Family Symmetry Research Articles

Theory for Neutrino Mixing

Since the discovery of neutrino oscillations, for which Takaaki Kajita and Arthur B. McDonald were awarded the 2015 Nobel prize in physics, tremendous progresses have been made in measuring the mixing angles which determine the oscillation pattern. A lot of theoretical efforts have been made to understand how neutrinos mix with each other. Present data show that in the standard parameterization of the mixing matrix, [Formula: see text] is close to [Formula: see text] and the CP violating phase is close to [Formula: see text]. In this talk I report results obtained in arXiv:1505.01932 (Phys. Lett. B750(2015)620) and arXive:1404.01560 (Chin. J. Phys.53(2015)100101) and discuss some implications for theoretical model buildings for such mixing pattern. Specific examples for neutrino mixing based on [Formula: see text] family symmetry are given.

AdjointSU(5)GUT model withT7flavor symmetry

We propose an adjoint $SU(5)$ GUT model with a $T_{7}$ family symmetry and an extra $Z_{2}\otimes Z_{3}\otimes Z_{4}\otimes Z_{4}^{\prime }\otimes Z_{12}$ discrete group, that successfully describes the prevailing Standard Model fermion mass and mixing pattern. The observed hierarchy of the charged fermion masses and the quark mixing angles arises from the $Z_{3}\otimes Z_{4}\otimes Z_{12}$ symmetry breaking, which occurs near the GUT scale. The light active neutrino masses are generated by type-I and type-III seesaw mechanisms mediated by the fermionic $SU(5)$ singlet and the adjoint $\mathbf{24}$-plet. The model predicts the effective Majorana neutrino mass parameter of neutrinoless double beta decay to be $m_{\beta \beta }=$ 4 and 50 meV for the normal and the inverted neutrino spectra, respectively. We construct several benchmark scenarios, which lead to $SU(5)$ gauge coupling unification and are compatible with the known phenomenological constraints originating from the lightness of neutrinos, proton decay, dark matter, etc. These scenarios contain TEV-scale colored fields, which could give rise to a visible signal or be stringently constrained at the LHC.

Testing constrained sequential dominance models of neutrinos

Constrained sequential dominance (CSD) is a natural framework for implementing the see-saw mechanism of neutrino masses which allows the mixing angles and phases to be accurately predicted in terms of relatively few input parameters. We analyze a class of CSD(n) models where, in the flavour basis, two right-handed neutrinos are dominantly responsible for the ‘atmospheric’ and ‘solar’ neutrino masses with Yukawa couplings to proportional to and respectively, where n is a positive integer. These coupling patterns may arise in indirect family symmetry models based on A4. With two right-handed neutrinos, using a χ2 test, we find a good agreement with data for CSD(3) and CSD(4) where the entire Pontecorvo–Maki–Nakagawa–Sakata mixing matrix is controlled by a single phase η, which takes simple values, leading to accurate predictions for mixing angles and the magnitude of the oscillation phase We carefully study the perturbing effect of a third ‘decoupled’ right-handed neutrino, leading to a bound on the lightest physical neutrino mass meV for the viable cases, corresponding to a normal neutrino mass hierarchy. We also discuss a direct link between the oscillation phase and leptogenesis in CSD(n) due to the same see-saw phase η appearing in both the neutrino mass matrix and leptogenesis.

Phenomenological implications of a minimal F-theory GUT with discrete symmetry

We discuss the origin of both non-Abelian discrete family symmetry and Abelian continuous family symmetry, as well as matter parity, from F-theory SUSY GUTs. We propose a minimal model based on the smallest GUT group SU(5), together with the non-Abelian family symmetry D 4 plus an Abelian family symmetry, where fluxes are responsible for doublet-triplet splitting, leading to a realistic low energy spectrum with phenomenologically acceptable quark and lepton masses and mixing. We show how a Z 2 matter parity emerging from F-theory can suppress proton decay while allowing neutron-antineutron oscillations, providing a distinctive signature of the set-up.

CP violation in neutrino mixing with [formula omitted] in [formula omitted] Type-II seesaw model

We study a class of models for neutrino mass matrix in Type-II seesaw with A4 family symmetry. The resulting neutrino mass matrix can be naturally made to respect a μ–τ exchange plus CP conjugate symmetry (GLS) with the CP violating phase δ and the mixing angle θ23 predicted to be ±π/2 and π/4, respectively. When GLS is explicitly broken by complex Yukawa couplings, the model predictions for δ and θ23 can be significantly modified. Should future experiments indeed determine θ23 and δCP away from the GLS limit values, one then had to consider models with broken GLS. We study several simple scenarios to show how the modifications arise w̱hen GLS is broken and how future experiments can test this class of models.

Stability and symmetry breaking in a three-Higgs-doublet model with lepton family symmetryO(2)⊗Z2

Motivated by the neutrino data, an extension of the Standard Model with three Higgs-boson doublets has been proposed. Imposing an O(2) x Z2 family symmetry, a neutrino mixing matrix with theta23 = pi/4 and theta13 = 0 appears in a natural way. Even though these values for the mixing matrix do not follow the recent experimental constraints, they are nevertheless a good approximation. We study the Higgs potential of this model in detail. We apply recent methods which allow for the study of any three-Higgs-boson doublet model. It turns out that for a variety of parameters the potential is stable, has the correct electroweak symmetry breaking, and gives the correct vacuum expectation value.

Invariant approach toCPin family symmetry models

We propose the use of basis invariants, valid for any choice of CP transformation, as a powerful approach to studying specific models of CP violation in the presence of discrete family symmetries. We illustrate the virtues of this approach for examples based on $A_4$ and $\Delta(27)$ family symmetries. For $A_4$, we show how to elegantly obtain several known results in the literature. In $\Delta(27)$ we use the invariant approach to identify how explicit (rather than spontaneous) CP violation arises, which is geometrical in nature, i.e. persisting for arbitrary couplings in the Lagrangian.

Δ(27) family symmetry and neutrino mixing

The observed neutrino mixing, having a near maximal atmospheric neutrino mixing angle and a large solar mixing angle, is close to tri-bi-maximal. This structure may be related to the existence of a discrete non-Abelian family symmetry. In this paper the family symmetry is the non-Abelian discrete group Δ(27), a subgroup of SU(3) with triplet and anti-triplet representations. Different frameworks are constructed in which the mixing follows from combining fermion mass terms with the vacuum structure enforced by the discrete symmetry. Mass terms for the fermions originate from familon triplets, anti-triplets or both. Vacuum alignment for the family symmetry breaking familons follows from simple invariants.

Deviation from bimaximal mixing and leptonic CP phases in S4 family symmetry and generalized CP

The lepton flavor mixing matrix having one row or one column in common with the bimaximal mixing up to permutations is still compatible with the present neutrino oscillation data. We provide a thorough exploration of generating such a mixing matrix from $S_4$ family symmetry and generalized CP symmetry $H_{CP}$. Supposing that $S_4\rtimes H_{CP}$ is broken down to $Z^{ST^2SU}_2\times H^{\nu}_{CP}$ in the neutrino sector and $Z^{TST^{2}U}_4\rtimes H^{l}_{CP}$ in the charged lepton sector, one column of the PMNS matrix would be of the form $\left(1/2, 1/\sqrt{2}, 1/2\right)^{T}$ up to permutations, both Dirac CP phase and Majorana CP phases are trivial in order to accommodate the observed lepton mixing angles. The phenomenological implications of the remnant symmetry $K^{(TST^2, T^2U)}_4\rtimes H^{\nu}_{CP}$ in the neutrino sector and $Z^{SU}_{2}\times H^{l}_{CP}$ in the charged lepton sector are studied. One row of PMNS matrix is determined to be $\left(1/2, 1/2, -i/\sqrt{2}\right)$, and all the three leptonic CP phases can only be trivial to fit the measured values of the mixing angles. Two models based on $S_4$ family symmetry and generalized CP are constructed to implement these model independent predictions enforced by remnant symmetry. The correct mass hierarchy among the charged leptons is achieved. The vacuum alignment and higher order corrections are discussed.

Neutrino Mixing: from experiment to theory

In first part of the talk we survey the experimental results on neutrino mixing, review phenomenological mixing patterns and discuss interpretations in terms of deviations from tri-bimaximal mixing, including mixing sum rules. In the second part of the talk we introduce theoretical approaches to neutrino mixing. We first consider the see-saw mechanism with right-handed neutrinos and sequential dominance. We then discuss approaches based on symmetry, in particular both GUT symmetry and discrete family symmetry, in constructing unified models of quark and lepton masses and mixing. Various different approaches to model building are described, denoted as direct, semi-direct and indirect, and the pros and cons of each approach is discussed.

Invariant approach to [formula omitted] in unbroken Δ(27)

The invariant approach is a powerful method for studying CP violation for specific Lagrangians. The method is particularly useful for dealing with discrete family symmetries. We focus on the CP properties of unbroken Δ(27) invariant Lagrangians with Yukawa-like terms, which proves to be a rich framework, with distinct aspects of CP, making it an ideal group to investigate with the invariant approach. We classify Lagrangians depending on the number of fields transforming as irreducible triplet representations of Δ(27). For each case, we construct CP-odd weak basis invariants and use them to discuss the respective CP properties. We find that CP violation is sensitive to the number and type of Δ(27) representations.

Discrete Symmetries and Models of Flavour Mixing

In this talk we shall give an overview of the role of discrete symmetries, including both CP and family symmetry, in constructing unified models of quark and lepton (including especially neutrino) masses and mixing. Various different approaches to model building will be described, denoted as direct, semi-direct and indirect, and the pros and cons of each approach discussed. Particular examples based on Δ(6n2) will be discussed and an A to Z of Flavour with Pati-Salam will be presented.

Observable dark radiation from a cosmologically safe QCD axion

We propose a QCD axion model that avoids the cosmological domain wall problem, introducing a global SU(3)_f family symmetry to which we embed the unwanted PQ discrete symmetry. The spontaneous breaking of SU(3)_f and PQ symmetry predicts eight NG bosons as well as axion, all of which contribute to dark radiation in the Universe. The derivation from the standard model prediction of dark radiation can be observed by future observations of CMB fluctuations. Our model also predicts a sizable exotic kaon decay rate, which is marginally consistent with the present collider data and would be tested by future collider experiments.

Towards a complete A4 × SU(5) SUSY GUT

We propose a renormalisable model based on $A_4$ family symmetry with an $SU(5)$ grand unified theory (GUT) which leads to the minimal supersymmetric standard model (MSSM) with a two right-handed neutrino seesaw mechanism. Discrete $\mathbb{Z}_9\times \mathbb{Z}_6$ symmetry provides the fermion mass hierarchy in both the quark and lepton sectors, while $\mathbb{Z}_4^R$ symmetry is broken to $\mathbb{Z}_2^R$, identified as usual R-parity. Proton decay is highly suppressed by these symmetries. We discuss both the $A_4$ and $SU(5)$ symmetry breaking sectors, including doublet-triplet splitting, Higgs mixing and the origin of the $\mu$ term. The model provides an excellent fit (better than one sigma) to all quark and lepton (including neutrino) masses and mixing with spontaneous CP violation. With the $A_4$ vacuum alignments, $(0,1,1)$ and $(1,3,1)$, the model predicts the entire PMNS mixing matrix with no free parameters, up to a relative phase, selected to be $2\pi/3$ from a choice of the nine complex roots of unity, providing a direct link between neutrino oscillations and leptogenesis.

Natural quasi-alignment with two Higgs doublets and RGE stability

In the context of two Higgs doublet models, we study the conditions required in order to have stable quasi-alignment in flavour space. We show that stability under the renormalisation group equations imposes strong constraints on the flavour structure of the Yukawa couplings associated to each one of the Higgs doublets. In particular, we find a novel solution, where all Yukawa couplings are proportional to the so-called democratic matrix. This solution is rather unique, since it is the only stable solution which is a good starting point for reproducing the observed pattern of quark masses and mixing. We also show that this stable solution can be obtained by imposing on the Lagrangian a \(Z_3 \times Z^\prime _3\) flavour symmetry. Quark masses of the lighter quark generations are generated through the breaking of this discrete symmetry, and, at this stage, scalar-mediated flavour-changing neutral-currents arise, but they are naturally suppressed by the smallness of the light quark masses. In this way, we relate Higgs alignment to the hierarchy of the quark masses through a discrete family symmetry.

Lepton mixing in A 5 family symmetry and generalized CP

We study lepton mixing patterns which can be derived from the A 5 family symmetry and generalized CP. We find five phenomenologically interesting mixing patterns for which one column of the PMNS matrix is $$ {\left(\sqrt{\frac{5+\sqrt{5}}{10},}\frac{1}{\sqrt{5+\sqrt{5}}},\frac{1}{\sqrt{5+\sqrt{5}}}\right)}^T $$ (the first column of the golden ratio mixing), $$ {\left(\sqrt{\frac{5-\sqrt{5}}{10}},\frac{1}{\sqrt{5-\sqrt{5}}},\frac{1}{\sqrt{5-\sqrt{5}}}\right)}^T $$ (the second column of the golden ratio mixing), $$ {\left(1,1,1\right)}^T/\sqrt{3} $$ or $$ {\left(\sqrt{5}+1,-2,\sqrt{5}-1\right)}^T/4 $$ . The three lepton mixing angles are determined in terms of a single real parameter θ, and agreement with experimental data can be achieved for certain values of θ. The Dirac CP violating phase is predicted to be trivial or maximal while Majorana phases are trivial. We construct a supersymmetric model based on A 5 family symmetry and generalized CP. The lepton mixing is exactly the golden ratio pattern at leading order, and the mixing patterns of case III and case IV are reproduced after higher order corrections are considered.

Bajc-Melfo vacua enable Yukawon ultraminimal grand unified theories

Bajc-Melfo ($\mathbf{BM}$) two-field ($S,\ensuremath{\phi}$) superpotentials define metastable F-term supersymmetry-breaking vacua suitable as hidden sectors for calculable and realistic family and grand unification models. The undetermined vacuum expectation value (VEV) $⟨{S}_{s}⟩$ of the Polonyi field that breaks supersymmetry can be fixed either by coupling to $N=1$ supergravity or by radiative corrections. $\mathbf{BM}$ hidden sectors extend to symmetric multiplets $(S,\ensuremath{\phi}{)}_{ab}$ of a gauged $O({N}_{g})$ family symmetry, broken at the GUT scale, so that the $O({N}_{g})$ charged component VEVs $⟨{\stackrel{^}{S}}_{ab}⟩$ are also undetermined before accounting for the $O({N}_{g})$ D-terms, which fix them by cancellation against D-term contributions from the visible sector. This facilitates Yukawon ultraminimal GUTs (YUMGUTs) proposed in [C. S. Aulakh and C. K. Khosa, Phys. Rev. D 90, 045008 (2014)] by relieving the visible sector from the need to give null D-terms for the family symmetry $O({N}_{g})$. We analyze symmetry breaking and spectra of the hidden-sector fields in the supergravity resolved case when ${N}_{g}=1,2,3$. Besides the Polonyi field ${S}_{s}$, most of the superfields ${\stackrel{^}{S}}_{ab}$ remain light, with fermions getting masses only from loop corrections. Such modes may yield novel dark matter lighter than 100 GeV. Possible Polonyi and moduli problems associated with the fields ${S}_{ab}$ call for detailed investigation of loop effects due to the Yukawa and gauge interactions in the hidden sector and of postinflationary field relaxation dynamics.

Quark mixing from Δ(6N2) family symmetry

We consider a direct approach to quark mixing based on the discrete family symmetry Δ(6N2) in which the Cabibbo angle is determined by a residual Z2×Z2 subgroup to be |Vus|=0.222521, for N being a multiple of 7. We propose a particular model in which unequal smaller quark mixing angles and CP phases may occur without breaking the residual Z2×Z2 symmetry. We perform a numerical analysis of the model for N=14, where small Z2×Z2 breaking effects of order 3% are allowed by model, allowing perfect agreement within the uncertainties of the experimentally determined best fit quark mixing values.

Local unitary symmetries of hypergraph states

Hypergraph states are multiqubit states whose combinatorial description and entanglement properties generalize the well-studied class of graph states. Graph states are important in applications such as measurement-based quantum computation and quantum error correction. The study of hypergraph states, with their richer multipartite entanglement and other nonlocal properties, has a promising outlook for new insight into multipartite entanglement. We present results analyzing local unitary symmetries of hypergraph states, including both continuous and discrete families of symmetries. In particular, we show how entanglement types can be detected and distinguished by certain configurations in the hypergraphs from which hypergraph states are constructed.

Self-intersecting fuzzy extra dimensions from squashed coadjoint orbits in N = 4 $$ \mathcal{N}=4 $$ SYM and matrix models

We find new vacuum solutions of $$ \mathcal{N}=4 $$ super-Yang-Mills with totally antisymmetric cubic soft SUSY breaking terms, or equivalently solutions of the IKKT matrix model of type $$ {\mathbb{R}}_{\theta}^4\times {\mathcal{K}}_N $$ with flux terms. The solutions can be understood in terms of 4- and 6-dimensional fuzzy branes $$ {\mathcal{K}}_N $$ in extra dimensions, describing self-intersecting projections of compact flag manifolds of SU(3). The 6-dimensional solutions provide a 6-fold covering of the internal space near the origin, while the 4-dimensional branes have a triple self-intersection spanning all 6 internal directions. The solutions have lower energy than the trivial vacuum, and we prove that there are no negative modes. The massless modes are identified explicitly. In particular there are chiral fermionic zero modes, linking the coincident sheets with opposite flux at the origin. They have a $$ {\mathbb{Z}}_3 $$ family symmetry, originating from the Weyl group rotations.