Papers - Higaki, Tetsutaro

Spontaneous CP violation and partially broken modular flavor symmetries

Tetsutaro Higaki, Tatsuo Kobayashi, Kaito Nasu, Hajime Otsuka

 2024.05

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We study the realization of spontaneous CP violation through moduli
stabilization. In modular flavor models, the source of CP violation is the
vacuum expectation values of the complex structure moduli of toroidal compact
space. We demonstrate that the combined effects of Type IIB flux
compactifications with modular invariant couplings between the moduli and
matter fields can induce spontaneous CP violation without or with supersymmetry
breaking. Furthermore, some general properties of CP and modular invariant
scalar potentials are presented. It is found that certain modifications or
partial breakings of modular symmetry are useful in generating spontaneous CP
violation.


Moduli stabilization in finite modular symmetric models

Yoshihiko Abe, Komei Goto, Testutaro Higaki, Tatsuo Kobayashi, Kaito Nasu

 2024.05

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We study vacua of moduli potential consisting of multiple contribution of modular forms in a finite modular symmetry. If the potential is given by a single modular form, the Minkowski vacuum is realized at the fixed point of the modular symmetry. We show that de Sitter vacuum is realized with a multiple modular form case and obtain a non-trivial vacuum which is away from the fixed point, i.e. a large modulus vacuum expectation value, depending on the choice of the weight and representation of the modular forms. We study these vacua by a numerical and analytically. It is also found that vacua obtained in this paper preserve CP symmetry.

Finite modular axion and radiative moduli stabilization

Tetsutaro Higaki, Junichiro Kawamura, Tatsuo Kobayashi

Journal of High Energy Physics (Journal of High Energy Physics)  2024 ( 4 )  2024.02

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We propose a simple setup which can stabilize a modulus field of the finite
modular symmetry by the Coleman-Weinberg potential. Our scenario leads to a
large hierarchy suppressing instanton-like corrections $e^{2\pi i\tau}$ and to
a light axion identified as $\mathrm{Re} \tau$, where $\tau$ is the modulus
field. This stabilization mechanism provides the axion solution to the strong
CP problem. The potential has a minimum at a large $\mathrm{Im}\tau$ which
suppresses explicit $U(1)_{\mathrm{PQ}}$ violation terms proportional to
$e^{-2\pi {\mathrm{Im}\tau}}$, and hence the quality of the axion is ensured by
the residual symmetry associated with the $T$-transformation, $\tau \to \tau
+1$, around the fixed point $\tau \sim i\infty$.


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Fermion Hierarchies in SU(5) Grand Unification from Γ6' Modular Flavor Symmetry

Yoshihiko Abe, Tetsutaro Higaki, Junichiro Kawamura, Tatsuo Kobayashi

Journal of High Energy Physics (Springer Science+Business Media)  08 ( 097 ) 1 - 25 2023.08

Research paper (scientific journal), Accepted

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We construct a model in which the hierarchies of the quark and lepton masses and mixing are explained by the Γ′6 modular flavor symmetry. The hierarchies are realized by the Froggatt-Nielsen-like mechanism due to the residual ZT6 symmetry, approximately unbroken at τ∼i∞. We argue that the Γ(′)6 symmetry is the minimal possibility to realize the up-type quark mass hierarchies, since the Yukawa matrix is symmetric. We find a combination of the representations and modular weights and then show numerical values of O(1) coefficients for the realistic fermion hierarchies.

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Quark and lepton hierarchies from S4' modular flavor symmetry

Yoshihiko Abe, Tetsutaro Higaki, Junichiro Kawamura, Tatsuo Kobayashi

Physics Letters B (Elsevier)  842 ( 137977 ) 1 - 9 2023.07

Research paper (scientific journal), Accepted

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We propose models in which the hierarchical structures of the masses and
mixing in both quark and lepton sectors are explained by the $S_4^\prime$
modular flavor symmetry near the fixed point $\tau \sim i\infty$. The model
provides the first explicit example which explains hierarchies of both quarks
and leptons. The hierarchies are realized by powers of $\epsilon = e^{2\pi i
\tau/4} = \mathcal{O}(0.01)$ and $2\,\mathrm{Im}\,\tau \sim 5$, where $\tau$
being the modulus. The small parameter $\epsilon$ plays a role of flavon in the
Froggatt-Nielsen mechanism under the residual $Z_4^T$ symmetry, and powers of
$2\,\mathrm{Im}\,\tau$ in the Yukawa couplings are controlled by modular
weights via the canonical normalization. The doublet quarks are identified to a
$S_4^\prime$ triplet to explain the hierarchical structure of the quark mixing
angles, while the doublet leptons are composed of three singlets for the large
mixing angles in the lepton sector. We show that the $S_4^\prime$ modular
symmetry alone can explain the hierarchies in both quark and lepton sectors by
$\mathcal{O}(1)$ coefficients.


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Moduli inflation from modular flavor symmetries

Yoshihiko Abe, Tetsutaro Higaki, Fumiya Kaneko, Tatsuo Kobayashi, Hajime Otsuka

Journal of High Energy Physics (Springer Science+Business Media)  6 ( 187 ) 1 - 39 2023.06

Research paper (scientific journal), Joint Work, Accepted

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We study slow-roll inflation model controlled by the modular flavor symmetry.
In the model, the modulus field plays a role of inflaton and the introduction
of the stabilizer field coupled to a modular form in the superpotential
produces the inflaton potential. In order to generate the flat direction for
the slow-roll inflation, we consider the K\"{a}hler potential corrected by the
modular form. It is noted that the modulus field perpendicular to the inflaton
direction is stabilized during the inflation. The model is turns out to be
consistent with the current observations and behaves similarly to the
$\alpha$-attractor models in some parameter spaces. The inflaton rolls down to
the CP-symmetric vacuum at the end of inflation.


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Quark masses and CKM hierarchies from S4′ modular flavor symmetry

Yoshihiko Abe, Tetsutaro Higaki, Junichiro Kawamurab, Tatsuo Kobayashi

e-Print: 2301.07439 [hep-ph] (European Physical Journal C)  83 ( 12 )  2023.01

Research paper (other academic), Joint Work,  ISSN  14346044

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We propose models to explain the hierarchies of the quark masses and mixing by utilizing the S4′ modular flavor symmetry. The hierarchy is realized by the modulus τ stabilized at Imτ≫1 , where the residual Z4T symmetry is approximately unbroken and the Froggatt–Nielsen mechanism works. It is found that the quark hierarchies are realized only in a few cases of quark representations. We study two models with assigning the modular weights, so that the observed quark hierarchies are explained in the cases of both small and large ratios of the top to bottom Yukawa couplings. We also argue that O(0.1) hierarchies of the O(1) coefficients and the spontaneous CP violation can be realized by imposing another S3 modular symmetry.

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Formation of Chiral Soliton Lattice

Tetsutaro Higaki, Kohei Kamada, Kentaro Nishimura

Physical Review D (American Physical Society)  106 ( 096022 )  2022.11

Research paper (scientific journal), Accepted

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The Chiral Soliton Lattice (CSL) is a lattice structure composed of domain
walls aligned in parallel at equal intervals, which is energetically stable in
the presence of a background magnetic field and a finite (baryon) chemical
potential due to the topological term originated from the chiral anomaly. We
study its formation from the vacuum state, with describing the CSL as a layer
of domain-wall disks surrounded by the vortex or string loop, based on the
Nambu-Goto-type effective theory. We show that the domain wall nucleates via
quantum tunneling when the magnetic field is strong enough. We evaluate its
nucleation rate and determine the critical magnetic field strength with which
the nucleation rate is no longer exponentially suppressed. We apply this
analysis to the neutral pion in the two-flavor QCD as well as the axion-like
particles (ALPs) with a finite (baryon) chemical potential under an external
magnetic field. In the former case, even though the CSL state is more
energetically stable than the vacuum state and the nucleation rate becomes
larger for sufficiently strong magnetic field, it cannot be large enough so
that the nucleation of the domain walls is not exponentially suppressed and
promoted, without suffering from the tachyonic instability of the charged pion
fluctuations. In the latter case, we confirm that the effective interaction of
the ALPs generically includes the topological term required for the CSL state
to be energetically favored. We show that the ALP CSL formation is promoted if
the magnetic field strength and the chemical potential of the system is
slightly larger than the scale of the axion decay constant.


4D effective action from non-Abelian DBI action with magnetic flux background

Yoshihiko Abe, Tetsutaro Higaki, Tatsuo Kobayashi, Shintaro Takada, Rei Takahashi

Physical Review D (American Physical Society)  104 ( 12 ) 126020-1 - 126020-15 2021.12

Research paper (other academic), Joint Work, Accepted,  ISSN  24700010

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We study a systematic derivation of four-dimensional N=1 supersymmetric effective theory from a ten-dimensional non-Abelian Dirac-Born-Infeld action compactified on a six-dimensional torus with magnetic fluxes on the D-branes. We find a new type of matter Kähler metric, while gauge kinetic function and superpotential are consistent with previous studies. For the ten-dimensional action, we use a symmetrized trace prescription and focus on the bosonic part up to O(F4). In the presence of the supersymmetry, four-dimensional chiral fermions can be obtained via index theorem. The new matter Kähler metric is independent of flavor but depends on the fluxes, 4D dilaton, Kähler moduli, and complex structure moduli and will be always positive definite if an induced Ramond-Ramond charge of the D-branes on which matters are living are positive. We read the superpotential from an F-term scalar quartic interaction derived from the ten-dimensional action and the contribution of the new matter Kähler metric to the scalar potential which we derive turns out to be consistent with the supergravity formulation.

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Implications of the weak gravity conjecture in anomalous quiver gauge theories

Yoshihiko Abe, Tetsutaro Higaki, Rei Takahashi

Physical Review D (American Physical Society)  102 ( 065004 )  2020.09

Research paper (scientific journal), Joint Work, Accepted,  ISSN  24700010

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© 2020 authors. Published by the American Physical Society. We argue a smallness of gauge couplings in Abelian quiver gauge theories, taking the anomaly cancellation condition into account. In theories of our interest there exist chiral fermions leading to chiral gauge anomalies, and an anomaly-free gauge coupling tends to be small, and hence can give a nontrivial condition of the weak gravity conjecture. As concrete examples, we consider U(1)k gauge theories with a discrete symmetry associated with cyclic permutations between the gauge groups, and identify anomaly-free U(1) gauge symmetries and the corresponding gauge couplings. Owing to this discrete symmetry, we can systematically study the models and we find that the models would be examples of the weak coupling conjecture. It is conjectured that a certain class of chiral gauge theories with too many U(1) symmetries may be in the swampland. We also numerically study constraints on the couplings from the scalar weak gravity conjecture in a concrete model. These constraints may have a phenomenological implication to model building of a chiral hidden sector as well as the visible sector.

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Note on a solution to domain wall problem with the Lazarides-Shafi mechanism in axion dark matter models

Chatterjee C., Higaki T., Nitta M.

Physical Review D (Physical Review D)  101 ( 7 )  2020.04

ISSN  24700010

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© 2020 authors. Published by the American Physical Society. Axion is a promising candidate of dark matter. After the Peccei-Quinn symmetry breaking, axion strings are formed and attached by domain walls when the temperature of the universe becomes comparable to the QCD scale. Such objects can cause cosmological disasters if they are long lived. As a solution for it, the Lazarides-Shafi mechanism is often discussed through introduction of a new non-Abelian (gauge) symmetry. We study this mechanism in detail and show configuration of strings and walls. Even if Abelian axion strings with a domain wall number greater than 1 are formed in the early universe, each of them is split into multiple Alice axion strings due to a repulsive force between the Alice strings even without domain wall. When domain walls are formed as the universe cools down, a single Alice string can be attached by a single wall because a vacuum is connected by a non-Abelian rotation without changing energy. Even if an Abelian axion string attached by domain walls is created due to the Kibble Zurek mechanism at the chiral phase transition, such strings are also similarly split into multiple Alice strings attached by walls in the presence of the domain wall tension. Such walls do not form stable networks since they collapse by the tension of the walls, emitting axions.

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A low-scale flavon model with a ℤ_N symmetry

Higaki T., Kawamura J.

Journal of High Energy Physics (Journal of High Energy Physics)  2020 ( 3 )  2020.03

ISSN  11266708

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© 2020, The Author(s). We propose a model that explains the fermion mass hierarchy by the Froggatt-Nielsen mechanism with a discrete ℤNF symmetry. As a concrete model, we study a super-symmetric model with a single flavon coupled to the minimal supersymmetric Standard Model. Flavon develops a TeV scale vacuum expectation value for realizing flavor hierarchy, an appropriate μ-term and the electroweak scale, hence the model has a low cutoff scale. We demonstrate how the flavon is successfully stabilized together with the Higgs bosons in the model. The discrete flavor symmetry ℤNF controls not only the Standard Model fermion masses, but also the Higgs potential and a mass of the Higgsino which is a good candidate for dark matter. The hierarchy in the Higgs-flavon sector is determined in order to make the model anomaly-free and realize a stable electroweak vacuum. We show that this model can explain the fermion mass hierarchy, realistic Higgs-flavon potential and thermally produced dark matter at the same time. We discuss flavor violating processes induced by the light flavon which would be detected in future experiments.

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Flavor structure, Higgs boson mass and dark matter in supersymmetric model with vector-like generations

Tetsutaro Higaki, Michinobu Nishida, Naoyuki Takeda

Progress of Theoretical and Experimental Physics 2017 ( 8 ) 1 - 36 2017.08

Research paper (scientific journal), Joint Work, Accepted

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Inflation from periodic extra dimensions

Tetsutaro Higaki, Yoshiyuki Tatsuta

Journal of Cosmology and Astroparticle Physics 2017 ( 07 ) 1 - 18 2017.07

Research paper (scientific journal), Joint Work, Accepted

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Cosmological relaxation and high scale inflation

Tetsutaro Higaki, Naoyuki Takeda, Yusuke Yamada

Physical Review D 95 ( 015009 )  2017.01

Research paper (scientific journal), Joint Work

Abelian tensor hierarchy in 4D N=1 conformal supergravity

Shuntaro Aoki, Tetsutaro Higaki, Yusuke Yamada, Ryo Yokokura

Journal of High Energy Physics 1609 ( 148 )  2016.09

Research paper (scientific journal), Joint Work

Topological Defects and nano-Hz Gravitational Waves in Aligned Axion Models

Tetsutaro Higaki, Kwang Sik Jeong, Naoya Kitajima, Toyokazu Sekiguchi, Fuminobu Takahashi

Journal of High Energy Physics 1608 ( 044 )  2016.08

Research paper (scientific journal), Joint Work

Quality of the Peccei-Quinn symmetry in the Aligned QCD Axion and Cosmological Implications

Tetsutaro Higaki, Kwang Sik Jeong, Naoya Kitajima, Fuminobu Takahashi

Journal of High Energy Physics 1606 ( 150 )  2016.06

Research paper (scientific journal), Joint Work

QCD Axion from Aligned Axions and Diphoton Excess

Tetsutaro Higaki, Kwang Sik Jeong, Naoya Kitajima, Fuminobu Takahashi

Physics Letters B (Elsevier)  755   13 - 16 2016.04

Research paper (scientific journal), Joint Work, Accepted

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Axion Landscape and Natural Inflation

Tetsutaro Higaki, Fuminobu Takahashi

Physics Letters B (Elsevier)  744   153 - 159 2015.05

Research paper (scientific journal), Joint Work, Accepted

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Elliptic Inflation: Interpolating from natural inflation to R2-inflation

Tetsutaro Higaki, Fuminobu Takahashi

Journal of High Energy Physics (Springer)  129 ( 03 )  2015.03

Research paper (scientific journal), Joint Work, Accepted

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Hidden axion dark matter decaying through mixing with QCD axion and the 3.5 keV X-ray line

Tetsutaro Higaki, Naoya Kitajima, Fuminobu Takahashi

Journal of Cosmology and Astroparticle Physics (IOPscience)  12 ( 004 )  2014.12

Research paper (scientific journal), Joint Work, Accepted

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Axion monodromy inflation with multi-natural modulations

Tetsutaro Higaki, Tatsuo Kobayashi, Osamu Seto, Yuya Yamaguchi

Journal of Cosmology and Astroparticle Physics (IOP Publishing)  10 ( 025 )  2014.10

Research paper (scientific journal), Joint Work, Accepted

Affleck-Dine baryogenesis with R-parity violation

Tetsutaro Higaki, Kazunori Nakayama, Ken'ichi Saikawa, Tomo Takahashi, Masahide Yamaguchi

Physical Review D (American Physical Society)  90 ( 4 ) 045001 2014.08

Research paper (scientific journal), Joint Work, Accepted

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Natural and Multi-Natural Inflation in Axion Landscape

Tetsutaro Higaki, Fuminobu Takahashi

Journal of High Energy Physics (Springer)  07 ( 074 )  2014.07

Research paper (scientific journal), Joint Work, Accepted

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Neutrinoful Universe

Tetsutaro Higaki, Ryuichiro Kitano, Ryosuke Sato

Journal of High Energy Physics (Springer)  07 ( 044 )  2014.07

Research paper (scientific journal), Joint Work, Accepted

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Multi-Natural Inflation in Supergravity and BICEP2

Michael Czerny, Tetsutaro Higaki, Fuminobu Takahashi

Physics Letters B (Elsevier)  734   167 - 172 2014.06

Research paper (scientific journal), Joint Work, Accepted

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Solving the Tension between High-Scale Inflation and Axion Isocurvature Perturbations

Tetsutaro Higaki, Kwang Sik Jeong, Fuminobu Takahashi

Physics Letters B (Elsevier)  734   21 - 26 2014.06

Research paper (scientific journal), Joint Work, Accepted

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The 7 keV axion dark matter and the X-ray line signal

Tetsutaro Higaki, Kwang Sik Jeong, Fuminobu Takahashi

Physics Letters B (Elsevier)  733   25 - 31 2014.06

Research paper (scientific journal), Joint Work, Accepted

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Multi-Natural Inflation in Supergravity

Michael Czerny, Tetsutaro Higaki, Fuminobu Takahashi

Journal of High Energy Physics (Springer)  05 ( 144 )  2014.05

Research paper (scientific journal), Joint Work, Accepted

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Cosmological constraints on axionic dark radiation from axion-photon conversion in the early Universe

Tetsutaro Higaki, Kazunori Nakayama, Fuminobu Takahashi

Journal of Cosmology and Astroparticle Physics (IOP Publishing)  09 ( 030 )  2013.09

Research paper (scientific journal), Joint Work, Accepted

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A Parallel World in the Dark

Tetsutaro Higaki, Kwang Sik Jeong, Fuminobu Takahashi

Journal of Cosmology and Astroparticle Physics (IOP Publishing)  08 ( 031 )  2013.08

Research paper (scientific journal), Joint Work, Accepted

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Moduli-Induced Axion Problem

Tetsutaro Higaki, Kazunori Nakayama, Fuminobu Takahashi

Journal of High Energy Physics (Springer)  07 ( 005 )  2013.07

Research paper (scientific journal), Joint Work, Accepted

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Hybrid inflation in high-scale supersymmetry

Tetsutaro Higaki, Kwang Sik Jeong, Fuminobu Takahashi

Journal of High Energy Physics (Springer)  12 ( 111 )  2012.12

Research paper (scientific journal), Joint Work, Accepted

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Dark Radiation and Dark Matter in Large Volume Compactifications

Tetsutaro Higaki, Fuminobu Takahashi

Journal of High Energy Physics (Springer)  11 ( 125 )  2012.11

Research paper (scientific journal), Joint Work, Accepted

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Supersymmetric Effective Theories of Axions

Tetsutaro Higaki, Ryuichiro Kitano

Physical Review D (American Physical Society)  86 ( 7 ) 075027 2012.10

Research paper (scientific journal), Joint Work, Accepted

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Higgs, Moduli Problem, Baryogenesis and Large Volume Compactifications

Tetsutaro Higaki, Kohei Kamada, Fuminobu Takahashi

Journal of High Energy Physics (Springer)  09 ( 043 )  2012.09

Research paper (scientific journal), Joint Work, Accepted

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Natural supersymmetric spectrum in mirage mediation

Masaki Asano, Tetsutaro Higaki

Physical Review D (American Physical Society)  86 ( 3 ) 035020 2012.08

Research paper (scientific journal), Joint Work, Accepted

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Note on moduli stabilization, supersymmetry breaking and axiverse

Tetsutaro Higaki, Tatsuo Kobayashi

Physical Review D (American Physical Society)  84 ( 4 ) 045021  2011.08

Research paper (scientific journal), Joint Work, Accepted

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Duality cascade of softly broken supersymmetric theories

Hiroyuki Abe, Tetsutaro Higaki, Tatsuo Kobayashi, Kazutoshi Ohta, Yuji Omura

Physical Review D (American Physical Society)  79 ( 4 ) 045003 2009.02

Research paper (scientific journal), Joint Work, Accepted

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Non-perturbative moduli superpotential with positive exponents

Hiroyuki Abe, Tetsutaro Higaki, Tatsuo Kobayashi, Osamu Seto

Physical Review D (American Physical Society)  78 ( 2 ) 025007 2008.07

Research paper (scientific journal), Joint Work, Accepted

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Moduli stabilization, F-term uplifting and soft supersymmetry breaking terms

Tetsutaro Higaki

Soryushiron Kenkyu Electron. 116 ( 1 ) A134 - A136 2008.04

Research paper (international conference proceedings), Single Work

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Dynamically sequestered F-term uplifting in extra dimension

Hiroyuki Abe, Tetsutaro Higaki, Tatsuo Kobayashi, Yuji Omura

Journal of High Energy Physics (Springer)  04 ( 072 )  2008.04

Research paper (scientific journal), Joint Work, Accepted

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Moduli stabilization, F-term uplifting and soft supersymmetry breaking terms

Tetsutaro Higaki

Soryushiron Kenkyu Electronics (素粒子論グループ)   2008.04

More about F-term uplifting

Hiroyuki Abe, Tetsutaro Higaki, Tatsuo Kobayashi

Physical Review D (American Physical Society)  76 ( 10 ) 105003  2007.11

Research paper (scientific journal), Joint Work, Accepted

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Moduli stabilization, F-term uplifting and soft supersymmetry breaking terms

Hiroyuki Abe, Tetsutaro Higaki, Tatsuo Kobayashi, Yuji Omura

Physical Review D (American Physical Society)  75 ( 2 ) 025019 2007.01

Research paper (scientific journal), Joint Work, Accepted

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Remark on integrating out heavy moduli in flux compactification

Hiroyuki Abe, Tetsutaro Higaki, Tatsuo Kobayashi

Physical Review D (Springer)  74 ( 4 ) 045012 2006.08

Research paper (scientific journal), Joint Work, Accepted

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Moduli-mixing racetrack model

Hiroyuki Abe, Tetsutaro Higaki, Tatsuo Kobayashi

Nuclear Physics B (Elsevier)  742 ( 1-3 ) 187 - 207 2006.05

Research paper (scientific journal), Joint Work, Accepted

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Kachru-Kallosh-Linde-Trivedi type models with moduli-mixing superpotential

Hiroyuki Abe, Tetsutaro Higaki, Tatsuo Kobayashi

Physical Review D (American Physical Society)  73 ( 4 ) 046005 2006.02

Research paper (scientific journal), Joint Work, Accepted

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Flavor structure and coupling selection rule from intersecting D-branes

Tetsutaro Higaki, Noriaki Kitazawa, Tatsuo Kobayashi, Kei-jiro Takahashi

Physical Review D (American Physical Society)  72 ( 8 ) 086003 2005.10

Research paper (scientific journal), Joint Work, Accepted

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D term inflation and nonperturbative Kahler potential of dilaton

Tetsutaro Higaki, Tatsuo Kobayashi, Osamu Seto

Journal of High Energy Physics (Springer)  07 ( 035 )  2004.08

Research paper (scientific journal), Joint Work, Accepted

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Anomalous U(1) D-term Contribution in Type I String Models

Tetsutaro Higaki, Yoshiharu Kawamura, Tatsuo Kobayashi, Hiroaki Nakano

Physical Review D (American Physical Society)  69 ( 8 ) 086004 2004.04

Research paper (scientific journal), Joint Work, Accepted

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Non-perturbative Kahler Potential, Dilaton Stabilization and Fayet-Iliopoulos Term

Tetsutaro Higaki, Yoshiharu Kawamura, Tatsuo Kobayashi, Hiroaki Nakano

Physics Letters B (Elsevier)  582 ( 3-4 ) 257 - 262 2004.03

Research paper (scientific journal), Joint Work, Accepted

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Twisted Moduli Stabilization in Type I String Models

Tetsutaro Higaki, Tatsuo Kobayashi

Physical Review D (American Physical Society)  68 ( 4 ) 046006 2003.08

Research paper (scientific journal), Joint Work, Accepted

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Wave function profile and SUSY breaking in 5-D model with Fayet-Iliopoulos term

Hiroyuki Abe, Tetsutaro Higaki, Tatsuo Kobayashi

Progress of Theoretical Physics (Yukawa Institute for Theoretical Physics and The Physical Society of Japan )  109 ( 5 ) 809 - 830 2003.05

Research paper (scientific journal), Joint Work, Accepted

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