1. The effective theory of cod-two branes Claudia de Rham McMaster/PI COSMO, 21 st August 2007 arXiv:0707.0884

2. Contents 1.Cosmology on Cod-two branes? Origin of problem 2.Observer on brane 3.Hierarchy problem 4.Conclusions Sidney Harris, the American Scientist

3. Cod-2 cosmology Cod-2 branes can provide a potential resolution of the c.c. problem (cf. Osamu Seto, Masato Minamitsuji and Tsutomu Kobayashi talks)

4. Cod-2 cosmology Cod-2 branes can provide a potential resolution of the c.c. problem (cf. Osamu Seto, Masato Minamitsuji and Tsutomu Kobayashi talks) Self-tuning mechanism: brane remains flat in presence of a tension Sundrum, hep-ph/9807348

5. Cod-2 cosmology Cod-2 branes can provide a potential resolution of the c.c. problem (cf. Osamu Seto, Masato Minamitsuji and Tsutomu Kobayashi talks) Self-tuning mechanism: brane remains flat in presence of a tension SLED: If susy is unbroken in the bulk, the c.c. could be the resulting Casmir energy of two mm-size extra-dim. Aghababaie et.al., hep-th/0304256Sundrum, hep-ph/9807348

6. Matter on a cod-2 defect It has been shown by Geroch&Traschen that cod-1 or pure tension cod-2 are the only meaningful distributional sources. Arbitrary matter on cod-2 and higher distributions lead to metric divergences on the defect. The defect should be regularized. Geroch & Traschen, 1987 Cline et.al., hep-th/0304147

7. Brane Regularization In the fundamental theory, the brane is regularized the defect arises as the result of other fields (e.g. Abelian-Higgs scalar and gauge fields ,A μ ),

8. Low-energy limit At low-energy, the physics should not be highly sensitive on the regularization scheme. The resulting low-energy theory is obtained by integrating out the regularizing fields ( ,A μ ). what is the resulting EFT?

9. Propagator on the brane Free scalar field  in 6d, If at least one of the point is taken in the bulk, D is finite. But D diverges log. when both points are on the brane:

10. Brane source A priori, there is nothing special at r,r’=0. In real space, the free propagator is finite. Problem arises as soon as a physical source is introduced at r=0: Consequences for an observer on the brane?

11. Consider a brane field     can be thought as a substitute for the graviton 2. Observer on the brane deficit angle: (1-a)

12. Effects on brane fields Without coupling to , the propagator for  is affects the propagator = +  ++ … D(0,0)

13. Classical Renormalization This divergence can be absorbed, in the brane coupling: Giving rise to the classical RG flows The Green’s function of  is well- defined despite its coupling with the bulk field which is itself ill-defined.

14. Further couplings Further interactions can be considered. In particular, for relevant and marginal operators,

15. Interactions Two different kinds of divergences: Loop-divergences, that can be renormalized in the standard way. Cod-2 divergences, arising at the tree-level, requiring a second type of renormalization     3  3     i  4  3 D(0,0) 2 

16. Interactions Two different kinds of divergences: Loop-divergences, that can be renormalized in the standard way. Cod-2 divergences, arising at the tree-level, requiring a second type of renormalization    div   div  n  div

17. Interactions Two different kinds of divergences: Loop-divergences, that can be renormalized in the standard way. Cod-2 divergences, arising at the tree-level, requiring a second type of renormalization With respect to the cod-2 div, the theory is renormalizable

18. 3.Brane/Bulk hierarchy The same kind of RG flow holds for more general couplings. They can be divided into 2 classes Bulk-Brane couplings  symbolizing interaction with gravity, that do not flow (in this case) Pure brane couplings , symbolizing interactions from the SM, which typically flow as

19. Natural hierarchy Couplings from the SM behave very differently from couplings with bulk fields (ie. gravity). Couplings between brane fields are naturally logarithmically enhanced compared to couplings with bulk fields.

20. Natural hierarchy This enhancement is not sufficient to solve the hierarchy problem by itself but acts constructively to the ADD scenario (1mm-size extra dims are already ruled out…) The enhancement could be more significant for higher codimension branes.

21. Conclusions and Perspectives Distributional sources on cod-2 objects only make sense as regularized objects. Bulk fields diverge logarithmically when evaluated on the brane. An effective theory can be made sense of when brane-bulk couplings are classically renormalized. Despite being coupled to bulk fields, brane fields are finite!

22. Conclusions and Perspectives A hierarchy between SM forces and gravity arises naturally. Can we imagine a dual 4d picture without gravity? Integrating out the dual field loops could also generate the same counterterms as those at tree level.