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Localization of gravity on Higgs vortices

Localization of gravity on Higgs vortices. with B. de Carlos. Hanoi, August 7th. Jesús M. Moreno IFT Madrid. hep-th/0405144. Planning. Topological defects & extra dimensions The Higgs global string in D=6 Numerical solutions Weak and strong gravity limits A BPS system Conclusions.

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Localization of gravity on Higgs vortices

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  1. Localization of gravity on Higgs vortices with B. de Carlos Hanoi, August 7th Jesús M. Moreno IFT Madrid hep-th/0405144

  2. Planning • Topological defects & extra dimensions • The Higgs global string in D=6 • Numerical solutions • Weak and strong gravity limits • A BPS system • Conclusions

  3. d=5domain wall d=6vortex d=7monopole, d=8instanton Topological defects & extra dimensions Our D=4 world: the internal space of a topological defect living in a higher dimensional space-time Rubakov & Shaposhnikov ´83 Akama ´83 Visser ´85

  4. Gravity localized in a 3-brane DW in D=5 • Graviton´s 0-mode reproduces Newton’s gravity on the brane • Corrections from the bulk under control • Need Lbulk< 0to balance positive tension on the brane Topological defects & extra dimensions Solitons in string theory (D-branes): ideal candidates for localizing gauge and matter fields Polchinski ´95 REVIVAL: Randall and Sundrum ´99

  5. Topological defects Gravitational field in D=4 V r 0 idomain walls: regular, non static gravitational field (or non-static DW in a static Minkowski space-time) Vilenkin ´83 Ipser & Sikivie ‘84 istrings: singular metric outside the core of the defect Cohen & Kaplan’88 Gregory ‘96 (non singular when we add time-dependence) imonopoles: static, well defined metric Barriola & Vilenkin’89 V x 0 Static DW, regular strings … (e.g. SUGRA models) Cvetic et al. 93 ….

  6. The string in D= 6 Local string/vortex iCompact transverse space (trapped magnetic flux, N vortices) Wetterich’85 Gibbons & Wiltshire ‘87 Sundrum ’99, Chodos and Poppitz ’00 previous work: iNon-compact transverse space: local string (Abelian Higgs model) Gherghetta & Shaposhnikov ´00 Gherghetta , Meyer & Shaposhnikov ´01 Global string iPlain generalization to D=6 still singular Cohen & Kaplan ‘99 iHowever, introducing L< 0 cures the singularity. Analytic arguments show that, in this case, there should be a non-singular solution Gregory ’00 Gregory & Santos ‘02

  7. The global string in D= 6 i Matter lagrangian: i Global U(1) symmetry |f| iLet us analyze this system in D=6 space-time

  8. The global string in D= 6 The action for the D=6 system is given by Metric: preserving covariance in D=4 compatible with the symmetries coordinates of the transverse space M(r), L(r) warp factors and we parametrize

  9. The global string in D= 6 i Equations i Gravity trapping

  10. The global string in D= 6 Equations imn iqq irr (constraint) ieom

  11. F(0) = 0 L(0) = 0 L’(0) = 1 ( F(r) = f1 r) (no deficit angle) The global string in D= 6

  12. QUESTION: Is it possible to match BOTH regions having a regular solution that confines gravity? ANSWER: YES! but for every value of v there is a unique value of Lthat provides such solution The global string in D= 6

  13. ODE finite-difference equations (mesh of points) RELAXATION Initial guess ( 5 x N variables) Iteration Improvement The global string in D= 6 Numerical method

  14. The global string in D= 6 Boundary conditions F(0) = 0 L(0) = 0 m(0) = 0 F’(0) = 0 L’(0) = 0 In general, there will be an angle deficit L’(0) = 1 L = Lc

  15. = 0.99577V V Numerical solutions Scalar-field profile M6 =V 6 L = -0.0671 V Coincides with the calculated value (no l dependence)

  16. Numerical solutions Cigar-like space-time metric Olasagasti & Vilenkin´00 De Carlos & J.M. ‘03 Asymptotically AdS5x S1

  17. Numerical solutions Dependence on the Higgs scale

  18. Numerical solutions Uniqueness of the solution: phase space Gregory ’00 Gregory & Santos ‘02 In the asymptotic region (far from the Higgs core) autonomous dynamical system

  19. Numerical solutions Flowing towards difficult because is next to a repellor (AdS6) Only one trajectory, corresponding toLc, ends up in which can be matched to a regular solution near the core 4 fixed points

  20. Numerical solutions We find a good fit Gregory´s estimate (v a M6) Plot + fit for small v values Numerical difficulties to explore the small v region

  21. Numerical solutions Super heavy limit: (vpM6) -V(0) < Lc < 0

  22. Numerical solutions Region explored by the Higgs field in the super heavy limit

  23. Numerical solutions Is it possible to generate a large hierarchy between M6 and the D=4 Planck mass ? From the numerical solutions : the hierarchy is d a few orders of magnitue (e.g. 1000 for v = 0.7) (increases for smaller vvalues) Gregory ’00 Problem: fine tuning stability under radiative corrections

  24. A BPS system Solving second order diff. eq. can be very hard and does not give analytical insight Is it possible to define a subsystem of first order (BPS-like) differential eqs. within the second order one? Carroll, Hellerman &Trodden ‘99

  25. BPS equations A BPS system

  26. A BPS system De Carlos & J.M. ’03 EXAMPLE No cosmological constant

  27. A BPS system Etotal= Egrav+ Ekin + Epot 0

  28. Conclusions We have analyzed the Higgs global string in a D=6 space time with a negative bulkLc trapping gravity solutions For every value ofvthere is a unique value of Lcthat that provides a regular solution. The critical cosmological constant is bounded by -V(0) < Lc < 0 It is difficult to get a hierarchy betweenM6andMPlanck Fine tuning, stability

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