The Practice of Computer-Assisted Planning and Navigation for Hip Arthroscopy

Despite significant developments in hip arthroscopic surgery in recent years, precise preoperative planning and accurate performance remain challenging. Preoperative planning is particularly important in cases of osteochondroplasty for cam-type femoroacetabular impingement (FAI), and can be aided by several computer-assisted tools, including three-dimensional imaging analysis or kinematic analysis. Initially, the exact point of the bony impingement is identified using simulation analysis; then, virtual osteochondroplasty is performed. Improvements in the range of motion can then be evaluated using computer simulation again. In this way, the required area and depth of bone resection can be assessed preoperatively. In addition, computed tomography-based navigation assistance can be used to complete the osteochondroplasty in accordance with the preoperative planning. After surgery, postoperative evaluation provides valuable feedback to improve future planning and procedures. In this chapter, we describe the practice of computer-assisted planning and navigation for hip arthroscopy

Modular flavor symmetries of three-generation modes on magnetized toroidal orbifolds

We study the modular symmetry on magnetized toroidal orbifolds with Scherk-Schwarz phases. In particular, we investigate finite modular flavor groups for three-generation modes on magnetized orbifolds. The three-generation modes can be the three-dimensional irreducible representations of covering groups and central extended groups of $\Gamma_N$ for $N=3,4,5,7,8,16$, that is, covering groups of $\Delta(6(N/2)^2)$ for $N=$ even and central extensions of $PSL(2,\mathbb{Z}_{N})$ for $N=$odd with Scherk-Schwarz phases. We also study anomaly behaviors.Comment: 34 page

Texture zeros of quark mass matrices at fixed point τ=ω\tau=\omega in modular flavor symmetry

We study systematically derivation of the specific texture zeros, that is the nearest neighbor interaction (NNI) form of the quark mass matrices at the fixed point $\tau=\omega$ in modular flavor symmetric models. We present models that the NNI forms of the quark mass matrices are simply realized at the fixed point $\tau=\omega$ in the $A_4$ modular flavor symmetry by taking account multi-Higgs fields. Such texture zero structure originates from the $ST$ charge of the residual symmetry $Z_3$ of $SL(2,Z)$. The NNI form can be realized at the fixed point $\tau = \omega$ in $A_4$ and $S_4$ modular flavor models with two pairs of Higgs fields when we assign properly modular weights to Yukawa couplings and $A_4$ and $S_4$ representations to three generations of quarks. We need four pairs of Higgs fields to realize the NNI form in $A_5$ modular flavor models.Comment: 37 page

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.Comment: 35 pages, 14 figure

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