generated by a scalar fermion coupling
 HΨ†Ψ
Can fit masses and mixings.
Acknowledgments
D.S. Acknowledges a CSU Fresno CSM
summer professional development grant.