Fermion Generations from “Apple Shaped” Extra Dimensions*

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 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

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 ε. • 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 2<b≤4 i.e. we need the 2D space to be apple shaped. For concreteness we take b=4

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 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.

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 p=±1. For last two terms we need p=±2. Carrying out the integrations gives: • It is possible to pick the constants such thatone gets something like the CKM hierarchy.

12. 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.

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