Predictive neutrino mass textures with origin of flavor symmetries

We investigate origins of predictive one-zero neutrino mass textures in a systematic way. Here we search Abelian continuous(discrete) global symmetries, and non-Abelian discrete symmetries, and show how to realize these neutrino masses. Then we propose a concrete model involving a dark matter candidate and an extra gauge boson, and show their phenomenologies.


I. INTRODUCTION
One of the most important issues in particle physics is to solve the mystery of the flavor structure of quarks and leptons such as the generation number, mass hierarchy, mixing angles, and CP phases. Indeed, a huge number of studies have been done through various approaches. The texture Ansatz is one of the interesting approaches. (See for a review, e.g. [1].) By assuming a certain mass texture, one can derive several predictions among masses and mixing angles as well as CP phases.
The experimental data on the neutrino sector has become more precise by neutrino oscillation experiments, although there remain still unknown aspects on the neutrino sector, e.g.
the absolute values of neutrino masses and the question whether neutrinos are Majorana or Dirac fermions. Thus, it would be interesting to apply the texture Ansatz to the lepton sector. Actually, a lot of authors have historically been analyzing neutrino mass textures in various setups. For example, it is known that only seven neutrino mass patterns (two-zero textures) can predict neutrino oscillation data without conflict of current neutrino oscillation data [2] in the case that neutrinos are Majorana fermions with rank three mass matrix [3].
Recently type-I seesaw models with maximally restricted texture zeros have been systematically classified and analyzed numerically in Refs. [4,5], where charged-lepton mass matrix is assumed to be diagonal and only two families of right-handed neutrinos have Dirac mass terms with three active neutrinos. Then, the active neutrino mass matrix has one texture zero, and obviously one of active neutrinos is massless. Such patterns with one texture zero lead to several interesting predictions among neutrino masses and mixing angles. Indeed, such predictions for the normal hierarchy are not compatible with the experimental data.
Also, some of patterns with one texture zero for the inverted hierarchy are already ruled out by experiments, while others are compatible.
Although the texture Ansatz is quite interesting as mentioned above, however, it is unclear why such a pattern of mass matrix is realized. Our purpose is to explore origins of the neutrino mass textures obtained in Refs. [4,5]. In this paper, in order to realize those textures, we apply flavor symmetries such as global U(1) symmetries, discrete Abelian symmetries Z N , and non-Abelian discrete symmetries. The flavor symmetries would provide a hint to explore underlying theory beyond the standard model (SM).
Indeed non-Abelian discrete flavor symmetries have been studied by a lot of authors in order to realize the lepton masses and mixing angles as well as CP phases. (See for review Refs. [6][7][8].) Furthermore, it has been shown that some non-Abelian discrete flavor symmetries appear in superstring theory with certain compactifications. Heterotic string theory on toroidal Z N orbifolds can lead non-Abelian flavor symmetries, e.g. D 4 , and ∆(54) [9]. (See also [10,11].) 1 Similarly flavor symmetries can be realized in magnetized D-brane models and intersecting D-brane models within the framework of type II superstring theory [13,14]. In addition, these flavor symmetries may be subgroups of the modular symmetry in superstring theory [15]. Thus, flavor symmetry would make a bridge between the neutrino physics and underlying high energy physics.
The minimal non-Abelian discrete symmetry is S 3 and the next one is D 4 . Thus, in this paper we consider these S 3 and D 4 flavor symmetries as well as global U(1) symmetry to realize the neutrino mass textures obtained in Refs. [4,5]. We will show that one can realize the desired textures by the D 4 flavor symmetry and U(1) symmetry, but not by the S 3 flavor symmetries. Also it will be found that the U(1) models need more Higgs fields than the D 4 flavor models. Then, we study the D 4 flavor model by using a concrete model. 2 This paper is organized as follows. In Sec. II, we give a brief review on the neutrino mass textures classified in Refs. [4,5]. In Sec. III, we study their realization by applying Abelian symmetries and non-Abelian discrete symmetries. In Sec. IV, we propose a concrete model, in which we formulate the boson sector, fermion sector, and dark matter sector (DM), and analyze collider physics based on an additional gauge symmetry. Then we discuss the DM candidate. Finally we conclude and discuss in Sec. V.

II. NEUTRINO MASS TEXTURES
In this section, we review the neutrino mass textures obtained in Refs. [4,5]. We consider the flavor basis, where charged lepton mass matrix is diagonal. Also we study the models, that only two families of right-handed neutrinos have Dirac mass terms with three families of left-handed neutrinos.
Dirac mass matrix and M N is (2 × 2) Majorana mass matrix that come from the following where m 1,2,3 are neutrino mass eigenvalues, which are positive real, c(s) 12 For the Dirac mass matrix m D , the maximally allowed number of texture zeros is one or two. Then, such matrices m D are classified as [4] T 1 : For the right-handed neutrino Majorana mass matrix M N , the maximally allowed number of texture zeros is one or two. Then, such matrices M N are classified as [4] R 1 : By combining these matrices, we can obtain the neutrino mass matrices m ν . Among all combinations, the realistic patterns of m ν are classified [4]: a :  These are one-zero textures. Explicitly, these patterns are realized by the following combi- where we can identify ∆m 2 21 = m 2 2 and |∆m 2 31 | = m 2 1 , since only inverted hierarchy is allowed for all the textures by the current neutrino oscillation data. Moreover, cos δ can be written in terms of observables and r ν by solving Eq.(II.6) directly, while cos α is also obtained in terms of the same parameters of cos δ by the fact that the imaginary part of Eq.(II.7) is zero.

REALIZATIONS OF TEXTURE ZEROS
Here, we study realization by use of global U(1) symmetry, S 3 and D 4 as well as Z N .

A. Abelian symmetries
Here we consider a global U(1) symmetry to realize predictive textures, where we fix the number of right-handed neutrinos to be two generations, i.e. N R 1,2 . A flavor-dependent U(1) symmetry in the lepton sector is useful to realize the diagonal mass matrix of the charged lepton sector. That is, the U(1) µ−τ , U(1) e−µ and U(1) e−τ would be good candidates. Here, let us study the realization of the Dirac mass texture T 4 by assuming the global U(1) µ−τ symmetry. 4 The assignment of U(1) µ−τ charges is shown in Table I. We also assign U(1) µ−τ charges, n 1 and n 2 to N R 1 and N R 2 . In order to realize Dirac neutrino mass terms, we have to introduce new SU(2) L doublet Higgs fields H i , and their minimal number is four, i.e., H i (i = 1, 2, 3, 4). Also, in order to realize the mass matrix M N , we have to introduce singlet scalar fields, ϕ 1,2,3 . Here the charges n 1 , n 2 should satisfy the condition, n 1 = n 2 and n 1 , n 2 = 0 in order to realize the desired Dirac texture of T 4 , and they should also satisfy (n 1 , n 2 ± 1, n 1 + 1) = ±1, +2 to forbid non-diagonal entries in the charged-lepton mass matrix. Under these symmetries and fields, one can write renormalizable coupling terms in the Lagrangian as follows: where several dangerous Goldstone bosons (GBs) can be evaded by introducing soft-breaking After the spontaneous symmetry breaking, the charged-lepton mass matrix and Dirac neutrino mass matrix are given by where v H and v Hi denote vacuum expectation values (VEVs) of the neutral components of H SM and H i , respectively. Then, the T 4 pattern of the Dirac neutrino mass matrix in Ref. [4] is derived. Also the right-handed neutrino mass matrix is given by where v ϕ i denote VEVs of ϕ i . From the above equation, one straightforwardly finds each of texture R 1 , R 2 , and R 3 in absence of ϕ 3 , ϕ 1 , and ϕ 2 .
We can realize the Dirac neutrino mass texture T 1 with the same charge assignment except replacing the charges of H 1 and H 2 such that H 1 and H 2 have U(1) µ−τ charges, n 2 and n 1 − 1. Then, we can realize the Dirac neutrino mass, Similarly, the patterns, T 5 and T 2 , are realized by U(1) e−µ instead of U(1) µ−τ . Also the patterns, T 6 and T 3 , can be realized by use of Once any global U(1) symmetries realize these predictive one-zero neutrino textures, discrete Abelian symmetries Z N are also possible in the same field contents, where N ≤ 19.

B. Non-Abelian discrete symmetries
Here we study the realization with non-Abelian discrete symmetries [7].

S 3 symmetry
First of all, we study the S 3 symmetry, which is the minimal group in the non-Abelian discrete symmetries. The irreducible representations of S 3 are the doublet 2, and the trivial singlet 1 and the non-trivial singlet 1 ′ . Here, we use the real representation [7] 5 , and their products are expanded as We assign (L L ℓ , ℓ R ) (ℓ = e, µ) to the S 3 doublets 2, and L Lτ , τ R to the S 3 trivial singlets 1.
In addition, we introduce four Higgs fields, which correspond to the S 3 doublet, H D ∼ 2, S 3 singlets, H 1 ∼ 1, and H 2 ∼ 1 ′ . Then the renormalizable coupling terms of the charged-lepton sector are given by After the spontaneously electroweak symmetry breaking, the charged-lepton mass matrix can be found as where VEVs are denoted by H 2 = 0, the diagonal charged-lepton mass matrix is realized; (III.11) However, from the above mass matrix, one cannot reproduce the mass difference between the masses of electron and muon. Thus, S 3 symmetry is not favorable. 5 Note here that the complex representations cannot construct the diagonal mass matrix of charged lepton.

D 4 symmetry
Next, we investigate the D 4 flavor symmetry that is the next minimal group in the non-Abelian discrete symmetries. The irreducible representations of D 4 symmetry are the doublet 2, and the trivial singlet 1, and three non-trivial singlets, 1 ′ , 1 ′′ , 1 ′′′ . 6 Here, we also use the real representation, and their productions are shown in Appendix. We assign (L L ℓ , ℓ R ) (ℓ = e, µ) to the D 4 doublets 2, and L Lτ , τ R to the D 4 trivial singlets 1. In addition, we introduce 6 Higgs fields, which correspond to all of the D 4 irreducible representations, Then the renormalizable coupling terms of the charged-lepton sector are given by After the spontaneously electroweak symmetry breaking, the charged-lepton mass matrix can be found as where their VEVs are denoted by H i ≡ v i / √ 2 (i = 1, · · · , 4) and H D i ≡ v D j / √ 2 for j = 1, 2. Once H D = H 2,3,4 = 0 and/or y ℓ 1,2,5,6 = 0, the diagonal charged-lepton mass matrix is realized; (III.14) From the above equation, one can reproduce the mass difference between the masses of electron and muon. Thus the D 4 flavor symmetry can be the minimal candidate to reproduce the desired textures. To realize the diagonal mass matrix of the charged lepton sector, we just need H 1 and H 2 , but we do not need H D or H 3,4 .
In the case of (N R 1 , N R 2 ) ∼ 2, the Majorana mass matrix is given by where these two masses are degenerated. Then the Dirac neutrino mass matrix is given by (III. 16) Hence one finds the desired Dirac mass matrix in the case of H 3,4 = 0 7 For this realization, we need H D , H 1 and H 2 , but not H 3 or H 4 . Now, let us study the models, that N R 1 and N R 2 are assigned to two D 4 singlets. If one assigns N R 1 and N R 2 into the same singlet representation under D 4 , the Majorana mass matrix does not give any vanishing elements without imposing additional symmetries.
Thus, we restrict ourselves to the models such that N R 1 and N R 2 are assigned to D 4 singlets different from each other.
When we assign N R 1 and N R 2 into different D 4 singlets such as ( (1 ′′ , 1 ′′′ ), etc. , the Majorana mass matrix is give by That is the R 1 form.
In the case of (N R 1 , N R 2 ) ∼ (1, 1 ′ ), the Dirac neutrino Yukawa mass matrix is given by This form cannot clearly reproduce any types of desired Dirac mass matrices, since y D 1 and y D 2 are located in the same column of upper (2 × 2) matrix. When we assign (N R 1 , N R 2 ) ∼ (1 ′′ , 1 ′′′ ), we obtain a similar result. Then, these two cases are not favorable, but the other cases are favorable.
To summarize results in this section, one can realize the desired textures by D 4 , but not by S 3 . Indeed, the D 4 flavor symmetry is interesting from the viewpoints of both high energy physic [9-11, 13-15, 22-24]. and bottom-up model building approach [16][17][18][19][20][21]. Similarly, we can discuss realization by using other non-Abelian discrete flavor symmetries. Also we can realize the desired textures by Abelian symmetries, U(1) and Z N . We need more Higgs fields in the Abelian models than the D 4 models. Thus, the D 4 flavor symmetry is useful to realize the desired textures. In the next section, we propose a concrete model with the D 4 flavor symmetry. Fields Also η and ϕ 8 respectively provide the Dirac and right-handed neutrino masses, η ′ and ϕ ′ 8 respectively provide the difference between the (1-1) and (2-2) elements of m D and M N , and η D gives the masses for the third row of Dirac mass matrix. ζ and ϕ 2 play a role in evading dangerous GBs due to accidental symmetries in the scalar potential. The D 4 symmetry assures diagonal mass matrices for charged leptons and right-handed neutrinos, and U(1) B−L plays a role in restricting (2×2) mass matrix for right-handed neutrinos which contribute to active neutrino masses. In addition, our U(1) B−L charge assignment makes N R 3 stable and it can be a DM candidate. All the field contents and their charge assignments are shown in Table. II. Under these contents with symmetries, one can write renormalizable Yukawa coupling terms and the Higgs potential as follows: 8 where V is the Higgs potential with non-trivial terms. These nontrivial terms forbid dangerous GBs arising from isospin doublets that spoil the model. In our model, we have two GBs that can be identified with CP-odd bosons of ϕ 2 and ϕ 8 (ϕ ′ 8 ). 9

A. Lepton sector
The resulting mass matrices are give by The above neutrino Dirac mass matrix m D corresponds to Eq. (III.17). Also the above Majorana mass matrix M N basically corresponds to Eq. (III.15). However, since there are two fields ϕ 8 and ϕ ′ 8 , then we obtain M 1 = M 2 . Then, we can obtain which corresponds to the pattern a in Eq. (II.5). Applying the discussion in Sec. II to our model, we find . (IV.7) Therefore, one obtains two relations from the above relation: Applying the current neutrino oscillation data [2] at 1σ confidence level, we find the predictions 0.485 |δ|/π 0.515 and 0.868 |α|/π 0.895, as can be seen in fig. 1, where the horizontal line is the best fit (BF) value of the neutrino oscillation data [32]. It implies that |δ| is in favor of not being BF value but being within 1σ confidence level.

B. Phenomenology
In this subsection, we discuss phenomenology of the model such as collider physics and dark matter physics. At the LHC Z ′ can be produced as it couples to the SM quarks, and can decay into the SM leptons providing clear di-lepton signal. On the other hand the signatures from exotic scalar bosons are more complicated containing more particles in final states and their branching ratios depend on the parameters in the scalar potential so that we have less predictability, although they can be also produced via Z ′ interaction and through electroweak interaction if an exotic scalar boson comes from iso-doublet. Thus we focus on Z ′ production in s-channel followed by decay mode of Z ′ → ℓ + ℓ − and estimate the constraints for new gauge coupling constant and mass of Z ′ . Then dark matter relic density is briefly discussed taking into account the constraint for Z ′ interaction.    (2) and (3). For exotic scalar modes, BRs for all components are summed up.

Collider physics and constraints
Here we explore collider physics focusing on Z ′ boson and provide constraints for its mass and gauge coupling constant. The relevant gauge interactions are given by where flavor indices for the SM fermions are omitted and Φ = {η 1 , η 1 ′ , η D , ϕ 8 , ϕ ′ 8 , ϕ 10 , ζ}; note that ϕ 2 is not included here since we assume its CP-odd component is Nambu-Goldstone boson absorbed by Z ′ . The mass of Z ′ is given by and its VEV respectively. The partial decay widths of Z ′ are estimated as where f SM denotes the SM fermions and {Φ 1 , Φ 2 } indicate components of Φ. We estimate branching ratios (BRs) for Z ′ decay in cases: Then we discuss constraint on g BL from the LHC experiments for three cases above.
Our Z ′ boson is produced via Z ′q q coupling and the production cross section is estimated using CalcHEP 3.6 [33] implementing relevant interactions. The most stringent constraint comes from the process pp → Z ′ → ℓ + ℓ − (ℓ = e, µ) and we estimate the corresponding cross section for each case. In Fig. 2, we compare ratio between σ · BR(pp → Z ′ → ℓ + ℓ − ) and σ · BR(pp → Z → ℓ + ℓ − ) in our model with the experimental constraints indicated by red curve [34] where solid, dashed and dotted curve correspond to cases (1), (2) and (3) respectively, and we apply g BL = 0.3(0.1) in left(right) plots. Thus lower limit of mass Z ′ is relaxed when the exotic scalar modes of Z ′ decay are kinematically allowed: the lower limit of m Z ′ is around 3300(2000) GeV for g BL = 0.3(0.1). For case (3), the Z ′ boson dominantly decays into exotic scalar bosons which further decay into SM particles via gauge interaction and/or couplings in the scalar potential providing multi-particle final states. The detailed analysis of the scalar modes is beyond the scope of our analysis.

Dark matter
In this subsection we discuss a dark matter candidate; X R ≡ N R 3 , whose stability is assured by the U(1) B−L symmetry with alternative charge assignment for the SM singlet fermions. Here, let us assume any contributions from the Higgs mediating interaction are negligibly small so as to avoid the constraints from direct detection searches as LUX [35], XENON1T [36], and PandaX-II [37]. Then DM annihilation processes are dominated by the gauge interaction with Z ′ and GB α G ≡ z ϕ 10 mainly originated from ϕ 10 , and their relevant Lagrangian in basis of mass eigenstate is found to be where Q X BL = 5, M X ≡ y N 3 v ϕ 10 / √ 2, v ϕ 10 << v ϕ 2 . Here we require Z ′ mass and gauge coupling g BL to satisfy the relation g BL /m Z ′ 1/(6.9 TeV) from LEP experiment [38] as well as the constraints from the LHC experiments as discussed in the previous subsection.
The relic density of DM is then given by [39,40] Ωh 2 ≈ 1.07 × 10 9 g * (x f )M P l J(x f ) [GeV] , (IV. 13) where g * (x f ≈ 25) is the degrees of freedom for relativistic particles at temperature T f = M X /x f , M P l ≈ 1.22 × 10 19 GeV, and J(x f )(≡ ∞ x f dx σv rel x 2 ) is given by [29,41] J where we assumed Z ′ boson and scalar bosons are heavier than X to forbid corresponding annihilation processes kinematically, for simplicity. Here decay width of Z ′ is given by Eq. (IV.11) where Z ′ can decay into 2X, if kinematically allowed. We find that two characterized solutions of measured relic density Ωh 2 ≈ 0.12 [42] in the above formula. The first one is a sharp region at around M X ∼ m Z ′ /2, that is a resonant solution from the contribution 2X → Z ′ → ff in Eq. (IV.15). The second one is the region in lighter mass of DM that mainly arises from the contribution 2X → 2α G in Eq.(IV.16). In the former case DM mass is around TeV scale to obtain right relic density due to the collider constraints for Z ′ mass while in the latter case DM mass can be O(10) GeV to O(100) GeV which depend on the coupling factor M X /v ϕ 10 ; for more details, see, e.g., Refs. [29,41].