論文

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.

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.

We study the moduli stabilization by the radiative corrections due to the moduli dependent vector-like masses invariant under the finite modular symmetry. The radiative stabilization mechanism can stabilize the modulus τ of the finite modular symmetry ΓN (N ∈ ℕ) at Im τ ≫ 1, where the shift symmetry τ → τ + 1 remains unbroken approximately. The shift symmetry can be considered as the residual ℤN symmetry which realizes the Froggatt-Nielsen mechanism with the hierarchy parameter e−2πIm τ/N ≪ 1. In this work, we study the stabilization of multiple moduli fields, so that various hierarchical values of the modular forms coexist in a model. For example, one modulus stabilized at Im τ1 ∼ 3 is responsible for the hierarchical structure of the quarks and leptons in the Standard Model, and another modulus stabilized at Im τ2 ∼ 15 can account for the flatness of the Re τ2 direction which may be identified as the QCD axion.

We study black hole thermodynamics using a two-dimensional effective theory obtained by dimensional reduction of four-dimensional Einstein--Maxwell theory. For spherically symmetric charged black holes, the resulting dilaton gravity has a nonlinear potential that reproduces the semiclassical phase structure of four-dimensional AdS black holes, including the Hawking--Page transition and the small/large Reissner--Nordström--AdS black hole transition. This shows that the two-dimensional theory before taking the near-horizon and near-extremal limits captures non-extremal thermodynamics beyond the Jackiw--Teitelboim gravity regime. We also include electromagnetic Kaluza--Klein modes on the internal sphere and integrate them out to derive the one-loop effective dilaton gravity. At leading order in the derivative expansion, these corrections appear as constant shifts in the black hole entropy and in the effective charge parameter of the dilaton potential. Therefore, the semiclassical phase structure is not qualitatively modified within this leading local approximation.

We propose a modular symmetric inflationary model based on a Coleman--Weinberg potential generated by integrating out heavy vector-like quarks that couple to the complex modulus field $τ$ through modular forms. In this framework, the imaginary part of modulus $τ$ plays the role of the inflaton, while the real part is identified with a heavy axion. We show that the model successfully explains the current cosmological observations. We further discuss reheating through modulus-dependent gauge kinetic functions and the cosmology of the axion. The axion oscillation dominates over the Universe after the reheating via inflaton decay, and then it decays before Big Bang Nucleosynthesis in the viable parameter region. The quantum fluctuation of the axion can be of order $\mathcal{O}(1)\% $ of that of the inflaton, which would induce isocurvature perturbations that may be detectable in future observations.

We construct finite-tension non-Abelian vortex solutions in a renormalizable $(3+1)$-dimensional $SO(3)$ gauge theory Higgsed to the tetrahedral group $A_4$ by a Higgs field in the spin-3 representation. Since the vacuum manifold is $SO(3)/A_4$, the vortices are characterized by the non-Abelian fundamental group $π_1(SO(3)/A_4)\simeq \widetilde{A}_4$, the binary tetrahedral group. We obtain explicit axisymmetric vortex solutions carrying holonomies corresponding to the order-two and order-three conjugacy classes of $A_4$, determine their tensions numerically, and show that they exhibit type-I, type-II, and Bogomol'nyi--Prasad--Sommerfield-like behavior depending on the Higgs and gauge boson mass ratios. The vortices are classified by conjugacy classes of $\widetilde{A}_4$, while their infrared descriptions are labeled by conjugacy classes of $A_4$. We further demonstrate that the smooth finite-tension vortices reduce in the infrared to Gukov--Witten surface operators of the $A_4$ discrete gauge theory, thereby establishing a finite-energy ultraviolet completion of non-invertible defects in a renormalizable gauge-Higgs theory.

We propose models to explain the hierarchies of the quark masses and mixing by utilizing the $S_4^\prime$ modular flavor symmetry. The hierarchy is realized by the modulus $τ$ stabilized at $\mathrm{Im}\,τ\gg 1$, where the residual $Z_4^T$ 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 $\mathcal{O}({0.1})$ hierarchies of the $\mathcal{O}({1})$ coefficients can be explained by imposing another $S_3$ modular symmetry.

We construct a model in which the hierarchies of the quark and lepton masses and mixing are explained by the $Γ_6^\prime$ modular flavor symmetry. The hierarchies are realized by the Froggatt-Nielsen-like mechanism due to the residual $Z^T_6$ symmetry, approximately unbroken at $τ\sim i\infty.$ We argue that the $Γ_6^{(\prime)}$ 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 $\mathcal{O}(1)$ coefficients for the realistic fermion hierarchies.

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 the 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 numerically and analytically. It is also found that the vacua obtained in this paper preserve CP symmetry.

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.

We study the vacua of modulus potentials consisting of multiple contributions 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 the de Sitter vacuum is realized with a multiple modular form case and obtain a nontrivial vacuum 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 numerically and analytically. It is also found that the vacua obtained in this paper preserve CP symmetry.

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 $\tau \sim i\infty$.

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.

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.

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.

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.

© 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.

© 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.

© 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.