1. Moduli phenomenology and cosmology Tatsuo Kobayashi
1. Introduction
2. Towards realization of SM
3. Moduli stabilization
4. Moduli/axion inflation
5 .Summary

2. 1. Introduction String phenomenology and cosmology Our purpose is
to realize the standard model of particle physics
　 to unravel mysteries of particle physics
and cosmology from superstring theory.
Realization of the SM
not just the gauge group, three generations,
but quantitative realization such as
gauge couplings, Yukawa couplings, etc.

3. 2. Towards realization of SM
10D superstring theories
E8xE8 Hetero.
SO(32) Hetero.
Type IIA
Type IIB
Type I　　　　　　
E-series is the GUT series,
E8,E7, E6, E5(SO(10)), E4(SU(5)), E3(SM)
E8 would be interesting.
Many people started string pheno. with E8xE8 hetero.

4. 2-1. E8xE8 hetero orbifold models
Compactification 10D  4Dx6D compact space
6D Calabi-Yau
6D Toroidal orbifold
………………
We can solve string theory on orbifolds.
In principle, We can carry out perturbative
calculations.
Calculability is not everything, but can tell us
something.

5. Examples of orbifolds S1/Z2 Orbifold There are two singular points,
which are called fixed points.

6. Orbifolds T2/Z3 Orbifold There are three fixed points on Z3 orbifold
(0,0), (2/3,1/3), (1/3,2/3) su(3) root lattice
Orbifold = D-dim. Torus /twist
Torus = D-dim flat space/ lattice

7. Closed strings on orbifold
Untwisted and twisted strings
Twisted strings are associated with fixed points.
“Brane-world” terminology:
untwisted sector bulk modes
twisted sector brane (localized) modes

8. Explicit Z6-II model: Pati-Salam
T.K. Raby, Zhang ’04
4D massless spectrum
Gauge group
Chiral fields
Pati-Salam model with 3 generations + extra fields
All of extra matter fields can become massive

9. Pati-Salam
ちなみに、Pati-Salam model 10D N=1 (4D N=4) E8 SYM だけでなく、 SU(8) SYM, SO(32) SYM にも含まれる

10. 2-2. Magnetized D-brane models
Torus/orbifold with magnetic flux The number of zero-modes, i.e. the generation number, is determined by magnetic flux. Magnetic flux ⇒ non-trival zero-mode profile We know analytical form of profiles ⇒ calculability

11. Quantum mechanics： Chap. 15 in Landau-Lifshitz book
particle in magnetic flux(Landau)
Harmonic oscillator at y=k/M 　　　　　M=integer
M degenerate ground states
k=0,1,2,…………,(M-1)

12. Chiral theory
We also apply this for spinor fields.  the degenerate (chiral) zero-modes with the same quantum numbers  generation numbers of quarks and leptons

13. Wave functions
For the case of M=3
Wave function profile on toroidal background
Zero-modes wave functions are quasi-localized far away each other in extra dimensions. Therefore the hierarchirally small Yukawa couplings may be obtained.

14. U(8) D-brane models Pati-Salam group
WLs along a U(1) in U(4) and a U(1) in U(2)R
=> Standard gauge group up to U(1) factors
U(1)Y is a linear combination.

15. PS => SM Zero modes corresponding to
three families of matter fields
remain after introducing WLs, but their profiles split
(4,2,1)
Q L

16. Symmetry breaking by flux and WL
Magnetic flux ⇒　non-trivial profile
GUT multiplet
unbroken gauge boson Wilson line breaking
flat profile Q L
Broken gauge boson non-trivial profile
Coupling between Q and L through heavy bosons is suppressed Hamada,T.K., ‘12

17. Orbifold with magnetic flux
ゼロモードが、ZNの固有値で分類
　　e.g. Z2 even, odd
世代数と波動関数の変更 Z2
H.Abe, Choi, T.K. Ohki, Oikawa, Sumita, Tatsuta,　‘08-
Z2,Z3,Z4,Z6 with discrete Wilson lines
T.-h.Abe, Fujimoto, T.K., Miura, Nishikawa,
Sakamoto, Tatsuta, ’13-

18. Magnetic flux background
Similarly, we can study E8xE8 hetero. on torus/orbifolds with magnetic flux Choi, T.K., Maruyama, Murata, Nakai, Ohki, Sakai, ‘10 SO(32) hetero. on torus/orbifold with magnetic flux Abe, T.K., Otsuka, Takano, Tatsuishi, ‘15

19. (Semi) realistic spectra
Visible sector
gauge groups SU(3)xSU(2)xU(1)Y
its extensions with breaking mechanism
3 chiral generations of quarks and leptons
no chiral exotics (no chiral extra matter)
vector-like exotics (extra matter) would be OK,
because they may have mass terms.
Usually, there appear lots of singlets.
Hidden sector
several types of gauge groups, charged matter, (singlets)
at high energy scale
Some of them may confine, condensate, get masses.　　

20. Compactification for realistic spectra
Examples (Our models in these ten years)
　E8xE8 heterotic string
on orbifold T.K., Raby, Zhang, ‘04,’05
SUSY Pati-Salam model with 3 generations
on asymmetric orbifold Beye, T.K., Kuwakino, ‘14
SUSY SM with 3 generations
SO(32) heterotic string on torus with magnetic fluxes 　
(SUSY) SM Abe, T.K., Ohtsuka, Takano, Tatsuishi, ’15
type IIB magnetized D-brane models on orbifold
SUSY SM with 3 generations
Abe, Choi, T.K., Ohki, Oikawa, Sumita, Tatsuta, ‘09-
type IIA intersecting D-brane models on torus
SM with 3 generations Hamada, T.K., Uemura, ‘14

21. Visible sector Other groups also have constructed other
(semi)realistic models
There are lots of (semi)realistic models in the market
for the gauge symmetry and matter content.
I cannot count. Maybe more than O(10,000-1,000,000).
Next issue to study is qualitative realization of the SM,
values of the gauge couplings, Yukawa couplings,
Higgs potential, CP, etc.
What about the sector other than the SM,
i.e. the hidden sector

22. 2-3. 4D effective theory integrate the compact space ⇒ 4D theory
Higher dimensional Lagrangian (e.g. 10D)
integrate the compact space ⇒　4D theory
Coupling is obtained by the overlap
integral of wavefunctions

23. Yukawa couplings The Yukawa couplings are obtained by ⇒ O(1) coupling
overlap integral of their zero-mode w.f.’s.
⇒　O(1) coupling
suppressed coupling

24. Stringy computation on Yukawa couplings among localized modes
modes localized far away　　 weak coupling
nearly localized moded 　　　 strong coupling
heterotic orbifold, intersecting/magnetized
u(1) su(2) models
su(3) H
Q stringy calculation
L Y~exp(-area)
u,d

25. Quark/lepton masses and mixing angles
Abe, T.K., Sumita, Tatsuta, ‘14
Example of U(8) D-brane models
Parameters two moduli ⇒　overall coupling, ratios of Y’s
Higgs 、　　２（up） 3（down） from 5 pair higgs
Flavor is still a challenging issue.

26. Quark/lepton masses and mixing angles
Abe, T.K., Otsuka, Takano, Tatsuishi, ‘16
Example from SO(32) heterotic string thoery
Similar results
Flavor is still a challenging issue.

27. 3. Moduli stabilization 10D ⇒ 4D our space-time + 6D space 10D tensor
4D tensor + 4D vector + 4D scalars
moduli
Moduli: geometrical aspects
Couplings depend on their VEVs.
Moduli is a characteristic feature
in superstring theory on compact space.

28. Couplings Gauge coupling
Gauge couplings in 4D depend on volume of compact space　as well as dilaton.
Other couplings like Yukawa couplings also
Depend on moduli.
We need proper values of moduli in order to
realize experimental values of gauge and
Yukawa couplings.

29. Moduli Perturbatively flat potential They should have potential,
otherwise there are massless scalar fields.
Non-perturbative effects generate potential.
 　
particle physics and cosmology
　　　low scale SUSY, Inflation, etc
Those provide us with characteristic
features of superstring on a compact space.
kind of prediction

30. 3-1. Moduli stabilization
Our scenario is based on 4D N=1 supergravity,
which could be derived from type IIB string.
Flux compactification
Giddings, Kachru, Polchinski, ‘01
The dilaton S and complex structure moduli U are
assumed to be stabilized by
the flux-induced superpotential
That implies that S and U can have heavy masses of O(Mp).
The Kaher moduli T remain not stabilized.

31. Non-perturbative effects
Non-perturbative effects such as D-brane instanton effects
　and gaugino condensation
moduli-depnedent superpotential terms
D-brane instanton
gaugino condensation

32. Gaugino condensation in hidden sector
Strong coupling dynamics
　gaugino condensation
b is one-loop beta function coefficient,
e.g. b=3N for SU(N) SYM.
T corresponds to

33. KKLT V Kachru,Kallosh,Linde,Trivedi,’03
SUSY breaking uplifting by SUSY
breaking
total potential
　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　T
before uplifting
SUSY point

34. Moduli particle phenomenology
For example、low energy SUSY breaking
　　　　　-> unique spectrum of superpartners
KKLT scenario  mirage mediation
　　　　　Choi,Falkowski,Nilles,Olchowski, ‘05
TeV-scale mirage mediation
Choi, Jeong,T.K.Okumura ‘05 T.K.Makino,Okumura,Shimomura,Takahashi, ‘12
Hagimoto, T.K., Makino,Okumura,Shimomura
Still (Ms=O(1TeV))　O(10)-O(1)% fine-tuning
Cf. CMSSM O(0.1)-O(0.01)% tuning

35. Racetrack potential
Two terms due to non-perturbative effects

36. 3-2. SM + moduli stabilization
これまでの研究
　SM visible sector の構築と　moduli stabilization は、
　　　独立に研究していた。
　e.g. Visible sector D3-brane にlocalize
moduli stabilization dynamics 　　D7 brane
離れている。　　
Visible sector と　moduli stabilization を同時に
１つの具体的な模型で考えてみたい。　　　

37. SM + moduli stabilization Asymmetric orbifold
Bye, T.K., Kuwakino, ‘16
heterotic string on asymmetric orbifold　
left-mover と　right-mover を独立に割る
（６D幾何学的描像がない）
　moduli は、dilaton しかない。

38. SM + moduli stabilization Asymmetric orbifold
Bye, T.K., Kuwakino, ‘16
starting point Narain lattice before Z3 orbifolding
SU(4)7xU(1) for left (gauge), E6 for right
具体的なstring massless spectrum
　　MSSM + hidden sector (SU(4) x SU(3)xSU(3))
SU(4)xSU(3) gaugino condensation
 racetrack superpotential
dilaton stabilization
SU(3) SUSY breaking F-term uplifting
　不満足　(hidden sector matter に適当なmass を仮定)

39. SM + moduli stabilization
このタイプの模型の特徴
Hetero は、visible と　hidden の　
gauge coupling が等しい　　　　　　　SM は、２倍。
gravitino mass 　(for MSSM)
N= MeV
N= GeV
　　 N= TeV

40. SM + moduli stabilization Magnetized D-brane models
Abe, T.K., Sumita, Uemura, ’17
type IIB magnetized D9 and D7 models on
Z2 x Z2 orbifold
D9, D7 sector  Pati-Salam models
MSSM-like models
magnetic fluxes  moduli-dependent FI-term
moduli の比が固定

41. SM + moduli stabilization Magnetized D-brane models
　　　　　　　　　　　　　　　Abe, T.K., Sumita, Uemura, ‘17
type IIB magnetized D9 and D7 models on
Z2 x Z2 orbifold
　 D9, D7 sector  Pati-Salam models
MSSM-like models
Dilaton, complex structure moduli は、flux により
　　　stabilize されていると仮定
　この設定で、 可能なD-brane instanton 効果を
　　具体的に計算 (zero-mode 積分)
KKLT 型のmoduli 固定

42. D-brane instanton: neutrino mass
new zero-modes appears
they couple with
neutrinos
Neutrino masses are induced.
Blumenhage, Cvetic, Kachru, Weigand, ’06
Ibanez, Uranga, ‘06

43. SUSY breaking and uplifting
Abe, T.K., Sumita, ‘16
type IIB magnetized D9 and D7 models on
Z2 x Z2 orbifold
　 D9, D7 sector
SUSY breaking sector の具体的構築、
 visible sector + moduli stabilization sector
+ SUSY breaking (F-term uplifting) sector
Abe, T.K., Sumita, Uemura, work in progress

44. 3-3 Radiative moduli stabilization
T.K.,Omoto,Otsuka,Tatsuishi, ‘17
We assume that one U remains light, while S, T and
the other U’s are stabilized with heavy masses
by 3-form fluxes and non-perturbative effects.
The real part can be stabilized at

45. 3-3 Radiative moduli stabilization
T.K.,Omoto,Otsuka,Tatsuishi, ‘17
The imaginary part is not stabilized
because it does not appear in the potential.
The F-term
SUSY breaking depends on Im(U).
SUSY vacuum and SUSY breaking vacuum are
degenerate.

46. 3-3 Radiative moduli stabilization
T.K.,Omoto,Otsuka,Tatsuishi, ‘17
F-term is determined by Im(U), which is not
stabilized by the tree-level potential.
If U couples to the visible sector, its F-term
induces the gaugino masses and sfermion masses.
These induce one-loop potential
The explicit form depends on details of
the visible sector.

47. 3-3 Radiative moduli stabilization
T.K.,Omoto,Otsuka,Tatsuishi, ‘17
simple illustration
one set of gauginos, whose masses are
a linear function of F
one set of sfermions, whose masses squared are
The visible sector
can stabilize moduli.
SUSY or SUSY
breaking vacuum depends on the visible sector.

48. 3-3 Radiative moduli stabilization
T.K.,Omoto,Otsuka,Tatsuishi, ‘17
We assume that S and U are stabilized
with heavy masses by 3-form fluxes.
Kahler moduli have the no-scale structure
We assume that the superpotential is independent
of Kahler moudli T.
Scalar potential is independent of F-terms of T
It independent of T if DIW=0.

49. 3-3 Radiative moduli stabilization
T.K.,Omoto,Otsuka,Tatsuishi, ‘17
F-terms of T can break SUSY, but the potential
is independent of F-term.
SUSY vacuum and SUSY breaking vacuum are
degenerate.
We consider two corrections.
　No-scale structure is violated.
One-loop corrections

50. 3-3 Radiative moduli stabilization
T.K.,Omoto,Otsuka,Tatsuishi, ‘17
Illustrating potential
The visible sector can stabilize F-term of T.
The visible sector can stabilize Re(T),
but axionic parts remain massless.
SUSY or SUSY breaking vacuum depends on the
visible sector.

51. 3-3 Radiative moduli stabilization
T.K.,Omoto,Otsuka,Tatsuishi, ‘17
At this stage, axionic parts of T’s remain massless.
Non-perturbative effects below this energy scale
would generate axion potentials.
⇒　　axion inflation, dark matter,
axiverse, etc.
some interesting aspects

52. 3-3 Radiative moduli stabilization
T.K.,Omoto,Otsuka,Tatsuishi, ‘17
Explicit potential depends on
the details of the visible sector.
It is interesting to study this scenario
explicitly in concrete models.

53. 4. Moduli/axion inflation
4D low-energy effective field theory respects
some symmetries of moduli, e.g.
geometrical symmetries of compact space,
gauge symmetries of n-form gauge fields.
Then, their potential is flat.
Non-perturbative effects break such symmetries,
and generate potential.
Some symmetries still remain, e.g.
continues shift symmetry  discrete one
modular symmetry, etc.
Moduli would be good candidates for inflaton. 　

54. 4. Moduli/axion inflation
　シナリオが可能か現在研究中。
　　moduli/axion inflation のバリエーションを考える。
　何が、string theory のinflation として可能か
　将来、先のrealistic visible sector +
moduli stabilization と合わせ、１つの模型で構成する
　ことを目標 　

55. A scenario for Inflation in string theory: moduli/axion
axion (imaginary part) has a continues shift symmetry
flat potential at tree level
non-perturbative effects, gaugino condensation
superpotential
discrete shift symmetry
natural inflation
Freese, Frieman, Olinto, ‘90

56. A scenario for Inflation in string theory: modular symmetry
Circle compactification with radius R
winding number
momentum
Stringy symmetry
moduli
Stringy non-perturbative effects
(such as world-sheet instanton effects ) would appear, but
4D low-energy effective field theory may respect
the modular symmetry somehow.

57. A scenario for Inflation in string theory:
Modular invariant inflation,
T.K. Nitta, Urakawa, ‘16
Two field inflations by ReT and Im T
by assuming that the others are heavy.
Inflaton trajectory along the edge of fundamental region
　　　　　 ns is small.

58. Threshold corrections
S: dilaton, T: Kahler moduli, U: complex structure moduli
Heterotic string theory on orbifold: certain twisted sector
Dixon, Kaplunovsky, Louis, ‘91
Type IIA intersecting D-brane models: certain parallel D-brane
Lust, Stieberger, ‘07
T-dual
Type IIB D-brane models
D3/D7-branes or D5/D9-branes
gaugino condensation

59. Eta function axion inflation in Type IIB
Abe, T.K., Otsuka, arXiv:
We assume that all of the moduli are stabilized
except the axion = Im(U).
Gaugino condensation
the decay constant can be large, f> 1.

60. Poly-instanton axion inflation
T.K., Uemura, Yamamoto, arXiv:
Non-perturbative corrections on the gauge kinetic function
gaugino condensation
Poly-instanton
Blumenhagen, Schmidt-Sommerfeld, ‘08

61. Poly-instanton axion inflation
T.K., Uemura, Yamamoto, arXiv:
Axion potential
One can realize very flat potential.

62. Poly-instanton axion inflation
T.K., Uemura, Yamamoto, arXiv:
Axion potential
One can realize very flat potential.

63. Inflation scenario in superstring theory
In the market, we have lots of inflation models in superstring
theory, which are consistent with the current experiments.
Future experiments would constrain more,
but many would remain.
It would be important to examine these stringy inflation
models by using other aspects, e.g.
thermal history after inflation,
consistency with construction of the visible sector, etc.

64. After inflation
We know the couplings of moduli to gauge bosons and matter.
Through such couplings, moduli (axion) =inflaton decays
into visible and hidden matter.
We can compute the reheating temperature of the visible
sector and abundance of the hidden sector.
The couplings are non-universal between the visible
and hidden sectors.

65. After inflation
We know the couplings of moduli to gauge bosons and matter.
After inflation, the moduli (axion)=inflaton oscillates.
Couplings and their phases vary after the inflation.
That would have some effects in the history of the Universe,
e.g. Baryogenesis
Akita, T.K., Otsuka,

66. Summary We have several types of realistic models
from superstring theories through compactifications,
SU(3)xSU(2)xU(1)Y and three generations of
quarks and leptons.
Next issues is to realize qualitatively particle physics,
e.g. Yukawa couplings.
Also it is important to study cosmological aspects.

67. Summary There are several mechanisms for moduli stabilization
and lots of inflation models.
The search of realistic spectra and study on
the inflation sector and moduli stabilization
have been studied separately.
It is important to study them in one concrete model
explicitly.