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 $\Gamma_6$ modular symmetry in the vicinity of the cusp $\tau=i\infty$ where $Z_6$ residual symmetry remains. We use only modular forms belonging to singlet representations of $\Gamma_6$ to make our analysis simple. Consequently, viable quark flavor models are obtained without fine-tuning.Comment: 29 page

Modular symmetry in magnetized T2gT^{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 $\Gamma_{g}=Sp(2g,\mathbb{Z})$. 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.Comment: 53 page

Quark mass hierarchies and CP violation in A4×A4×A4A_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 $\tau = i\infty$ and $\omega$, $A_4$ is broken to $Z_3$ residual symmetry. When the modulus $\tau$ 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 $\tau$. 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.Comment: 41 pages, 3 figure

Sp(6,Z)Sp(6,Z) modular symmetry in flavor structures: quark flavor models and Siegel modular forms for Δ~(96)\widetilde{\Delta}(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 $(\omega_1,\omega_2,\omega_3)$, which are multiplets of finite modular groups. We show several examples. As one of examples, we study Siegel modular forms for $\widetilde{\Delta}(96)$ in detail. Then, as a phenomenological applicantion, we study quark flavor models using Siegel modular forms for $\widetilde{\Delta}(96)$. Around the cusp, $\omega_1=i\infty$, the Siegel modular forms have hierarchical values depending on their $T_I$-charges. We show the deviation of $\omega_1$ from the cusp can generate large quark mass hierarchies without fine-tuning. Furthermore CP violation is induced by deviation of $\omega_2$ from imaginary axis.Comment: 54 page

Zero-modes in magnetized T6/ZNT^6/\mathbb{Z}_N orbifold models through Sp(6,Z)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.Comment: 37 pages, 2 figure

Effectiveness of forward obstacles collision warning system based on deceleration for collision avoidance

In the authors previous study, the authors proposed deceleration for collision avoidance (DCA) as an index to evaluate collision risks against forward obstacles and examined the effectiveness of their forward obstacles collision warning system (FOCWS) based on DCA. In the present manuscript, they improve the visual interface of the FOCWS, and conduct driving simulator experiments to quantitatively evaluate the effectiveness of the improved FOCWS in situations where a preceding vehicle decelerates abruptly. The experimental results revealed that the FOCWS based on DCA was effective in assisting drivers to shorten the reaction time and to avoid collisions. Moreover, in the subjective assessment questionnaire, a significant number of experimental participants reported that the FOCWS based on DCA could evaluate collision risks more properly compared with the FOCWS based on a time-to-collision

Quark mass hierarchies and CP violation in A 4 × A 4 × A 4 modular symmetric flavor models

Abstract We study A 4 × A 4 × 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∞ 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 × A 4 × A 4, the residual symmetry is Z 3 × Z 3 × Z 3 which can generate sufficient hierarchies to realize quark mass ratios and absolute values of the CKM matrix |V 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 CP, quark mass ratios and CKM matrix |V CKM | simultaneously, if (10) adjustments in coefficients of Yukawa couplings are allowed or moduli values are non-universal

Sp(6, Z) modular symmetry in flavor structures: quark flavor models and Siegel modular forms for Δ~(96)\widetilde{\Delta }\left(96\right)

Abstract 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 $$\widetilde{\Delta }\left(96\right)$$ in detail. Then, as a phenomenological applicantion, we study quark flavor models using Siegel modular forms for $$\widetilde{\Delta }\left(96\right)$$ . Around the cusp, ω 1 = i∞, 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