In modular-invariant models of flavour, hierarchical fermion mass matrices may arise solely due to the proximity of the modulus $τ$ to a point of residual symmetry. This mechanism does not require flavon fields, and modular weights are not analogous to Froggatt-Nielsen charges. Instead, we show that hierarchies depend on the decomposition of field representations under the residual symmetry group. We systematically go through the possible fermion field representation choices which may yield hierarchical structures in the vicinity of symmetric points, for the four smallest finite modular groups, isomorphic to $S_3$, $A_4$, $S_4$, and $A_5$, as well as for their double covers. We find a restricted set of pairs of representations for which the discussed mechanism may produce viable fermion (charged-lepton and quark) mass hierarchies. We present two lepton flavour models in which the charged-lepton mass hierarchies are naturally obtained, while lepton mixing is somewhat fine-tuned. After formulating the conditions for obtaining a viable lepton mixing matrix in the symmetric limit, we construct a model in which both the charged-lepton and neutrino sectors are free from fine-tuning.
We discuss fermion mass hierarchies within modular invariant flavour models. We analyse the neighbourhood of the self-dual point $τ=i$, where modular invariant theories possess a residual $Z_4$ invariance. In this region the breaking of $Z_4$ can be fully described by the spurion $ε\approx τ- i$, that flips its sign under $Z_4$. Degeneracies or vanishing eigenvalues of fermion mass matrices, forced by the $Z_4$ symmetry at $τ=i$, are removed by slightly deviating from the self-dual point. Relevant mass ratios are controlled by powers of $|ε|$. We present examples where this mechanism is a key …
We propose models to explain the hierarchies of the quark masses and mixing by utilizing the $S_4^\prime$ modular flavor symmetry. The hierarchy is realized by the modulus $τ$ stabilized at $\mathrm{Im}\,τ\gg 1$, where the residual $Z_4^T$ symmetry is approximately unbroken and the Froggatt-Nielsen mechanism works. It is found that the quark hierarchies are realized only in a few cases of quark representations. We study two models with assigning the modular weights, so that the observed quark hierarchies are explained in the cases of both small and large ratios of the top to bottom Yukawa coup…
We show how quark and lepton mass hierarchies can be reproduced in the framework of modular symmetry. The mechanism is analogous to the Froggatt-Nielsen (FN) mechanism, but without requiring any Abelian symmetry to be introduced, nor any Standard Model (SM) singlet flavon to break it. The modular weights of fermion fields play the role of FN charges, and SM singlet fields with non-zero modular weight called weightons play the role of flavons. We illustrate the mechanism by analysing $A_4$ (modular level 3) models of quark and lepton (including neutrino) masses and mixing, with a single modulus…
We investigate the neutrino masses and mixing patten in a version of the $SU(3)_c\otimes SU(3)_L\otimes U(1)_X$ model with one extra exotic charged lepton per family as introduced by Ozer. It is shown that an extended scalar sector, together with a discrete $Z_2$ symmetry, is able to reproduce a consistent lepton mass spectrum without a hierarchy in the Yukawa coupling constants, the former as a carefull balance between one universal see-saw and two radiative mechanisms.
We discuss the issue of fermion mass hierarchies between lepton and quark families in toy models of dynamical generation of fermion masses through minimal Lorentz Invariance Violation. Realistic hierarchies in chiral theories necessitate an extended gauge structure, which includes both vector and axial-vector interactions.
The mass hierarchies of quarks and charged leptons as well as a large $\n_\m$-$\n_\t$ mixing angle are naturally explained by the Frogatt-Nielsen mechanism with a nonparallel family structure of chiral charges. We extend this mechanism to right-handed neutrinos. Their out-of-equilibrium decay generates a cosmological baryon asymmetry whose size is quantized in powers of the hierarchy parameter $\e^2$. For the simplest hierarchy pattern the neutrino mass $\bar{m}_\n= (m_{\n_\m}m_{\n_\t})^{1/2} \sim 10^{-2}$ eV, which is inferred from present indications for neutrino oscillations, implies a bary…
Two guiding new phenomenological flavor and flavor-electroweak ideas are expounded in this paper: (I) Oppositeness relation between neutrino and Charged Lepton Mass-Degeneracy-Deviation quantities. With inputs from the mass and neutrino oscillation data, it enables two independent estimations of the neutrino masses m =(0.11-0.30)eV and m =(0.11-0.16) eV. Small value of the neutrino oscillation hierarchy parameter is a consistency condition. (II) Essential connection between lepton mass hierarchies and low energy electroweak coupling constants. It enables an independent estimation of the QD-neu…
We will show that an extension of quantum field theory, recently proposed to solve the hierarchy problem, can give an elegant explanation of quark/lepton family replications. This scenario prefers fermion mass models based on a family permutation discrete symmetry.
Lepton-flavour symmetry in the Standard Model is broken by small masses for charged leptons and neutrinos. Introducing neutrino masses via dimension-5 operators associated to lepton-number violation at a very high scale, the corresponding coupling matrix may still have entries of order 1, resembling the situation in the quark sector with large top Yukawa coupling. As we have shown recently, in such a situation one may introduce the coupling matrices between lepton and Higgs fields as non-linear representations of lepton-flavour symmetry within an effective-theory framework. This allows us to s…
Extended scalar and fermion sectors offer new opportunities for generating the observed strong hierarchies in the fermion mass and mixing patterns of the Standard Model (SM). In this work, we elaborate on the prospects of a particular extension of the Inert Higgs doublet model where the SM hierarchies are generated sequentially by radiative virtual corrections in a fully renormalisable way, i.e. without adding any non-renormalisable Yukawa terms or soft-breaking operators to the scalar potential. Our model has a potential to explain the recently observed $R_{K}$ and $R_{K^{\ast }}$ anomalies, …
Flavour symmetries of Froggatt-Nielsen type can naturally reconcile the large quark and charged lepton mass hierarchies and the small quark mixing angles with the observed small neutrino mass hierarchies and their large mixing angles. We point out that such a flavour structure, together with the measured neutrino mass squared differences and mixing angles, strongly constrains yet undetermined parameters of the neutrino sector. Treating unknown O(1) parameters as random variables, we obtain surprisingly accurate predictions for the smallest mixing angle, sin^2(2 theta_13) = 0.07 (+0.11)(-0.05),…
Local SU(5) F-theory models lead naturally to Yukawa couplings for the third generation of quarks and leptons, but inducing Yukawas for the lighter generations has proven elusive. Corrections coming from gauge fluxes fail to generate the required couplings, and naively the same applies to instanton effects. We nevertheless revisit the effect of instantons in F-theory GUT constructions and find that contributions previously ignored in the literature induce the leading non-perturbative corrections to the Yukawa couplings. We apply our results to the case of 10 x 5* x 5* couplings in local SU(5) …
We discuss a one loop model for neutrino masses which leads to a seesaw-like formula with the difference that the charged lepton masses replace the unknown Dirac mass matrix present in the usual seesaw case. This is a considerable reduction of parameters in the neutrino sector and predicts a strong hierarchical pattern in the right handed neutrino mass matrix that is easily derived from a $U(1)_H$ family symmetry. The model is based on the left-right gauge group with an additional $Z_4$ discrete symmetry which gives vanishing neutrino Dirac masses and finite Majorana masses arising at the one …
We construct a supersymmetric model based on $T'\otimes Z_3\otimes Z_9$ flavor symmetry. At the leading order, the charged lepton mass matrix is not diagonal, $T'$ is broken completely, and the hierarchy in the charged lepton masses is generated naturally. Nearly tri-bimaximal mixing is predicted, subleading effects induce corrections of order $λ^2$, where $λ$ is the Cabibbo angle. Both the up quark and down quark mass matrices textures of the well-known U(2) flavor theory are produced at the leading order, realistic hierarchies in quark masses and CKM matrix elements are obtained. The vacuum …
We study the connection between quark and lepton mass matrices in a supersymmetric SO(10) GUT model in six dimensions, compactified on an orbifold. The physical quarks and leptons are mixtures of brane and bulk states. This leads to a characteristic pattern of mass matrices and high-energy CP violating phases. The hierarchy of up and down quark masses determines the CKM matrix and most charged lepton and neutrino masses and mixings. The small hierarchy of neutrino masses is a consequence of the mismatch of the up and down quark mass hierarchies. The effective CP violating phases in the quark s…
We study the soft supersymmetry (SUSY) breaking terms due to the modulus F-term in the modular flavor models of leptons. It is found that the soft SUSY breaking terms are constrained by the modular forms, and specific patterns are derived. Those phenomenological implications are discussed in such as the lepton flavor violation, $μ\rightarrow e + γ$ decay. In order to examine numerically, two modular flavor $A_4$ models are taken. The SUSY breaking scale is significantly constrained by inputting the observed upper bound of the $μ\rightarrow e + γ$ decay. The SUSY mass scale is larger than aroun…
Motivated by effective low energy models of string origin, we discuss the neutrino masses and mixing within the context of the Minimal Supersymmetric Standard Model supplemented by a U(1) anomalous family symmetry and additional Higgs singlet fields charged under this extra U(1). In particular, we interpret the solar and atmospheric neutrino data assuming that there are only three left-handed neutrinos which acquire Majorana masses via a lepton number violating dimension-five operator. We derive the general form of the charged lepton and neutrino mass matrices when two different pairs of singl…
We explore a mechanism for generating the Yukawa hierarchies by displacing the left- and right-handed components of the Standard Model fermions in a higher-dimensional space. We find a unique configuration of displacements which fits all quark and lepton masses and mixing angles, with a prediction for the strange quark mass m_s = 120 MeV.
The minimal supersymmetric standard model with soft breaking has a large landscape of supersymmetric particle mass hierarchies. This number is reduced significantly in well-motivated scenarios such as minimal supergravity and alternatives. We carry out an analysis of the landscape for the first four lightest particles and identify at least 16 mass patterns, and provide benchmarks for each. We study the signature space for the patterns at the CERN Large Hadron Collider by analyzing the lepton + (jet $\geq 2$) + missing $P_T$ signals with 0, 1, 2 and 3 leptons. Correlations in missing $P_T$ are …
We show that a special type of colliding 7-brane configuration of a codimension-two singularity realizes not only a six-dimensional spectrum with exactly the same quantum numbers as that of the three-generation E7/(SU(5) x U(1)^3) coset family unification model, but also the three sets of nonchiral singlet pairs with precisely the correct U(1) charges needed for explaining the Yukawa hierarchies and large lepton-flavor mixings in a well-known seesaw scenario.
We propose a Clifford algebra approach to chiral symmetry breaking and fermion mass hierarchies in the context of composite Higgs bosons. Standard model fermions are represented by algebraic spinors of six-dimensional binary Clifford algebra, while ternary Clifford algebra-related flavor projection operators control allowable flavor-mixing interactions. There are three composite electroweak Higgs bosons resulted from top quark, tau neutrino, and tau lepton condensations. Each of the three condensations gives rise to masses of four different fermions. The fermion mass hierarchies within these t…
We study the lepton flavor models, whose flavor symmetries are finite subgroups of the modular group such as $S_3$ and $A_4$. In our models, couplings are also nontrivial representations of these groups and modular functions of the modulus. We study the possibilities that these models realize realistic values of neutrino masses and lepton mixing angles.
We investigate models of charged lepton and neutrino masses and lepton mixing based on broken modular symmetry. The matter fields in these models are assumed to transform in irreducible representations of the finite modular group $Γ_4 \simeq S_4$. We analyse the minimal scenario in which the only source of symmetry breaking is the vacuum expectation value of the modulus field. In this scenario there is no need to introduce flavon fields. Using the basis for the lowest weight modular forms found earlier, we build minimal phenomenologically viable models in which the neutrino masses are generate…
We study a double covering of modular $A_4$ flavor symmetry in which we construct lepton models in cases of canonical seesaw and radiative seesaw models. Through $χ$ square numerical analysis, we show some predictions for the cases, depending on normal and inverted hierarchies.
We study quark and lepton mass matrices derived from magnetized $T^2/\mathbb{Z}_2$ orbifold models. Quark and lepton masses have a large hierarchy. In addition, mixing angles are large in the lepton sector, while those are small in the quark sector. We find that this structure can be realized in certain flavor models, which are identified by the zero points of the zero-mode wave functions of fermions and Higgs modes. We classify such realistic flavor models. Fixed points $τ=i$, $e^{2πi/3}$ and $i\infty$ of the modulus $τ$ play a role in realizing a large mass hierarchy through our scenario, wh…
It is argued taking into account the observed mass spectra of the leptons and quarks that the phenomenon of flavor mixing is intimately related to the mass spectra. We discuss a particularly interesting way to describe the flavor mixing which is particularly suited for models of quark mass matrices based on flavor symmetries. CP violated is maximal or class to being maximal in our approach.
We perform a comprehensive analysis of the homogeneous finite modular group $Γ'_4\equiv S'_4$ which is the double covering of $S_4$ group. The weight 1 modular forms of level 4 are constructed in terms of Dedekind eta function, and they transform as a triplet $\mathbf{\hat{3}'}$ of $S'_4$. The integral weight modular forms until weight 6 are built from the tensor products of weight 1 modular forms. We perform a systematical classification of $S'_4$ modular models for lepton masses and mixing with/without generalized CP, where the left-handed leptons are assigned to triplet of $S'_4$ and right-…
In SO(10) models with a vectorial fourth generation of down quarks and leptons, the structure of the neutrino Majorana and Dirac mass matrices generically coincide with those of up- and down-quarks, respectively. Then the small neutrino mass hierarchy follows from the mismatch of the up and down quark mass hierarchies and we can accommodate naturally a normal hierarchy. The effective CP violating phases in the quark sector, neutrino oscillations and leptogenesis are unrelated.
We propose a quark-lepton model via leptoquarks and modular $A_4$ symmetry. Since the neutrino mass is induced at one-loop level mediated by down quarks as well as leptoquarks, we have to explain lepton and quark masses and mixings with a single modulus $τ$. Here, we find predictions for lepton and quark sectors with unified modulus $τ$, and show several constraints originating from leptoquarks.
We study a flavor model that the quark sector has the $S_3$ modular symmetry,while the lepton sector has the $A_4$ modular symmetry. Our model leads to characteristic quark mass matrices which are consistent with experimental data of quark masses, mixing angles and the CP violating phase. The lepton sector is also consistent with the experimental data of neutrino oscillations. We also study baryon and lepton number violations in our flavor model.
We study models of lepton masses and mixing based on broken modular invariance. We consider invariance under the finite modular group $Γ_4 \simeq S_4$ and focus on the minimal scenario where the expectation value of the modulus is the only source of symmetry breaking, such that no flavons need to be introduced. After constructing a basis for the lowest weight modular forms, we build two minimal models, one of which successfully accommodates charged lepton masses and neutrino oscillation data, while predicting the values of the Dirac and Majorana CPV phases.
We propose a lepton model under modular $A_4$ and gauged $U(1)_R$ symmetries, in which the neutrino masses are induced at one-loop level. Thanks to the modular $A_4$ symmetry, we have several predictions on the lepton sector, especially, on fixed points of $τ=i,ω\equiv e^{2πi/3},i\times \infty$ each of which has remnant symmetry; $Z_2$ for $τ=i$ and $Z_3$ for $τ=ω, i\infty$. These points are favored by a string theory and phenomenologically interesting.
In the presence of independent generations of leptons, I show that the same type of ambiguity in the mass spectrum arises as was discussed in ref.[1] for neutral kaons. It results from the freedom to add to their Majorana mass matrix, usually taken to be symmetric, an antisymmetric term which vanishes as soon as fermions belonging to different generations anticommute. In the simple examples proposed, dealing with two generations, this procedure introduces an extra (mass) parameter $ρ$, which is shown to connect the (CP violating) mixing angle to the hierarchy of neutrino masses. We use this op…
We consider for the first time level 7 modular invariant flavour models where the lepton mixing originates from the breaking of modular symmetry and couplings responsible for lepton masses are modular forms. The latter are decomposed into irreducible multiplets of the finite modular group $Γ_7$, which is isomorphic to $PSL(2,Z_{7})$, the projective special linear group of two dimensional matrices over the finite Galois field of seven elements, containing 168 elements, sometimes written as $PSL_2(7)$ or $Σ(168)$. At weight 2, there are 26 linearly independent modular forms, organised into a tri…
We perform a comprehensive analysis of the most distinctive and important phenomenological implications of the recently proposed mechanism of sequential loop generation of strong hierarchies in the Standard Model (SM) fermion mass spectra. This mechanism is consistently realized at the level of renormalizable interactions in an extended variant of the Inert Higgs Doublet model, possessing the additional $Z_{2}^{(1)}\times Z_{2}^{(2)}$ discrete and $U_{1X}$ gauge family symmetries, while the matter sectors of the SM are extended by means of $SU_{2L}$-singlet scalars, heavy vector-like leptons a…
We perform a systematical analysis of the $A_4$ modular models with generalized CP for the masses and flavor mixing of quarks and leptons, and the most general form of the quark and lepton mass matrices is given. The CP invariance requires all couplings real in the chosen basis and thus the vacuum expectation value of the modulus $τ$ uniquely breaks both the modular symmetry and CP symmetry. The phenomenologically viable models with minimal number of free parameters and the results of fit are presented. We find 20 models with 7 real free parameters that can accommodate the experimental data of…
Large mixing parameters of neutrino mass eigenstates in the neutrino flavor eigenstates are determined, and interpreted as closely related to the neutrino and charged lepton mass-ratio patterns. The three known seemingly different charged lepton and quasi-degenerate neutrino deviation-from-extreme hierarchies are shown to be three particular manifestations of one unifying quadratic hierarchy-rule in lepton flavor physics.
We classify a supersymmetric extension of the Standard Model by discrete symmetries originating from finite modular symmetries $Γ_N$. Since all the couplings in supersymmetric theories of finite modular symmetries $Γ_N$ are described by holomorphic modular forms with even modular weights, renormalizable and non-renormalizable operators such as baryon- and/or lepton-number violating operators are severely constrained. From the modular transformation of matter multiplets with modular weight $1/M$, we find $\mathbb{Z}_{2M}$ symmetries, including the generalized baryon and lepton parities, $R$-par…
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 $\mathbb{Z}_3^{ST}$ residual symmetry, $τ\simeq e^{2πi/3}$. 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.
We study quark and lepton masses and mixings in a double covering of modular $A_4$ flavor symmetry in which we search for common solution of a single modulus $τ$, applying chi-square numerical analysis under the as minimum framework as possible. We have found the common region of $τ$ within $5σ$ interval that is narrow in case of normal hierarchy, but not found in case of inverted hierarchy.
We study quark and lepton mass matrices with the common modulus $τ$ in the $A_4$ modular symmetry. The viable quark mass matrices are composed of modular forms of weights $2$, $4$ and $6$. It is remarked that the modulus $τ$ is close to $i$, which is a fixed point in the fundamental region of SL$(2,Z)$, and the CP symmetry is not violated. Indeed, the observed CP violation is reproduced at $τ$ which is deviated a little bit from $τ=i$. The charged lepton mass matrix is also given by using modular forms of weights $2$, $4$ and $6$, where five cases have been examined. The neutrino mass matrix i…
