We generalize the modular invariance approach to include the half-integral weight modular forms. Accordingly the modular group should be extended to its metaplectic covering group for consistency. We introduce the well-defined half-integral weight modular forms for congruence subgroup $Γ(4N)$ and show that they can be decomposed into the irreducible multiplets of finite metaplectic group $\widetildeΓ_{4N}$. We construct concrete expressions of the half-integral/integral modular forms for $Γ(4)$ up to weight 6 and arrange them into the irreducible representations of $\widetildeΓ_4$. We present three typical models with $\widetildeΓ_4$ modular symmetry for neutrino masses and mixing, and the phenomenological predictions of each model are analyzed numerically.
Modular symmetries have been impeccable in neutrino and quark sectors. This motivated us, therefore, to propose a variant of scotogenic model based on modular $A_4$ symmetry to realize the neutrino mass generation at one-loop level through radiative mechanism. Alongside, the lepton flavour violating process $μ\to e γ$ and the muon $g-2$ anomaly are also addressed. The lightest Majorana fermions turn out to be potential dark matter candidates, made stable by suitable assignment of modular weights. The relic density of the same has been computed with annihilations mediated by inert scalars and n…
We propose a model with radiatively induced neutrino mass at two-loop level, applying modular $A_4$ symmetry. The neutrino mass matrix is formulated where the structure of associated couplings are restricted by the symmetry. Then we show several predictions in the lepton sector, satisfying lepton flavor violations as well as neutrino oscillation data. We also discuss muon anomalous magnetic moment and briefly comment on dark matter candidate.
In this paper, we investigate lepton mass spectra, flavor mixing and CP violation with a modular $S^{}_4$ symmetry under the assumption that neutrinos are Dirac fermions. We find that the Majorana mass term can be forbidden by adjusting the weight of the right-handed neutrino superfields. We study the scenarios where modular forms in the neutrino sector take the lowest non-trivial weight two, while those in the charged-lepton sector vary from two to six. The corrections from renormalization-group running effects to our model have also been discussed in detail. We totally obtain twelve differen…
We extend the even weight modular forms of modular invariant approach to general integral weight modular forms. We find that the modular forms of integral weights and level $N$ can be arranged into irreducible representations of the homogeneous finite modular group $Γ'_N$ which is the double covering of $Γ_N$. The lowest weight 1 modular forms of level 3 are constructed in terms of Dedekind eta-function, and they transform as a doublet of $Γ'_3 \cong T'$. The modular forms of weights 2, 3, 4, 5 and 6 are presented. We build a model of lepton masses and mixing based on $T'$ modular symmetry.
We present a neutrino mass model based on modular symmetry with the fewest input parameters to date, which successfully accounts for the 12 lepton masses and mixing parameters through 6 real free parameters including the modulus. The neutrino masses are predicted to be normal ordering, the atmospheric angle $θ_{23}$ is quite close to maximal value and the Dirac CP phase $δ_{CP}$ is about $1.34π$. We also study the soft supersymmetry breaking terms due to the modulus $F$-term in this minimal model, which are constrained to be the non-holomorphic modular forms. The radiative lepton flavor violat…
We present a comprehensive analysis of neutrino mass and lepton mixing in theories with $A_5$ modular symmetry. We construct the weight 2, weight 4 and weight 6 modular forms of level 5 in terms of Dedekind eta-functions and Klein forms, and their decomposition into irreducible representation of $A_5$. We construct all the simplest models based on $A_5$ modular symmetry, including scenarios of models with and without flavons in the charged lepton sectors. For each case, the neutrino masses can be generated through either the Weinberg operator or the type I seesaw mechanism. We perform an exhau…
We propose a predictive lepton model under a modular $S_4$ symmetry, where the neutrino mass matrix arises from a radiative seesaw at one-loop level. The tree-level mass matrix is forbidden by well-assigned modular weights, which also play an important role in stabilizing dark matter candidate due to a remnant $Z_2$ symmetry even after breaking the modular symmetry. Supposing three families of the Majorana neutrinos, right-handed charged-leptons and left-handed charged-leptons to be embedded respectively into singlet, doublet, and triplet under $S_4$, we obtain the predictive mass matrices in …
We investigate a modular $A_4$ invariant two-loop neutrino mass model in a supersymmetric framework, where we introduce new fields as minimum as possible, expecting contributions of superpartners to the neutrino masses. We successfully reproduce the neutrino oscillation data thanks to the superpartner contributions in case of normal hierarchy, and predict several observables such as phases and neutrino masses, concentrating on three specific regions at nearby fixed points of modulus $τ=i,ω,i\times \infty$, where $ω\equiv e^{2πi/3}$. These points are statistically favored in flux compactificati…
Discrete symmetries being preferred to explain the neutrino phenomenology, we chose the simplest $S_3$ group and explore the implication of its modular form on neutrino masses and mixing. Non-trivial transformations of Yukawa couplings under this symmetry, make the model phenomenologically interesting by reducing the requirement of multiple scalar fields. This symmetry imposes a specific flavor structure to the neutrino mass matrix within the framework of less frequented type III seesaw mechanism and helps to explore the neutrino mixing consistent with the current observation. Apart, we also e…
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 $A_4$ symmetry. Degenerate heavy Majorana neutrino masses can be naturally realized in an appropriate assignments under modular $A_4$ with large imaginary part of modulus $τ$, and it can induce measured baryon number via…
We propose a model based on modular $A_4$ symmetry containing a dark matter candidate, realizing radiatively induced neutrino mass at one-loop level. One finds that stability of dark matter candidate can be assured by nonzero value of modular weight and heavy neutral fermion mass hierarchies, which include dark matter under the $A_4$ triplet, are uniquely determined; $M_X\ll M_2<M_3$. Therefore we clearly identify single dark matter field which can discriminate from the other models with $A_4$ modular symmetry. Then we discuss several phenomenological aspects and show predictions on the lepton…
We study the phenomenological implications of the modular symmetry $Γ(3) \simeq A_4$ of lepton flavors facing recent experimental data of neutrino oscillations. The mass matrices of neutrinos and charged leptons are essentially given by fixing the expectation value of modulus $τ$, which is the only source of modular invariance breaking. We introduce no flavons in contrast with the conventional flavor models with $A_4$ symmetry. We classify our neutrino models along with the type I seesaw model, the Weinberg operator model and the Dirac neutrino model. In the normal hierarchy of neutrino masses…
We present a comprehensive analysis of neutrino mass and lepton mixing in theories with $A_4$ modular symmetry, where the only flavon field is the single modulus field $τ$, and all masses and Yukawa couplings are modular forms. Similar to previous analyses, we discuss all the simplest neutrino sectors arising from both the Weinberg operator and the type I seesaw mechanism, with lepton doublets and right-handed neutrinos assumed to be triplets of $A_4$. Unlike previous analyses, we allow right-handed charged leptons to transform as all combinations of $\mathbf{1}$, $\mathbf{1}'$ and $\mathbf{1}…
In this paper, we study the implications of the modular $A^{}_4$ flavor symmetry in constructing a supersymmetric minimal type-(I+II) seesaw model, in which only one right-handed neutrino and two Higgs triplets are introduced to account for the tiny neutrino masses, flavor mixing and CP violation. The right-handed neutrino as well as the Higgs triplets in this model are assigned into the trivial one-dimensional irreducible representation of the modular group $A^{}_{4}$. We show that the individual contributions to the neutrino masses from the right-handed neutrino and the Higgs triplet are com…
We construct phenomenologically viable models of lepton masses and mixing based on modular $A_4$ invariance broken to residual symmetries $\mathbb{Z}^{T}_3$ or $\mathbb{Z}^{ST}_3$ and $\mathbb{Z}^S_2$ respectively in the charged lepton and neutrino sectors. In these models the neutrino mixing matrix is of trimaximal mixing form. In addition to successfully describing the charged lepton masses, neutrino mass-squared differences and the atmospheric and reactor neutrino mixing angles $θ_{23}$ and $θ_{13}$, these models predict the values of the lightest neutrino mass (i.e., the absolute neutrino …
We discuss a linear seesaw model with as minimum field content as possible, introducing a modular $S_4$ with the help of gauged $U(1)_{B-L}$ symmetries. Due to rank two neutrino mass matrix, we have a vanishing neutrino mass eigenvalue, and only the normal mass hierarchy of neutrinos is favored through the modular $S_4$ symmetry. In our numerical $Δχ^2$ analysis, we especially find rather sharp prediction on sum of neutrino masses to be around $60$ meV in addition to the other predictions.
We discuss a linear seesaw model with local $U(1)_{B-L}$ and modular $A_4$ symmetries. The neutrino mass matrix for linear seesaw mechanism is realized by $U(1)_{B-L}$ charge assignment and the nature of modular $A_4$ symmetry. We formulate neutrino mass and carry our numerical analysis showing some predictions for observables in neutrino sector. Remarkably, the case of inverted neutrino mass ordering (IO) is realized by a specific region around $τ=ω=e^{2πi/3}$, which is favored by a string theory. Thus, our prediction would be very strong in case of IO.
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 $SU(2)_L$ 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 compati…
In this work, we have proposed a modular $A_4$ symmetric model of neutrino mass which, simultaneously, explains observed baryon asymmetry of the Universe(BAU). In minimal extension of the standard model(SM) with two right-handed neutrinos we work in a supersymmetric framework. At Type-I seesaw level, the model predicts scaling in the neutrino mass matrix. In order to have correct low energy phenomenology, we propose two possible scenarios of scale-breaking in the neutrino mass matrix emanating from Type-I seesaw. Scenario-1 is based on the dimension-5 Weinberg operator whereas scenario-2 imple…
Two-loop neutrino mass model with modular $S_4$ symmetry
Universidad Técnica Federico Santa María（费德里科·圣塔玛丽亚技术大学）；Centro Científico-Tecnológico de Valparaíso（瓦尔帕莱索科技中心）；Millennium Institute for Subatomic Physics at High-Energy Frontier (SAPHIR)（高能前沿亚原子物理千年研究所（SAPHIR））；Universidad Andres Bello（安德烈斯·贝略大学）；Vellore Institute of Technology（维洛尔理工学院）；Dzhelepov Laboratory of Nuclear Problems, JINR（杜布纳联合核研究所捷列波夫核问题实验室）；Secretaría de Ciencia, de Humanidades, Tecnología e Innovación（科学、人文、技术与创新秘书处）
We show how to realize two-zero textures of the Majorana neutrino mass matrix $M_ν$ based on modular $A_4$ invariant models without flavons. In these models, all matter fields are assigned to three inequivalent singlets, ${\bf 1}$, ${\bf 1^\prime}$ and ${\bf 1^{\prime\prime}}$, of the finite modular group $Γ_3 \simeq A_4$. Considering tensor products of the $A_4$ group, it is easy to make the charged lepton mass matrix $M_\ell$ diagonal. Since not all modular forms of a specific weight and level 3 can be arranged into three inequivalent singlets of $A_4$ simultaneously, we can always make some…
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.
We study a Zee-Babu model in a modular $A_4$ flavor symmetry, in which we search for several predictions such as phases, sum of neutrino masses, and neutrinoless double beta decay, satisfying neutrino oscillation data in a minimum framework of the charge assignments of modular weight. We perform $Δχ^2$ analysis to get our results and find $τ$ is localized nearby at one of the fixed points of $i\times \infty$ for both of normal and inverted mass hierarchies.
We propose a minimal extention of the Standard Model where neutrino masses are generated radiatively at one-loop level via Scotogenic scanario. The model is augmented with $A_4$ modular symmetry as a scotogenic and flavor symmetry. With minimal number of parameters, the model makes predictions for neutrino oscillation data, Majorana and Dirac phases, dark matter characteristics, and neutrinoless double beta decay.
We study the prediction for leptogenesis in two renormalizable supersymmetric $SO(10)\times A_4$ 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 fe…
We discuss a minimal flavour model with twin modular symmetries, leading to trimaximal TM$_1$ lepton mixing in which the first column of the tri-bimaximal lepton mixing matrix is preserved. The model involves two modular $S_4$ groups, one acting in the neutrino sector, associated with a modulus field value $τ_{SU}$ with residual $Z^{SU}_2$ symmetry, and one acting in the charged lepton sector, associated with a modulus field value $τ_{T}$ with residual $Z^{T}_3$ symmetry. Apart from the predictions of TM$_1$ mixing, the model leads to a new neutrino mass sum rule which implies lower bounds on …
We consider a flavor model based on A4 modular group to account for both lepton and quark parameters (masses and mixing). The inverse seesaw mechanism is considered to produce the light neutrino masses. Lepton masses and mixing are obtained in terms of Yukawa coupling ratios and values of the modulus τ nearby some fixed points for inverted neutrino mass hierarchy. The quark masses and mixing are arisen at the same τ values used in inverted neutrino mass hierarchy and are in agreements with the recent data.
We discuss an inverse seesaw model based on right-handed fermion specific $U(1)$ gauge symmetry and $A_4$-modular symmetry. These symmetries forbid unnecessary terms and restrict structures of Yukawa interactions which are relevant to inverse seesaw mechanism. Then we can obtain some predictions in neutrino sector such as Dirac-CP phase and sum of neutrino mass, which are shown by our numerical analysis. Besides the relation among masses of heavy pseudo-Dirac neutrino can be obtained since it is also restricted by the modular symmetry. We also discuss implications to lepton flavor violation an…
We consider the modular symmetry associated with the compactification of extra dimensions as the flavor symmetry on lepton sector. Especially, we propose a model based on the modular $A_4$ symmetry with three right-handed neutrinos and a gauge singlet Higgs, which works well in the so-called large volume limit of the extra dimensions, i.e., $\mbox{Im}τ\to \infty$ for a modulus $τ$. The right-handed neutrinos are introduced to realize the seesaw mechanism for tiny neutrino masses observed in oscillation experiments. The vacuum expectation value of the singlet Higgs gives the Majorana masses for…
We discuss type-II seesaw models adopting modular $A_4$ symmetry in supersymmetric framework. In our approach, the models are classified by the assignment of $A_4$ representations and modular weights for leptons and triplet Higgs fields. Then neutrino mass matrix is characterized by modulus $τ$ and two free parameters. Carrying out numerical analysis, we find allowed parameter sets which can fit the neutrino oscillation data. For the allowed parameter sets, we obtain the predictions in neutrino sector such as CP violating phases and the lightest neutrino mass. Finally we also show the predicti…
We make an investigation of modular $Γ^{\prime}_5 \simeq A^{\prime}_5$ group in inverse seesaw framework. Modular symmetry is advantageous because it reduces the usage of extra scalar fields significantly. Moreover, the Yukawa couplings are expressed in terms of Dedekind eta functions, which also have a $q$ expansion form, utilized to achieve numerical simplicity. Our proposed model includes six heavy fermion superfields i.e., $\mathcal{N}_{Ri}$, $\mathcal{S}_{Li}$ and a weighton. The study of neutrino phenomenology becomes simplified and effective by the usage of $A^\prime_5$ modular symmetry…
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.
In the framework of the see-saw mechanism the normal hierarchy is favorable for the neutrino mass spectrum. For this spectrum we present a detailed calculation of the half-lives of neutrinoless double beta decay for several nuclei of experimental interest. The half-lives are evaluated by considering the most comprehensive nuclear matrix elements, which were obtained within the renormalized QRPA by the Bratislava-Caltech-Tuebingen group. The dependence of the half-lives on sin^2 theta_{13}$ and the lightest neutrino mass is studied. We present also the results of the calculations of the half-li…
Predictions of Modular Symmetry Fixed Points on Neutrino Masses, Mixing, and Leptogenesis
Central University of Himachal Pradesh（喜马偕尔邦中央大学）；Indian Institute of Science Education and Research Berhampur（印度科学教育研究所布兰普分校）；Sichuan University（四川大学）
