Fermion Generations from “Apple Fermion Generations from “Apple Shaped” Extra Dimensions*Shaped” Extra Dimensions*

Douglas SingletonDouglas SingletonCSU FresnoCSU Fresno

QUARKS 2008QUARKS 2008Sergiev Posad, Russia May 23-29, 2008Sergiev Posad, Russia May 23-29, 2008

*Work in collaboration with M. Gogberashvili, P. Midodashvili, and S. Aguilar [JHEP *Work in collaboration with M. Gogberashvili, P. Midodashvili, and S. Aguilar [JHEP 08(2007) 033; PRD 73, 085007 (2006)] 08(2007) 033; PRD 73, 085007 (2006)]

The family/generation puzzleThe family/generation puzzle

Why are there three (and only three?) Why are there three (and only three?) families/generations of fermions?families/generations of fermions?

Why do they have the masses that they do?Why do they have the masses that they do?

Why do they have the CKM mixings that they do?Why do they have the CKM mixings that they do?

Brane world model for the family puzzleBrane world model for the family puzzle

With a 6D brane world theory we construct a With a 6D brane world theory we construct a toy model to explain the family puzzle. toy model to explain the family puzzle.

One 6D fermion gives rise to three zero modes One 6D fermion gives rise to three zero modes which become the three effective 4D fermions.which become the three effective 4D fermions.

Masses Masses andand mixings are given by coupling to mixings are given by coupling to a 6D scalar field.a 6D scalar field.

Unlike the cartoon there is only one brane. Unlike the cartoon there is only one brane.

6D metric and sources6D metric and sourcesThe 6D field equations The 6D field equations [M.Gogberashvili and D.Singleton, (Phys. Rev. D69, [M.Gogberashvili and D.Singleton, (Phys. Rev. D69, 026004 (2004) and 026004 (2004) and ibid.ibid. (Phys. Lett., B582, 95 (2004) ]. (Phys. Lett., B582, 95 (2004) ].

6D Metric 6D Metric 4D warp factor plus 2D cylindrical geometry. For b<1 we 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 have rugby ball geometry; for b=1 soccer ball geometry; for b>1 we have apple shaped geometry. apple shaped geometry.

The 4D and 2D sources The 4D and 2D sources

Warp factors and matter sourcesWarp factors and matter sources The warp factor is given by:The warp factor is given by:

The determinant goes to zero at the “north “ and The determinant goes to zero at the “north “ and “south poles”. The radius of the 2D space is “south poles”. The radius of the 2D space is εε..

The solutions for the 4D and 2D matter sources are:The solutions for the 4D and 2D matter sources are:

0 .5 1 .0 1 .5 2 .0 2 .5 3 .0r

0 .2

0 .4

0 .6

0 .8

1 .0r

0 .5 1 .0 1 .5 2 .0 2 .5 3 .0r

5

1 0

1 5

2 0

2 5

3 0

1r

6D Fermions6D FermionsWe now place a 6D fermion into this background.We now place a 6D fermion into this background.

Action:Action:

Spinor:Spinor:

Gamma Matrices:Gamma Matrices:

6D Dirac equation:6D Dirac equation:

Effective 4D FermionsEffective 4D Fermions

The 6D fermion has solutions of the form:The 6D fermion has solutions of the form:

The quantum number The quantum number ll is integer valued and acts at the family number. is integer valued and acts at the family number.

We are looking for zero mode solutions: We are looking for zero mode solutions:

The 2D part of the spinors are: The 2D part of the spinors are:

Three zero-modesThree zero-modesWe require the 6D fermions be normalizable separately in the 4D and 2D parts i.e. 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 For this last integral to be convergent one needs

In order to have only three zero modes (In order to have only three zero modes (ll=-1, 0, +1) we need 2<b=-1, 0, +1) we need 2<b≤≤4 i.e. we need the 2D 4 i.e. we need the 2D space to be apple shaped. For concreteness we take b=4space to be apple shaped. For concreteness we take b=4

6D scalar field6D scalar fieldThese three zero mode fermions have zero mass and are orthogonal (i.e. do not These three zero mode fermions have zero mass and are orthogonal (i.e. do not mix) with one another.mix) with one another.

To generate masses and mixings we introduce a 6D scalar field. To generate masses and mixings we introduce a 6D scalar field.

The solutions of the scalar field equations in the background areThe solutions of the scalar field equations in the background are

By introducing a coupling between scalar and fermion fields and integrating By introducing a coupling between scalar and fermion fields and integrating over the 2D space we findover the 2D space we find

MassesMasses

The UThe Ull’ ll’ are mass (mixing) terms if are mass (mixing) terms if l=l’ l=l’ ((ll≠l’≠l’)).. For For UUll’ ll’ to be non-zero one needs the condition to be non-zero one needs the condition p-p-l+l’=l+l’=0.0.

Explicitly these terms are Explicitly these terms are

For the mass case (For the mass case (l=l’l=l’) we can choose the constants so as to reproduce the “down” family ) we can choose the constants so as to reproduce the “down” family

The 2D radius has been set as 1/The 2D radius has been set as 1/εε=1 TeV to push the non-zero modes to higher masses.=1 TeV to push the non-zero modes to higher masses.

MixingsMixings

When When ll≠l’ ≠l’ one has mixing terms between the different families. one has mixing terms between the different families.

To get mixings for the first four terms above we need either p=To get mixings for the first four terms above we need either p=±±1. For last two terms we 1. For last two terms we need p=need p=±±2.2. Carrying out the integrations gives: Carrying out the integrations gives:

It is possible to pick the constants such thatIt is possible to pick the constants such that one gets something like the CKM hierarchy.one gets something like the CKM hierarchy.

Summary and ConclusionsSummary and Conclusions

Examining a Dirac field placed in the 6D, nonsingular brane Examining a Dirac field placed in the 6D, nonsingular brane background one finds 3, m=0 modes for a steep enough background one finds 3, m=0 modes for a steep enough φφ(r)(r)

The role of the family number is played by The role of the family number is played by ll

Masses and Mixings are both generated by a scalar fermion coupling Masses and Mixings are both generated by a scalar fermion coupling H HΨΨ††ΨΨ

Can fit masses and mixings.Can fit masses and mixings.

AcknowledgmentsAcknowledgments

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