4. Issues of Moduli
Stabilization
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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
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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
Gukov-Vafa-Witten ’99
(3,0) form
• complex moduli
• quantization of fluxes
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 superpotential for complex moduli (z) and dilaton (t)
• 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-KachruPolchinski 02
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KKLT set-up
Kachru-Kallosh-Linde-Trivedi 03
• 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
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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
Kallosh-Linde
Balance among the three terms 
– Gravitino mass can be arbitrarily small.
– different scenario from mirage mediation
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KK modes of Warped String
compactification
Noguchi,Yamashita&MY, 05
Yamashita, in preparation
• 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
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RS model.
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
V
Re T
How to avoid runaway?
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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
Coughlan et al 83
de Carlos-Casas-Quevedo-Roulet 93
Banks-Kaplan-Nelson 93
• 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)
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Life is not that easy!
– Overproduction of neutralino LSPs from
moduli decay
(Moroi-MY-Yanagida 95,
Kawasaki-Moroi-Yanagida 96)
• efficient annihilation among LSPs is required
• moduli mass >106-107 GeV (sensitive to LSP
nature)
– Overprodution of gravitinos from moduli decay
Nakamura-MY 06
Endo-Hamaguchi-Takahashi 06
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Moduli Decay into Gravitino Pair
Lagrangian (in Planck unit)
Nakamura-MY 06
Endo,Hamaguchi,Takahashi 06
total Kaehler potential
Interaction of X with gravitino bi-linear
Auxiliary field (SUSY breaking)
expectation for moduli:
Decay amplitude
helicity ½ component
 enhancement
(no such enhancement for helicity 3/2)
Decay Rate
Nakamura-MY 06
Endo,Hamaguchi,Takahashi 06
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REMARKS
Possibile mixing with SUSY breaking field (Polonyi field) Z
couples to gravitino pair.
scalar partner of goldstino
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.
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We do not consider this possible suppression.
Note: Decay into other particles
e.g. decay into gauge bosons & gauginos
Note: gaugino mass is a function of moduli fielld
Decay Rate
Nakamura-MY 06
Endo-Hamaguchi
-Takahashi 06
Dine et al 06
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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
Namakura-MY 06
Scenario
Endo-Hamaguchi
-Takahashi 06
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)
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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
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
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LSP abundance
Nakamura-MY 06
case study: LSP=neutral Wino
(largest annihilation cross section)
Gravitino mass must be
heavier than ~106 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)
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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
Asaka-MY in preparation
Introduction of Singlet S
(a la deflected anomaly mediation) Pomaral-Rattazzi
Solution to m-Bm 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
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REMARK:
Gravitino Production from Inflaton Decay
Kawasaki-Takahashi-Yanagida 06
Asaka-Nakamura-MY 06
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
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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!!
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