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4. Issues of Moduli Stabilization

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4.1. Moduli fields

• Moduli fields:– characterize size and shape of extra dimensions in su

perstring theory

• Why moduli important?– Structure of extra dimensions

• gauge & Yukawa structure …– light moduli

• SUSY breaking• Cosmology

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• 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

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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)

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• 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

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　 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

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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

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• Kaehler potential:• superpotential:• Assume in Planck unit (low-energy SUSY)

　　 (Note: runaway potential if )

• Potential minimum:

• moduli mass 100 times heavier than gravitino

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• However the vacuum is SUSY AdS

• One needs to add something to obtain flat Minkowski (vanishing potential energy)

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• 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)

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Landscape

• Many possible vacua with different sets of fluxes

• No principle (so far) to single out a particular vacuum

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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

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• 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

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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 re

gion sensitive probe

• KK modes can be used as a probe to discriminate stringy compactification from RS model.

Noguchi,Yamashita&MY, 05Yamashita, in preparation

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KK modes Mass and Coupling

Coupling Universality is Violated cf. RS model

Mass spectrum is busier.

Two geometries are Distinguishable Experimentally

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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

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exponetial superpotential runaway scalar potential

How to avoid runaway?

V

Re T

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• A way to avoid runway:

add constant to superpotential

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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).

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4.3. Cosmology 　 of Moduli

• Cosmological implications:

– cosmological moduli problem

– modular inflation (not discussed here)• moduli as inflaton

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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 83de Carlos-Casas-Quevedo-Roulet 93Banks-Kaplan-Nelson 93

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Life is not that easy!– Overproduction of neutralino LSPs from modu

li decay

• efficient annihilation among LSPs is required• moduli mass >106-107 GeV (sensitive to LSP natur

e)

– Overprodution of gravitinos from moduli decay

(Moroi-MY-Yanagida 95, Kawasaki-Moroi-Yanagida 96)

Nakamura-MY 06Endo-Hamaguchi-Takahashi 06

Moduli Decay into Gravitino Pair

Lagrangian (in Planck unit)

total Kaehler potential

Nakamura-MY 06Endo,Hamaguchi,Takahashi 06

Interaction of X with gravitino bi-linear

Auxiliary field (SUSY breaking)

expectation for moduli:

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Decay amplitude

helicity ½ component

enhancement

(no such enhancement for helicity 3/2)

Decay Rate Nakamura-MY 06Endo,Hamaguchi,Takahashi 06

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REMARKSPossibile 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

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Note: Decay into other particles

e.g. decay into gauge bosons & gauginos

Note: gaugino mass is a function of moduli fielld

Nakamura-MY 06Endo-Hamaguchi-Takahashi 06Dine et al 06

Decay Rate

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Summary Sheet on Moduli Decay

total decay rate

reheat temperature

decay rate into gravitino pair

branching ratio into gravitino

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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 06Endo-Hamaguchi-Takahashi 06

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Constraints from BBN Kawasaki-Kohri

-Moroi 04

Gravitino yield from moduli decay

BBN constraint pushes gravitino heavier than ~ 105 GeV

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Constraint from LSP abundance

GravitinoLSP

– 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 (lat

er epoch)

lower bound on the gravitino mass

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LSP abundance case study: LSP=neutral Wino (largest annihilation cross section)

Gravitino mass must beheavier than ~106 GeVto escape overclosureconstraint. (wino case)

Even severer constrainton gravitino mass for other neutralino case

Low energy SUSY maybe disfavored in the presenceof moduli. (unstable gravitino)

Nakamura-MY 06

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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)

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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

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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 06Asaka-Nakamura-MY 06

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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– ….

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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

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LHC will open New Paradigm of Particle Physics.

Hope: Bottom-up & top-down approaches will meet there!!