A hybrid seesaw model and hierarchical neutrino flavor structures based on $A_{4}$ symmetry

We propose a hybrid seesaw model based on $A_{4}$ flavor symmetry, which generates a large hierarchical flavor structure. In our model, tree-level and one-loop seesaw mechanisms predict different flavor structures in the neutrino mass matrix, and generate a notable hierarchy among them. We find that such a hierarchical structure gives a large effective neutrino mass which can be accessible by next-generation neutrinoless double beta decay experiments. Majorana phases can also be predictable. The $A_{4}$ flavor symmetry in the model is spontaneously broken to the $Z_{2}$ symmetry, leading to a dark matter candidate which is assumed to be a neutral scalar field. The favored mass region of the dark matter is obtained by numerical computations of the relic abundance and the cross section of the nucleon. We also investigate the predictions of the several hierarchical flavor structures based on $A_{4}$ symmetry for the effective neutrino mass and the Majorana phases, and find the characteristic features depending on the hierarchical structures.


I. INTRODUCTION
Discovery of neutrino oscillations shows that neutrinos are mixed with each other and have tiny masses. Since neutrinos are massless particles in the Standard Model (SM), new physics beyond the SM which has some mechanism to generate the neutrino masses are required. Type-I seesaw mechanism [1-5] is one of the attractive ways to generate such tiny neutrino masses at the treelevel, which requires an introduction of right-handed neutrinos. Another attractive way to explain the tiny masses is a radiative seesaw mechanism in which neutrino masses are generated by loop effects (see [6][7][8][9][10][11][12] for early works and also [13] for a latest review). In the radiative seesaw models involving right-handed neutrinos [9][10][11] where a discrete symmetry is imposed to forbid the Type-I seesaw mechanism. This symmetry is also responsible for the stability of the dark matter (DM).
The neutrino mass matrix is diagonalized by the lepton flavor mixing matrix, so-called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix which is parameterized as where c ij = cos θ ij and s ij = sin θ ij for i, j = 1, 2, 3. The parameter δ is a Dirac phase, while α 2 and α 3 denote Majorana phases. The data obtained in neutrino oscillation experiments [14][15][16][17][18] show that the neutrino mixing angles are θ 12 33 • , θ 23 [19]. The mixing matrix has two large mixings, which is very different from the quark mixing. Apart from the tiny masses of neutrinos, such flavor structures will give us hints of physics behind the SM.
One candidate behind the lepton sector is non-Abelian discrete flavor symmetries, such as S 3 , A 4 and S 4 (see [20][21][22][23][24] for reviews). 1 In particular, the study of the A 4 models has received considerable interest. It has been shown in [26] that the A 4 flavor symmetry leads naturally to the neutrino mass matrix which gives the tri-bimaximal flavor mixing, M Tri (i.e. s 12 = 1/ √ 3, s 23 = 1 √ 2, s 13 = 0) [27]. It is known that M Tri is given by a linear combination of three flavor structures as However, since the observed value of θ 13 is small but non-zero, the neutrino mass matrix should be modified from M Tri so as to realize the non-zero (1,3) off-diagonal element in the flavor mixing matrix. One possible form of the neutrino mass matrix which derives the non-zero θ 13 is given by adding another new flavor structure as [28] M ν = a     1 0 0 where the coefficients of each flavor structure a, b, c and d are the arbitrary mass dimensionful parameters. The non-vanishing d term in the models with the A 4 symmetry is discussed in [21,[28][29][30][31][32][33]. It is expected that the relation among these four flavor structures provides us an important information on the flavor symmetry in the neutrino mass generation mechanism.
In this paper, we propose a hybrid seesaw model based on the non-Abelian symmetry in our model is broken into the Z 2 subgroup by the vacuum expectation value (VEV) of the A 4 triplet scalar field. Therefore, a lightest neutral Z 2 -odd field, where we assume it a CP-even neutral scalar field, is stable and becomes a DM candidate. We compute the relic abundance and the spin-independent cross section of the DM, and show the plausible mass region of the DM.

II. MODEL
The non-Abelian A 4 flavor symmetry has four irreducible representations which are three singlets 1, 1 and 1 and one triplet 3. The A 4 symmetry is generated by two elements S and T , (L e , L µ , L τ ) 1, 1 , 1 2 (+, +, +) (l e R , l µ R , l τ R ) 1, 1 , 1 1 (+, +, +) where ω = e and an A 4 singlet H is a Higgs doublet field. In this model, the Yukawa sectors of neutrinos are described by where y 1 , y 1 and y 1 are the Yukawa couplings andη i = iσ 2 η * i . In this work we assume y ≡ y 1 = y 1 = y 1 , where y is real. Majorana mass terms of right-handed neutrinos are given by where M R , M R , M R and M 23 are the Majorana masses of right-handed neutrinos. We note the second, third and fourth therms in Eq. (5) break the A 4 symmetry. 3 Because of the fourth term, the neutrinos N 2 and N 3 are mixed each other. The mass matrix of right-handed neutrinos is 3 These three terms can be generated by A4 singlet 1 , 1 and triplet 3 scalar fields, respectively.
diagonalized by the complex mixing angle tan 2θ R ≡ 2M 23 (ω−ω 2 )(M R −M R ) , where we define the diagonal elements as (M 1 , M 2 e iδ R 2 , M 3 e iδ R 3 ). Throughout this paper, we work in the basis where the charged lepton mass matrix is diagonal.
Based on the A 4 symmetry, the scalar potential is given by [46] We assume that the couplings in the scalar potential are real and λ 4 = λ 4 for simplicity. When one of the A 4 triplet field η 1 , in addition to H, has the VEV, the A 4 symmetry breaks to the subgroup Z 2 symmetry whose charge assignments are also shown in Here the VEVs are real and satisfy v 2 h + v 2 η = v = 246 GeV. The physical scalar states in the Z 2 -even sector can be obtained by the mixing angles β, where tan β ≡ v η /v h , and α as Here h 1 is the SM-like Higgs particle whose mass is m h 1 = 125 GeV. The masses of the charged scalar field H ± and the CP-odd scalar field A 1 are described as m 2 The Z 2 -odd fields η 2 and η 3 , which do not have the VEVs, are defined as These two states are mixed through the λ 12 , λ 13 , λ 14 and λ 15 terms in Eq. (6) and the mixing angle between them is π/4. The neutral CP-even (-odd) states give the mass eigenstates η 0 2 and η 0 3 (A 2 and A 3 ) with masses m η 0 2 and m η 0 3 (m A 2 and m A 3 ) as where The masses of the charged scalar fields η ± 2 and η ± 3 are given by where λ x 4 ≡ −3λ 3 − 4λ 4 − 2λ 5 + λ 8 . Note that the mass differences between m η 0 2 and m η 0 The lightest Z 2 -odd particle is stable and can be a DM if it is neutral. In our model, the right-handed neutrinos are heavy as shown later, so that the DM candidates are η 0 2,3 and A 2,3 . Hereafter, we assume that η 0 2 is a DM candidate.

III. NEUTRINO MASSES AND FLAVOR STRUCTURES
In this model, the neutrinos obtain their masses via the tree-level and the one-loop seesaw mechanisms. Since the Yukawa interactions L αH N i (α = e, µ, τ ) are forbidden by the A 4 symmetry, the usual Type-I seesaw mechanism does not work. However, the neutrinos can obtain their masses via the other tree-level seesaw mechanism due to the existence of the Yukawa interactions L αη1 N 1 in Eq. (4) with the nonzero VEV of η 1 . The neutrino mass matrix M tree ν which is generated by the tree-level seesaw mechanism in Fig. 1 is given by The flavor structure in M tree ν is the same form as that of b term in Eq. (3). We note that the rank of M tree ν is one, so that the other contributions to the neutrino mass generation should be necessary.
The neutrino masses are also generated by the one-loop seesaw mechanisms which are shown in Fig. 2 and Fig. 3. The Z 2 -even right-handed neutrino N 1 contributes to the mass generation in Fig. 2, while the Z 2 -odd right-handed neutrinos N 2 , N 3 and their mixing contribute in Fig. 3.
Feynman diagram for neutrino mass via the tree-level seesaw mechanism.
and m A ≡ m A 2 ≈ m A 3 , the neutrino mass matrix via the one-loop diagrams in Fig. 2 and Fig. 3 is described as Here It can be seen that the four flavor structures in Eq. (3) are generated. 4 We find that the one-loop diagrams with N 1 in Fig. 2 generate only b term. On the other hand, the contributions of N 2 and N 3 in Fig. 3 give all four flavor structures. In particular, the mixing between N 2 and N 3 realizes the nonzero a term, while the origin of the nonzero d term (i.e. nonzero θ 13 ) comes from the difference between Λ 2 and Λ 3 . From Eq. (17) and Eq. (18), the neutrino mass matrix in our model 4 Even when the assumption λx 3 1 is removed, the four flavor structures are derived.
Feynman diagrams for neutrino masses via the one-loop seesaw mechanism with N 1 .
Feynman diagrams for neutrino masses via the one-loop seesaw mechanism with N 2 and N 3 .
is given by where the flavor structures are the same as those in Eq. (3). In this model, the ratios |b|/|c| and |b|/|d| are naively given by the inverse of the loop suppression factor ≈ 16π 2 when M i ∼ m η , since the b term contains the contributes from the tree-level diagrams whereas the c and d terms are generated by the one-loop diagrams. However, such large hierarchies between b and c, d are not plausible by the current experimental data, as will be shown later. On the other hand, when . As the mass difference between M i and m η becomes larger, the ratio M tree ν /Λ i becomes smaller. Therefore, the milder (but large) hierarchies between b and c, d, such as |b|/|c| ∼ |b|/|d| ∼ O(10), can be possible. For the a term, although it is also generated by the one-loop diagrams, its magnitude is controlled by the mixing between N 2 and N 3 .
The neutrino mass matrix in Eq. (3) (and thus in Eq. (24)) is diagonalized with the PMNS matrix which is formed by the tri-bimaximal mixing, the (1,3) mixing and the Majorana phase matrix [28]: The shaded regions are excluded by the constraint of the sum of light neutrino masses [47].
Here the mixing angleθ is the complex parameter. Comparing to the standard parametrization of U PMNS in Eq.
(1), we obtain sinθ = 3/2 sin θ 13 e −iδ . Using the coefficient parameters a, b, c and d in Eq. (3), the angleθ is given by The neutrino masses m 1 , m 2 and m 3 and the Majorana phases α 2 and α 3 are written as The observed values of the neutrino oscillation parameters ∆m 2 31 , ∆m 2 21 , θ 13 and δ give the constraints on the relationship among the coefficients a, b, c and d. In The Dirac phase is taken as δ = 0 and the coefficients a, b and c are assumed to be real for simplicity.
There are two solutions for the |b|, which are shown by |b + | and |b − | in Fig. 4. The shaded regions 5 We use the center values in v4.1 of Ref. [19].
are excluded by the constraint of the sum of light neutrino masses [47]: In the NO (left panel of Fig. 4), the |a| decreases and the |c| increases as the parameter d larger.
We note that the coefficients |c| and |d| are comparable but the |c| is larger than |d| due to the relation of Eq. (26). The hierarchy |a|, |b + | > |c|, |d| is shown for the smaller d (d 0.008 eV), while the hierarchy |b + |, |c|, |d| > |a| appears for the larger d (d 0.02 eV). On the other hand, the hierarchy |a|, |c|, |d| > |b − | is obtained for d ∼ 0.03 eV, where b − changes its sign. In the IO (right panel), the similar hierarchies among the coefficients can be seen except for the vanishment of |b − |. 6 We note that, in both of the NO and the IO, the large hierarchies between b and c, d such as |b|/|c|, |b|/|d| ≈ 16π 2 are disfavored by current data.
In our hybrid seesaw model, the milder but large hierarchy as |b|/|d| ≈ π 2 can be realized for From Eqs. (33) and (35), we find that the large hierarchy |b|/|d| ≈ π 2 is realized for the masses of the right-handed neutrinos M i ∼ O(10 10 ) GeV and the scalar fields m η i ∼ O(10 2-3 ) GeV for y ∼ 10 −2 . Furthermore, both of the BP NO and the BP IO satisfy |a| |c|, |d|, so that the Majorana phase α 3 is expected to be close to zero as can be seen from Eq. (29).

PHASES
In this section, we discuss the predictions of the effective neutrino mass m ee and the Majorana CP phases. The m ee is defined as The other parameters |a|, |b|, |c| can be determined so as to satisfy the observed values of ∆m 2 21 , ∆m 2 31 and θ 13 in Eq. (30). Furthermore, the arg(c) is fixed through the relation in Eq. (26). In Fig. 5 [19] and the sensitivity of next-generation 0νββ experiment by nEXO [48].
In Fig. 5, the hierarchical case with |b|/|d| > 1 (green) does not constrain the parameter space comparing to the cyan points, while the other three cases constraint the parameter space. For the hierarchical case with |b|/|c| > 1 (yellow), the predicted regions of the effective neutrino mass m ee and the lightest neutrino mass m 1 are m ee 0.001 eV and m 1 0.007 eV. In this case, the Majorana phase α 2 ∼ 0 is excluded by the constraint from Eq. (31) and the α 3 is constrained as |α 3 |/rad. 2.0. The hierarchical cases with |d|/|a| > 1 (orange) and |b|/|a| > 1 (black) have similar predictions, but the former is more constrained, such as giving m 1 0.015 eV and |α 3 |/rad. 2.2. respectively the hierarchy case with |b|/|d| > 1, |b|/|a| > 1, |d|/|a| > 1 and |b|/|c| > 1.
Here the |α 3 | π radians is obtained for |d| |a| as can be seen from Eq. (29). Figure 6 shows the results for the IO. For the hierarchical case with |b|/|c| > 1, the predicted regions for m 3 and α 3 are wider than those in the NO. In particular, the α 3 is not constrained for 0.015 m ee /eV 0.04.
Similarly, in those range of m ee , the α 3 is unconstrained for the case with |b|/|a| > 1 (although it is not shown in the lower right panel in Fig. 6 as it is behind the yellow region). The predictions for the cases with |d|/|a| > 1 show the similar feature to that in the NO. The next-generation 0νββ experiment nEXO can explore all predicted regions of m ee for the IO, and thus there is the possibility to obtain hints of the Majorana phases for some hierarchical cases. In this analysis, the predicted values for θ 23 and θ 12 are 0.4 sin 2 θ 23 0.6 and sin 2 θ 12 ≈ 0.34, respectively, which are allowed within 3σ [19]. We note that the predicted regions of the hierarchies with |b|/|c| > 1 and |d|/|a| > 1 are included in those with |b|/|d| > 1 and |c|/|a| > 1, respectively, because of the relation |c| > |d| obtained by Eq. (26).
Next, we discuss the larger hierarchical case with |b|/|d| > π 2 which can be applied to our model. We display in Fig. 7 and Fig. 8 the predictions of m ee for |b|/|d| > π 2 by the red points. The green points are the predictions for |b|/|d| > 1 which are the same as those in Fig. 5 and Fig. 6. We also show the predictions of the BP NO and the BP IO in our hybrid seesaw model by the black triangles in Fig. 7 and Fig. 8, respectively. In Figure 7, we can see that the predicted regions for |b|/|d| > π 2 are strictly constrained : m ee 0.02 eV, m 1 0.06 eV, π/2 |α 2 |/rad. ≤ π, |α 3 |/rad.

0.2.
Such large m ee is within the sensitivity reach of the next-generation 0νββ experiments. Note that, for the larger hierarchy between the coefficients |b| and |d|, the larger hierarchy between the |a| and |d| is expected and thus the Majorana phase |α 3 | gets closer to zero. In our hybrid seesaw model, where the nonsignificant CP violations by the Majorana phases are expected.

V. DARK MATTER
The A 4 flavor symmetry in our model is spontaneously broken to the Z 2 symmetry via the VEV of the A 4 triplet scalar field, which predicts the DM candidates and we assume that the Z 2 -odd scalar field η 0 2 is the DM. 7 The main annihilation processes of the DM in our scenario are shown in Fig. 9. Note that the processes are almost the same as those in the Inert Doublet Model [53]. In our model, the mass splitting between η 0 2 and η 0 3 is small because of the small λ x 3 coupling, so that we also consider the annihilation of η 0 3 and the relic density is computed for the sum of η 0 2 and η 0 3 . 8 The rate of DM annihilation depends on the scalar couplings λ 1 , λ x 5 , λ x 6 and λ x 7 , except for the gauge couplings, where λ x 5 ≡ λ 9 + λ 10 + 2λ 11 , λ x 6 ≡ λ 2 + λ 3 + 2λ 4 + λ 5 and λ x 7 ≡ 2λ 2 − λ 3 − 2λ 4 + λ 5 + 2λ 6 + λ 7 + λ 8 . For sin(β − α) = 1, the relevant scalar couplings λ x 5 , λ x 6 and λ 1 are written by the masses of the Z 2 -even neutral scalar fields m h 1 , m h 2 and the mixing 7 Such DM (so-called "discrete DM") is discussed in [37,38,46,[49][50][51][52]. 8 We note that the η 0 3 decays into the η 0 2 through e.g. η 0 3 → η 0 2 γ after its decoupling.
angle β as The scalar coupling λ x 7 can be determined to satisfy the relic abundance Ωh 2 ≈ 0.12 [47].
The spin-independent cross section of the nucleon is given by wheref ≈ 0.3 is the usual nucleonic matrix element [54], m N is the nucleon mass, λ DD = λ x 7 sin 2 β + λ x 5 cos 2 β and λ DD = λ x 7 sin 2β − λ x 5 sin 2β. Since the contributions from the h 1 and h 2 mediations give a relative negative sign, there is the possibility for destructive interference In Fig. 10, the spin-independent cross section of DM is shown as a function of the DM mass, where the relic abundance of the DM satisfies Ωh 2 ≈ 0.12 [47]. We here have fixed the masses of the Z 2 -even scalar fields as m h 2 = 200 GeV and m H ± = m A 1 = 250 GeV. For the Z 2 -odd scalar fields, the masses are taken as m A = m η ± = m η 0 + 20 GeV. The cyan, red, blue and green lines show the results for tan β = 1, 2, 3 and 4, respectively, where the dotted lines are excluded by the unitarity condition λ i < 4π (i = 1 ∼ 15) or the bounded-from-below condition on the scalar potential [46].
As the reference, we show the prediction of the BP NO and the BP IO by the black triangle. Above region of black dashed line are excluded by XENON1T [55]. In Fig. 10, we can see the cancellations between the contributions of h 1 and h 2 . When the DM mass is smaller than about 400 GeV, the relic abundance of the DM is smaller than the observed value Ωh 2 < 0.12. For tan β 5, the unitarity condition cannot be satisfied. We find that the allowed ranges for the DM mass are m η 0 2 520 − 540 GeV, 490 − 580 GeV and 400 − 500 GeV for tan β = 2, 3 and 4, respectively.
The future sensitivity of the direct detection experiment XENONnT is σ SI ∼ O(10 −47 ) cm 2 [55], which can probe our DM scenario.

VI. CONCLUSION
We have studied the neutrino mass matrix which is composed of the four flavor structures in We have also performed the model-independent analysis of the neutrino mass matrix in Eq. (3), particularly for the cases with some hierarchical flavor structures. It has been found that the hierarchical cases with |b|/|a| > 1, |b|/|c| > 1 and |d|/|a| > 1 reduce the allowed parameter space and show the characteristic predictions for the Majorana phases. On the other hand, the hierarchical case with |b|/|d| > 1 does not show the specific predictions. However, the larger hierarchy with |b|/|d| > π 2 , which can be realized in our hybrid seesaw model, can reduce the predicted parameter region, which can be testable by the future 0νββ experiments.