Finite flavour groups of fermions

Abstract

We present an overview of the theory of finite groups, with regard to their application as flavour symmetries in particle physics. In a general part, we discuss useful theorems concerning group structure, conjugacy classes, representations and character tables. In a specialized part, we attempt to give a fairly comprehensive review of finite subgroups of SO(3) and SU(3), in which we apply and illustrate the general theory. Moreover, we also provide a concise description of the symmetric and alternating groups and comment on the relationship between finite subgroups of U(3) and finite subgroups of SU(3). Though in this review we give a detailed description of a wide range of finite groups, the main focus is on the methods which allow the exploration of their different aspects.

89 pages, 6 figures, some references added, rearrangement of part of the material, section on SU(3) subgroups substantially extended, some minor revisions. Version for publication in J. Phys. A. Table 12 corrected to match eq.(256), table 14 and eq.(314) corrected to match the 2-dimensional irreps defined on p.67

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• Indication of Electron Neutrino Appearance from an Accelerator-produced Off-axis Muon Neutrino Beam
• Stringy origin of non-Abelian discrete flavor symmetries
• Neutrino tri-bi-maximal mixing from a non-Abelian discrete family symmetry
• The Horizontal Symmetry for Neutrino Mixing
• Tri-Bimaximal Neutrino Mixing and the Family Symmetry Z_7 x Z_3
• The Flavor Group Delta(3n^2)
• Fermion Masses and Mixings from Dihedral Flavor Symmetries with Preserved Subgroups
• Near Tribimaximal Neutrino Mixing with Delta(27) Symmetry
• The Flavor Group Delta(6n^2)
• Is S4 the horizontal symmetry of tri-bimaximal lepton mixing?
• Golden Ratio Neutrino Mixing and $A_5$ Flavor Symmetry
• Lepton Mixing and Cancellation of the Dirac Mass Hierarchy in SO(10) GUTs with Flavor Symmetries T7 and Sigma(81)
• Simple Finite Non-Abelian Flavor Groups
• Simplified Renormalizable $T^{{}^{\prime }}$ Model for Tribimaximal Mixing and Cabibbo Angle
• A new family symmetry for SO(10) GUTs
• Tribimaximal Mixing From Small Groups
• Observable T_7 Lepton Flavor Symmetry at the Large Hadron Collider
• The discrete flavor symmetry D5
• Lepton Family Symmetry and Possible Application to the Koide Mass Formula
• Lepton Masses in a Minimal Model with Triplet Higgs Bosons and $S_3$ Flavor Symmetry
• A direct link between neutrinoless double beta decay and leptogenesis in a seesaw model with $S_4$ symmetry
• Tri-bimaximal Mixing and Cabibbo Angle in S4 Flavor Model with SUSY
• A three-parameter model for the neutrino mass matrix
• A Note on Embedding Nonabelian Finite Flavor Groups in Continuous Groups
• Neutrino Masses and a Fourth Generation of Fermions

Cited by in corpus (70)

• Neutrino Mass and Mixing with Discrete Symmetry
• Lepton Mixing Patterns from a Scan of Finite Discrete Groups
• Trimaximal TM1 neutrino mixing in S4 with spontaneous CP violation
• A Grand Delta(96) x SU(5) Flavour Model
• Lepton Mixing from Delta (3 n^2) and Delta (6 n^2) and CP
• $A_4$ Flavour Model for Dirac Neutrinos: Type I and Inverse Seesaw
• Discrete symmetries in the three-Higgs-doublet model
• Generalised CP and $\Delta (96)$ Family Symmetry
• A Realistic U(2) Model of Flavor
• Origin of Constrained Maximal CP Violation in Flavor Symmetry
• Neutrino masses and mixing in A4 models with three Higgs doublets
• Mixing Patterns from the Groups Sigma (n phi)
• GUT Physics in the era of the LHC
• Lepton and Quark Mixing Patterns from Finite Flavor Symmetries
• Flavor and CP symmetries for leptogenesis and 0nubb decay
• Lepton Flavor Mixing and CP Symmetry
• CP Violation in the Lepton Sector and Implications for Leptogenesis
• Models with three Higgs doublets in the triplet representations of A4 or S4
• Generation of Universal Linear Optics by Any Beamsplitter
• A Bayesian comparison of U(1) lepton flavour models
• Neutrino Masses and Hubble Tension via a Majoron in MFV
• Golden ratio lepton mixing and nonzero reactor angle with $A_5$
• B-L Model with ${\mathbf{S}}_3$ Symmetry: Nearest Neighbor Interaction Textures and Broken $\mu \leftrightarrow \tau$ Symmetry
• Constraining multi-Higgs flavour models
• Horizontal symmetries of leptons with a massless neutrino
• CP-odd invariants for multi-Higgs models: applications with discrete symmetry
• Discrete flavour symmetries for degenerate solar neutrino pair and their predictions
• Discrete symmetries, roots of unity, and lepton mixing
• Common Origin of Non-zero ${\theta }_{13}$ and Baryon Asymmetry of the Universe in a TeV scale Seesaw Model with $A_4$ Flavour Symmetry
• Perturbations to $\mu -\tau$ Symmetry, Leptogenesis and Lepton Flavour Violation with Type II Seesaw
• Neutrino Mixing and Discrete Symmetries
• Predictivity of Neutrino Mass Sum Rules
• Status of Neutrino Mass and Mixing
• A left-right symmetric flavor symmetry model
• Flavored Non-Minimal Left-Right Symmetric Model Fermion Masses and Mixings
• Lepton masses and mixing in a two-Higgs-doublet model
• Constraining Type I Seesaw with $A_4$ Flavor Symmetry From Neutrino Data and Leptogenesis
• Neutrino masses and mixing in $A_5$ with flavour antisymmetry
• Predictive Leptogenesis from Minimal Lepton Flavour Violation
• Non-zero ${\theta }_{13}$ with Unbroken $\mu -\tau$ Symmetry of the Active Neutrino Mass Matrix in the Presence of a Light Sterile Neutrino
• Cutting the Scotogenic loop: Adding flavor to Dark Matter
• On the characterization of the SU(3)-subgroups of type C and D
• Breaking classical Lie groups to finite subgroups - an automated approach
• Texture zeros of low-energy Majorana neutrino mass matrix in 3+1 scheme
• Lepton and Quark Mixing from Stepwise Breaking of Flavor and CP
• Flavour and CP symmetries in the inverse seesaw
• Common Origin of Non-zero ${\theta }_{13}$ and Dark Matter in an $S_4$ Flavour Symmetric Model with Inverse Seesaw
• Neutrino masses and lepton mixing from $A_5\times \mathrm{CP}$
• Realization of the minimal extended seesaw mechanism and the $T{M}_2$ type neutrino mixing
• FCNC-free multi-Higgs-doublet models from broken family symmetries
• GAP listing of the finite subgroups of U(3) of order smaller than 2000
• Neutrino Mass Sum Rules and Symmetries of the Mass Matrix
• Froggatt-Nielsen models with a residual Z_4^R symmetry
• Lepton Mixing, Residual Symmetries, and Trigonometric Diophantine Equations
• Projective symmetry of partons in Kitaev's honeycomb model
• Low-scale leptogenesis with flavour and CP symmetries
• Data Driven Flavour Model
• Relative scale separation in orbifolds of $S^2$ and $S^5$
• High scale mixing relations as a natural explanation for large neutrino mixing
• Radiative Generation of Non-zero ${\theta }_{13}$ in MSSM with broken $A_4$ Flavor Symmetry
• The Freedman group: a physical interpretation for the SU(3)-subgroup D(18,1,1;2,1,1) of order 648
• Reconstructing Effective Lagrangians Embedding Residual Family Symmetries
• Planck Scale Origin of Non-zero ${\theta }_{13}$ and Super-WIMP Dark Matter
• Signatures of $A_4$ symmetry in the charged lepton flavour violating decays in a neutrino mass model
• A bottom-up approach to the strong CP problem
• Coxeter groups and the PMNS matrix
• The discrete family symmetry as a possible solution to the flavour problem
• Flavour anomalies meet flavour symmetry
• PSL(2,7) Representations and their relevance to Neutrino Physics
• A comparative study between the modified Fritzsch and nearest neighbor interaction textures