1. 1 4. Issues of Moduli Stabilization

2. 2 4.1. Moduli fields Moduli fields: –characterize size and shape of extra dimensions in superstring theory Why moduli important? –Structure of extra dimensions gauge & Yukawa structure … –light moduli SUSY breaking Cosmology

3. 3 Moduli stabilization –long standing problem Flux compactifications –most of moduli are stabilized –new and important insights KKLT set-up –simple and interesting framework phenomenology & cosmological studies initiated

4. 4 4.2. Flux Compactifications Moduli: Difficulty in generating potentials for moduli –known mechanism: very limited gaugino condensate worldsheet instantons Recent developments –Calabi-Yau compactifications with fluxes (type IIB superstring theory)

5. 5 IIB superstring –2-form potential (NS-NS, RR) –3-form field strength superpotential for IIB theory complex moduli quantization of fluxes (3,0) form Gukov-Vafa-Witten ’99

6. 6  superpotential for complex moduli (z) and dilaton (  ) Consistent solution for flux compactifications in IIB –fluxes  warped throat –W stabilizes complex moduli as well as dilaton –Kaehler moduli are not stabilized by fluxes Giddinge-Kachru- Polchinski 02

7. 7 KKLT set-up Potential for Kaehler moduli:  non-perturbative effects e.g. gaugino condensate on D7 brane case of single overall moduli: gauge kinetic function on D7: superpotential from gaugino condensate Kachru-Kallosh-Linde-Trivedi 03

8. 8 Kaehler potential: superpotential: Assume in Planck unit (low-energy SUSY) (Note: runaway potential if ) Potential minimum: moduli mass  100 times heavier than gravitino

9. 9 However the vacuum is SUSY AdS One needs to add something to obtain flat Minkowski (vanishing potential energy)

10. 10 Up-lifting of the scalar potential –KKLT: anti-D3 on top of warped throat (Dynamical SUSY breaking sector on D-branes may also be OK)

11. 11 Landscape Many possible vacua with different sets of fluxes No principle (so far) to single out a particular vacuum

12. 12 Many possibilities to phenomenology KKLT set-up –with small constant term in superpotential –SM brane localized far from warped throat  low-energy SUSY with mirage mediation

13. 13 KKLT set-up –with large constant term in W –with SM brane at the top of warped throat  warped extra dimensions a la RS model Many more scenarios Balance among the three terms  –Gravitino mass can be arbitrarily small. –different scenario from mirage mediation Kallosh-Linde

14. 14 KK modes of Warped String compactification Geometry of warped throat: –Klebanov-Strassler geometry –very different from AdS5 in infra-red region –KK modes of graviton: localized at infra-red region  sensitive probe KK modes can be used as a probe to discriminate stringy compactification from RS model. Noguchi,Yamashita&MY, 05 Yamashita, in preparation

15. 15 KK modes Mass and Coupling Coupling Universality is Violated cf. RS model Mass spectrum is busier. Two geometries are Distinguishable Experimentally

16. 16 Small SUSY breaking scale ? Once we have low-energy SUSY, then EW scale is protected from radiative corrections. Q. Can we obtain small SUSY breaking scale? A. promising way: dimensional transmutation in string theory: gauge coupling is dynamical

17. 17 exponetial superpotential  runaway scalar potential How to avoid runaway? V Re T

18. 18 A way to avoid runway: add constant to superpotential

19. 19 flux compactification in type IIB superstring Small SUSY breaking scale  small w_0 cancellation among big numbers??? Bousso-Polchinski At this moment we do not yet understand the ultimate reason of why SUSY breaking scale is much smaller than the fundamental scale (even if low-energy SUSY is correct).

20. 20 4.3. Cosmology of Moduli Cosmological implications: –cosmological moduli problem –modular inflation (not discussed here) moduli as inflaton

21. 21 Cosmological Moduli Problem Coherent oscillation of moduli fields would dominate the energy density of the universe. Late decay  reheating of the universe  disaster for big-bang nucleosynthesis (BBN) Hope: may be OK if moduli is heavy (Moroi-MY-Yanadiga 95) Coughlan et al 83 de Carlos-Casas-Quevedo-Roulet 93 Banks-Kaplan-Nelson 93

22. 22 Life is not that easy! –Overproduction of neutralino LSPs from moduli decay efficient annihilation among LSPs is required moduli mass >10 6 -10 7 GeV (sensitive to LSP nature) –Overprodution of gravitinos from moduli decay (Moroi-MY-Yanagida 95, Kawasaki-Moroi-Yanagida 96) Nakamura-MY 06 Endo-Hamaguchi-Takahashi 06

23. Moduli Decay into Gravitino Pair Lagrangian (in Planck unit) total Kaehler potential Nakamura-MY 06 Endo,Hamaguchi,Takahashi 06 Interaction of X with gravitino bi-linear Auxiliary field (SUSY breaking) expectation for moduli:

24. 24 Decay amplitude helicity ½ component  enhancement (no such enhancement for helicity 3/2) Decay Rate Nakamura-MY 06 Endo,Hamaguchi,Takahashi 06

25. 25 REMARKS Possibile mixing with SUSY breaking field (Polonyi field) Z Mass diagonalization is needed. Coupling of heavy “X” field with gravitino is suppressed if 1) absence of certain coupling (e.g. ZZX*) 2) sufficiently light Z These conditions are easily violated. Furthermore, light Z would cause conventional moduli (or Polony) problem. We do not consider this possible suppression. Dine-Kitano- Morisse-Shirman 06 couples to gravitino pair. scalar partner of goldstino

26. 26 Note: Decay into other particles e.g. decay into gauge bosons & gauginos Note: gaugino mass is a function of moduli fielld Nakamura-MY 06 Endo-Hamaguchi -Takahashi 06 Dine et al 06 Decay Rate

27. 27 Summary Sheet on Moduli Decay total decay rate  reheat temperature decay rate into gravitino pair  branching ratio into gravitino

28. 28 Cosmology of Heavy Moduli Scenario Consider the case: Planck mass >> moduli mass>> gravitino mass >> soft masses Assume that the LSP is a neutralino. Moduli decay into  SM particles  reheating  SM sparticles  LSPs  gravitinos  hadronic/EM showers: BBN  LSPs Constraints: 1) BBN constraint on gravitino decay 2) Overclosure (LSP overproduction) Namakura-MY 06 Endo-Hamaguchi -Takahashi 06

29. 29 Constraints from BBN Kawasaki-Kohri -Moroi 04 Gravitino yield from moduli decay  BBN constraint pushes gravitino heavier than ~ 10 5 GeV

30. 30 Constraint from LSP abundance Gravitino  LSP –Overabundance of the LSPs Relic abundance of the LSPs –LSPs are produced by gravitino decay –Annihilation of LSPs –Annihilation is not very efficient at low temperature (later epoch)  lower bound on the gravitino mass

31. 31 LSP abundance case study: LSP=neutral Wino (largest annihilation cross section) Gravitino mass must be heavier than ~10 6 GeV to escape overclosure constraint. (wino case) Even severer constraint on gravitino mass for other neutralino case Low energy SUSY may be disfavored in the presence of moduli. (unstable gravitino) Nakamura-MY 06

32. 32 Ways Out? lighter LSP (such as axino) –how to realize lighter LSP? in particular within mirage mediation Stable Gravitino: (Asaka,Nakamura&MY 06) –Constraint on gravitino relic abundance: less constrained –possibility of gravitino warm dark matter –give up mirage mediation? Dilution by entropy production –thermal inflation (Lyth & Stewart 96)

33. 33 Thermal inflation in mirage mediation Introduction of Singlet S (a la deflected anomaly mediation) Solution to  -B  problem in MSSM S field: very flat potential with TeV mass  flaton: Thermal inflation occurs - Dilutes the primordial moduli - Neutralino Dark matter: may come from the flaton decay, or moduli/graviitno decay Asaka-MY in preparation Pomaral-Rattazzi

34. 34 REMARK: Gravitino Production from Inflaton Decay Inflaton decay into gravitino pair: proportional to Fx: model dependent modular inflation: (inflaton=moduli) severely constrained other inflation scenarios (chaotic inflation/new inflation/hybrid inflation) – somewhat model dependent (VEV of Fx) –very severe constraints on inflationary scenario Kawasaki-Takahashi-Yanagida 06 Asaka-Nakamura-MY 06

35. 35 5. Conclusions Standard Model –good fit with electroweak data –flavor: described by CKM –Higgs (or something else) will be found near future. Motivations for Beyond Stnadard Model –cosmology –neutrino oscillations –naturalness problem with EW scale –….

36. 36 FCNC  mediation mechanisms of SUSY breaking –gravity mediation –gauge mediation –anomaly mediation –mirage mediation  different phenomenology and cosmology Moduli Stabilization –flux compactification –landscape? –many solutions to naturalness problem –KKLT type set up? or something else

37. 37 LHC will open New Paradigm of Particle Physics. Hope: Bottom-up & top-down approaches will meet there!!