Generalized CP Symmetries Research Articles

Generalized CP symmetries in three-Higgs-doublet models

We study the scalar and Yukawa sectors of three Higgs doublets models with a generalized CP symmetry. Imposing the symmetry on the quadratic and quartic couplings of the scalar potential, we show that there are only four classes of scalar potentials, merely one more than in two-Higgs-doublet models (2HDM). Two 3HDM cases are analogs of two 2HDM cases, while the other 2HDM case splits here into two distinct potentials. In 2HDM with generalized CP symmetries extended to the Yukawa sector, there are only two possible cases: the usual CP, with 18 real Yukawa couplings; and a minimal generalized CP model, with 12 real Yukawa parameters. In contrast, with three Higgs there is a rich variety of allowed models. We classify all possible Yukawa textures, showing that there are 40 possibilities, several of which have only 10 real Yukawa couplings. Published by the American Physical Society 2024

Non-holomorphic modular flavor symmetry

The formalism of non-holomorphic modular flavor symmetry is developed, and the Yukawa couplings are level N polyharmonic Maaß forms satisfying the Laplacian condition. We find that the integer (even) weight polyharmonic Maaß forms of level N can be decomposed into multiplets of the finite modular group ΓN′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Gamma}_N^{\\prime } $$\\end{document} (ΓN). The original modular invariance approach is extended by the presence of negative weight polyharmonic Maaß forms. The non-holomorphic modular flavor symmetry can be consistently combined with the generalized CP symmetry. We present three example models for lepton sector based on the Γ3 ≅ A4 modular symmetry, the charged lepton masses and the neutrino oscillation data can be accommodated very well, and the predictions for the leptonic CP violation phases and the effective Majorana neutrino mass are studied.

Neutrino mass matrices with generalized CP symmetries and texture zeros

We investigate the properties of neutrino mass matrices that incorporate texture zeros and generalized CP symmetries associated with tribimaximal mixing. By combining these approaches, we derive predictive neutrino mass matrices and explore their implications for mass hierarchies, mixing angles, and CP-violating phases. We find that the three angles defining the generalized CP symmetries have narrow allowed ranges. We also obtain distinct correlations between the three mixing angles and the CP-violating phases that distinguish the various texture patterns from one another. Moreover, we compute the effective neutrino mass for neutrinoless double beta decay and the sum of neutrino masses. Our results highlight the predictability and testability of neutrino mass matrices with generalized CP symmetry.

A simplest modular S3 model for leptons

We present minimalist constructions for lepton masses and mixing based on flavour symmetry under the modular group ΓN of lowest level N = 2. As opposed to the only existing model of Γ2 ≅ S3 formulated in a SUSY framework, the only non-SM field is the modulus τ, and a generalised CP symmetry is implemented. Charged-leptons masses are reproduced through symmetry arguments, without requiring fine-tuning of the free parameters. As a result, all lepton observables (masses and mixing) are reproduced within 1σ experimental range using a minimum of nine free real parameters (including the real and imaginary parts of the modulus). A normal ordering for the neutrino masses is predicted. We also obtain predictions for the CP violating phases: the Dirac CP phase is predicted around 1.6π, the Majorana phases lie in narrow regions near ±π. The sum of neutrino masses is within the current bound at ∼ 0.09 eV. Furthermore, we provide predictions for the neutrinoless double beta decay and tritium decay effective masses, around 20 meV. Given the reduced number of free input parameters as compared to the existing literature on modular S3, this work renews interest for a unified predictive model of quark-lepton sectors based on Γ2 ≅ S3.

A general formula by LDLT decomposition for the type-I seesaw mechanism

Abstract By performing an approximate spectral decomposition on the inverse mass matrix of the right-handed neutrinos M−1, we obtain a basis-independent formula for the type-I seesaw mechanism. Mathematically, it is based on the generalized Cholesky (or LDLT) decomposition of the symmetric matrix M−1 = LDLT, with a diagonal matrix D and a lower unitriangular matrix L. Since the diagonalization of L can be inverted without solving cubic equations, the formula will be useful in investigating the general properties of the mechanism, such as flavor symmetries, generalized CP symmetries, and fine-tunings.

Conditions of naturalness and fine-tuning for the type-I seesaw mechanism with four-zero texture

Abstract In this paper, we search for conditions under which the mass matrix of light neutrinos mν is not a result of large cancellations for the type-I seesaw mechanism with four-zero texture. For the Yukawa matrix of neutrinos Yν and heavy Majorana mass matrix MR, these conditions are written as $(Y_{\nu })_{i2} \propto (m_{\nu })_{i2} \Rightarrow (Y_{\nu })_{i2} \propto (M_{R})_{i2}$. We call them alignment conditions because they align certain rows or columns of the three neutrino mass matrices. If these conditions do not hold, the large mixing in mν is a result of fine-tuning due to the cancellation of several terms. Then they are required from the viewpoint of naturalness. They give an explanation of the seesaw invariance of four-zero texture, and place rough restrictions on flavor structures of neutrinos. Under these conditions, Yν must have a cascade hierarchy. For MR, the 12 submatrix has a similar hierarchy to Yν and mν. However, the 23 submatrix has a waterfall hierarchy without some fine-tuning. Therefore, it is likely that Yν and MR have qualitatively different flavor structures. Furthermore, since the conditions restrict CP phases of the matrix elements, they imply the existence of a universal generalized CP symmetry in the neutrino sector.

Almost exact diagonal reflection symmetries and three-zero texture

In this paper, we consider the three-zero texture with diagonal reflection symmetries in the SM. The three-zero texture has two less assumptions ((Mu)11,(Mν)11≠0) than the universal four-zero texture of mass matrices (Mf)11=(Mf)13,31=0 for f=u,d,ν,e. The texture allows diagonal reflection symmetries to be almost exact and d-e unification. They reproduce the CKM and MNS matrices with accuracies of O(10−4) and O(10−3). By using perturbative approximations of the diagonalization, two relations with good accuracy for quark mass and mixing are obtained. Although this calculation is done in a special basis, it is a general result in a sense, because other textures and generalized CP symmetries exist by some weak basis transformation.By assuming a d-e unified relation (Md∼Me), we obtain the lightest neutrino mass m1≃2.12−5.64[meV] and the effective mass of the double beta decay |mee|≃1.23−2.15[meV].

Fermion masses and mixing from the double cover and metaplectic cover of the A5 modular group

We perform a comprehensive study of the homogeneous finite modular group $A'_5$ which is the double covering of $A_5$. The integral weight and level 5 modular forms have been constructed up to weight 6 and they are decomposed into the irreducible representations of $A'_5$. Then we perform a systematical analysis of the $A'_5$ modular models for lepton masses and mixing. The phenomenologically viable models with minimal number of free parameters and the results of fit are presented. We find out 15 models with 9 real free parameters which can accommodate the experimental data of lepton sector. After including generalized CP symmetry, 9 viable models with 7 free parameters are found out. We apply $A'_5$ modular symmetry to the quark sector, and a quark-lepton unification model is given. The framework of modular invariance is extended to include the rational weight modular forms of level 5. The ring of modular forms at level 5 can be generated by two algebraically independent weight $1/5$ modular forms denoted by $F_1(\tau)$ and $F_2(\tau)$. We give the expressions of the rational weight modular forms of level 5 up to weight $3$ and arrange them into the irreducible multiplets of finite metaplectic group $\widetilde{\Gamma}_5\cong A'_5\times Z_5$. A neutrino mass model with $\widetilde{\Gamma}_5$ modular symmetry is presented, and the phenomenological predictions of the model are analyzed numerically.

Modular invariant A4 models for quarks and leptons with generalized CP symmetry

We perform a systematical analysis of the A4 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 lepton sector. We then apply A4 modular symmetry to the quark sector to explain quark masses and CKM mixing matrix, the minimal viable quark model is found to contain 10 free real parameters. Finally, we give two predictive quark-lepton unification models which use only 16 real free parameters to explain the flavor patterns of both quarks and leptons.

A natural mechanism for approximate Higgs alignment in the 2HDM

The 2HDM possesses a neutral scalar interaction eigenstate whose tree-level properties coincide with the Standard Model (SM) Higgs boson. In light of the LHC Higgs data which suggests that the observed Higgs boson is SM-like, it follows that the mixing of the SM Higgs interaction eigenstate with the other neutral scalar interaction eigenstates of the 2HDM should be suppressed, corresponding to the so-called Higgs alignment limit. The exact Higgs alignment limit can arise naturally due to a global symmetry of the scalar potential. If this symmetry is softly broken, then the Higgs alignment limit becomes approximate (although still potentially consistent with the current LHC Higgs data). In this paper, we obtain the approximate Higgs alignment suggested by the LHC Higgs data as a consequence of a softly broken global symmetry of the Higgs Lagrangian. However, this can only be accomplished if the Yukawa sector of the theory is extended. We propose an extended 2HDM with vector-like top quark partners, where explicit mass terms in the top sector provide the source of the soft symmetry breaking of a generalized CP symmetry. In this way, we can realize approximate Higgs alignment without a significant fine-tuning of the model parameters. We then explore the implications of the current LHC bounds on vector-like top quark partners for the success of our proposed scenario.

Spontaneous CP violation by modulus \u03c4 in A4 model of lepton flavors

We discuss the modular A4 invariant model of leptons combining with the generalized CP symmetry. In our model, both CP and modular symmetries are broken spontaneously by the vacuum expectation value of the modulus τ. The source of the CP violation is a non-trivial value of Re[τ] while other parameters of the model are real. The allowed region of τ is in very narrow one close to the fixed point τ = i for both normal hierarchy (NH) and inverted ones (IH) of neutrino masses. The CP violating Dirac phase δCP is predicted clearly in [98°, 110°] and [250°, 262°] for NH at 3 σ confidence level. On the other hand, δCP is in [95°, 100°] and [260°, 265°] for IH at 5 σ confidence level. The predicted ∑mi is in [82, 102] meV for NH and ∑mi = [134, 180] meV for IH. The effective mass 〈mee〉 for the 0νββ decay is predicted in [12.5, 20.5] meV and [54, 67] meV for NH and IH, respectively.

Lepton Mixing Patterns from PSL2(7) with a Generalized CP Symmetry

Lepton mixing patterns from the modular group PSL2(7) with generalized CP symmetries are studied. The residual symmetries in both charged lepton and neutrino sectors are Z2×CP. Seven types of mixing patterns at the 3σ level of the new global fit data are obtained. Among these patterns, three types of patterns can give the Dirac CP phase which is in the 1σ range of the global fit data. The effective mass of neutrinoless double-beta decay for these patterns is also examined.

New Partial Symmetries from Group Algebras for Lepton Mixing

Recent stringent experiment data of neutrino oscillations induces partial symmetries such as Z2 and Z2×CP to derive lepton mixing patterns. New partial symmetries expressed with elements of group algebras are studied. A specific lepton mixing pattern could correspond to a set of equivalent elements of a group algebra. The transformation which interchanges the elements could express a residual CP symmetry. Lepton mixing matrices from S3 group algebras are of the trimaximal form with the μ−τ reflection symmetry. Accordingly, elements of S3 group algebras are equivalent to Z2×CP. Comments on S4 group algebras are given. The predictions of Z2×CP broken from the group S4 with the generalized CP symmetry are also obtained from elements of S4 group algebras.

CP symmetries as guiding posts: Revamping tribimaximal mixing. II.

In this follow up of arXiv:1812.04663 we analyze the generalized CP symmetries of the charged lepton mass matrix compatible with the complex version of the Tri-Bi-Maximal (TBM) lepton mixing pattern. These symmetries are used to `revamp' the simplest TBM \textit{Ansatz} in a systematic way. Our generalized patterns share some of the attractive features of the original TBM matrix and are consistent with current oscillation experiments. We also discuss their phenomenological implications both for upcoming neutrino oscillation and neutrinoless double beta decay experiments.

Generalised CP symmetry in modular-invariant models of flavour

The formalism of combined finite modular and generalised CP (gCP) sym-metries for theories of flavour is developed. The corresponding consistency conditions for the two symmetry transformations acting on the modulus τ and on the matter fields are derived. The implications of gCP symmetry in theories of flavour based on modular invariance described by finite modular groups are illustrated with the example of a modular S4 model of lepton flavour. Due to the addition of the gCP symmetry, viable modular models turn out to be more constrained, with the modulus τ being the only source of CP violation.

Testing generalized CP symmetries with precision studies at DUNE

We examine the capabilities of the DUNE experiment in probing leptonic CP violation within the framework of theories with generalized CP symmetries characterized by the texture zeros of the corresponding CP transformation matrices. We investigate DUNE's potential to probe the two least known oscillation parameters, the atmospheric mixing angle $\theta_{23}$ and the Dirac CP-phase $\delta_{\rm CP}$. We fix theory-motivated benchmarks for ($ \sin^2\theta_{23}, \delta_{\rm CP} $) and take them as true values in our simulations. Assuming 3.5 years of neutrino running plus 3.5 years in the antineutrino mode, we show that in all cases DUNE can significantly constrain and in certain cases rule out the generalized CP texture zero patterns.

CP symmetries as guiding posts: revamping tri-bi-maximal mixing. Part I

We analyze the possible generalized CP symmetries admitted by the Tri-Bi-Maximal (TBM) neutrino mixing. Taking advantage of these symmetries we construct in a systematic way other variants of the standard TBM Ansatz. Depending on the type and number of generalized CP symmetries imposed, we get new mixing matrices, all of which related to the original TBM matrix. One of such “revamped” TBM variants is the recently discussed mixing matrix of arXiv:1806.03367. We also briefly discuss the phenomenological implications following from these mixing patterns.

Symmetries and mass degeneracies in the scalar sector

We explore some aspects of models with two and three SU(2) scalar doublets that lead to mass degeneracies among some of the physical scalars. In Higgs sectors with two scalar doublets, the exact degeneracy of scalar masses, without an artificial fine-tuning of the scalar potential parameters, is possible only in the case of the inert doublet model (IDM), where the scalar potential respects a global U(1) symmetry that is not broken by the vacuum. In the case of three doublets, we introduce and analyze the replicated inert doublet model, which possesses two inert doublets of scalars. We then generalize this model to obtain a scalar potential, first proposed by Ivanov and Silva, with a CP4 symmetry that guarantees the existence of pairwise degenerate scalar states among two pairs of neutral scalars and two pairs of charged scalars. Here, CP4 is a generalized CP symmetry with the property that (CP4)n is the identity operator only for integer n values that are multiples of 4. The form of the CP4-symmetric scalar potential is simplest when expressed in the Higgs basis, where the neutral scalar field vacuum expectation value resides entirely in one of the scalar doublet fields. The symmetries of the model permit a term in the scalar potential with a complex coefficient that cannot be removed by any redefinition of the scalar fields within the class of Higgs bases (in which case, we say that no real Higgs basis exists). A striking feature of the CP4-symmetric model is that it preserves CP even in the absence of a real Higgs basis, as illustrated by the cancellation of the contributions to the CP violating form factors of the effective ZZZ and ZWW vertices.

The 10−3 eV frontier in neutrinoless double beta decay

The observation of neutrinoless double beta decay would allow to establish lepton number violation and the Majorana nature of neutrinos. The rate of this process in the case of 3-neutrino mixing is controlled by the neutrinoless double beta decay effective Majorana mass |〈m〉|. For a neutrino mass spectrum with normal ordering, which is favoured over the spectrum with inverted ordering by recent global fits, |〈m〉| can be significantly suppressed. Taking into account updated data on the neutrino oscillation parameters, we investigate the conditions under which |〈m〉| in the case of spectrum with normal ordering exceeds 10−3(5×10−3)eV: |〈m〉|NO>10−3(5×10−3)eV. We analyse first the generic case with unconstrained leptonic CP violation Majorana phases. We show, in particular, that if the sum of neutrino masses is found to satisfy Σ>0.10eV, then |〈m〉|NO>5×10−3eV for any values of the Majorana phases. We consider also cases where the values for these phases are either CP conserving or are in line with predictive schemes combining flavour and generalised CP symmetries.

Neutrino predictions from generalized CP symmetries of charged leptons

We study the implications of generalized CP transformations acting on the mass matrices of charged leptons in a model-independent way. Generalized e − μ, e − τ and μ−τ symmetries are considered in detail. In all cases the physical parameters of the lepton mixing matrix, three mixing angles and three CP phases can be expressed in terms of a restricted set of independent “theory parameters” that characterize a given choice of CP transformation. This leads to implications for neutrino oscillations as well as neutrinoless double beta decay experiments.