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 $U_{PMNS} \sim V_{CKM}^{\dagger} U_0$, where structure of $U_0$ is determined by the flavor symmetry. The key feature of the framework is that apart from the Dirac mass matrices $m_D$, the portal mass matrix $M_D$ and the mass matrix of singlets $M_S$ are also involved in generation of the lepton mixing. This opens up new possibilities to realize the flavor symmetries and explain the data. Using $A_4 \times Z_4$ 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 $U_0 \sim U_{TBM}$ or $U_{BM}$. 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.


Introduction
There are several hints in favor of a framework with Grand unification, hidden sector at the string/Planck scale, and flavor symmetries acting both in visible and hidden sectors: ‡ xchu@ictp.it § smirnov@mpi-hd.mpg.de 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), 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 [48][49][50].
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 [11,51], (iii) stability of this screening with respect to renormalization group (RG) running [51], renormalization of the relation (1) [52], (iv) symmetry origins (groups T 7 and Σ 81 have been used) of the screening, as well as partial screening [53], (v) lepton mixing from the symmetries of the hidden sector [45].
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 sect. 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 sect. 3 we present the simplest viable schemes with fermion singlets S transforming under three-dimensional representation 3 of A 4 . In sect. 4 the schemes are considered with S transforming under one-dimensional representations 1, 1 , 1 of A 4 . Sect. 5 is devoted to various generalizations: effect of linear seesaw, contribution from additional singlets, elements of the CKM physics. We summarize our results in sect. 6. Some technical details are presented in the Appendix.
2 Neutrino mass and mixing matrices 2

.1 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: • 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, which produce masses of fermions: 1). The visible (Dirac) sector Yukawa interactions 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 sect. 5.3).
2). The portal interactions, generate mass matrices of Dirac type for the LH and RH neutrinos from 16 F i and singlets 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 .., k = 10, 16, 1.
In what follows we will take the lowest-order approximation in Φ /Λ f . The mass matrix of neutral leptons generated by interactions (3), (5) and (7) in the basis (ν, N, S), where N ≡ ν c , equals Given that m ν ∼ O(0.1) eV and v L < 246 GeV, it is possible to define a critical mass scale M cr = 4 × 10 14 v L · Y 16 246 GeV 2 GeV , (11) 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 P l the "double-seesaw" dominates and the "linear-seesaw" term may lead to small corrections.
According to Eq. (4) 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. (8), 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. (13). 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 [54][55][56][57][58].

Screening and partial screening
The double seesaw allows to realize the screening of the Dirac structures [11,51], which will be used in many examples presented in this paper. If then according to Eq. (10) 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 (15) the mixing is determined exclusively by M S .
If the screening condition (14) 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 (14) 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. (16) produces a matrix of special form, which then leads to the required form of total mixing matrix (see sect. 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 . 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).

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: 1. F 1 ∼ 3, F 2 ∼ 3. The most general coupling with flavons invariant under A 4 is 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 (17) lead to the mass terms 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 (19) are generated by different flavons. Depending on additional symmetry assignments one, or two, or all three terms can contribute. Explicitly, 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 (20) can be written as where 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 (23) leads to the hierarchy of masses: The hierarchy between the lighter states, m 2 m 1 , can be obtained by special tuning of relations in Eq. (23) or introduction of other contributions, e.g. from the linear seesaw terms.
In general, it is difficult to get correlation (23) 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 as a consequence of special forms of the potential.
In Eq. (19) the off-diagonal matrix equals It has the form of the Zee-Wolfenstein mass matrix [59,60] with the eigenvalues satisfying the equality m 1 + m 2 + m 3 = 0. For arbitrary values of u i , the diagonalization matrix and eigenvalues of M ξ 33 are presented in Appendix. C, and here we consider some special cases. If u 1 = u 2 , the matrix can be diagonalized by maximal 2-3 rotation and vanishing θ 13 . Then for the 1-2 mixing we obtain sin 2 2θ 12 = 8u 2 1 /(8u 2 1 + u 2 2 ). If all VEVs are equal, u 1 = u 2 = u 3 , then sin 2 2θ 12 = 8/9, which corresponds to the TBM mixing.
Let us consider a special case of the total matrix M 33 (19), 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 [61]. 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 T BM , 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 equal eigenstates. In particular, the matrix (25) can be unitarily diagonalized by the magic matrix as 1 U * ω M special
The diagonalization gives λ 1 = µ + 2β and λ 2 = λ 3 = µ − β, which coincides with Eq. (26) up to permutation. The ambiguity is removed when the degeneracy of eigenvalues is broken. Let us consider correction matrices of the form of which both can arise from a Z 3 symmetry, with different charge assignments. The former symmetric matrix fixes the diagonalization matrix to U T BM , 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. F 1 ∼ 3 and F 2 ≡ (F 2 , F 2 , F 2 ) ∼ (1, 1 , 1 ). The most general couplings with flavons generate the mass term F 1 M 31 F T 2 with Obviously, only flavon triplets contribute to M 31 [62,63]. This matrix can be represented in the form where D u ≡ diag(u 1 , u 2 , u 3 ) and D h ≡ diag(h, h , h ). 1 Note that in terms of the SM components, the Dirac mass terms read where l L and (l c ) L are the left and the right components, and m D is a symmetric matrix. Then the unitary diagonalization U * ω m D U ω = diag(m 1 , m 2 , m 3 ), would imply that l L and (l c ) L transform differently, but l L and l R transform in the same way.
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 (29) 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.

F
). 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 It leads to the mass terms For all free parameters, M 11 is an arbitrary symmetric matrix. One can get from Eq. They are satisfied if, e.g. m 0 =m 0 , while all the couplings and all the VEVs are equal, respectively. 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. (32) 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. (34) 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 T BM or U ω . The latter appears from various interactions, and together with additional maximal 1-3 rotation, can again lead to U T BM . In this sense A 4 has features which are close to the TBM mixing.  11 . Correspondingly, the light neutrino mass matrix is given by and now M D is usually not symmetric. Using representation (30) we have The screening can arise from M  Table. 1. In what follows we mostly focus on the first two possibilities with 16 F ∼ 3 and S transforming as either 3 or (1, 1 , 1 ). 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 sect. 5.2.
Finally, in the case of 16 F being singlets of A 4 , m D = M (D) 11 in the form of Eq. (32), 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. (34). 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 the SO(10) GUT and the CKM physics. The rest, 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 P M N S 0 .
Flavor features which lead to TBM matrix have been extensively studied before (see [64,65] 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 T BM , without using the magic matrix U ω directly, is U max 23 U 12 (θ T BM ), where sin 2 θ T BM = 1/3. However, mass matrices diagonalized by U 12 (θ T BM ) 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. (37) 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 sect. 3.
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) 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 [45] 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 obtained, as a consequence of symmetry of the potential [66][67][68]. 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 where a = h/ √ 3y. To reach the correlation (39) we introduce a dimensionless auxiliary field A with the VEVs A = (a, a, a)/ 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 (41) 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 (18). 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. (42). 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 = 0the 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 Graphic representation of the scheme is shown in Fig. 1. The most general A 4 × Z 4 symmetric Lagrangian has the form This is enough for the screening since the same flavon fields contribute to both mass matrices. The proportionality of Eq. (44) is rather a generic requirement for several schemes, so we discuss it in some details. This proportionality can be obtained in various ways as follows: (i) All the coupling constants for 10 H and independently 16 H are equal to each other (or approximately equal): y 10 = y 10 = y 10 , y 16 = y 16 = y 16 .
In this case correlations are not needed. The equalities can be residuals of embedding of A 4 into SO(3), so that the singlet of A 4 (1, 1 , 1 ) steam from the triplet representation of SO(3). The (approximate) equality of the coupling constants is also needed to obtain the hierarchy of masses of quarks and charged leptons.
(ii) The proportionality can be a consequence of further unification of 16 F and S in 27-plets of E 6 .
In both cases (i) and (ii) the Z 4 charges of 16 F and S should be equal, as given in Eq. (43). (iii) The proportionality can be a consequence of symmetry with respect to permutation 16 F ↔ S and 10 H ↔ 16 H . In this case also Z 4 charges of 10 H and 16 H should be the same, which is satisfied in the symmetry assignment.
Under condition (44) 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 sect. 5.2.
2). Another modification of the scheme which would be equivalent to the correction ∆M (27) 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 (45) 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 the required mass spectrum. Figure 2: The same as in Fig. 1, but for the triplet -triplet flavon scheme. Grey box with 1 corresponds to flavonless couplings.

Triplet -triplet flavon scheme
The Lagrangian is given in Eq. (42) with y 10 = y 16 = 0 and graphic representation of the scheme is shown in Fig. 2. It allows to realize a scenario in which the mixing originates from the charged leptons. Vanishing singlet couplings y 10 and y 16 can be obtained by imposing the Z 4 assignment Now only the triplet and operators without flavons contribute to the visible and portal masses. Therefore m D = m 0 where m 0 D = y b 10 10 H , m 0 D = y b 16 16 H and r ≡ h 10 10 H /h 16 16 H . Diagonalizing m D we find that masses of charged leptons can be reproduced if 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. (47) 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 Complete screening, m D (M D ) −1 = c · 1, requires that y b 10 : y b 16 = h 10 : h 16 . In this case we obtain from Eq. (49) m in the flavor basis.
In the hidden sector due to Z 4 symmetry given by Eq. (46) 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. (50) 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 P M N S 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 (24) 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 (48) 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 diag 33 .

Singlet -triplet flavon scheme
The Lagrangian of this scheme is given by Eq. (42) with h 10 = y 16 = 0, and the graphic representation is shown in Fig. 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 The key feature is that 16 F and S have different Z 4 charges. The Z 4 symmetry also forbids the low-dimension (flavonless) contribution of the mass M 0 D , so that where m 0 D = y b 10 H . The matrix m D has enough free parameters to reproduce the charged lepton masses.
With the assignment in Fig. 3, only triplet ξ contributes to the hidden sector matrix, so that M S ∝ M ξ 33 . Furthermore, since the same ξ contributes to both M S and M D we have Effectively the expression for mass looks like a single seesaw and only the overall scale of the RH neutrino masses imprints information about the presence of the hidden sector.
In spite of strong hierarchy in m diag l the neutrino mass matrix in (52) can be made in agreement with data due to the special form of M ξ 33 , such that its inverse matrix, has quadratic hierarchy in u i . To show this we take m diag l = m τ diag(λ 6 , λ 2 , 1), where λ ≡ sin θ C 0.22 is the Cabibbo angle. Then for VEV the hierarchy ξ = (u 1 , u 2 , u 3 ) = (−1, λ 3 , λ 5 ) with coefficients of the order 1 (which is even weaker than the hierarchy in m diag l ) Eqs. (52) and (53) lead to 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.

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 Fig. 4. The Lagrangian

Scheme with mixed contributions: the TBM mixing
Here and in the next subsection we consider general situations when both triplet and singlet flavons interact in the visible sector. The model in this subsection realizes the scenario in which 13 . 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 T BM . The symmetry assignments of the field content we use is under Z 4 , and the full graphic representation are shown in Fig. 5. The Lagrangian has the following terms The Z 4 symmetry forbids all the interactions without flavons including the bare mass terms.
That is, now couplings of ϕ and ξ should also correlate. Possible origins of this proportionality have been discussed in sect. 3.1. As a consequence, The eigenvalues of M S equal y 1 v 1 ±h u and y 1 v 1 , and consequently, the light neutrino masses equal This gives normal mass ordering with m 1 : m 2 : m 3 1 : 1 : 3, if h 1 u ∼ −2y 1 v 1 . The inverted mass ordering can be obtained if two additional flavon singlets, φ and φ , are introduced in the hidden sector with couplings satisfying y 1 φ = ωy 1 φ . This adds one free parameter to M S , and thus, allows arbitrary neutrino masses without changing maximal 1-3 mixing.
The model can be simplified if certain high-order corrections to m D are added. Assume that ϕ = ϕ = 0, or simply that ϕ and ϕ are removed from the scheme. In practice, this is equivalent to the case y 10  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. (27)). In particular, one can use the anti-symmetric correction Figure 6: The same as in Fig. 1  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. (57), 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 Fig. 6, where the Z 4 charges are These equalities imply the permutation symmetry ( ↔ ) in the visible and portal sectors, and therefore lead to maximal 2-3 mixing. Under conditions (58) we obtain The eigenvalues of this matrix equal To obtain mass hierarchy we should arrange that y 10 v ≈ −0.5y 10 v, which leads to 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.
The visible interactions are the same as before, so that the Dirac mass matrices have two contributions: one from singlets ϕ and the other from triplet ξ of A 4 . Since the fermion singlets S transform as the three one-dimensional representations of A 4 , the portal term has only one contribution: from triplet ξ. Therefore, in contrast to the schemes in the previous section with S ∼ 3, now we can only have two possibilities for zero couplings related to the Dirac terms: h 10 = 0 or y 10 = 0. For both possibilities, according to Eqs. (30)(31)(32) the portal and hidden sector mass matrices are given by (in the case of only one triplet flavon) For D h ∝ 1 and m diag l (D u ) −1 ∝ 1 (partial screening), the expression (62) is reduced to 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 fixed completely. More contributions are required to break it. A straightforward possibility is to use additional contributions to M 11 in order to obtain maximal 1-3 mixing from M S , so that U 0 = U ω U max 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.
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 Fig. 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) allowed by symmetry in the lowest order are Notice that Z 4 forbids the low-dimension flavonless term 16 F 16 F 10 H since 10 H has the charge −1. It also ensures that triplet flavons do not couple with 16 F 16 F , and therefore only singlets contribute to the Dirac mass matrix, leading to m D = M ϕ 33 . The symmetry Z 4 also forbids the bare mass terms (SS) and (S S ).
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. (63) only generates the same diagonal Dirac mass matrices m D = M ϕ 33 ( y 10 , v) to all SM fermion components as in Eq. (21) 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. (23).
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. (65) can be written as in Eq. (30): The mass matrix of singlet fermions generated by Eq. (63) 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. (36) the light neutrino mass matrix To get the TBM mixing the following two conditions should be fulfilled: which leads to partial screening, and 5 The diagonal matrix m D can also be written in the form of product of matrices as 10 H Λ f U * ω (y 10 v1 + y 10 v C + y 10 v C T )U ω , where C is the cyclic matrix with only non-zero (1,3), (2,1), and (3,2) unit elements.
to get maximal 1-3 mixing from the central matrix of Eq. (67).
The first condition (68) is the correlation of the singlet and triplet VEVs, which can be obtained in the same way as it was done in Eq. (41), 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. (69) can be satisfied if, e.g.
These equalities then produce the masses of light neutrinos as Strong mass hierarchy requires 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 sect. 2.5). If S is odd and other particles of the hidden sector are even, then M S will have the form (66) with zero 12-and 23-elements. This additional symmetry is broken in the visible and portal sectors, which have substantially lower energy scale M GU T . So, one expects that corrections to the Z 2 symmetric results are of the order M GU T /M P l , 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 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 and it converts to in the flavor basis. In the case of two Higgs 10-plets, there is actually m diag D = m u . In the hidden sector the bare mass terms, M 0 S andM 0 S , are forbidden.
If D u = 1, the matrix M 33 has special form M special

33
, which can be diagonalized by U 33 = U ω . In this case Eq. (71) gives Then for partial screening, m diag With these conditions we essentially reproduce complete screening. Then M S should have the TBM form, which can be obtained in the way described by Eqs. (32,33) in sect. 2.4.

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. (9). Under the condition M S M cr , where M cr is defined by Eq. (11), this contribution is subleading, but may provide certain required corrections to some observables.
On the contrary, if M S M cr , the linear seesaw contribution is dominant. As we have mentioned in sect. 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 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 where n = 6. Taking into account the A 4 symmetry assignment we can write different mass matrices involved as where M N is the effective mass matrix of RH neutrinos: Using expressions (74) 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.
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. (81) becomes gU ω 1 U ω , which reproduces exactly the correction ∆M as in Eq. (27). 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 [45] for more details. To be in agreement with observations, three aspects should be addressed. 1). Difference of the up-and down-type fermions mass hierarchies. There are various ways to obtain different mass hierarchies of the "upper" and "down" fermions (in the SM doublets) and generate the CKM mixing.
(i) A second 10-plet of Higgses, 10 d H , can be introduced to give masses of the down fermions, whereas the first one, 10 u H , generates masses of the upper fermions: up-type quarks and neutrinos. Eq. (3) then results in 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 operator This operator also breaks the mass degeneracy between up-and down-type quarks [69].
(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. The interactions which provide this mixing can originate from renormalizable operators 10 F 16 F 16 H .
2). Difference of masses of down quarks and charged leptons. To breaks the mass degeneracy between quarks and leptons one can use non-renormalizable terms with additional Higgs multiplets. One option is to introduce 45-plet Higgs scalars 45 H and the dimension-5 with 45 H and 10 d H only [71]. 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 (83) 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 [29,72].
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) 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), which implies that both matrices should be diagonalized by the same rotation U l U d [45].
One solution to this problem was described before in sect. 3.2. The mass matrix can be the sum of a diagonal matrix proportional to the unit matrix and an off-diagonal matrix: The mixing is determined by M ξ 33 only, whereas the eigenvalues are given by the total matrix, which depends on m 0 and M ξ 33 . So, by varying m 0 , we can change the eigenvalues without modifying the mixing.

Conclusion
1. We have considered the lepton mixing and neutrino masses in the framework of SO (10) GUTs with the hidden sector composed of fermionic and bosonic singlets of SO (10), and flavor symmetries realized in both visible and hidden sectors. Two sectors communicate via the portal interactions provided by 16 H of Higgses.
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: 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.
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 Dthe 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).
U 0 in Eq. (1) can be factorized as U 0 = U 1 U 2 ..., where U i are the matrices of maximal two-generation rotation or matrices of special type, such as the magic matrix U ω . Then different mass matrices can be responsible for different rotations U i . In this connection we studied properties of matrices m D , M D , M S generated by different SO(10) Higgs multiplets. Depending on A 4 assignments for fermions they can be of 5 types: M diag 33 , M ξ 33 , M 31 , M 11 and M 0 · 1.
For a complete set of flavons (triplet and singlets) with arbitrary couplings and VEVs, any (symmetric) structure of M 33 and M 11 (and consequently, m D and M S ) and M 31 can be obtained. As a consequence, the flavor structure of light neutrino mass matrix is arbitrary.
The matrices acquire certain flavor structures if only specific flavons (e.g. only triplet) contribute to their generation. If the same flavons contribute to both m D and M D , the Dirac screening can be obtained. This can be achieved by additional symmetry, e.g. Z 4 . This symmetry forbids certain couplings of flavons, ensuring that different flavons participate in different types of interactions. It can then explains that some flavons appear in the hidden sector only. Or on the contrary, a common symmetry assignment ensures that the same flavons contribute to different mass matrices.
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 T BM , 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 non-zero VEV of only one component of the triplet; -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 T BM , 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 T BM ; -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.

.
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. 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 , 1 = (ab) = a 1 b 1 + ωa 2 b 2 + ω 2 a 3 b 3 , 3 = a · b = h 1 · (a 2 b 3 , a 3 b 1 , a 1 b 2 ) + h 2 · (a 3 b 2 , a 1 b 3 , a 2 b 1 ), 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 .

D the TBM and BM mixings
The TBM mixing