In a class of neutrino mass models with modular flavor symmetries, it has been observed that CP symmetry is preserved at the fixed point (or stabilizer) of the modulus parameter $τ= {\rm i}$, whereas significant CP violation emerges within the neighbourhood of this stabilizer. In this paper, we first construct a viable model with the modular $A^\prime_5$ symmetry, and explore the phenomenological implications for lepton masses and flavor mixing. Then, we introduce explicit perturbations to the stabilizer at $τ= {\rm i}$, and present both numerical and analytical results to understand why a small deviation from the stabilizer leads to large CP violation. As low-energy observables are very sensitive to the perturbations to model parameters, we further demonstrate that the renormalization-group running effects play an important role in confronting theoretical predictions at the high-energy scale with experimental measurements at the low-energy scale.
Sum rules in the lepton sector provide an extremely valuable tool to classify flavour models in terms of relations between neutrino masses and mixing parameters testable in a plethora of experiments. In this manuscript we identify new leptonic sum rules arising in models with modular symmetries with residual symmetries. These models simultaneously present neutrino mass sum rules, involving masses and Majorana phases, and mixing sum rules, connecting the mixing angles and the Dirac CP-violating phase. The simultaneous appearance of both types of sum rules leads to some non-trivial interplay, fo…
We present the first complete model of the Littlest Modular Seesaw, based on two right-handed neutrinos, within the framework of multiple modular symmetries, justifying the use of multiple moduli fields which take their values at 3 specific stabilizers of $Γ_4 \simeq S_4$, including a new phenomenological possibility. Using a semi-analytical approach, we perform a $χ^2$ analysis of each case and show that good agreement with neutrino oscillation data is obtained, including predictive relations between the leptonic mixing angles and the ratio of light neutrino masses, which non-trivially agree …
We present a model of leptonic mixing based on two $A_5$ modular symmetries using the Weinberg operator. The two modular symmetries are broken to the respective diagonal $A_5$ subgroup. At the effective level, the model behaves as a model with a single $A_5$ 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.
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 …
We propose an $A_4$ modular invariant flavor model of leptons, in which both CP and modular symmetries are broken spontaneously by the vacuum expectation value of the modulus $τ$. The value of the modulus $τ$ is restricted by the observed lepton mixing angles and lepton masses for the normal hierarchy of neutrino masses at $3σ$ confidence level. The predictive Dirac CP phase $δ_{CP}$ is in the ranges $[0^\circ,50^\circ]$, $[170^\circ,175^\circ]$ and $[280^\circ,360^\circ]$ for ${\rm Re}\, [τ]<0$, and $[0^\circ,80^\circ]$, $[185^\circ,190^\circ]$ and $[310^\circ,360^\circ]$ for ${\rm Re}\,[τ]>0…
We present a flavor model with the $S_3$ modular invariance in the framework of SU(5) GUT. The $S_3$ modular forms of weights $2$ and $4$ give the quark and lepton mass matrices with a common complex parameter, the modulus $τ$. The GUT relation of down-type quarks and charged leptons is imposed by the VEV of adjoint 24-dimensional Higgs multiplet in addition to the VEVs of $5$ and $\bar 5$ Higgs multiples of SU(5). The observed CKM and PMNS mixing parameters as well as the mass eigenvalues are reproduced properly. We discuss the leptonic CP phase and the effective mass of the neutrinoless doub…
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 combine $SU(5)$ Grand Unified Theories (GUTs) with $A_4$ modular symmetry and present a comprehensive analysis of the resulting quark and lepton mass matrices for all the simplest cases. Classifying the models according to the representation assignments of the matter fields under $A_4$, we find that there are seven types of $SU(5)$ models with $A_4$ modular symmetry. We present 53 benchmark models with the fewest free parameters. The parameter space of each model is scanned to optimize the agreement between predictions and experimental data, and predictions for the masses and mixing paramet…
Modular TM$_1$ mixing in light of precision measurement in JUNO
School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院基础物理与数学学院，UCAS）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）
Minimalistic Dirac radiative neutrino mass model based on modular symmetry is proposed. We predict maximum number of observables possible including neutrino mass splittings, neutrino mass scale, lepton mixing angles, and Dirac phases in the leptonic sector with minimum number of input parameters possible. Model is capable of accommodating multicomponent dark matter, thanks to the $\emph{R}-$parity and accidental scotogenic $\mathbb{Z}_2$ discrete symmetry. Furthermore, even-though neutrinos are Dirac in our model, matter-antimatter asymmetry of the Universe is achieved via neutrinogenesis mech…
We present the first example of a grand unified theory (GUT) with a modular symmetry interpreted as a family symmetry. The theory is based on supersymmetric $SU(5)$ in 6d, where the two extra dimensions are compactified on a $T_2/\mathbb{Z}_2$ orbifold. We have shown that, if there is a finite modular symmetry, then it can only be $A_4$ with an (infinite) discrete choice of moduli, where we focus on $τ= ω=e^{i2π/3}$, the unique solution with $|τ|=1$. The fields on the branes respect a generalised CP and flavour symmetry $A_4\ltimes \mathbb{Z}_2$ which is isomorphic to $S_4$ which leads to an e…
We combine $SO(10)$ Grand Unified Theories (GUTs) with $A_4$ modular symmetry and present a comprehensive analysis of the resulting quark and lepton mass matrices for all the simplest cases. We focus on the case where the three fermion families in the 16 dimensional spinor representation form a triplet of $Γ_3\simeq A_4$, with a Higgs sector comprising a single Higgs multiplet $H$ in the ${\mathbf{10}}$ fundamental representation and one Higgs field $\overlineΔ$ in the ${\mathbf{\overline{126}}}$ for the minimal models, plus and one Higgs field $Σ$ in the ${\mathbf{120}}$ for the non-minimal m…
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 $Γ_N$ 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, $Γ_3$ 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 a…
Radiative Dirac Neutrino Masses from Modular $S_3$ Symmetry in an Axion Model
Seoul National University of Science and Technology（首尔科学技术大学）；Institute for Convergence of Basic Studies, Seoul National University of Science and Technology（首尔科学技术大学基础融合研究所）；Henan Normal University（河南师范大学）
