arXiv subjects

Cai-Chang Li

Publications and source records attributed to Cai-Chang Li.

At least 19 recordsLinked to original sources

arXiv2026

Lepton mixing from the $\Delta(96)$ Modular Littlest Seesaw

We perform the first comprehensive and model independent study of Modular Littlest Seesaw models based on the finite modular group $\Delta(96)$. We construct the vector-valued modular forms (VVMFs) for all irreducible representations of modular $\Delta(96)$, classify the inequivalent symmetry-preserving fixed points, and derive the corresponding alignments of the low-weight and next-to-lowest-weight triplet VVMFs. These results allow an exhaustive scan over the residual symmetries in the charged lepton, atmospheric neutrino, and solar neutrino sectors. We identify 35 phenomenologically viable and inequivalent breaking patterns, including 21 with normal ordering and 14 with inverted ordering. The resulting Dirac neutrino mass matrices go beyond the conventional CSD$(n)$ structure, yielding new fixed PMNS columns and novel correlations among the lepton mixing parameters beyond the TM$_1$ paradigm. The viable models are highly predictive, giving narrow ranges for neutrino masses, mixing parameters and CP phases, and can be stringently tested by upcoming experiments such as JUNO, DUNE and T2HK.

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arXiv2026

Non-holomorphic $S^{\prime}_{4}$ modular symmetry for leptons and leptogenesis

We perform a comprehensive and systematic investigation of lepton models based on the non-holomorphic $S^{\prime}_{4}$ modular symmetry, by using level 4 polyharmonic Maa{\ss} forms spanning integer weights from $-4$ to $6$. The light neutrino masses are generated by the type-I seesaw mechanism with two right-handed neutrinos, no flavon fields other than the modulus $\tau$ is introduced, and the generalized CP symmetry is not imposed. An exhaustive numerical analysis yields 36 viable models with only four real couplings besides the modulus $\tau$ when neutrino masses are normal ordering. They are classified into three categories, each containing twelve models which yield quite similar predictions for lepton observables and are distinguished by the assignment of $E^c_1$. Furthermore, we perform a detailed numerical analysis for one representative model from each category. These representative models are found to yield very sharp predictions for neutrino masses and mixing parameters, and they are distinguished by the predictions for the atmospheric mixing angle $\theta_{23}$, the Dirac CP phase $\delta_{CP}$ and the Majorana CP phase $\alpha_{21}$. Furthermore, we find that only two of these three representative models accommodate successful thermal leptogenesis in the unflavored regime, reproducing the observed baryon asymmetry with the identical parameter values that satisfy neutrino oscillation data. In these models, the real part of the modulus $\tau$ is the unique source of CP violation in both lepton mixing and leptogenesis.

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arXiv2025

Discrete flavour and CP symmetries in light of JUNO and neutrino global fit

Working within the reference three-neutrino mixing framework, we confront the lepton mixing predictions derived using non-Abelian discrete flavour and CP symmetries with the first JUNO data on the solar neutrino mixing parameters $\sin^2θ_{12}$ and with the results of the latest global neutrino data analysis. We focus on symmetry breaking patterns for which the lepton PMNS mixing matrix depends only on one or two free real parameters. Performing a comprehensive statistical analysis in each of the considered cases, we report the best fit values, the $3σ$ C.L. allowed ranges and the $χ^2$-distributions of the lepton mixing observables - the three mixing angles and the three CP-violation phases. We find that the JUNO measurements can disfavour or rule out a number of the mixing patterns associated with specific types of breaking of the discrete flavour and CP symmetries. The synergy of JUNO, DUNE and T2HK data can provide an exhaustive test of the considered approach to lepton mixing based on non-Abelian discrete lepton flavour symmetries combined with the CP symmetry.

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arXiv2025

Minimal lepton models with non-holomorphic modular $A_{4}$ symmetry

We present a comprehensive bottom-up analysis of lepton mass and mixing based on the non-holomorphic $A_{4}$ modular symmetry. Neutrinos are assumed to be Majorana particles and the light neutrino masses are generated through the Weinberg operator. In this framework, we construct all phenomenologically viable models with minimal number of free parameters, where the Yukawa couplings are expressed in terms of polyharmonic Maaß forms of weights $\pm4$, $\pm2$ and $0$ at level $N=3$. Without imposing generalized CP (gCP) symmetry, we identify 147 (6) viable models with seven real free parameters that successfully reproduce the current experimental data of lepton sector for the normal (inverted) mass ordering. When gCP symmetry consistent with $A_{4}$ modular symmetry is included, the number of free parameters is reduced by one, yielding 47 (5) phenomenologically viable models in the normal (inverted) mass ordering. Finally, we present detailed numerical analyses of a representative model for both mass orderings to illustrate these results.

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arXiv2025

Neutrino mixing parameters and masses from $Δ(96)\rtimes H_{CP}$ in the tri-direct CP approach

We present a comprehensive model independent analysis of all breaking patterns resulting from $Δ(96)\rtimes H_{CP}$ in the tri-direct CP approach of the minimal seesaw model with two right-handed neutrinos. The three generations of left-handed lepton doublets are assumed to transform as the irreducible triplet $\bm{3_{0}}$ of $Δ(96)$, and the two right-handed neutrinos are assigned to singlets. In the case that both flavon fields $ϕ_{\text{atm}}$ and $ϕ_{\text{sol}}$ transform as triplet $\bm{\bar{3}_{0}}$, only one phenomenologically viable lepton mixing pattern is obtained for normal ordering neutrino masses. The lepton mixing matrix is predicted to be the TM1 pattern, with neutrino masses, mixing angles, and CP violation phases depending on only three real input parameters. When $ϕ_{\text{sol}}$ is assigned to the $\bm{\bar{3}_{1}}$ representation, an additional real parameter $x$ must be included. Then we find 42 (12) independent phenomenologically interesting mixing patterns for normal (inverted) ordering neutrino masses, and the corresponding predictions for lepton mixing parameters and neutrino masses are obtained. Furthermore, we perform a detailed numerical analysis for five (one) example breaking patterns with some benchmark values of $x$ for normal (inverted) ordering. For the five normal examples, the absolute values of the first columns of the PMNS matrix are fixed to be $\left(\sqrt{\frac{2}{3}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}\right)^{T}$, $\frac{1}{5}\left(\sqrt{17},2,2\right)^{T}$, $\frac{1}{\sqrt{38}}\left(5,2,3\right)^{T}$, $\frac{1}{\sqrt{57}}\left(\sqrt{37},\sqrt{10},\sqrt{10}\right)^{T}$ and $\frac{1}{3}\left(\sqrt{6},1,\sqrt{2}\right)^{T}$, respectively. For the inverted example, the absolute value of the third column of the PMNS matrix is $\frac{1}{2\sqrt{11}}\left(1,5,3\sqrt{2}\right)^{T}$.

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arXiv2025

Lepton models from non-holomorphic $A^{\prime}_{5}$ modular flavor symmetry

In the framework of non-holomorphic modular invariance approach, we have systematically constructed all minimal lepton models based on the non-holomorphic $A^{\prime}_{5}$ modular symmetry from a bottom-up approach. In these models, the Yukawa couplings are described by polyharmonic Maaß forms of integer weights at level $N=5$. Under the assumption of Majorana neutrinos, both the Weinberg operator and the type-I seesaw mechanism are considered for neutrino mass generation. All minimal models are found to be based on generalized CP (gCP) symmetry, and each of them depends on five real dimensionless parameters and two overall scales. Through comprehensive numerical scanning, we obtain 6 (4) phenomenologically viable Weinberg operator models and 94 (76) phenomenologically viable seesaw models for normal (inverted) ordering neutrino masses. For each viable model, we present predictions for key neutrino properties, such as lepton masses, CP violation phases, mixing angles, effective Majorana mass for neutrinoless double beta decay and the kinematical mass in beta decay. Furthermore, we provide detailed numerical analysis for two representative models to illustrate our results.

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arXiv2024

Minimal eclectic flavor group $Q_{8}\rtimes S_3$ and neutrino mixing

We perform a comprehensive analysis of the minimal eclectic flavor group $Q_{8}\rtimes S_3$ which is isomorphic to $GL(2,3)$, and all its irreducible representations are induced from the irreducible representations of $Q_{8}$ and $S_{3}$. The consistency conditions between EFG and generalized CP (gCP) symmetry are revisited, and we find the gCP symmetry compatible with the minimal EFG $Q_{8}\rtimes S_3$. The most general forms of Kähler potential and superpotential based on $Q_{8}\rtimes S_3$ are discussed, and the corresponding fermion mass matrices are presented. A concrete lepton model invariant under $Q_{8}\rtimes S_3$ and gCP is constructed, in which the experimental data of all six lepton masses and six mixing parameters can be successfully described through seven real input parameters. The model predicts a vanishing effective mass $m_{ββ}$ in neutrinoless double beta decay.

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arXiv2024

Non-holomorphic modular $A_{5}$ symmetry for lepton masses and mixing

We perform a comprehensive bottom-up study of all the simplest lepton models based on non-holomorphic $A_{5}$ modular flavor symmetry, in which neutrinos are assumed to be Majorana particles and their masses are generated by the Weinberg operator or the type I seesaw mechanism. In the case that the generalized CP (gCP) symmetry is not considered, we find that 21 Weinberg operator models and 174 seesaw models can accommodate the experimental data in lepton sector, and all of them depend on six dimensionless free parameters and two overall scales. If gCP symmetry compatible with $A_{5}$ modular symmetry is imposed, one more free parameter would be reduced. Then only 4 of the 21 Weinberg operator models and 100 of the 174 seesaw models agree with the experimental data on lepton masses and mixing parameters. Furthermore, we perform a detailed numerical analysis for two example models for illustration.

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arXiv2024

Eclectic flavor group $Δ(27)\rtimes S_3$ and lepton model building

We have performed a systematical study of the eclectic flavor group $Δ(27)\rtimes S_3$ which is the extension of the traditional flavor symmetry $Δ(27)$ by the finite modular symmetry $S_3$. Consistency between $Δ(27)$ and $S_3$ requires that the eight nontrivial singlet representations of $Δ(27)$ should be arranged into four reducible doublets. The modular transformation matrices are determined for various $Δ(27)$ multiplets, and the CP-like symmetry compatible with $Δ(27)\rtimes S_3$ are discussed. We study the general form of the Kähler potential and superpotential invariant under $Δ(27)\rtimes S_3$, and the corresponding fermion mass matrices are presented. We propose a bottom-up model for lepton masses and mixing based on $Δ(27)\rtimes S_{3}$, a numerical analysis is performed and the experimental data can be accommodated.

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arXiv2023

Neutrino Mass and Mixing Models with Eclectic Flavor Symmetry $Δ(27) \rtimes T'$

The Kähler potentials of modular symmetry models receive unsuppressed contributions which may be controlled by a flavor symmetry, where the combination of the two symmetry types is referred to as eclectic flavor symmetry. After briefly reviewing the consistency conditions of eclectic flavor symmetry models, including with generalised (g)CP, we perform a comprehensive bottom-up study of eclectic flavor symmetry models based on $Ω(1)\cong Δ(27)\rtimes T^\prime$, consisting of the flavor symmetry $Δ(27)$ in a semi-direct product with the modular symmetry $T^\prime$. The modular transformations of different $Δ(27)$ multiplets are given by solving the consistency condition. The eight nontrivial singlets of $Δ(27)$ are related by $T'$ modular symmetry, and they have to be present or absent simultaneously in any $Ω(1)$ model. The most general forms of the superpotential and Kähler potential invariant under $Ω(1)$ are discussed, and the corresponding fermion mass matrices are presented. Based on the eclectic flavor group $Ω(1)$, two concrete lepton models which can successfully describe the experimental data of lepton masses and mixing parameters are constructed. For the two models without gCP, all six mixing parameters vary in small regions. A nearly maximal atmospheric mixing angle $θ_{23}$ and Dirac CP phase $δ_{CP}$ are obtained in the first model. After considering the compatible gCP symmetry and the assumption of $\Re τ=0$ in the first model, the $μ-τ$ reflection symmetry is preserved in the charged lepton diagonal basis. As a consequence, the atmospheric mixing angle and Dirac CP phase are predicted to be maximal, and two Majorana CP phases are predicted to be $π$.

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arXiv2021

Modular symmetry at level 6 and a new route towards finite modular groups

We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as $Γ(N')/Γ(N")$, and the modular group $SL(2,\mathbb{Z})$ is extended to a principal congruence subgroup $Γ(N')$. The original modular invariant theory is reproduced when $N'=1$. We perform a comprehensive study of $Γ'_6$ modular symmetry corresponding to $N'=1$ and $N"=6$, five types of models for lepton masses and mixing with $Γ'_6$ modular symmetry are discussed and some example models are studied numerically. The case of $N'=2$ and $N"=6$ is considered, the finite modular group is $Γ(2)/Γ(6)\cong T'$, and a benchmark model is constructed.

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arXiv2020

Modular Invariant Models of Leptons at Level 7

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 triplet, a septet and two octets of $Γ_7$. A full list of modular forms up to weight 8 are provided. Assuming the absence of flavons, the simplest modular-invariant models based on $Γ_7$ are constructed, in which neutrinos gain masses via either the Weinberg operator or the type-I seesaw mechanism, and their predictions compared to experiment.

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arXiv2020

A New Littlest Seesaw Model

We propose and discuss a new Littlest Seesaw model, realised in the tri-direct CP approach, in which the couplings of the two right-handed neutrinos to the lepton doublets are proportional to $(0,-1,1)$ and $(1,5/2,-1/2)$ respectively with the relative phase $η=-π/2$. This model can give an excellent description of lepton flavour mixing, including an atmospheric neutrino mixing angle in the second octant, in terms of only two input parameters. We show that the observed baryon asymmetry can be generated for the lightest right-handed neutrino mass $M_{1}=1.176\times 10^{11}$ GeV in SM and $M_{1}=3.992\times 10^{10}$ GeV in MSSM with $\tanβ=5$. We construct an explicit Littlest Seesaw model based on the flavour symmetry $S_4\times Z_4\times Z_9$ in which the desired alignments and the phase $η=-π/2$ are achieved.

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arXiv2018

Tri-Direct CP in the Littlest Seesaw Playground

We discuss spontaneously broken CP symmetry in two right-handed neutrino models based on the idea of having a {\it different residual flavour symmetry}, together with a {\it different residual CP symmetry}, associated with each of the two right-handed neutrinos. The charged lepton sector also has a {\it different residual flavour symmetry}. In such a {\it tri-direct CP approach}, we show that the combination of the three residual flavour and two residual CP symmetries provides a new way of fixing the parameters. To illustrate the approach, we revisit the Littlest Seesaw (LSS) model based on $S_4$ and then propose new variants which have not so far appeared in the literature, with different predictions for each variant. We analyse numerically the predictions of the new variants, and then propose an explicit model which can realise one of the successful benchmark points, based on the atmospheric flavon vacuum alignment $(1, ω^2 , ω)$ and the solar flavon vacuum alignment $(1, -7/2, -7/2 )$.

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arXiv2018

Lepton Mixing Predictions from $S_4$ in the Tri-Direct CP approach to Two Right-handed Neutrino Models

We perform an exhaustive analysis of all possible breaking patterns arising from $S_4\rtimes H_{CP}$ in a new {\it tri-direct CP approach} to the minimal seesaw model with two right-handed neutrinos, and construct a realistic flavour model along these lines. According to this approach, separate residual flavour and CP symmetries persist in the charged lepton, `atmospheric' and `solar' right-handed neutrino sectors, i.e. we have {\it three} symmetry sectors rather than the usual two of the {\it semi-direct CP approach} (charged leptons and neutrinos). Following the {\it tri-direct CP approach}, we find twenty-six kinds of independent phenomenologically interesting mixing patterns. Eight of them predict a normal ordering (NO) neutrino mass spectrum and the other eighteen predict an inverted ordering (IO) neutrino mass spectrum. For each phenomenologically interesting mixing pattern, the corresponding predictions for the PMNS matrix, the lepton mixing parameters, the neutrino masses and the effective mass in neutrinoless double beta decay are given in a model independent way. One breaking pattern with NO spectrum and two breaking patterns with IO spectrum corresponds to form dominance. We find that the lepton mixing matrices of three kinds of breaking patterns with NO spectrum and one form dominance breaking pattern with IO spectrum preserve the first column of the tri-bimaximal (TB) mixing matrix, i.e. yield a TM1 mixing matrix.

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arXiv2017

Implications of residual CP symmetry for leptogenesis in a model with two right-handed neutrinos

We analyze the interplay between leptogenesis and residual symmetry in the framework of two right-handed neutrino model. Working in the flavor basis, we show that all the leptogenesis CP asymmetries are vanishing for the case of two residual CP transformations or a cyclic residual flavor symmetry in the neutrino sector. If a single remnant CP transformation is preserved in the neutrino sector, the lepton mixing matrix is determined up to a real orthogonal matrix multiplied from the right side. The $R$-matrix is found to depend on only one real parameter, it can take three viable forms, and each entry is either real or purely imaginary. The baryon asymmetry is generated entirely by the CP violating phases in the mixing matrix in this scenario. We perform a comprehensive study for the $Δ(6n^2)$ flavor group and CP symmetry which are broken to a single remnant CP transformation in the neutrino sector and an abelian subgroup in the charged lepton sector. The results for lepton flavor mixing and leptogenesis are presented.

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arXiv2017

Toward a unified interpretation of quark and lepton mixing from flavor and CP symmetries

We discussed the scenario that a discrete flavor group combined with CP symmetry is broken to $Z_2\times CP$ in both neutrino and charged lepton sectors. All lepton mixing angles and CP violation phases are predicted to depend on two free parameters $θ_{l}$ and $θ_ν$ varying in the range of $[0, π)$. As an example, we comprehensively study the lepton mixing patterns which can be derived from the flavor group $Δ(6n^2)$ and CP symmetry. Three kinds of phenomenologically viable lepton mixing matrices are obtained up to row and column permutations. We further extend this approach to the quark sector. The precisely measured quark mixing angles and CP invariant can be accommodated for certain values of the free parameters $θ_{u}$ and $θ_{d}$. A simultaneous description of quark and lepton flavor mixing structures can be achieved from a common flavor group $Δ(6n^2)$ and CP, and accordingly the smallest value of the group index $n$ is $n=7$.

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arXiv2017

Golden Littlest Seesaw

We propose and analyse a new class of Littlest Seesaw models, with two right-handed neutrinos in their diagonal mass basis, based on preserving the first column of the Golden Ratio mixing matrix. We perform an exhaustive analysis of all possible remnant symmetries of the group $A_5$ which can be used to enforce various vacuum alignments for the flavon controlling solar mixing, for two simple cases of the atmospheric flavon vacuum alignment. The solar and atmospheric flavon vacuum alignments are enforced by {\em different} remnant symmetries. We examine the phenomenological viability of each of the possible Littlest Seesaw alignments in $A_5$, which preserve the first column of the Golden ratio mixing matrix, using figures and extensive tables of benchmark points and comparing our predictions to a recent global analysis of neutrino data. We also repeat the analysis for an alternative form of Golden Ratio mixing matrix.

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