arXiv subjects

Tatsuo Kobayashi

Publications and source records attributed to Tatsuo Kobayashi.

At least 55 records · Page 3Linked to original sources

arXiv2023

$Sp(6,Z)$ modular symmetry in flavor structures: quark flavor models and Siegel modular forms for $\widetildeΔ(96)$

We study an approach to construct Siegel modular forms from $Sp(6,Z)$. Zero-mode wave functions on $T^6$ with magnetic flux background behave Siegel modular forms at the origin. Then $T$-symmetries partially break depending on the form of background magnetic flux. We study the background such that three $T$-symmetries $T_I$, $T_{II}$ and $T_{III}$ as well as the $S$-symmetry remain.Consequently, we obtain Siegel modular forms with three moduli parameters $(ω_1,ω_2,ω_3)$, which are multiplets of finite modular groups. We show several examples. As one of examples, we study Siegel modular forms for $\widetildeΔ(96)$ in detail. Then, as a phenomenological applicantion, we study quark flavor models using Siegel modular forms for $\widetildeΔ(96)$. Around the cusp, $ω_1=i\infty$, the Siegel modular forms have hierarchical values depending on their $T_I$-charges. We show the deviation of $ω_1$ from the cusp can generate large quark mass hierarchies without fine-tuning. Furthermore CP violation is induced by deviation of $ω_2$ from imaginary axis.

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arXiv2023

Modular flavor models with positive modular weights: a new lepton model building

We propose an interesting assignment of positive modular wights for fields in a modular non-Abelian discrete flavor symmetry. By this assignment, we can construct inverse seesaw and linear seesaw models without any additional symmetries which possess good testability in current experiments. At first, we discuss probabilities for positive modular wights from a theoretical point of view. Then we show concrete examples of inverse seesaw and linear seesaw scenarios applying modular $A_4$ symmetry as examples and demonstrate some predictions as well as being consistent with experimental results such as their masses and mixings.

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arXiv2023

Modular symmetry in magnetized $T^{2g}$ torus and orbifold models

We study the modular symmetry in magnetized $T^{2g}$ torus and orbifold models. The $T^{2g}$ torus has the modular symmetry $Γ_{g}=Sp(2g,\mathbb{Z})$. Magnetic flux background breaks the modular symmetry to a certain normalizer $N_{g}(H)$. We classify remaining modular symmetries by magnetic flux matrix types. Furthermore, we study the modular symmetry for wave functions on the magnetized $T^{2g}$ and certain orbifolds. It is found that wave functions on magnetized $T^{2g}$ as well as its orbifolds behave as the Siegel modular forms of weight $1/2$ and $\widetilde{N}_{g}(H,h)$, which is the metapletic congruence subgroup of the double covering group of $N_{g}(H)$, $\widetilde{N}_{g}(H)$. Then, wave functions transform non-trivially under the quotient group, $\widetilde{N}_{g,h}=\widetilde{N}_{g}(H)/\widetilde{N}_{g}(H,h)$, where the level $h$ is related to the determinant of the magnetic flux matrix. Accordingly, the corresponding four-dimensional (4D) chiral fields also transform non-trivially under $\widetilde{N}_{g,h}$ modular flavor transformation with modular weight $-1/2$. We also study concrete modular flavor symmetries of wave functions on magnetized $T^{2g}$ orbifolds.

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arXiv2023

Moduli trapping mechanism in modular flavor symmetric models

We discuss how the moduli in modular flavor symmetric models dynamically select enhanced symmetry points at which the residual modular symmetry renders extra matter fields massless. The moduli dynamics non-perturbatively produces the extra matter particles, which gives (time-dependent) effective potential that traps the moduli to enhanced symmetry points. We show analytic estimates of particle production rate consistent with numerical results, and the dynamics of moduli based on the analytic estimates.

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arXiv2023

Modular flavor symmetric models

We review the modular flavor symmetric models of quarks and leptons focusing on our works. We present some flavor models of quarks and leptons by using finite modular groups and discuss the phenomenological implications. The modular flavor symmetry gives interesting phenomena at the fixed point of modulus. As a representative, we show the successful texture structure at the fixed point $τ= ω$. We also study CP violation, which occurs through the modulus stabilization. Finally, we study SMEFT with modular flavor symmetry by including higher dimensional operators.

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arXiv2023

Fermion Hierarchies in $SU(5)$ Grand Unification from $Γ_6^\prime$ Modular Flavor 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.

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arXiv2023

Quark mass hierarchies and CP violation in $A_4\times A_4\times A_4$ modular symmetric flavor models

We study $A_4 \times A_4 \times A_4$ modular symmetric flavor models to realize quark mass hierarchies and mixing angles without fine-tuning. Mass matrices are written in terms of modular forms. At modular fixed points $τ= i\infty$ and $ω$, $A_4$ is broken to $Z_3$ residual symmetry. When the modulus $τ$ is deviated from the fixed points, modular forms show hierarchies depending on their residual charges. Thus, we obtain hierarchical structures in mass matrices. Since we begin with $A_4\times A_4 \times A_4$, the residual symmetry is $Z_3 \times Z_3 \times Z_3$ which can generate sufficient hierarchies to realize quark mass ratios and absolute values of the CKM matrix $|V_{\textrm{CKM}}|$ without fine-tuning. Furthermore, CP violation is studied. We present necessary conditions for CP violation caused by the value of $τ$. We also show possibilities to realize observed values of the Jarlskog invariant $J_{\textrm{CP}}$, quark mass ratios and CKM matrix $|V_{\textrm{CKM}}|$ simultaneously, if $\mathcal{O}(10)$ adjustments in coefficients of Yukawa couplings are allowed.

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arXiv2023

Moduli inflation from modular flavor symmetries

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 Kähler 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 turns out to be consistent with the current observations and behaves similarly to the $α$-attractor models in some parameter spaces. The inflaton rolls down to the CP-symmetric vacuum at the end of inflation.

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arXiv2023

Number of zero-modes on magnetized $T^4/Z_N$ orbifolds analyzed by modular transformation

We study fermion zero-mode wavefunctions on $T^4/Z_N$ orbifold with background magnetic fluxes. The number of zero-modes is analyzed by use of $Sp(4,\mathbb{Z})$ modular transformation. Conditions needed to realize three generation models are clarified. We also study parity transformation in the compact space which leads to better understanding of relationship between positive and negative chirality wavefunctions.

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arXiv2023

Zero-modes in magnetized $T^6/\mathbb{Z}_N$ orbifold models through $Sp(6,\mathbb{Z})$ modular symmetry

We study of fermion zero-modes on magnetized $T^6/\mathbb{Z}_N$ orbifolds. In particular, we focus on non-factorizable orbifolds, i.e. $T^6/\mathbb{Z}_7$ and $T^6/\mathbb{Z}_{12}$ corresponding to $SU(7)$ and $E_6$ Lie lattices respectively. The number of degenerated zero-modes corresponds to the generation number of low energy effective theory in four dimensional space-time. We find that three-generation models preserving 4D $\mathcal{N}=1$ supersymmetry can be realized by magnetized $T^6/\mathbb{Z}_{12}$, but not by $T^6/\mathbb{Z}_7$. We use $Sp(6,\mathbb{Z})$ modular transformation for the analyses.

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arXiv2023

Remark on modular weights in low-energy effective field theory from type II string theory

We revisit the modular weights in type IIB magnetized D-brane models. The simple analysis of wave function shows that the four-dimensional matter fields have the modular weight -1/2, but it may shift as one in type IIA intersecting D-brane models. For example, the localized gauge flux as well as the localized curvature can shift the modular weight in the magnetized D-brane models. Such corrections do not affect physical couplings such as physical Yukawa couplings. However, it leads to differences in supersymmetry breaking sfermion masses, which depend on the modular weights, although the $A$-term coefficients and the sum of sfermion masses squared seem to be the same between two models.

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arXiv2023

Generalized Matter Parities from Finite Modular Symmetries

We classify a supersymmetric extension of the Standard Model by discrete symmetries originating from finite modular symmetries $Γ_N$. Since all the couplings in supersymmetric theories of finite modular symmetries $Γ_N$ are described by holomorphic modular forms with even modular weights, renormalizable and non-renormalizable operators such as baryon- and/or lepton-number violating operators are severely constrained. From the modular transformation of matter multiplets with modular weight $1/M$, we find $\mathbb{Z}_{2M}$ symmetries, including the generalized baryon and lepton parities, $R$-parity, $\mathbb{Z}_3$ baryon triality and $\mathbb{Z}_6$ proton hexality. Such $\mathbb{Z}_{2M}$ symmetries are enlarged to $\mathbb{Z}_{2M} \rtimes \mathbb{Z}_2^{\text{CP}}$ symmetries together with the CP transformation.

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arXiv2023

Quark and lepton hierarchies from $S_4^\prime$ modular flavor symmetry

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 $τ\sim i\infty$. The model provides the first explicit example which explains hierarchies of both quarks and leptons. The hierarchies are realized by powers of $ε= e^{2πi τ/4} = \mathcal{O}(0.01)$ and $2\,\mathrm{Im}\,τ\sim 5$, where $τ$ being the modulus. The small parameter $ε$ plays a role of flavon in the Froggatt-Nielsen mechanism under the residual $Z_4^T$ symmetry, and powers of $2\,\mathrm{Im}\,τ$ 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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arXiv2023

Quark masses and CKM hierarchies from $S_4^\prime$ modular flavor symmetry

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.

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arXiv2023

Quark hierarchical structures in modular symmetric flavor models at level 6

We study modular symmetric quark flavor models without fine-tuning. Mass matrices are written in terms of modular forms, and modular forms in the vicinity of the modular fixed points become hierarchical depending on their residual charges. Thus modular symmetric flavor models in the vicinity of the modular fixed points have a possibility to describe mass hierarchies without fine-tuning. Since describing quark hierarchies without fine-tuning requires $Z_n$ residual symmetry with $n\geq 6$, we focus on $Γ_6$ modular symmetry in the vicinity of the cusp $τ=i\infty$ where $Z_6$ residual symmetry remains. We use only modular forms belonging to singlet representations of $Γ_6$ to make our analysis simple. Consequently, viable quark flavor models are obtained without fine-tuning.

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arXiv2022

Zero-mode wave functions by localized gauge fluxes

We study chiral zero-mode wave functions on blow-up manifolds of $T^2/Z_N$ orbifolds with both bulk and localized magnetic flux backgrounds. We introduce a singular gauge transformation in order to remove $Z_N$ phases for $Z_N$ twisted boundary condition of matter fields. We compute wave functions of not only bulk zero modes but also localized modes at the orbifold singular points, which correspond to new zero modes induced by localized flux. By studying their Yukawa couplings, it turns out that only three patterns of Yukawa couplings are allowed. Our theory has a specific coupling selection rule.

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arXiv2022

Index theorem on magnetized blow-up manifold of $T^2/\mathbb{Z}_N$

We investigate blow-up manifolds of $T^2/{\mathbb{Z}}_N\,(N=2,3,4,6)$ orbifolds with magnetic flux $M$. Since the blow-up manifolds have no singularities, we can apply the Atiyah-Singer index theorem to them. Then, we establish the zero-mode counting formula $n_{+}-n_{-}=(M-V_{+})/N+1$, where $V_{+}$ denotes the sum of winding numbers at fixed points on the $T^2/{\mathbb{Z}}_N$ orbifolds, as the Atiyah-Singer index theorem on the orbifolds, and clarify physical and geometrical meanings of the formula.

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arXiv2022

Quark and lepton flavor structure in magnetized orbifold models at residual modular symmetric points

We study quark and lepton mass matrices derived from magnetized $T^2/\mathbb{Z}_2$ orbifold models. Quark and lepton masses have a large hierarchy. In addition, mixing angles are large in the lepton sector, while those are small in the quark sector. We find that this structure can be realized in certain flavor models, which are identified by the zero points of the zero-mode wave functions of fermions and Higgs modes. We classify such realistic flavor models. Fixed points $τ=i$, $e^{2πi/3}$ and $i\infty$ of the modulus $τ$ play a role in realizing a large mass hierarchy through our scenario, where residual $S$, $ST$, and $T$ symmetries remain and the lightest Higgs modes can correspond to eigenstates of residual symmetries at the leading order. As a result, we find that there are 24 flavor models in total which can be realistic in a vicinity of $S$-symmetric vacuum but no flavor models for $ST$ and $T$-symmetric vacua.

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