Modular Symmetry Research Articles

Confronting the prediction of leptonic Dirac CP -violating phase with experiments

We update and improve past efforts to predict the leptonic Dirac CP-violating phase with models that predict perturbatively modified tribimaximal or bimaximal mixing. Simple perturbations are applied to both mixing patterns in the form of rotations between two sectors. By translating these perturbed mixing matrices to the standard parameterization for the neutrino mixing matrix we derive relations between the Dirac CP-phase and the oscillation angles. We use these relations together with current experimental results to constrain the allowed range for the CP-phase and determine its probability density. Furthermore, we elaborate on the prospects for future experiments probing on the perturbations considered in this work. We present a model with $A_4$ modular symmetry that is consistent with one of the described perturbed scenarios and successfully predicts current oscillation parameter data.

Mass matrices with CP phase in modular flavor symmetry

Abstract We study the CP violation and the CP phase of quark mass matrices in modular flavor symmetric models. The CP symmetry remains at τ = e2πi/3 by a combination of the T-symmetry of the modular symmetry. However, T-symmetry breaking may lead to CP violation at the fixed point τ = e2πi/3. We study such a possibility in magnetized orbifold models as examples of modular flavor symmetric models. These models, in general, have more than one candidate for Higgs modes, while generic string compactifications also lead to several Higgs modes. These Higgs modes have different behaviors under the T-transformation. The light Higgs mode can be a linear combination of those modes so as to lead to realistic quark mass matrices. The CP phase of the mass matrix does not appear in a certain case, which is determined by the T-transformation behavior. Deviation from this is important to realize the physical CP phase. We discuss an example leading to a non-vanishing CP phase at the fixed point τ = e2πi/3.

Leptogenesis in SO(10) models with A4 modular symmetry

We study the prediction for leptogenesis in two renormalizable supersymmetric SO(10) × A4 modular models in which the neutrino mass is dominantly generated by the type I seesaw mechanism. The evolution of the lepton asymmetries are described in terms of the three-flavored density matrix equations for three heavy Majorana neutrinos, where both vanishing initial condition and thermal initial condition of the right-handed neutrinos are considered. We also present an analytical approximation based on the Boltzmann equations. We find regions of parameter space compatible with the measured fermion masses and mixing parameters as well as the baryon asymmetry of the Universe. The predictions for the light neutrino masses, the effective mass in neutrinoless double beta decay and the leptonic CP violation phases are discussed.

Lepton mass matrix from double covering of A 4 modular flavor symmetry* *The work of H. O. is supported by the Junior Research Group (JRG) Program at the Asia-Pacific Center for Theoretical Physics (APCTP) through the Science and Technology Promotion Fund and Lottery Fund of the Korean Government and was supported by the Korean Local Governments-Gyeongsangbuk-do Province and Pohang City.

We study a double covering of modular flavor symmetry. To this end, we construct lepton models for canonical and radiative seesaw scenarios. Using irreducible doublet representations, heavier Majorana fermion masses are characterized by one free parameter that would differentiate from symmetry. symmetry. Through χ square numerical analysis, we demonstrate that both scenarios produce some predictions in case of normal hierarchy reproducing neutrino oscillation data. However, no solution satisfies the neutrino oscillation data in case of radiative seesaw of inverted hierarchy.

The first string-derived eclectic flavor model with realistic phenomenology

Eclectic flavor groups arising from string compactifications combine the power of modular and traditional flavor symmetries to address the flavor puzzle. This top-down scheme determines the representations and modular weights of all matter fields, imposing strict constraints on the structure of the effective potential, which result in controlled corrections. We study the lepton and quark flavor phenomenology of an explicit, potentially realistic example model based on a \U0001d54b6/ℤ3× ℤ3 orbifold compactification of the heterotic string that gives rise to an Ω(2) eclectic flavor symmetry. We find that the interplay of flavon alignment and the localization of the modulus in the vicinity of a symmetry-enhanced point leads to naturally protected fermion mass hierarchies, favoring normal-ordered neutrino masses arising from a see-saw mechanism. We show that our model can reproduce all observables in the lepton sector with a small number of parameters and deliver predictions for so far undetermined neutrino observables. Furthermore, we extend the fit to quarks and find that Kähler corrections are instrumental in obtaining a successful simultaneous fit to the quark and lepton sectors.

Two A5 modular symmetries for Golden Ratio 2 mixing

We present a model of leptonic mixing based on two A5 modular symmetries using the Weinberg operator. The two modular symmetries are broken to the respective diagonal A5 subgroup. At the effective level, the model behaves as a model with a single A5 modular flavour symmetry, but with two moduli. Using both as stabilisers, different residual symmetries are preserved, leading to Golden Ratio mixing that is perturbed by a rotation.

Lepton flavor violation, lepton (g − 2)μ, e and electron EDM in the modular symmetry

We study the lepton flavor violation (LFV), the leptonic magnetic moments (g − 2)μ, e and the electric dipole moment (EDM) of the electron in the Standard-Model Effective Field Theory with the ΓN modular flavor symmetry. We employ the stringy Ansatz on coupling structure that 4-point couplings of matter fields are written by a product of 3-point couplings of matter fields. We take the level 3 finite modular group, Γ3 for the flavor symmetry, and discuss the dipole operators at nearby fixed point τ = i, where observed lepton masses and mixing angles are well reproduced. Suppose the anomaly of the anomalous magnetic moment of the muon to be evidence of the new physics (NP), we have related it with (g − 2)e, LFV decays, and the electron EDM. It is found that the NP contribution to (g − 2)e is proportional to the lepton masses squared likewise the naive scaling. We also discuss the correlations among the LFV processes μ → eγ, τ → μγ and τ → eγ, which are testable in the future. The electron EDM requires the tiny imaginary part of the relevant Wilson coefficient in the basis of real positive charged lepton masses, which is related to the μ → eγ transition in our framework.

4D modular flavor symmetric models inspired by a higher-dimensional theory

We study a scenario to derive four-dimensional modular flavor symmetric models from a higher-dimensional theory by assuming the compactification consistent with the modular symmetry. In our scenario, wave functions in extra-dimensional compact space are modular forms. That leads to constraints on combinations between modular weights and ${\mathrm{\ensuremath{\Gamma}}}_{N}$ (${\mathrm{\ensuremath{\Gamma}}}_{N}^{\ensuremath{'}}$) representations of matter fields. We also present illustrating examples.

Natural mass hierarchy among three heavy Majorana neutrinos for resonant leptogenesis under modular A4 symmetry

It is clear that matter is dominant in the Universe compared to antimatter. We call this problem baryon asymmetry. The baryon asymmetry is experimentally determined by both cosmic microwave background and big bang nucleosynthesis measurements. To resolve the baryon number asymmetry of the Universe as well as neutrino oscillations, we study a radiative seesaw model in a modular A4 symmetry. Degenerate heavy Majorana neutrino masses can be naturally realized in an appropriate assignments under modular A4 with large imaginary part of modulus τ, and it can induce measured baryon number via resonant leptogenesis that is valid in around TeV scale energy theory. We also find that the dominant contribution to the CP asymmetry arises from Re[τ] through our numerical analysis satisfying the neutrino oscillation data.

Modular symmetry anomaly and nonperturbative neutrino mass terms in magnetized orbifold models

We study the modular symmetry anomaly in magnetized orbifold models. The non-perturbative effects such as D-brane instanton effects can break tree-level symmetry. We study which part of the modular symmetry is broken explicitly by Majorana mass terms with three generations of neutrinos. The modular weight of neutrino mass terms does not match with other coupling terms in tree-level Lagrangian. In addition, the $Z_N$ symmetry of the modular flavor symmetry is broken and a certain normal subgroup of the modular flavor symmetry remains in neutrino mass terms.

Implications of A4 modular symmetry on neutrino mass, mixing and leptogenesis with linear seesaw

The present work is inspired by the application of A 4 modular symmetry in the linear seesaw framework, which restricts the use of multiple flavon fields. Linear seesaw is realized with six heavy S U ( 2 ) L singlet fermion superfields and a weighton in a supersymmetric framework. The non-trivial transformation of Yukawa couplings under the A 4 modular symmetry helps to explore the neutrino phenomenology with a specific flavor structure of the mass matrix. We discuss the phenomena of neutrino mixing and show that the obtained mixing angles and CP violating phase in this framework are compatible with the observed 3 σ range of the current oscillation data. In addition, we also investigate the non-zero CP asymmetry from the decay of lightest heavy fermion superfield to explain the preferred phenomena of baryogenesis through leptogenesis including flavor effects.

Lepton sector in modular A4 and gauged U(1)R symmetry

We propose a lepton model under modular A4 and gauged U(1)R symmetries in a supersymmetric framework, in which the neutrino masses are induced at one-loop level. Thanks to the modular A4 symmetry, we have several predictions on the lepton sector, especially, on fixed points of the vacuum expectation value of the modulus τ=i,ω(≡e2πi/3),i∞ each of which has remnant symmetry; Z2 for τ=i and Z3 for τ=ω,i∞. These points are favored by a string theory and are phenomenologically interesting. We examine favored parameter spaces in the models and also show that the lepton flavor violation in the models is smaller enough than the current experimental constraints.

Linear Seesaw in A5′ Modular Symmetry With Leptogenesis

In this study, we investigate the implication of modular Γ5′≃A5′ symmetry on neutrino oscillation phenomenology in the linear seesaw framework. In order to achieve the well-defined mass structure for the light active neutrinos as dictated by the linear seesaw mechanism, we introduce six heavy fermion fields along with a pair of weightons to retain the holomorphic nature of the superpotential. The notable feature of modular symmetry is that it reduces the usage of flavon fields significantly. In addition, the Yukawa couplings transform non-trivially under the flavor symmetry group and are expressed in terms of the Dedekind eta functions, the q expansion of which renders numerical simplicity in calculations. We demonstrate that the model framework diligently accommodates all the neutrino oscillation data. Alongside, we also investigate the effect of CP asymmetry generated from the decay of lightest heavy fermions to explain the observed baryon asymmetry through the phenomenon of leptogenesis.

Two A4 modular symmetries for Tri-Maximal 2 mixing

We construct a lepton flavour model based on two A4 modular symmetries. The two A4 are broken by a bi-triplet field to the diagonal A4 subgroup, resulting in an effective modular A4 flavour symmetry with two moduli. We employ these moduli as stabilisers, that preserve distinct residual symmetries, enabling us to obtain Tri-Maximal 2 (TM2) mixing with a minimal field content, flavonless at the effective scale, below the breaking to the single A4.

Cationic vacancies as defects in honeycomb lattices with modular symmetries

Layered materials tend to exhibit intriguing crystalline symmetries and topological characteristics based on their two dimensional (2D) geometries and defects. We consider the diffusion dynamics of positively charged ions (cations) localized in honeycomb lattices within layered materials when an external electric field, non-trivial topologies, curvatures and cationic vacancies are present. The unit (primitive) cell of the honeycomb lattice is characterized by two generators, J_1, J_2 in mathrm{SL}_2({mathbb {Z}}) of modular symmetries in the special linear group with integer entries, corresponding to discrete re-scaling and rotations respectively. Moreover, applying a 2D conformal metric in an idealized model, we can consistently treat cationic vacancies as topological defects in an emergent manifold. The framework can be utilized to elucidate the molecular dynamics of the cations in exemplar honeycomb layered frameworks and the role of quantum geometry and topological defects not only in the diffusion process such as prediction of conductance peaks during cationic (de-)intercalation process, but also pseudo-spin and pseudo-magnetic field degrees of freedom on the cationic honeycomb lattice responsible for bilayers.

Modular symmetry in the SMEFT

We study the modular symmetric standard model effective field theory. We employ the stringy Ansatz on coupling structure that 4-point couplings ${y}^{(4)}$ of matter fields are written by a product of 3-point couplings ${y}^{(3)}$ of matter fields, i.e., ${y}^{(4)}={y}^{(3)}{y}^{(3)}$. In this framework, we discuss the flavor structure of bilinear fermion operators and 4-fermion operators, where the holomorphic and antiholomorphic modular forms appear. From the Ansatz, the ${A}_{4}$ modular-invariant semileptonic four-fermion operator $[{\overline{E}}_{R}\mathrm{\ensuremath{\Gamma}}{E}_{R}][{\overline{D}}_{R}\mathrm{\ensuremath{\Gamma}}{D}_{R}]$ does not lead to the flavor changing (FC) processes since this operator would be constructed in terms of gauge couplings $g$ as ${y}^{(3)}\ensuremath{\sim}g$. The chirality flipped bilinear operator $[{\overline{D}}_{R}\mathrm{\ensuremath{\Gamma}}{D}_{L}]$ also does not lead FC if the mediated mode corresponds to the Higgs boson ${H}_{d}$. In this case, the flavor structure of this operator is the exactly same as the mass matrix. On the other hand, if the flavor structure of the operator is not the exactly same as the mass matrix, the situation would change drastically. Then, we obtain the nontrivial relations of FC transitions at nearby fixed points $\ensuremath{\tau}=i,\ensuremath{\omega},i\ensuremath{\infty}$, which are testable in the future. As an application, we discuss the relations of the lepton flavor violation processes $\ensuremath{\mu}\ensuremath{\rightarrow}e\ensuremath{\gamma},\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\mu}\ensuremath{\gamma}$ and $\ensuremath{\tau}\ensuremath{\rightarrow}e\ensuremath{\gamma}$ at nearby ${\ensuremath{\tau}}_{e}=i$, where the successful lepton mass matrix was obtained. We also study the flavor changing 4-quark operators in the ${A}_{4}$ modular symmetry of quarks. As a result, the minimal flavor violation could be realized by taking relevant specific parameter sets of order one.

Top-down anatomy of flavor symmetry breakdown

A top-down approach to the flavor puzzle leads to eclectic flavor groups which include modular and traditional flavor symmetries. Based on examples of semirealistic T2/Z3 orbifold compactifications of heterotic string theory, we discuss the breakdown patterns of the eclectic flavor group via the interplay of vacuum expectation values (vevs) of moduli and flavon fields. This leads to an attractive flavor scheme with various possibilities to obtain "flavor hierarchies" through the alignment of these vevs. Despite the fact that the top-down approach gives strong restrictions for bottom-up flavor model building, it seems to be well suited to provide a realistic flavor pattern for quarks and leptons.

Linear seesaw model with a modular S 4 flavor symmetry * *Supported by an appointment to the JRG Program at the APCTP through the Science and Technology Promotion Fund and Lottery Fund of the Korean Government

We discuss a linear seesaw model with a field content as minimum as possible, introducing a modular using gauged symmetries. Owing to the rank two neutrino mass matrix, we obtain a vanishing neutrino mass eigenvalue, and only the normal mass hierarchy of neutrinos is favored via the modular symmetry. In our numerical analysis, we especially determine a relatively sharp prediction on the sum of neutrino masses to be approximately meV, in addition to other predictions.

Modular flavor symmetry and vector-valued modular forms

We revisit the modular flavor symmetry from a more general perspective. The scalar modular forms of principal congruence subgroups are extended to the vector-valued modular forms, then we have more possible finite modular groups including ΓN and {Gamma}_N^{prime } as the flavor symmetry. The theory of vector-valued modular forms provides a method of differential equation to construct the modular multiplets, and it also reveals the simple structure of the modular invariant mass models. We review the theory of vector-valued modular forms and give general results for the lower dimensional vector-valued modular forms. The general finite modular groups are listed up to order 72. We apply the formalism to construct two new lepton mass models based on the finite modular groups A4× Z2 and GL(2, 3).

Modular flavour symmetries and modulus stabilisation

We study the problem of modulus stabilisation in the framework of the modular symmetry approach to the flavour problem. By analysing simple UV-motivated CP-invariant potentials for the modulus τ we find that a class of these potentials has (non-fine-tuned) CP-breaking minima in the vicinity of the point of {mathrm{mathbb{Z}}}_3^{ST} residual symmetry, τ ≃ e2πi/3. Stabilising the modulus at these novel minima breaks spontaneously the CP symmetry and can naturally explain the mass hierarchies of charged leptons and possibly of quarks.