Neutrino mixing and masses in SO(10) GUTs with hidden sector and flavor symmetries

We consider the neutrino masses and mixing in the framework of SO(10) GUTs with hidden sector consisting of fermionic and bosonic SO(10) singlets and flavor symmetries. The framework allows to disentangle the CKM physics responsible for the CKM mixing and different mass hierarchies of quarks and leptons and the neutrino new physics which produces smallness of neutrino masses and large lepton mixing. The framework leads naturally to the relation UPMNS ∼ VCKM†U0, where structure of U0 is determined by the flavor symmetry. The key feature of the framework is that apart from the Dirac mass matrices mD , the portal mass matrix MD and the mass matrix of singlets MS are also involved in generation of the lepton mixing. This opens up new possibilities to realize the flavor symmetries and explain the data. Using A4 × Z4 as the flavor group, we systematically explore the flavor structures which can be obtained in this framework depending on field content and symmetry assignments. We formulate additional conditions which lead to U0 ∼ UTBM or UBM. They include (i) equality (in general, proportionality) of the singlet flavons couplings, (ii) equality of their VEVs; (iii) correlation between VEVs of singlets and triplet, (iv) certain VEV alignment of flavon triplet(s). These features can follow from additional symmetries or be remnants of further unification. Phenomenologically viable schemes with minimal flavon content and minimal number of couplings are constructed.


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6. The double seesaw mechanism allows to explain difference of mixing patterns in the quark and lepton sectors and reproduce the relation (1.1). It reveals new possibilities to realize flavor symmetries. Striking differences in the quark and lepton sectors follow essentially from the mixing of new fermion singlets S with neutrino (left-and righthanded) components of 16-plet, thus providing the main source of the lepton mixing.
Origin of singlets S can be related to further extension of the gauge symmetry: SO(10) → E 6 , in which new singlets are additional neutral components of 27-plets [50]. Alternatively, singlet fermions S can be components of the hidden sector composed of the fermionic and bosonic singlets of SO (10). Hidden sector may have its own gauge symmetries. Sterile neutrinos, if exist, particles of dark matter, axions, etc., can also originate from this sector.
It seems that the existing data on masses and mixing are too complicated to be explained all at once, from a few principles and a few parameters. So, according to eq. (1.1), probably two different (but somehow connected) types of new physics are involved: the "CKM new physics" responsible for the CKM mixing and difference of hierarchies of Dirac masses, and the "neutrino new physics", which explains smallness of neutrino masses and large mixing with its form dictated by symmetry. The neutrino new physics is related to the hidden sector and properties of the portal interactions which connect the visible sector and the hidden one [51][52][53].
This framework has been explored to some extent previously. That includes (i) the seesaw enhancement of lepton mixing due to coupling with new singlets [11], (ii) screening of the Dirac structures, which means that the mixing of light neutrinos is essentially related to mixing in the singlet sector S: m ν ∝ M S [11,54], (iii) stability of this screening with respect to renormalization group (RG) running [54], renormalization of the relation (1.1) [55], (iv) symmetry origins (groups T 7 and Σ 81 have been used) of the screening, as well as partial screening [56], (v) lepton mixing from the symmetries of the hidden sector [48].
In this paper we further elaborate on the framework with emphasis on new mechanisms of getting flavor structures from symmetries. We systematically explore the fermion masses and mixing matrices which can be obtained in the framework. Interplay of the flavor and GUT symmetries is crucial. We mainly discuss here the symmetry aspects and formulate conditions of the vacuum alignment which eventually should be realized in complete model buildings.
The paper is organized as follows. In section 2 we study properties of the mass matrices involved in generation of the neutrino masses and mixing in this framework. We elaborate on the main elements and building blocks of the schemes of lepton mixing. In section 3 we present the simplest viable schemes with fermion singlets S transforming under three-dimensional representation 3 of A 4 . In section 4 the schemes are considered with S transforming under one-dimensional representations 1, 1 , 1 of A 4 . Section 5 is devoted to various generalizations: effect of linear seesaw, contribution from additional singlets, elements of the CKM physics. We summarize our results in section 6. Some technical details are presented in the appendix.

The framework
Let us first summarize main ingredients of the framework outlined in the introduction. We consider the SO(10) gauge model with three generations of fermions and the following additional features: • All the chiral fermions (including the right-handed neutrino) of each generation are accommodated in a 16-dimensional spinor representation, 16 F i (i = 1, 2, 3).
• Three or more singlet fermions S i exist. They mix with neutrinos due to Yukawa interactions with 16-plets of Higgs scalars 16 H .
• Only low-dimension representations of Higgs fields, 10 H , 16 H and 1 H , give masses to fermions. High dimensional representations 126 H and 120 H are absent.
• Approximate flavor symmetry exists with fermions 16 F i , S j carrying non-zero flavor charges. For definiteness we consider the simplest discrete group which has irreducible three-dimensional representations, A 4 . Various features obtained from this symmetry can be also reproduced with other groups.
• Three generations of the matter fields, 16 F i , transform as triplet of A 4 in most of the cases. We explore different symmetry assignments for three singlets S i , (3, 1, 1 , 1 ).
• A number of flavons (scalar fields, SO(10) singlets) exist, which have non-trivial transformations with respect to A 4 and acquire vacuum expectation values (VEV), thus breaking the symmetry.
• The Higgs multiplets, 10 H and 16 H , are singlets of A 4 , and therefore factor out of the flavor structure in the simplest cases. This reduces the complexity in obtaining certain form of mass matrices and mixing from flavor symmetries. Still, 10 H and 16 H may carry charges of additional symmetries, in which case they will affect the flavor.
We do not specify here the SO(10) symmetry breaking, although the corresponding Higgs multiplets can also take part in generation of the fermion masses. Rather than constructing complete models, we will explore systematically the flavor structures which can be obtained in the formulated framework.

Yukawa couplings and mass matrices
The Lagrangian has three types of Yukawa couplings, generate after symmetry breaking the usual Dirac masses of fermions. With just one 10 H these couplings produce Dirac mass matrices of the up-type quark, down-type quark, charge lepton mass matrices and neutrinos of the same form: where v u v d are the VEVs of 10 H . No quark mixing is generated at this level. Nevertheless, such a mixing can be produced if two (or more) different 10 a H (a = u, d) are introduced (see section 5.3).

The portal interactions,
Here v L , v R are the VEVs of the SU(2) L and SU(2) R doublets from 16 H . As long as only one Higgs 16-plet generates M D and m , the two mass matrices are proportional to each other: 3. The hidden sector interactions, produce the mass matrix of singlets Here 1 H fields are singlets under both flavon symmetry and SO(10), which may not be necessary and are assumed to be absorbed into flavons from now on. The masses of singlets can be generated at the renormalizable level, and also bare mass terms M 0 S can appear for some or all singlets if it is not forbidden by flavor symmetry.
Notice that here the Yukawa couplings are effective couplings -functions of VEVs of flavon fields, Φ: where Λ f is a scale of flavor physics, the constants can be expanded as In what follows we will take the lowest-order approximation in Φ /Λ f .

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The mass matrix of neutral leptons generated by interactions (2.2), (2.4) and (2.6) in the basis (ν, N, S), where N ≡ ν c , equals GeV , (2.10) so that for M S M cr the "linear-seesaw" dominates, whereas for M S M cr , the "doubleseesaw" gives the main contribution. In our framework with M S ∼ M Pl the "doubleseesaw" dominates and the "linear-seesaw" term may lead to small corrections.
According to eq. (2.3) and m D ∝ m l , no mixing is produced by the Dirac mass matrices. In contrast, due to extended structure of the mass matrix of neutral leptons eq. (2.7), m ν differs from m l , and so the lepton mixing is generated. In the basis where the Dirac mass matrices are diagonal, m D = m diag D , the origin of lepton mixing is the structure In the lowest order, the Majorana mass term of right-handed neutrinos, N T C −1 N , is absent, since we do not introduce 126 Higgs multiplet. Such a term can be generated by higher-dimension operator (1/Λ)16 F i 16 F j (16 H 16 H ) ij , or effectively generated by the decoupling of heavy particles with mass ∼ Λ. It then gives M N = v 2 R /Λ. In our framework, the mass term N T C −1 N appears after integrating out S, which gives We assume that this contribution to M N dominates. Other contributions either are forbidden by certain auxiliary symmetry or have the same flavor structure as eq. (2.12). Notice that since in our framework there is m D M D , the RG running of Yukawa couplings may modify the expressions for neutrino masses non-negligibly, and thus may need to be taken into account [57][58][59][60][61].

Screening and partial screening
The double seesaw allows to realize the screening of the Dirac structures [11,54], which will be used in many examples presented in this paper. If then according to eq. (2.9) m ν ∝ M S , (2.14) i.e. structure of the light neutrino mass matrix is given by that of M S , thus disentangling properties of neutrinos and quarks completely. The form of M S can strongly differ from those of m D and M D , thus leading to very different lepton mixing. Suppose that the screening condition is fulfilled in the basis where m D ∝ M T D are diagonal due to some certain symmetry, i.e. in the flavor basis, and that in this basis the singlet mass matrix is M S . Then according to (2.14) the mixing is determined exclusively by M S .
If the screening condition (2.13) is fulfilled in some other basis (again determined by symmetry) where m l = m D is non-diagonal and the singlet matrix is M S , then m ν ∝ M S . But now the lepton mixing gets contributions both from m D and M S . We present schemes which realize both possibilities.
Let us consider the screening condition in more details. We define the unitary transformations U l , V R , V S and V N which diagonalize the mass matrices m D and M D , so that Then the screening matrix (factor) equals The screening (2.13) implies that If only the first condition (the same rotations of the RH neutrino components) and the second one are fulfilled we obtain This partial screening gives additional contribution to the neutrino mixing. Due to symmetry the matrices U l and V S can be related in such a way that eq. (2.15) produces a matrix of special form, which then leads to the required form of total mixing matrix (see section 3.5). The screening can be due to further embedding of singlets S in extended gauge symmetry or due to common flavor symmetries. Some additional physics (which we call the CKM physics) can lead to the misalignment of mass matrices for up-and down-type quarks. Due to GUT similar misalignment is expected between the neutrino Dirac matrix m D and the charged lepton matrix m l = m D .

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Then in the presence of screening m D ∝ M T D = diag the PMNS mixing gets the CKM type contribution ∼ V CKM from m l and another one, U S , from M S : This allows to maximally disentangle the CKM and neutrino new physics and realize the scenario in eq. (1.1).

Symmetry and flavor structures
The flavor symmetry imposes restrictions on the Yukawa couplings Y ij k , and thus forbids some of them or leads to relations between them. For definiteness we use A 4 flavor group (see appendix A for details). Various structures we obtain in this paper can be reproduced in models with other symmetry groups which include representations 3, 1, 1 , 1 . On the other hand, with additional symmetries it is possible to produce in our framework features from other symmetry groups.
We introduce flavon triplets ξ and singlets ϕ = (ϕ, ϕ , ϕ ), which acquire the VEV ξ ≡ (u 1 , u 2 , u 3 ) and ( ϕ , ϕ , ϕ ) ≡ (v, v , v ). The general form of a fermion mass term is F i F j , where the fermionic multiplets F i = 16 F , S. Depending on symmetry assignment of F i and for general sets of the flavon fields, which include the flavon triplets ξ and singlets ϕ, we obtain three types of mass matrices: where we use notations Here y ≡ (y, y , y ) and (Q) ϕ (Q = F 1 F 2 in this case) is the field operator which produces one-dimensional representations of A 4 together with a given ϕ.
The mass term m 0 (F 1 F 2 ) without flavons can be forbidden by symmetries. The interactions (2.16) lead to the mass terms F 1 M 33 F T 2 + h.c. with where the off-diagonal part is generated by flavon triplet, whereas the diagonal one is due to coupling with the flavon singlets of A 4 and flavonless contribution. Three terms in (2.18) are generated by different flavons. Depending on additional symmetry assignments one, or two, or all three terms can contribute. Explicitly,

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Notice that for all equal values (yv = y v = y v ), M diag 33 = diag(m 0 + 3yv, m 0 , m 0 ), which can be a good first approximation in some cases. For real parameters µ ≡ m 0 + yv, µ ≡ y v and µ ≡ y v two eigenstates are degenerate: This important feature together with certain forms of the off-diagonal part can lead to maximal mixing. General situation is analyzed in the appendix B.
In the absence of the bare mass m 0 , the eigenvalues of matrix (2.19) can be written as is the magic matrix. The matrix U ω is unitary, U ω U † ω = I and the inverse U −1 ω = U † ω = U * ω , can be obtained by substitution ω ↔ ω 2 . Then the condition yv ≈ ω 2 y v ≈ ωy v (2.22) leads to the hierarchy of masses: The hierarchy (mass splitting) between the lighter states, m 2 m 1 , can be obtained by special tuning of relations in eq. (2.22) or introduction of other contributions, e.g. from the linear seesaw terms.
In general, it is difficult to get correlation (2.22) between VEVs and Yukawa couplings. The simplest realization would be two separate equalities: as a result of additional symmetry, or the fact that singlets originate from breaking of representation 3 and v ≈ ω 2 v ≈ ωv , as a consequence of special forms of the potential.
In eq. (2.18) the off-diagonal matrix equals (2.23) It has the form of the Zee-Wolfenstein mass matrix [62,63] with the eigenvalues satisfying the equality m 1 + m 2 + m 3 = 0. For arbitrary values of u i , the diagonalization JHEP05(2016)135 matrix and eigenvalues of M ξ 33 are presented in appendix C, and here we consider some special cases.
Let us consider a special case of the total matrix M 33 (2.18), which will appear often in our constructions. If u 1 = u 2 = u 3 ≡ u and also the diagonal elements are all equal, we have where β ≡ hu [64]. Such a matrix can be obtained when ϕ and ϕ do not contribute to masses, or when the bare mass terms or flavonless operators dominate. This matrix having the TBM form is diagonalized by U TBM , see appendix D. The eigenvalues equal of which two are equal and the sum is given by For µ = 0 we have i λ i = 0, and therefore M special 33 (µ = 0) can not be used to describe masses of quarks or charged leptons.
Degeneracy of the two eigenstates implies that the diagonalization matrix is not unique, and actually there is infinite ambiguity in diagonalization related to rotation in the space of eigenstates with equal masses. In particular, the matrix (2.24) can be unitarily diagonalized by the magic matrix as 1 The diagonalization gives λ 1 = µ + 2β and λ 2 = λ 3 = µ − β, which coincides with eq. (2.25) up to permutation.
The ambiguity is removed when the degeneracy of eigenvalues is broken. Let us consider correction matrices of the form 1 Note that in terms of the SM components, the Dirac mass terms read where lL and (l c )L are the left and the right components, and mD is a symmetric matrix. Then the unitary diagonalization U * ω mDUω = diag(m1, m2, m3), would imply that lL and (l c )L transform differently, but lL and lR transform in the same way.

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of which both can arise from a Z 3 symmetry, with different charge assignments. The former symmetric matrix fixes the diagonalization matrix to U TBM , whereas the latter one fixes it to U ω . Note that U ω can only be obtained from non-symmetric corrections, such as ∆M .
It is easy to get maximal mixing from M 33 , which will be often needed for our constructions below. For instance, the maximal 2-3 mixing can be obtained if ξ = (u 1 , 0, 0) and y v = y v , or y = 0. For maximal 1-3 mixing one needs ξ = (0, u 2 , 0) and y v = ωy v .
In general, by varying yv, y v , y v as well as hu i , one can obtain an arbitrary symmetric matrix for M 33 . (2.28) Obviously, only flavon triplets contribute to M 31 [65,66]. This matrix can be represented in the form Additional symmetries related to transformations of components of F 2 may lead, e.g. to h = h = h , thus further reducing the number of free parameters in D h . If two or all three fermions in F 2 have the same symmetry assignments, including charges under A 4 , e.g. F 2 ∼ (1, 1 , 1 ), we would obtain a singular mass matrix M 31 with two or three columns proportional to each other.
In general, the matrix (2.28) is non-symmetric. It can be made symmetric if D h ∝ D u (or equivalently, u i ∝ h i ), which can be obtained when h i themselves are given by VEVs of some new fields.
For the same set of the fermionic singlets, i.e. F 2 = F 1 , the most general coupling with flavons ϕ, ϕ , ϕ is given by
If the bare mass term dominates, it produces the structure with maximal 2-3 mixing which can be used to construct the TBM or BM mixing.
If all the couplings in eq. (2.31) are equal and the bare mass terms are absent being forbidden by additional symmetry, the matrix M 11 acquires a form where a = hv, b = hv and c = hv . The matrix M ω is diagonalized by the magic matrix U ω as U T ω M ω U ω . The eigenvalues of M ω in this case equal and there is no mass (eigenvalue) degeneracy provided that none of α, ωα, and ω 2 α is real, where α ≡ ab * +bc * +ca * . Notice that eq. (2.33) is the only form of symmetric matrix which produces the mixing U l = U * ω . 2 Thus, A 4 with additional conditions (i.e. equalities of couplings and VEVs, which in turn imply some additional symmetries) allows to get U TBM or U ω . The latter appears from various interactions, and together with additional maximal 1-3 rotation, can again lead to U TBM . In this sense A 4 has features which are close to the TBM mixing.  11 . Correspondingly, the light neutrino mass matrix is given by The screening can arise from M In general, any structure of m ν can be obtained. So, additional restrictions are needed in order to produce specific flavor structures. Further structuring can be achieved if not all possible flavons are introduced when some couplings and VEVs of flavons are forbidden (zero) or are related to each other. The latter implies an additional symmetry, such as Z 2 or Z 4 . If more than three fermionic singlets S exist we can obtain more flavor structures assigning S in triplet and singlet representations of A 4 : S ∼ 3 + (1, 1 , 1 ). This extension will be considered in some details in section 5.2.
Finally, in the case of 16 F being singlets of A 4 , m D = M (D) 11 in the form of eq. (2.31), and some new possibilities appear: 1. As we have marked already, equal Yukawa couplings (without bare masses) lead to the Dirac mass matrix of the form eq. (2.33). The later is diagonalized by U ω . Then maximal 1-3 mixing can follow from the portal and hidden sector interactions to produce the TBM mixing.
2. If S are transforming as triplets of A 4 , partial screening, i.e. m D /M D ∝ U ω , can be obtained since in this case M However, such a construction usually corresponds to a diagonal m D , which is difficult to generate from m D = M 11 .

Factorization of mixing matrix
In general, the lepton mixing matrix can be constructed as where different factors come from diagonalization of different mass matrices involved. In our approach we take V 0 ∼ V † CKM . This common part for the quarks and leptons is due to

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the SO(10) GUT and the CKM physics. The rest, U 0 ≡ U PMNS 0 = U 1 × U 2 . . ., reproduces U TBM or U BM , i.e. matrices of special type dictated by the flavor symmetry. Here we focus on construction of U 0 from the double seesaw mechanism, and discuss its TBM or BM forms. The CKM mixing, the non-zero 1-3 mixing, as well as other deviations from the TBM mixing, are probably related to each other and have some other origin, which we refer to as the CKM new physics. In view of small CKM mixing, U 0 can be considered as the zero-order approximation of the PMNS mixing, U PMNS 0 .
Flavor features which lead to TBM matrix have been extensively studied before (see [67,68] and references therein). The real TBM matrix can be factorized as This factorization is not unique: in general any product of U ω (also with permuted rows and columns) and 2-state maximal mixings, U ω U max 12 , U ω U max 13 , or U ω U max 23 , leads to the TBM matrix with permuted rows and columns up to rephasing. This, however, does not affect physics: flavor content of the eigenstates with certain eigenvalues is the same for all the cases.
It is straightforward to generalize the factorization via U ω and U maxT where U A is arbitrary unitary matrix, and D α is arbitrary diagonal phase matrix and the phases can be absorbed into the wave functions of the charged leptons. The matrix D β contributes to the Majorana phases of neutrinos. In the generalized form, one may use and one of the factors is maximal 1-2 rotation instead of 1-3 rotation. However, nowŨ ω = D α U ω U maxT 13 U max 12 , which is not reduced to U ω with permuted columns (rows), and so it is difficult to obtain U ω as a result of symmetry.
Another possible factorization of U TBM , without using the magic matrix U ω directly, is U max 23 U 12 (θ TBM ), where sin 2 θ TBM = 1/3. However, mass matrices diagonalized by U 12 (θ TBM ) do not naturally appear from A 4 symmetry.
According to the double seesaw, there are three mass matrices relevant for generation of the lepton mixing: where the first matrix is the mass matrix of charged leptons, the second one is the screening factor and the third one is the mass matrix of singlet fermions. In general all these matrices provide contributions to the lepton mixing matrix. Different mass matrices in eq. (2.36) can be responsible for generation of different factors in the TBM matrix. Let us mention some possibilities mostly related to U ω and U max 13 , which will be realized in specific schemes in section 3.

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1. U ω may follow from the charged leptons, and diagonalize the matrix m D , whereas U max 13 comes from the neutrino mass matrix, so that In the SO(10) framework U l = U * ω implies that m l = m D = M ω defined in eq. (2.33).
2. U l = 1, which means that m D is diagonal, U ω follows from the portal factor:

Additional symmetries and VEV alignments
In what follows we construct schemes for masses and mixing using mass matrices elaborated in this section as building blocks. It is easy to show that in the formulated framework without additional constraints any mass matrix of the light neutrinos can be obtained. Therefore to get certain flavor structures we will use -minimal possible number flavon fields; -minimal number of (different) couplings; -the most symmetric situations when as many as possible couplings are either zero or equal. The latter implies certain extended symmetries and for definiteness we will use additional symmetry Z 4 .
Existence of different energy scales in this framework, m D M D M S , allows to realize the idea that different symmetries are realized at different scales with larger symmetry at higher mass scale. Higher scale has larger symmetry, which is then broken explicitly or spontaneously at lower scale. Thus, M S may have additional symmetry which is not realized in portal and visible interactions. Feedback of this breaking onto higher scale physics is expected to be suppressed by factors M D /M S ∼ 10 −2 . In particular, due to additional symmetries M S can have the TBM form. Although the symmetry is not realized at the portal scale, the corrections to TBM mixing can be of the order 10 −2 . See also [48] for some realizations of this idea.
We formulate conditions on couplings and VEVs which should be satisfied to get the required fermion masses and mixing. We provide some hints of how the conditions can be realized and where they can originate from.
Mechanisms have been elaborated which allow to get relations between VEVs of multiplets. In particular for triplets the alignments v(1, 1, 1), or v(0, 1, 1), v(0, 0, 1) can be JHEP05 (2016)135 obtained, as a consequence of symmetry of the potential [69][70][71]. Usually such potentials have several degenerate vacua, and so some mechanisms of selection should exist. Construction of models which realize the conditions is beyond the scope of this paper.
In various cases we need also correlations of VEVs of singlet and triplet flavons: Multiplied by U * ω , the relation can be inverted to: For equal constants y = y = y, it can be written as For non-equal constants (y, y , y ) we should substitute v → v y /y and v → v y /y in the left-hand sides of these equalities. In turn, the relation (2.40) can be induced e.g. by scalar potentials of the form Consequently, A · ξ should have the same Z 4 charges as ϕ.
3 Schemes with S transforming as a triplet of A 4 For single complete set of the flavons ξ ∼ 3, ϕ ∼ 1 the most general A 4 symmetric Yukawa Lagrangian (at the lowest order) is Here etc., with general definitions of these combinations given in (2.17). The terms L S contain renormalizable couplings. The symmetry allows also to introduce renormalizable flavonless couplings of 16 F and S. If several sets of flavons ϕ a and ξ b exist, for each of them one should introduce interactions similar to those in eq. (3.1).

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All the mass matrices (m D , M D and M S ) generated by this Lagrangian are of the M 33 type plus matrices proportional to unit matrix. (The latter can be forbidden by additional symmetry.) If the same flavons contribute to all three mass matrices, these matrices are not completely independent being functions of the same set of VEVs. On the other hand, sets of coupling constants for three types of interactions, y i ≡ (y i , y i , y i ) and h i (i = 10, 16, 1), can be totally different. With this one can get arbitrary matrices m D , M D and M S , and consequently, arbitrary symmetric mass matrix of the light neutrinos.
In what follows we will study the situations that some interactions are forbidden by additional symmetry Z 4 . One possibility is to keep minimal number of couplings in visible and portal sectors, and to obtain all the rest from the hidden sector. Among four types of couplings in the visible sector and portal interactions, y 10 , h 10 , y 16 and h 16 , at most two can be zero (one for 10 H and one for 16 H if we do not take into account low-dimension interaction couplings y b ). Next complication is to assume that some couplings among y 10 and y 16 are zero, which would require more symmetries. The couplings of the hidden sector y 1 and h 1 , are also restricted by Z 4 . Thus, we have four possibilities determined by representations of flavons that contribute to the corresponding couplings/mass terms: h 10 = h 16 = 0, which we call the singlet-singlet scheme; y 10 = y 16 = 0 -the triplet-triplet scheme; h 10 = y 16 = 0 -the singlet-triplet scheme; and y 10 = h 16 = 0 -the triplet-singlet scheme. We consider these possibilities in order and then present schemes in which both singlets and triplets contribute to the same mass terms.

Singlet-singlet flavon scheme
The scheme with h 10 = h 16 = 0 allows to realize the following scenario: where U l diagonalizes m D , and U S diagonalizes M S . Here the TBM mixing comes from solely the hidden sector. 3 Only singlet flavons and possibly flavonless operators generate visible and portal masses. The triplet flavon couplings, h 10 and h 16 , are forbidden by additional Z 4 symmetry. We introduce flavon singlets ϕ, which operate in the visible sector and portal terms. One flavon triplet ξ of A 4 act in the hidden sector. For symmetry assignment under Z 4 we use Graphic representation of the scheme is shown in figure 1.
The most general A 4 × Z 4 symmetric Lagrangian has the form JHEP05(2016)135 and there is enough freedom to get any spectrum. 4 The light neutrino masses satisfy that m i ∝ λ i . There are several ways to obtain the additional correction ∆M S in the form of ∆M : 1. A correction proportional to ∆M but directly contributing to m ν can come from additional fermion singlets S, S , S , as we will discuss in section 5.2.
2. Another modification of the scheme which would be equivalent to the correction ∆M (2.26) is to introduce flavons φ, φ , φ which modify the diagonal elements of M S . Suppose that φ, φ , φ acquire VEVs (v 1 , v 1 , v 1 ) which satisfy together with couplings the equality y 1 v 1 = y 1 v 1 = y 1 v 1 . And differently from above, ξ = (u 1 , u 2 , u 3 ). Then the mass matrix of singlets can have the form In the case of complete screening it gives the light neutrino mass matrix: m ν = b·M S . Imposing further equality u 3 ≡ u 2 we obtain from (3.4) maximal 2-3 mixing, and still enough freedom is left to fit all the data. In particular, equality h 1 u 1 = h 1 u 2 + 3y 1 v 1 leads to the TBM mixing and to the mass eigenvalues which can reproduce essentially any neutrino mass spectrum, and therefore normal or inverted mass hierarchy.

Triplet-triplet flavon scheme
The Lagrangian is given in eq. (3.1) with y 10 = y 16 = 0 and graphic representation of the scheme is shown in figure 2. It allows to realize a scenario The second and third conditions are satisfied if, e.g. u 1 ≈ u 2 ≈ u 3 ≈ m τ /(3h 10 ). One important feature of this case is that the matrices m D and M D in eq. (3.6) have different eigenvalues but the same mixing. Since off-diagonal parts of both matrices are generated by the same flavon field ξ, they are diagonalized by the same rotation U 33 which diagonalizes
In the hidden sector due to Z 4 symmetry given by eq. (3.5) h 1 = 0, and so the flavon triplet does not contribute to the singlet mass matrix. Therefore M S = M ϕ 33 and the neutrino matrix in eq. (3.9) is diagonalized by U † 33 . Consequently, the neutrino mixing U 0 = U † 33 . This result can be immediately obtained by noticing that in the original symmetry basis and in the presence of complete screening the neutrino mass matrix is diagonal whereas the lepton mass matrix is diagonalized by U l = U 33 .
In turn, the matrix U 33 can reproduce the TBM or BM mixing. Indeed, if the 1-3 mixing is generated by the CKM physics, the 1-3 mixing, which originates from U 33 , should be zero. So, in terms of the standard parametrization of U PMNS we have to have U 33 = U 12 (θ 12 ) T U 23 (θ 23 ) T . This requires equality u 2 = u 1 in M ξ 33 (2.23) which leads to the maximal 1-2 mixing, θ 12 = π/4. Then for the 2-3 mixing we obtain For u 1 u 3 this equation would give nearly maximal 2-3 mixing. However, this is not consistent with (3.7) which has for u 2 = u 1 two solutions: u 3 = u 1 and u 3 = (2 √ 2i)u 1 . In the first (best) case we obtain θ 23 = 35 • , which can be slightly bigger if u 2 = u 1 , but the latter corresponds to non-zero 1-3 mixing. Thus, the scheme produces the bi-large mixing from the neutrino sector. Neutrino masses are given by eigenvalues of M ϕ 33 . According to the discussion in section 2.4 (see eq. (2.19) there and also appendix B) any type of neutrino mass spectrum and any hierarchy can be obtained, depending on the Yukawa couplings and VEVs of ϕ.

Singlet-triplet flavon scheme
The Lagrangian of this scheme is given by eq. (3.1) with h 10 = y 16 = 0, and the graphic representation is shown in figure 3. Only flavon singlets contribute to the Dirac mass matrix and only triplet flavons to the portal mass matrix. Here we can realize the scenario with cancellation of the portal and hidden sector matrices in the double seesaw mechanism. Vanishing h 10 and y 16 can be obtained with Z 4 symmetry assignment (3.10) The key feature is that 16 F and S have different Z 4 charges. The This matrix produces a zero (small) θ 13 , maximal 2-3 mixing and large 1-2 mixing. It only corresponds to the normal mass hierarchy, which is due to the connection of neutrino masses with the mass hierarchy of charged fermions.

Triplet-singlet flavon scheme
This case is in some sense opposite to the previous one: the triplet ξ produces the Dirac mass matrix, whereas singlets generate the portal mass matrix, see figure 4. The Lagrangian is given by eq.  So, U ω appears from the charged leptons mass matrix whereas the hidden sector gives maximal 1-3 mixing, and there is a complete screening. As a result, U 0 ∝ U ω U max 13 = U TBM . The symmetry assignments of the field content we use is under Z 4 , and the full graphic representation are shown in figure 5.
Both the singlet ϕ and triplet ξ flavons, with arbitrary VEVs, participate in visible and portal interactions. The others, φ and ξ , exclusively couple with fermion singlets.
The Lagrangian has the following terms

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It is diagonalized as U † l m D V R , where U l = U * ω and V R = U ω . The eigenvalues of the matrix equal (3.14) That is, now couplings of ϕ and ξ should also correlate. Possible origins of this proportionality have been discussed in section 3.1. As a consequence,

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To break the degeneracy of eigenvalues of m D and fix U l = U ω , one has to introduce a non-symmetric correction δm D (see e.g. ∆M in eq. (2.26)). In particular, one can use the anti-symmetric correction with non-degenerate masses of charged leptons given by To keep screening, similar corrections should also be introduced for M D . It is straightforward to see that the masses of light neutrinos are the same as in eq. (3.16), because M S is unchanged.

Scheme with mixed contributions: the BM mixing
Our framework also allows for new possibilities to produce the BM mixing (see appendix D). The BM mixing can be factorized by two maximal mixing rotations: U BM = U max 23 U max 12 . In this connection we present a scenario in which maximal 2-3 rotation comes from the charged leptons and the maximal 1-2 one arises from the hidden sector: The particle content and the charge assignments are given in figure 6, where the Z 4 charges are The eigenvalues of this matrix equal To obtain mass hierarchy we should arrange that y 10 v ≈ −0.5y 10 v, which leads to and m D1 m D2 . Additional relation h 10 u ≈ 3 2 y 10 v gives m D2 m D3 . To realize complete screening, m D ∝ M D , we impose the proportionality of both singlet and triplet couplings y and h with 10 H -plets and 16 H -plets as eq. (3.14).
The maximal 1-2 mixing from M S requires that the second flavon triplet, interacting in the hidden sector, has VEV ξ → (0, 0, u ), which implies the permutation symmetry of the first and second generations in hidden sector. Equal diagonal elements of M S are generated by singlet φ with VEV φ = v 1 . Under these conditions Since m ν ∝ M S , an agreement with observed mass spectrum can be achieved for the inverted mass hierarchy when |y 1 v 1 | |h 1 u |. As a result, the three light neutrino masses equal m 3 ∝ y 1 v 1 , m 2 ≈ m 1 ∝ h u and ∆m 2 21 ∝ 4y 1 v 1 h 1 u . To obtain normal mass hierarchy we can introduce also φ and φ with couplings y 1 φ = ωy 1 φ , which actually allows to obtain arbitrary values of neutrino masses. Recall that conditions we impose here can be consequences of additional symmetries operating in the hidden sector.
If the bare mass term dominates in M S we would get where M dem is the matrix with all the elements equal 1. It can be diagonalized by the TBM mixing but has two equal neutrino masses. Such a degeneracy means that the mixing is not JHEP05(2016)135 In what follows we will consider another way of getting the required corrections, which also justifies some assumptions we have made before. We elaborate on a scenario which realizes relations Here, as in the previous case, the first relation means that U l = 1, the second relation corresponds to partial screening of the Dirac structures which generates the magic matrix. The singlet matrix together with D h produces maximal 1-3 mixing. Then U 0 ∝ U ω U max 13 = U TBM . We introduce Z 4 to forbid h 10 and some other couplings. The field content, symmetry assignments and graphic representation of the scheme are given in figure 7. The features of the scheme (as compared with minimal structure discussed in the beginning of the subsection) are (i) different Z 4 charges of fermion singlets, (ii) two flavon triplets participating in the portal interactions, (iii) additional flavon singlet φ , (iv) auxiliary flavon triplet A.
The most general expressions for the three terms of the Lagrangian (2.1) allowed by symmetry in the lowest order are There is a generic problem here: on the one hand, obtaining U ω from M D requires (in the simplest case) that only one flavon triplet participates in the portal interactions. On the other hand, to get maximal 1-3 mixing from the hidden sector one needs to distinguish S from S and S , so that the terms (SS )ϕ and (S S )ϕ are forbidden. To forbid such terms, singlets S and S should have equal Z 4 charges which differ from the charge of S . Therefore in L N S two flavon triplets ξ and ξ carrying different Z 4 charges are necessary. Moreover, a new flavon singlet, φ , with Z 4 charge different from ϕ, should be introduced to generate the mass of S . Now in the visible sector, both the bare mass and off-diagonal elements of M 33 are absent, so L D in eq. (4.4) only generates the same diagonal Dirac mass matrices m D = M ϕ 33 ( y 10 , v) to all SM fermion components as in eq. (2.20) So the expressions of mass matrices are in the flavor basis from the beginning. 5 The hierarchy of mass eigenvalues m D1 , m D2 , m D3 can be obtained in the same way as eq. (2.22). The portal mass matrix generated by the terms L N S equals where the VEVs of two flavon triplet are denoted as ξ = (u 1 , u 2 , u 3 ), ξ = (u 1 , u 2 , u 3 ).
To get exactly the form of M 31 , their VEVs need to be aligned in the same way: ξ = ξ (i.e. u i = u i ). With this equality, the mass matrix of eq. (4.6) can be written as in eq. (2.29): The mass matrix of singlet fermions generated by eq. (4.4) has the form of M 11 with zeros due to symmetry assignment: where v 1 ≡ φ . Combining M D and M S we obtain from eq. (2.35) the light neutrino mass matrix The diagonal matrix mD can also be written in the form of product of matrices as 10 H Λ f U * ω (y10v1 + y 10 v C + y 10 v C T )Uω, where C is the cyclic matrix with only non-zero (1,3), (2,1), and (3,2) unit elements.

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To get the TBM mixing the following two conditions should be fulfilled: which leads to partial screening, and to get maximal 1-3 mixing from the central matrix of eq. (4.8).
The first condition (4.9) is the correlation of the singlet and triplet VEVs, which can be obtained in the same way as it was done in eq. (2.40), by introducing the auxiliary A 4 triplet field A with VEVs A = (a, a, a)/ √ 3. The inequality m D M D implies a very small VEV: a 1. Therefore the auxiliary field does not change other structures of the model. Provided that y 10 = y 10 = y 10 , the Dirac masses can be rewritten as The second condition, eq. (4.10) can be satisfied if, e.g.
These equalities then produce the masses of light neutrinos as Strong normal (inverted) mass hierarchy requires y 1 /h 16 ≈ −y 1 /h 16 (y 1 /h 16 ≈ y 1 /h 16 ). There exist different ways to avoid the introduction of the second triplet and tuning of its VEVs. One possibility is to use single ξ with Z 4 charge i and assign to S the same charge i as other singlets have. Then to get nearly maximal 1-3 mixing from the hidden sector additional symmetry, e.g. Z 2 , can be introduced and made conserved in the hidden sector only (see also discussion in section 2.5). If S is odd and other particles of the hidden sector are even, then M S will have the form (4.7) with zero 12-and 23-elements. This additional symmetry is broken in the visible and portal sectors, which have substantially lower energy scale M GUT . So, one expects that corrections to the Z 2 symmetric results are of the order M GUT /M Pl , which are not important.
Instead of second flavon triplet ξ , second 16 H with Z 4 charge i can be introduced in the portal interaction The VEVs of 16-plets should be equal, 16 H = 16 H , to get the matrix factor U ω . Alternatively, to replace ξ , one can also introduce a flavon singlet φ with Z 4 charge −i, which leads to a dimension-6 portal interaction (16 F ξ) 16 H φS .
Notice that this scheme can be made more logical if three 16 H (or three ξ) with different Z 4 charges are introduced. The neutrino mass matrix equals

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and it converts to , which can be diagonalized by U 33 = U ω . In this case eq. (4.12) gives

Generalizations
We have assumed above that the light neutrino masses are mainly determined by the double seesaw mechanism with three fermion singlets S. Here we consider other possible contributions to masses and mixing, including the CKM physics effects.

Effects of linear seesaw
The direct mixing between left-handed neutrinos and fermion singlets S via the 16 H Higgs gives the linear seesaw contribution to the light neutrino masses: which are the last terms of eq. As an example, we will briefly describe the case when the linear seesaw gives the main contribution to neutrino mass. We assume that 16 F and S are A 4 triplets. Then m l = m D can have the form of M ω , as eq. (2.33), which is diagonalized by U ω . Also complete screening can be realized, For certain VEV alignment as in the scheme of section 3.5, m has the form of eq. (2.37), producing the maximal 1-3 rotation, so that U 0 = U ω U max 13 = U TBM . Realization of partial screening is different from the case of double seesaw. Let us take a scheme with 16 F ∼ 3 and S ∼ (1, 1 , 1 ω , and m has the form of M 31 . Then the neutrino mass matrix is given by

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Under conditions u 3 = u 2 in D u and h 3 = h 2 in D h , which imply the 2-3 permutation symmetry, this matrix gives with maximal 2-3 and zero 1-3 mixings. Additional assumption h 2 = −h 1 is then required to produce the TBM mixing, since U l = 1. Three parameters h 1 , u 1 and u 2 allow to get arbitrary masses for the three light neutrinos.

Extended hidden sector
Introduction of more than three fermionic singlets of SO(10) opens up new possibilities to obtain the observed mixing and masses too. And in some cases it allows to simplify the conditions on couplings and VEVs we imposed in the previous sections. On the other hand, the Dirac screening may not be straightforward any more. Let us consider for definiteness effects of three additional SO(10) singlets, so that the hidden fermion sector consists of S ∼ (3, 1) of A 4 . The light neutrino mass matrix is given by  3) where M N is the effective mass matrix of RH neutrinos: Using expressions (5.2) we find in terms of 3 × 3 matrices: where the subscript S means that all the matrices in the bracket belong to the hidden sector. Inserting it into eq. (5.4) and then into eq. where g is a coefficient, which takes care of the proportionality and normalization of matrices, thus it is irrelevant for this discussion. Finally, the proportionality M (S) 11 ∝ 1 can be imposed by certain symmetries that only apply to hidden sector interactions. In this case the second term in eq. (5.9) becomes gU ω 1 U ω , which reproduces exactly the correction ∆M as in eq. (2.26). This correction removes ambiguity in diagonalization of the first term and fixes the TBM mixing. There are enough parameters left to reproduce the observed neutrino masses.

On the CKM new physics
So far we have discussed situations that no CKM mixing is generated and all the Dirac masses of SM fermions are the same: m u = m d = m l , contradicting observations. Additional physics (new fields, interactions) is therefore needed to explain properties of the quark sector. This additional physics should not destroy the constructions in the neutrino sector. Here we outline how this can be done, and refer to [48] for more details. To be in agreement with observations, three aspects should be addressed.
Therefore, different Yukawa coupling matrices Y u 10 and Y d 10 produce different mass hierarchies and also CKM mixing.
To keep the neutrino part unchanged we assume that in the symmetry basis the CKM mixing comes from the down fermions. That is, couplings of 10 d H further violate A 4 , explicitly or spontaneously.
(ii) Another possibility is to keep the single 10 H but introduce the dimension-5 This operator also breaks the mass degeneracy between up-and down-type quarks [72].
(iii) Vector-like families of fermions 10 F , 10 F can exist. Their mixing with usual 16 F break the degeneracy between up-type and down-type masses of quarks. The reason is that 10 F only contains down-type quarks and lepton doublets, but no up-type quarks.  [74]. The product of 10 d H and 45 H has the decomposition 10×45 = 10+120+320, which contains 120 H . If 45 H acquires its VEV in the direction of (B-L), where B (L) is the baryon (lepton) number, then the effective VEV of 120 H leads via interactions (5.11) to difference of masses of the second fermion generation (e.g. m µ and m s ).
If more than one 45-plet exist, the dimension-6 operators give contributions to the masses of the first generation of fermions [29,75].
3. Nearly the same CKM-type mixing of down-type quarks and charged leptons despite the different mass hierarchies of them. In our framework the CKM matrix is generated by m d , and the CKM type correction to lepton mixing which lead to expression (1.1)

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are generated by m l . Since masses of the down-type quarks and charged leptons are different, the mass matrices m d and m l should be different, and therefore in general, one expects that diagonalization matrices U d and U l that diagonalize them are also different. This may invalidate eq. (1.1), which implies that both matrices should be diagonalized by the same rotation U l U d [48].
One solution to this problem was described before in section 3.2. The mass matrix can be the sum of a diagonal matrix proportional to the unit matrix and an off-diagonal matrix:  The framework allows to disentangle the CKM new physics and the neutrino new physics. The neutrino new physics explains the smallness of neutrino masses and generates the large lepton mixing. The CKM new physics, being common for quarks and leptons due to GUT, is responsible for the CKM mixing and difference of mass hierarchies of the up-type and down-type quarks, as well as the charged leptons. The framework naturally realizes the relation between the quark and lepton mixings: U PMNS ∼ V † CKM U 0 , where the structure of matrix U 0 is determined by the flavor symmetry.
In the lowest-order approximation with neutrino physics only: (i) quark and lepton masses are generated, (ii) no quark mixing appears, (iii) large lepton mixing is produced, whose form is dictated by the flavor symmetry, (iv) small neutrino masses are generated via the double seesaw mechanism, or its generalizations are related to more than three fermionic singlets.
The CKM physics gives corrections (or perturbations) to this lowest-order picture and the corrections are related to further violation of flavor symmetry effects and higher-order operators.
2. We have mainly focussed on the neutrino new physics and generation of U 0 in this paper. Using A 4 × Z 4 as the simplest example of flavor group, we have systematically explored the flavor structures (masses, mixing) which can be obtained in this framework, depending on the field content and symmetry assignment. The interplay of GUT SO(10) symmetry and flavor symmetries is crucial in obtaining certain flavor structures.

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In this framework three different mass matrices are involved in generation of lepton mixing: m D -the Dirac mass matrix of SM fermions (charged leptons and neutrinos), M D -the portal mass matrix, or m D M −1 T D -the portal factor, and the matrix of fermion singlets M S . This leads to new possibilities to explain the observed mixing. One of the key elements, which allows to disentangle the quark masses and mixing from neutrino masses and mixing, is the Dirac screening (either complete or partial).
Generically, even without flavor symmetries, the framework explains no or small quark mixing, large lepton mixing and small neutrino masses. It leads to the relation between the quark and lepton mixings of the form (1.1).  It is also possible that certain symmetries exist in the hidden sector only, being broken (spontaneously or explicitly) by low scale interactions in the visible and portal interactions.
To have a complete determination of masses and mixing, in particular, to obtain U 0 ∼ U TBM , or U BM , further restrictions are needed. We found the following generic conditions, of which all or some should be satisfied in specific schemes: -equality (or proportionality) of the flavon singlet couplings y in the visible sector (i.e. with 10 H ) and in portal (with 16 H ) interactions; -equality of flavon singlet VEVs; -correlation between VEVs of flavon singlets and triplet; -certain VEV alignment of triplet(s) ξ including equality of three VEVs, or nonzero VEV of only one component of the triplet;

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-corrections to special matrices which break degeneracy of mass eigenstates and thus fix the mixing.
These relations can be manifestations of further unification, additional symmetries, or selection of certain minimum of the potential from several (degenerate) possibilities.
In some cases correlations of the bare mass terms or flavonless contributions are also required.
3. We have presented several schemes with minimal possible number of flavons and couplings in the visible and portal sectors. Interesting realizations are when U TBM , U BM , or certain factors of these matrices follow from different sectors: -the TBM mixing from the hidden sector M S (with complete screening); -approximate TBM mixing from charged leptons; -approximate TBM mixing from the portal and hidden sectors; -the magic matrix from charged leptons and maximal 1-3 mixing from the hidden sector, thus reproducing U TBM ; -the magic matrix from the portal (partial screening) and maximal 1-3 mixing from the hidden sector; -the 2-3 maximal mixing from charged leptons and the 1-2 maximal mixing from M S , thus reproducing U BM .
In many cases explanations of masses and mixing decouple from each other. This is because of the special (maximal, magic) form of the mixing.
We also commented on possible effects of the linear seesaw and additional fermion singlets in the hidden sector. They can provide required corrections to the zero-order structures.
We outlined the CKM physics, indicating additional elements which could be introduced to obtain the quark sector. We showed that they can be introduced in a consistent way without destroying the part related to neutrino new physics.
4. This is an initial study -the first step toward the model building, which shows, nevertheless, interesting potential to explain fermion masses and mixing. In complete models one needs to further introduce the scalar potential explicitly and find its minima, to consider effects of high-order operators, and to take into account renormalization group effects, etc. Other flavor symmetry groups can be studied. All these issues are beyond the scope of the present paper.

5.
The key challenge is the possibility of testing this framework experimentally. In some cases (or with additional assumptions) one can obtain certain predictions for the neutrino mass hierarchy, the type of mass spectrum, value of the Dirac CP phase and values of Majorana phases. Recall that in most of the scenarios considered here there exist enough free parameters (couplings, VEVs) to accommodate any neutrino JHEP05(2016)135 mass spectrum and therefore any type of mass hierarchy. As usual, A 4 symmetry restricts mixing but not masses, and additional restrictions on the field content and some additional symmetries should be introduced to fix the spectrum. Discovery of the proton decay would be a partial confirmation of the GUT framework. The leptogenesis can give some probes of the framework too.
The hidden sector can provide stable particles to act as dark matter candidates. So, detection of dark matter particles may shed some light on the hidden sector, and consequently, the framework studied in this paper. Sterile neutrinos, if exist, can also originate from such a hidden sector.

A A 4 group
In the context of A 4 flavor symmetry, we use decomposition of product of two triplets a, b as follows 1 = (ab) = a 1 b 1 + a 2 b 2 + a 3 b 3 , 1 = (ab) = a 1 b 1 + ω 2 a 2 b 2 + ωa 3 b 3 , where ω = e 2πi/3 , and h 1 and h 2 are both free number parameters. We further assume that h 1 = h 2 is protected by some underlying mechanism, and in practice choose the value of h 1 to be 1 because it can be absorbed into relevant Yukawa couplings. Generally, this assumption does not affect the forms of m D and M S , which are required to be symmetric anyway. But in the case that both 16 F i and S i are triplets under A 4 , this assumption leads to a symmetric M D , and thus helps to preserve the screening condition which links M D to m D . and α ≡ µµ * + µ µ * + µ µ * . The arguments of M 2 and M 3 are given by 2 arctan respectively. If α is real, the moduli of two eigenstates are degenerate.