Flavor Symmetry Research Articles

Discrete global symmetries: gauging and twisted compactification

Discrete global symmetries of 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs are studied via two operations: gauging and twisted compactification. We consider gauging of discrete symmetries in several well-known 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs, including SU(n) SQCD with 2n flavors, theories of class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document} of type A2n−1, and Argyres-Douglas theories of type (AN, AN), as well as propose new 4d SCFTs as a result. The wreathing technique, which involves gauging a subgroup of the automorphism group of the quiver diagram of the corresponding 3d mirror theory, is exploited. This allows us to understand several properties of discretely gauged theories, including moduli spaces and how discrete gauging affects the mixed ’t Hooft anomaly between the 1-form symmetry and the 0-form flavor symmetry. Many examples are viewed through the lens of the Argyres-Seiberg duality and its generalization. We also examine discrete gauging of SU(2) SQCD with 4 flavors by various ℤ2 and ℤ2 × ℤ2 subgroups of the permutation group S4 using the superconformal index. Regarding compactification, we propose a magnetic quiver for 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SU(n) SQCD with 2n flavors compactified on a circle with a ℤ2 twist. The twisted compactification by non-invertible symmetries of the 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM theory with gauge group SU(N) is revisited. The non-invertible symmetry naturally gives rise to a ℤk action on the scalar fields parametrizing the moduli space. Upon examining the ℤk invariant chiral ring of the Higgs branch, we find that, in addition to the largest branch of the moduli space that is expected to be captured by the ABJ(M) theory, there exist in general nilpotent operators that lead to a branch of the moduli space which is a radical ideal.

Higher-derivative corrections to flavoured BPS black hole thermodynamics and holography

A Cardy-like regime of the four-dimensional superconformal index has been shown to be governed by ’t Hooft anomalies and to single out a large-N saddle carrying the Bekenstein-Hawking entropy of dual supersymmetric black holes in AdS5. For the universal index where no flavour fugacities are turned on, this correspondence has been improved by matching the first subleading corrections to the saddle-point action with the four-derivative corrections to the black hole action in minimal gauged supergravity, as well as the respective corrected entropies. Here, we extend this match by including flavour symmetries. We consider five-dimensional gauged supergravity with vector multiplet and four-derivative couplings, and provide an effective theory reproducing the ’t Hooft anomalies of the R- and flavour symmetries of generic holographic superconformal field theories at next-to-leading order in the large-N expansion. Then we focus on a specific model dual to ℂ3/ℤν quiver gauge theories, where the ’t Hooft anomaly coefficients receive simple but sufficiently generic corrections. In this model, we evaluate the four-derivative corrections to the on-shell action of the supersymmetric multi-charge black hole, showing agreement with the flavoured Cardy-like formula from the index. We give a prediction for the corrected entropy of the supersymmetric black hole and discuss the general validity of our results. Taking the limit of infinite AdS5 radius, we also obtain four-derivative corrections to the action and entropy of supersymmetric asymptotically flat black holes.

Nonleptonic three-body charmed baryon weak decays with H(15)

We study the nonleptonic three-body charmed baryon weak decays of Bc→BnPP′ under the SU(3)F flavor symmetry, where Bc denotes the antitriplet charmed baryon, comprising (Ξc0,−Ξc+,Λc+), and Bn and P(P′) represent octet baryon and pseudoscalar meson states, respectively. In addition to 12 parameters from the contributions of the color-antisymmetric part of the effective Hamiltonian, denoted as H(6¯), there are 4 parameters from the color-symmetric one, H(15), which were not included in the previous study. With 16 parameters in total and 28 experimental data points, we obtain the minimal χ2 over degree of freedom of χ2/d.o.f=1.5, which is a great improvement comparing to that without H(15). With the better fitting values, we evaluate the branching ratios and up-down asymmetries of Bc→BnPP′, which present some interesting results such as B(Λc+→(Ξ(1690)0→Σ+K−)K+)≡(1.5±0.4)×10−3 and potential SU(3) breaking effects in Ξc+→pπ+K− and Λc+→Σ+π−K+ to be verified by the experiments at BESIII, Belle-II, and LHCb. Published by the American Physical Society 2024

Quark and lepton modular models from the binary dihedral flavor symmetry

Inspired by the structure of top-down derived models endowed with modular flavor symmetries, we investigate the yet phenomenologically unexplored binary dihedral group 2D3. After building the vector-valued modular forms in the representations of 2D3 with small modular weights, we systematically classify all (Dirac and Majorana) mass textures of fermions with fractional modular weights and all possible 2 + 1-family structures. This allows us to explore the parameter space of fermion models based on 2D3, aiming at a description of both quarks and leptons with a minimal number of parameters and best compatibility with observed data. We consider the separate possibilities of neutrino masses generated by either a type-I seesaw mechanism or the Weinberg operator. We identify a model that, besides fitting all known flavor observables, delivers predictions for six not-yet measured parameters and favors normal-ordered neutrino masses generated by the Weinberg operator. It would be interesting to figure out whether it is possible to embed our model within a top-down scheme, such as \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb{T}}^{2}/{\\mathbb{Z}}_{4}$$\\end{document} heterotic orbifold compactifications.

3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 theories from M-theory on CY4 and IIB brane box

We study 3D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetric field theories geometrically engineered from M-theory on non-compact Calabi-Yau fourfolds (CY4). We establish a detailed dictionary between the geometry and topology of non-compact CY4 and the physics of 3D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 theories in three different regimes. The first one is the Coulomb branch description when the CY4 is smooth. The second one is non-abelian gauge theory when the CY4 has a degenerate ℙ1-fibration structure. The third one is the strongly coupled SCFT from a CY4 singularity. We find interesting flavor symmetry enhancements in the singular limit of CY4, as well as an interesting and previously unexplored phenomenon in 3D, termed “flavor symmetry duality”. Many examples are analyzed with an emphasis on toric CY4s and ℂ4 orbifolds with crepant resolutions. We develop a new brane box method to study the physics of Coulomb branch of 3D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 theory that admits a toric construction. Via IIB/M-theory duality we find that the brane box diagram living in ℝ3 can be physically realized as a configuration of intersecting 4-branes which are extended objects in 8D maximal supersymmetric theory, which is shown to be consistent via various chains of dualities. The rank, effective gauge coupling and certain hints to flavor symmetry enhancement of the 3D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 theory are read off from the brane box and cross-checked against the results obtained from geometric engineering. The exotic branes in 8D maximal supersymmetric theory and the 4-string junctions thereof are shown to play a crucial role in the construction of the brane box.

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 [Formula: see text]. We also study CP violation, which occurs through the modulus stabilization. Finally, we study SMEFT with modular flavor symmetry by including higher dimensional operators.

Spectrum of the molecular pentaquarks

We investigate the mass spectrum of the molecular pentaquarks composed of a baryon and a meson. We establish the underlying relations among the near-threshold interactions of the molecular tetraquark and pentaquark systems. We find the existence of the molecule candidates in the ΣcD¯(*), DD*, and DD¯* systems indicates a substantial presence of the hadronic molecules in the heavy baryon plus heavy meson systems ( refers to the hadrons with the c and/or s quarks). We make an exhaustive prediction of the possible bound/virtual molecular states in the systems: Σc(*)D¯(*), Σc(*)D(*), Ξc(′,*)D¯(*), Ξc(′,*)D(*), Σc(*)K*, Σc(*)K¯*, Ξc(′,*)K*, Ξc(′,*)K¯*, Ξcc(*)D¯(*), Ξcc(*)D(*), Ξcc(*)K*, Ξcc(*)K¯*, Σ(*)D¯(*), Σ(*)D(*), Ξ(*)D¯(*), Ξ(*)D(*), Σ(*)K*, Σ(*)K¯*, Ξ(*)K*, Ξ(*)K¯*. Hunting for the predicted states in experiments will significantly deepen our understanding of the formation mechanism of the hadronic molecules, and shed light on the manifestation of flavor symmetry in the low-energy residual strong interactions. Published by the American Physical Society 2024

Flavor's Delight.

Discrete flavor symmetries provide a promising approach to understand the flavor sector of the standard model of particle physics. Top-down (TD) explanations from string theory reveal two different types of such flavor symmetries: traditional and modular flavor symmetries that combine to the eclectic flavor group. There have been many bottom-up (BU) constructions to fit experimental data within this scheme. We compare TD and BU constructions to identify the most promising groups and try to give a unified description. Although there is some progress in joining BU and TD approaches, we point out some gaps that have to be closed with future model building.

The flavor puzzle: Textures and symmetries

We discuss aspects of a promising top-down origin of flavor symmetries in particle physics. Modular transformations originating from string theory dualities are shown to play a crucial role. We introduce the notion of an “eclectic” flavor scheme that unifies traditional flavor symmetries, modular symmetries and [Formula: see text]-transformations. It exhibits the phenomenon of “Local Flavor Unification” with enhanced flavor symmetries at fixed points or lines in moduli space. Successful fits of masses and mixing angles of quarks and leptons are found in the vicinity of these points and lines.

Flavor invariance of leptonic Yukawa terms in the 3HDM

As an extension of the Standard Model (SM), the 3HDM (Three-Higgs-Doublet Model) defines additional relationships among the fermions. In the visible leptonic Yukawa sector of the 3HDM, we investigate the existence of flavor symmetries under discrete non-abelian groups up to order 1032. When the VEVs are not allowed to depart from their alignment imposed by the condition of minimal potential in the Higgs scalar sector, there will be severe mass degeneracies and a lack of lepton flavor mixing for any nontrivial flavor group. Yet we propose to study the Yukawa terms of the lepton sector for in case the VEV alignment would not be protected, hence leaving the VEV ratios as free parameters. Then, as it turns out, mass splitting for the charged leptons and the neutrinos is obtained although it is still impossible to obtain correct fits to the experimentally known lepton mass spectrum and to the PMNS matrix simultaneously.

Neutrino model with broken [formula omitted] -[formula omitted] symmetry and unflavored leptogenesis with dihedral flavor symmetry

We propose a new neutrino flavor model based on a D4×U(1) flavor symmetry providing predictions for neutrino masses and mixing along with a successful generation of the observed Baryon Asymmetry of the Universe (BAU). After the spontaneous breaking of the flavor symmetry, the type I seesaw mechanism leads to a light neutrino mass matrix with broken μ−τ symmetry. By performing a numerical analysis, we find that the model favors a normal mass hierarchy with the lightest neutrino mass lies in the range m1∈[2.516,21.351] meV. The phenomenological implications of the neutrino sector are explored in detail and the results are discussed. Moreover, the generation of BAU is addressed via the leptogenesis mechanism from the decay of three right-handed neutrinos Ni. Through analytical and numerical analysis of the baryon asymmetry parameter YB, a successful unflavored leptogenesis takes place within the allowed parameter space obtained from neutrino phenomenology. We also examine interesting correlations between YB and low energy observables and provide a comprehensive discussion of the results.

U(2) Is Right for Leptons and Left for Quarks.

We posit that the distinct patterns observed in fermion masses and mixings are due to a minimally broken U(2)_{q+e} flavor symmetry acting on left-handed quarks and right-handed charged leptons, giving rise to an accidental U(2)^{5} symmetry at the renormalizable level without imposing selection rules on the Weinberg operator. We show that the symmetry can be consistently gauged by explicit examples and comment on realizations in SU(5) unification. Via a model-independent analysis of a standard model viewed as an effective field theory, we find that selection rules due to U(2)_{q+e} enhance the importance of charged lepton flavor violation as a probe, where significant experimental progress is expected in the near future.

Bs → μ+μ− in a two-Higgs-doublet model with flavour-changing up-type Yukawa couplings

We present a Two-Higgs-Doublet Model in which the structure of the quark Yukawa matrices is governed by three spurions breaking the flavour symmetries of the quark Yukawa sector. The model naturally suppresses flavour-changing neutral current (FCNC) amplitudes in the down-type sector, but permits sizable FCNC couplings in the up sector. We calculate the branching ratio of Bs → μ+μ− to leading and next-to-leading order of QCD for the case with FCNC couplings of the heavy neutral Higgs bosons to up-type quarks and verify that all counterterms follow the pattern dictated by the spurion expansion of the Yukawa matrices. We find correlations between Bs → μ+μ−, b → sγ, and the Higgs masses. The \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${B}_{s}-{\\overline{B} }_{s}$$\\end{document} mixing amplitude is naturally suppressed in the model but can probe a portion of the parameter space with very heavy Higgs bosons.

Triplet scalar flavored leptogenesis with spontaneous CP violation

The inclusion of two triplet scalars in the Standard Model (SM) enables to accommodate neutrino mass generation as well as baryogenesis through leptogenesis. One of theessential ingredients of leptogenesis is the violation ofcharge conjugation and parity (CP) symmetry in lepton number violating decays of the triplet scalars. We work on the promising sector of spontaneous CP violation (SCPV) which is manifested by the involvement of one scalar singlet and two scalar fields, added to the SM. The predictiveaspect of the model is accomplished by imposing A 4 × Z 4 symmetry which results in the traditional tribimaximal mixing pattern. With updated data on neutrino oscillation, we study the parameter space of the model.The phase of the complex vacuum expectation value (VEV) of the singlet scalar acts as the common source ofCP violation in both low and high energy sectors.Due to the flavor symmetry of the model, required baryon asymmetry cannot be accomplished via unflavored leptogenesis. In the temperature regime, [109, 1012] GeV when flavor effects become important in the study of leptogenesis, it is shown that baryogenesis is achievable. The rich flavor interplay is explored through the study of the density matrix equations. We also study the interplay of hierarchical branching ratios of the decay of the triplet scalars and SCPV phase to accommodate the required CP asymmetry to account for the final baryon asymmetry in the observational range. Considering all possible mass hierarchies among the triplet scalars, the flavor structure of the triplet Yukawa couplings results in different scales of leptogenesis.

Standard and Non-Standard Aspects of Neutrino Physics

This review provides a succinct overview of the basic aspects of neutrino physics. The topics covered include neutrinos in the standard model and the three-neutrino mixing scheme; the current status of neutrino oscillation measurements and what remains to be determined; the seesaw mechanisms for neutrino mass generation and the associated phenomenology, including the leptogenesis mechanism to explain the observed matter–antimatter asymmetry of the Universe; and models for the origin of the pattern of neutrino mixing and lepton masses based on discrete flavour symmetries and modular invariance.

The conformal manifold of S-folds in string theory

We continue the holographic exploration of the conformal manifold of 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 S-fold SCFTs constructed by gauging the flavor symmetry of the Gaiotto-Witten T[U(N)] theory. We show how to uplift the two-parameter family of AdS4 vacua dual to this conformal manifold to 10d backgrounds of type IIB supergravity. We use these uplifted solutions to shed new light on the mysterious nature of the infinite distance limit on the conformal manifold and to study probe strings and D3-branes. This analysis uncovers an intriguing structure of the S3 partition function of the S-fold SCFTs which resembles the giant graviton expansion of the superconformal index of 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM. We also show how to each member of the family of supersymmetric AdS4 vacua one can associate a consistent truncation to 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 gauged supergravity and use this result, in conjunction with holography, to calculate the large N partition function of the 3d S-fold SCFT on compact Euclidean manifolds. Finally, we generalize the supersymmetric AdS4 vacua to a four-parameter family of non-supersymmetric AdS4 solutions.

Atmospheric Neutrino octant from flavor symmetry

Atmospheric Neutrino octant from flavor symmetry

Generalized symmetries and anomalies of 3d $\mathcal{N}=4$ SCFTs

We study generalized global symmetries and their ’t Hooft anomalies in 3d \mathcal{N}=4𝒩=4 superconformal field theories (SCFTs). Following some general considerations, we focus on good quiver gauge theories, comprised of balanced unitary nodes and unbalanced unitary and special unitary nodes. While the global form of the Higgs branch symmetry group may be determined from the UV Lagrangian, the global form of Coulomb branch symmetry groups and associated mixed ’t Hooft anomalies are more subtle due to potential symmetry enhancement in the IR. We describe how Coulomb branch symmetry groups and their mixed ’t Hooft anomalies can be deduced from the UV Lagrangian by studying center charges of various types of monopole operators, providing a concrete and unambiguous way to implement ’t Hooft anomaly matching. The final expression for the symmetry group and ’t Hooft anomalies has a concise form that can be easily read off from the quiver data, specifically from the positions of the unbalanced and flavor nodes with respect to the positions of the balanced nodes. We provide consistency checks by applying our method to compute symmetry groups of 3d \mathcal{N}=4𝒩=4 theories corresponding to magnetic quivers of 4d Class S theories and 5d SCFTs. We are able to match these results against the flavor symmetry groups of the 4d and 5d theories computed using independent methods. Another strong consistency check is provided by comparing symmetry groups and anomalies of two theories related by 3d mirror symmetry.

Electron EDM and LFV decays in the light of Muon (g-2)μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(g-2)_\\mu $$\\end{document} with U(2) flavor symmetry

We study the interplay of New Physics (NP) among the lepton magnetic moment, the lepton flavor violation (LFV) and the electron electric dipole moment (EDM) in light of recent data of the muon (g-2)μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(g-2)_\\mu $$\\end{document}. The NP is discussed in the leptonic dipole operator with the U(2) flavor symmetry of the charged leptons, where possible CP violating phases of the three family space are taken into account. It is remarked that the third-family contributes significantly to the LFV decay, μ→eγ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu \\rightarrow e\\gamma $$\\end{document}, and the electron EDM. The experimental upper-bound on μ→eγ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu \\rightarrow e\\gamma $$\\end{document} decay gives a severe constraint on the parameters of the flavor model. The predicted electron EDM is rather large due to the CP violating phases in the three family space. In addition, we also study (g-2)e,τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(g-2)_{e,\ au }$$\\end{document} of the electron and tauon, and EDMs of the muon and tauon as well as the τ→eγ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au \\rightarrow e \\gamma $$\\end{document} and τ→μγ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au \\rightarrow \\mu \\gamma $$\\end{document} decays. The τR→μLγ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _R \\rightarrow \\mu _L \\gamma $$\\end{document} decay is predicted to be close to the experimental upper-bound.

Eclectic flavor group ∆(27) ⋊ S3 and lepton model building

We have performed a systematical study of the eclectic flavor group ∆(27) ⋊ S3 which is the extension of the traditional flavor symmetry ∆(27) by the finite modular symmetry S3. Consistency between ∆(27) and S3 requires that the eight nontrivial singlet representations of ∆(27) should be arranged into four reducible doublets. The modular transformation matrices are determined for various ∆(27) multiplets, and the CP-like symmetry compatible with ∆(27) ⋊ S3 are discussed. We study the general form of the Kähler potential and superpotential invariant under ∆(27) ⋊ S3, and the corresponding fermion mass matrices are presented. We propose a bottom-up model for lepton masses and mixing based on ∆(27) ⋊ S3, a numerical analysis is performed and the experimental data can be accommodated.