Natural neutrino sector in a 331-model with Froggatt-Nielsen mechanism

The extensions of the Standard Model based on the SU(3)c × SU(3)L × U(1)X gauge group (331-models) have been advocated to explain the number of fermion families in nature. It has been recently shown that the Froggatt-Nielsen mechanism, a popular way to explain the mass hierarchy of the charged fermions, can be incorporated into the 331-setting in an economical fashion (FN331). In this work we extend the FN331-model to include three right-handed neutrino singlets. We show that the seesaw mechanism is realized in this model. The scale of the seesaw mechanism is near the SU(3)L×U(1)X breaking scale. The model we present here simultaneously explains the mass hierarchy of all the fermions, including neutrinos, and the number of families. ∗E-mail: Katri.Huitu@helsinki.fi †E-mail: Niko.Koivunen@helsinki.fi ‡E-mail: Timo.j.Karkkainen@helsinki.fi ar X iv :1 90 8. 09 38 4v 1 [ he pph ] 2 5 A ug 2 01 9


Introduction
After the discovery of the Higgs boson at the Large Hadron Collider, the last elementary particle predicted by the Standard Model (SM) has been confirmed. Today, particle physics has moved on to a new era, where we attempt to answer the problems plagueing the SM with economical extensions. The problems include number of generations, nonzero neutrino mass, neutrino mixing and fermion mass hierarchies.
In Nature three generations of quarks and leptons have been observed. Number of neutrino flavours is 2.984 ± 0.008 [1]- [7], which is a statistical fit to SM using LEP data. This is a strong indication for exactly three generations of matter, which, however, is not imposed by SM itself. We know from neutrino oscillation experiments that at least two of the three SM neutrinos are massive, with masses less than 0.1 eV and the sum of their masses is less than 0.12 eV from cosmological constraints by the PLANCK experiment. Neutrino masses are not included in the Standard Model, and they are six orders of magnitude lighter than the next lightest massive particle, electron, and twelve orders of magnitudes lighter than the heaviest particle, top quark. This huge range of different masses gives birth to the flavour problem.
Extensions of the Standard Model based on the SU (3) c × SU (3) L × U (1) X gauge group (331-models) have been proposed in the literature to explain the number of fermion families in Nature. In the traditional 331-models [8]- [21] the gauge anomalies cancel only if the number of fermion familes is three. The SU (3) c ×SU (3) L ×U (1) X gauge group contains one additional diagonal generator compared to the SM-gauge group SU (3) c × SU (2) L × U (1) Y . This means that the electric charge can be defined in multiple different ways in the 331-models: where the parameter β defines the particle content of the model. The models with β = ± √ 3 [8]- [12] and β = ±1/ √ 3 [13]- [21] are extensively studied in the literature. The models with β = ± √ 3 contain particles with exotic electric charges such as doubly charged scalars and gauge bosons. They also contain a very large scalar sector, composed of three SU (3) L -triplets and an SU (3) L -sextet. The models based on the β = ±1/ √ 3 on the other hand have simpler scalar sector, composed from only three SU (3) L -triplets. The models based on β = ±1/ √ 3 do not contain particles with exotic electric charges. Also the models with β = 0 have been studied [22].
Even though the 331-models can shed light on the number of fermion familes, the fermion mass hierarchy is left unexplained in the traditional models. Recently it was shown that the Froggatt-Nielsen mechanism [23] can be incorporated into the 331-models with β = ±1 √ 3 without extending the scalar sector [24,25]. The Froggatt-Nielsen mechanism (FN) is a well established method to explain the mass hierarchy of the fermions, and a 331-model with incorporated FN-mechanism (FN331) can therefore simultaneously explain both the number of fermion families and the mass hierarchy of the charged fermions. The neutrino masses and mixings are not naturally explained in FN331-model, however. The neutrino mass matrix is antisymmetric in the FN331-model and therefore one of the neutrinos is massless and the two other mass degenerate at tree-level. Loop corrections are needed to lift the one eigenvalue from zero and to break the degeneracy of the other two [14]. This neutrino sector is identical to the one presented in [13]- [21].
Our aim is to extend the neutrino sector to make it natural and explain the neutrino masses and mixings without fine-tuning at tree-level. We propose an extension of the FN331model where we add three right-handed neutrino singlets to the model. This allows the tree-level masses for all the neutrinos and implementation of seesaw mechanism [26]- [31] for the neutrino sector. Here the seesaw is combined with the FN-mechanism, which allows the seesaw scale to be low thanks to suppression in the neutrino Yukawa couplings due to the FN-mechanism. The seesaw scale is essentially the same as the SU (3) L × U (1) X -breaking scale. We study as low SU (3) L × U (1) X -breaking scale as possible. The SU (3) L × U (1) Xbreaking scale can be as low as 5 TeV as was demonstrated in [25]. We aim to generate singlet neutrinos at medium-energy scale and sterile neutrinos at TeV scale, utilizing seesaw mechanism. By medium-energy we refer to a mass scale between the active neutrinos and mass of electron. We will present the lowest possible SU(3) L ×U(1) X breaking scale consistent with experimental data, which turns out to be approximately 5 TeV.
The hierachical structure of the neutrino Yukawa couplings is determined by the FNmechanism. The hierarchy of the neutrino Yukawa couplings can be arranged so that the neutrino mixings are generated without fine-tuning. All the PMNS-matrix elements are experimentally known to be order-one-numbers. This kind of pattern of mixing can be achieved in FN-setting by assigning all the FN-charges of the left-handed leptons to be equal. We show that the correct sub-eV neutrino masses, mass square differences and mixing angles are produced within experimental limits.
The paper is structured as follows. We present the particle content of the model in Section 2. In Section 3 we review the Froggatt-Nielsen mechanism in the 331-setting, the FN331model. In Sections 4 to 7 we study the lepton mass matrices and mixings and finally in the Section 8 we present numerical example for the neutrino masses and mixings.

Particle content
We propose a model where the gauge group of the Standard Model is extended to We define the electric charge as 1 : where the T 3 and T 8 are the diagonal SU (3) L generators. We also introduce global U (1) F Nsymmetry, under which fermions and some of the scalars are charged. 1 The choice β = + 1 √ 3 would result in essentially a same model.

Fermion representations
Let us now write down the fermion representations. The left-handed leptons are assigned to SU (3) L -triplets and the right-handed leptons are assigned to SU (3) L -singlets: The numbers in the parantheses label the transformation propeties under the gauge group The ν L,i and N R,i are new leptons with electric charges 0. The three right-handed neutrinos N R,i are not present in the model studied in [24,25] and they allow the tree-level masses for all the neutrinos. The cancellation of anomalies requires the number of fermion triplets to be the same as antitriplets. This is achieved by assigning two quark families to SU (3) L antitriplets and one family to a triplet. We choose to assign first quark generations into triplet and the second and the third into an antitriplet: We have introduced new quarks D 1 and D 2 with electric charge −1/3 and U with electric charge 2/3, which will mix with the SM quarks. All the fermions are also charged under the global Froggatt-Nielsen U (1) F N symmetry. This will be discussed in detail in the dedicated Section 3.
When we take into account the colour, there are six fermion triplets and six antriplets, ensuring the cancellation of pure SU (3) L -anomaly. All the gauge anomalies will cancel with this particle content. The anomaly cancellation forces one quark generation to be placed in a different representation than the other two. The unequal treatment of quark generations inevitably leads to scalar mediated flavour changing neutral currents (FCNC) at tree-level, which is a feature plaguing the traditional 331-models [8]- [21]. It was however recently shown that tree-level scalar mediated FCNCs of quarks are suppressed in the FN331-model [25]. This is in contrast to the traditional 331-models, which offer no natural suppression mechanism for the tree-level scalar mediated FCNCs.

Scalar sector
The 331-models with β = − 1 √ 3 contain only two types of scalar triplets with neutral entries: X = 2/3 and X = −1/3. One must include at least two X = −1/3 triplets with X = −1/3 and one triplet with X = 2/3 in order to generate the masses for all the charged fermions at tree level. We choose to have this minimal scalar sector: ).
All the neutral fields can in general develop a nonzero vacuum expectation value (VEV). The minima are related to each other by SU (3) L rotation. We choose to rotate one of the X = −1/3 triplet VEVs so that the upper component VEV goes to zero. This rotation will leave the rest of the VEVs general. So we have vacuum structure: The VEVs v 2 and u break the SU (3) L -symmetry, and we assume them to be: v 2 , u O(TeV). The VEVs v and v 1 break the SU (2) L symmetry and we take them to be of the order of the electroweak scale. The scalar triplets in Eq. (9) are charged under the global symmery U (1) F N with the charge assignment presented in the Table 1. Note that since the scalar triplets ρ and χ are in the same representation, the combination ρ † χ is gauge invariant. Also according to Eq. (10) and the Table 1, the gauge invariant combination ρ † χ carries a non-zero FN-charge and has a non-zero VEV. Therefore the ρ † χ-combination can play the role of the flavon in the Froggatt-Nielsen mechanism, as was demonstrated in [24,25]. The Froggatt-Nielsen mechanism can thus be implemented without introducing new scalar degrees of freedom into the model.
The scalar potential is greatly simplified due to inclusion of global U (1) F N -symmetry.
Particle η ρ χ FN-charge −1 1 0 The most general U (1) F N -symmetric scalar potential is, However the global U (1) F N -symmetry is spontaneously broken by the scalar field VEVs. This leaves one Golstone boson to the physical spectrum. In order to give it a mass we add the following soft FN-breaking term to the potential: All the complex phases in the scalar potential can be absorbed into the fields and therefore all the parameters in the scalar potential are real. The real and imaginary parts of the scalars will therefore not mix. We choose the parameter f to be comparable to the SU (3) L × U (1) Xbreaking scale. The soft-breaking term is also chosen to be large, b ∼ −(v heavy ) 2 , in order to decouple the pseudo-Goldstone boson in the low energies.
The scalar sector has five CP-even, five CP-odd and four charged scalars. One CPeven, three CP-odd and two charged scalars are massless would-be-Goldstone bosons that are absorbed by the gauge bosons of the model, namely the Z, W ± of the SM, new heavy charged gauge boson V ± , new heavy neutral gauge boson Z and a non-hermitian heavy neutral gauge boson X 0 . Thus there are four CP-even, two CP-odd and two charged scalars left in the physical spectrum. All the physical scalars, except the 125 GeV scalar, have their masses around the SU (3) L × U (1) X -breaking scale and they are very heavy. The details of the scalar mass matrices are provided in the Appendix A.

Gauge sector
As previously mentioned the gauge sector of 331-model is enlarged compared to the SM. The 331-models will contain five additional gauge bosons compared to the SM. The covariant derivative for SU (3) L triplet is: where g 3 and g x are the SU (3) L and U (1) X gauge couplings respectively. The T a = λ a /2 are the SU (3) L generators, where λ a are the Gell-Mann matrices. The SU (3) L gauge bosons are, where we have denoted, The fields W 3µ , W 8µ , B µ and W 4µ will form neutral mass eigenstates: photon, Z-boson and new heavy gauge bosons Z and W 4µ . The field W 5µ does not mix with the other neutral gauge bosons and is a mass eigenstate, with same mass as W 4µ . These fields are identified as a physical neutral non-hermitian gauge boson X 0 . Details of the neutral gauge boson masses are given in the Appendix B. The off-diagonal gauge bosons W ± µ and V ± µ will form the SM gauge bosons W ± µ and the heavy new gauge bosons V ± µ .

Charged gauge bosons
The mass term for the charged gauge bosons is given by, where Y T = (W + µ , V + µ ) and, is the charged gauge boson mass matrix. The eigenvalues of the matrix are, where v heavy = v 2 , u and v light = v 1 , v . According to Eq. (15) the SM Higgs VEV is related to the triplet VEVs through the relation where v sm = 246 GeV.
The mass eigenstates are defined as where the mixing angle θ is defined as: The mixing angle between The experimental bound for the mixing is |θ| O(10 −2 ) [1], and has been taken into account in our numerical analysis (Section 8).

The Yukawa sector and the Froggatt-Nielsen mechanism in the 331-framework
Next we study the Yukawa sector of the model. We are employing the Froggatt-Nielsen mechanism to generate the hierarchical structure of the fermion Yukawa couplings. The original Froggatt-Nielsen model extends the Standard Model with a flavour symmetry (FN symmetry), whose symmetry group in the simplest case is global or local U (1) or a discrete Z N symmetry. The FN-framework introduces a new complex scalar field, the flavon, which is a singlet under standard model gauge group The SM fermions, the SM Higgs and the flavon are charged under the FN symmery. The key property of the FN-symmetry is to forbid the SM Yukawa couplings, save perhaps the top quark. The SM Yukawa couplings are generated as effective couplings instead. The FN mechanism can be economically incorporated into a 331-model. The scalar sector we have introduced in section 2.2 contains the neccesary incredients to house Froggatt-Nielsen mechanism, as was demonstrated in [24,25]. The addition of complex scalar field to act as a flavon is thus unneccesary. Here a gauge singlet combination, ρ † χ, will act as the flavon instead of single complex scalar field. The effective flavon, ρ † χ, obtains a nonzero vacuum expectation value, ρ † χ = (v 2 u)/2, as can be seen from the Eq. (10).
The effective operator that generates the Yukawa couplings of the charged fermions is 2 : where (c f s ) ij is a dimensionless order-one number, Λ is the scale of the new physics, S denotes any of the three scalar triplets η, ρ or χ. Theψ f L,i and f R,j represent here the fermion triplets, antitriplets and singlets that were introduced in section 2.1. The power (n s f ) ij is positive integer number 3 and determined by the FN charge conservation (see Table 2): Particle The usual 331-model Yukawa terms are generated as effective couplings when the scalar triplets ρ and χ acquire VEVs: = where only the renormalizable contributions are kept. The first term in Eq. (21) gives the usual Yukawa terms of the model, as in the original FN model. The Yukawa coupling is now defined as: Hierarchical Yukawa couplings are produced by assuming that = (v 2 u)/(2Λ 2 ) < 1. The FN-charges of the SM fermions determine the power (n f s ) ij and therefore the amount of suppression each Yukawa coupling obtains. One can obtain the observed fermion mass hierarchy by assigning larger FN charges to the lighter fermions compared to the heavier ones. This is in contrast to the Standard Model where the hierarchy is obtained only by fine-tuning the couplings themselves. 2 The effective operator that generates the neutrino Yukawa couplings is presented later in Eq. (28). 3 If (n s f ) ij were negative, we would have to include operator (c f s ) ij The second term in Eq. (21) is not proportional to the Yukawa matrix and is therefore flavour violating. This flavour violating part is suppressed by the scale of the SU (3) L ×U (1) Xbreaking. We assume that the SU (3) L × U (1) X -breaking scale v scale is 5 TeV, which was shown in the [25] to be the lowest scale the quark FCNCs are safely suppressed. We will safely ignore the flavour violating contributions as they are heavily suppressed.

Charged lepton Yukawa couplings and masses
Charged lepton Yukawa couplings are: where i, j = 1, 2, 3. The first term is the traditional 331 Yukawa term for the charged leptons whereas the second term is the additional Yukawa interaction term due to the FN-mechanism.
As stated earlier, this additional term is suppressed by the v heavy and we will ignore it in the following. The charged lepton Yukawa matrix is given as follows: We will specify the FN-charges we use later in section 8, when we study numerical examples.
The charged leptons acquire masses as the scalar triplet η obtains a VEV: L ⊃ y e ijL L,i η e R,j + h.c. = m e ijē L,i e R,j + h.c., where the charged lepton mass matrix is, The charged lepton mass matrix is diagonalized as: The charged lepton mass matrix proportional to the Yukawa matrix will be diagonalized simultaneously with the Yukawa coupling. Therefore there will be no flavour changing couplings in the standard Yukawa couplings. The only flavour violation to the charged leptons is coming from the Froggatt-Nielsen mechanism which is however suppressed.

Neutrino mass matrix
The neutrino Yukawa couplings originate from effective operators of two types. The first type was already presented in Eq. (19) and the operator of the second kind is: The operators in Eqs. (19) and (28) produce the following Yukawa couplings for neutrinos: where i, j = 1, 2, 3 and the Yukawa couplings are, The Yukawa coupling e ij is antisymmetric: e ij = −e ji , due to presence of the antisymmetric tensor αβγ in the Eq. (29). The first line in (29) contains the standard Yukawa interactions for the neutrinos and the two last lines contain the additional Yukawa interactions originating from the Froggatt-Nielsen mechanism, which we will ignore due to them being suppressed.
The neutrino masses will be generated by the Yukawa terms in the first line of Eq. (29) as the scalars obtain VEVs. The right-handed neutrino singlet N R,i will also have a Majorana mass term which is generated by the following operator, where the mass scale M 0 is in principle a free parameter. We choose the mass scale to be same as the FN-messengers, in order not to introduce new mass scales into the model. The Majorana mass term becomes, where the Majorana mass matrix is, The messenger scale is related to the SU (3) L × U (1) X -breaking VEVs by, The full contribution to the neutrino masses is finally given by the following terms: The neutrino masses can be written in a 9 × 9 matrix form as: where the 3 × 3 sub-matrices are: We next determine the pattern of FN-charges for the leptons using the experimental values of the PMNS matrix as guidance. Once the FN-charges are known, the exact hierarchy of the neutrino mass matrix becomes clear, and we can proceed with the block diagonalization of the neutrino mass matrix.

Neutrino masses and eigenstates
The Froggatt-Nielsen charges determine the hierarchy of the fermion mass matrices. The fermion FN-charges should be chosen so that the order of magnitude of the fermion masses becomes right, thus the mass hierarchy is explained without fine-tuning. The FN charges also determine the structure of the matrices that diagonalize the fermion mass matrix. This is important as the left-handed fermion diagonalization matrices enter the two physical observables: the CKM-matrix and the PMNS-matrix. Proper choice of the left-handed fermion FN-charges can ensure that also the hierarchy of the CKM-and PMNS-matrices are produced correctly, and no fine-tuning is required. The quark sector of our model is identical to the one in [24,25], where it was studied in great detail. We will therefore not consider it here. We instead concentrate on lepton sector which differs from the model presented in [24,25] only by the additional neutrino singlets N R,i .
where the value of each entry is given at 3σ confidence level [42]. The PMNS-matrix elements are O(1) numbers in contrast to CKM-matrix where distinct hierarchy is present. The PMNS-matrix is given schematically by the left-handed charged lepton diagonalization matrix U e L and the neutrino diagonalization matrix U ν as 4 : The observed PMNS-hierarchy is naturally obtained, if the left-handed charged lepton rotation matrix U e L and neutrino diagonalization matrix U ν , also have this anarchical texture. This is the method we adopt here. The hierarchy of U e L and U ν depend on the FN-charges of the left-handed leptons. The anarchical hierarchy is achieved when all the lepton families are treated equally under the FN-symmetry. We will therefore choose from now on all the lepton triplets to have equal FN-charges: The FN-charges of the right-handed neutrino singlets do not affect the hierarchy of the light-neutrinos. We will choose the FN-charge of the right-handed neutrinos to be zero for simplicity: q(N R,1 ) = q(N R,2 ) = q(N R,3 ) = 0.
We can now see the order of magnitude in the neutrino mass matrix elements and proceed with the block diagonalization of the neutrino mass matrix.

Neutrino mass matrices
The neutrino mass matrix M ν in Eq. (36) will have nine eigenvalues corresponding to nine Majorana neutrinos. The neutrino mass matrix M ν can be written in the following notation: where The order of magnitude of the sub-matrices are given by where v light = v , v 1 and v heavy = u, v 2 . Note that sub-matrices in Eq. (44) do not have an internal hierarchy, but distinct hierarchy is present between sub-matrices M D and M R . The entries in the sub-matrix M R are proportional to the SU (3) L × U (1) X breaking VEVs, u and v 2 , whereas the entries in the sub-matrix M D are proportional to SU (2) L × U (1) Y breaking VEVs v and v 1 . Therefore the eigenvalues of the M R are much larger than the entries in the M D . This hierarchy is reflected in the eigenvalues of the matrix: it has three "light" and six heavier eigenvalues. The elements in the heavier block M R also have different orders of magnitude: matrix m N is heavily suppressed compared to M by L . The eigenvalues of the heavier block will therefore be in two distinct scales we call "medium" and "heavy". Our neutrino sector is subject to kind of "double-seesaw". The neutrino mass matrix M ν will have in total three "light" eigenvalues, three "medium" eigenvalues and three "heavy" eigenvalues.
The neutrino mass matrix M ν can be block-diagonalized into three blocks, each corresponding to these eigenvalue-types according to: where unitary matrix W separates the three "light"-neutrinos from the six heavier ones, and unitary matrix Z further block diagonalizes the block of heavier neutrinos into block of "medium"-mass neutrinos and "heavy" neutrinos. The matrices W and Z are to the leading order: and, with, and, The light, medium and heavy blocks can be written at lowest order as: The order of magnitude of light-, medium-and heavy-neutrino masses can now be estimated using Eq. (50) with Eq. (44): The light-neutrino masses are proportional to v 2 light /v heavy , where v light is the electroweak scale and v heavy is the scale of new physics, which is characteristic to the seesaw-mechanism. Additional suppression factor, 2L+2 , is however present, due to the Froggatt-Nielsen mechanism. The masses of the "medium"-neutrinos are heavily suppressed compared to SU (3) L × U (1) Xbreaking scale, making them typically lighter than m Z /2. They are therefore subject to the LEP bound [1]- [7] on the number of light neutrinos. However, suppression on their couplings to Z boson make their contribution to the invisible decay with of Z boson tiny, as will be demonstrated later for our benchmark points. The heavy neutrinos have their masses around the SU (3) L × U (1) X -breaking scale and will therefore decouple.

Neutrino eigenstates
The neutrino mass eigenstates are obtained once the light, medium and heavy neutrino blocks are diagonalized. The neutrino mass matrix M ν in Eq. (36) is fully diagonalized according to: with unitary matrix U is given by, The unitary matrices U ν , U n and U N diagonalize light, medium and heavy neutrino blocks respectively. The U ν , U n and U N are anarchical in nature, as the blocks m light , m medium and m heavy have no internal hierarchy.
According the Eq. (52), the mass eigenstate neutrinos are: The mixing between the neutrinos can be estimated with the use of FN-textures of the neutrino Yukawa couplings as: (56)

Neutrino coupling to charged gauge bosons and PMNSmatrix
Our model includes additional charged gauge boson V ± µ , that mixes with the W ± µ boson as shown in the Section 2.3.1. The mixing between the charged gauge bosons is however tiny. The neutrino gauge eigenstates couple to the physical charged gauge bosons as: With the use of Eqs. (27) and (55) the coupling of W ± µ to light neutrinos can be writen as, from which we can identify the PMNS matrix: The term proportional to cos θB 1 B † 1 induces nonunitarity effects to neutrino oscillations, which is an expected effect due to inclusion of sterile neutrinos in the model. Deviation from the unitarity is suppressed by the factor O(v 2 light /v 2 heavy ) and is significantly smaller than the current bounds [33,34,35]. In any case, nonunitary mixing strength of ≥ 10 −2 is ruled out.
The term proportional to sin θB † 1 is similarly suppressed by a factor O(v 2 light /v 2 heavy ), but since B 1 is not Hermitian, the anti-Hermitian part of it induces neutrino decay. Since the nonunitarity and unstability effects are both small, we shall ignore them in the remainder of this paper.
The PMNS matrix therefore is: We have chosen the v heavy 5 TeV, which makes the mixing angle θ very small and cos θ ≈ 0.9 ≈ 1. As stated in Eq. (54) the light-neutrino diagonalization matrix U ν is anarchical. Since the lepton triplet FN-charges are identical also the left-handed charged lepton diagonalization matrix U e L is without a hierarchy 5 : The texture for the PMNS is therefore anarchical as well, which is compatible with the experimental measurements presented in Eq. (38). We note here that this is the extent which Froggatt-Nielsen setting can predict the structure of PMNSmatrix. This is in contrast to many models involving more elaborate flavour symmetries in the neutrino sector [36,37,38]. As of now, the PMNS matrix is consistent with anarchical mixing. It is up to the numerics to acquire the order-one coefficients that produce the correct lepton masses and the PMNS-matrix within the experimental limits, which is the focus of section 8.

Constraints and numerical examples
There are many important experimental constraints that have to be taken into account when considering the neutrino sector of any model. Constraints for active neutrinos are the most well-known and restrictive. Least model-dependent is the direct detection bound of m(ν e ) from electron energy spectrum of tritium β decay [39] and data from supernova SN1987a burst. Also, neutrinoless double beta decay experiments [40], cosmic microwave background and growth of large scale structures in the early universe [41] all constrain the upper limits of flavour neutrino masses, and their sum. Cosmological constraints are stricter by one order of magnitude, but are dependent on the cosmological model. In addition, neutrino 5 When charged lepton mass matrix satisfies m e i,j ≤ m e i+1,j , the left-handed diagonalization matrix satisfies: oscillation experiments provide neutrino mass squared differences, ∆m 2 21 and |∆m 3j | 2 (with j = 1, 2 corresponding to inverted and normal mass orderings, respectively) [42]. From these, a lower bound for two heavier light neutrinos can be deduced, being approximately 9 meV and 50 meV. The lightest neutrino state may be massless. Cosmological constraints are m ν < 0.12 eV. The existence of medium-mass sterile neutrinos at eV and keV scale would distort the electron energy spectrum, and different sterile neutrino mass ranges of this distortion can be detected via unstable nuclei, such as 3 H, 20 F, 35 S, 63 Ni and 187 Re. Searches for these distortions, i.e. kink searches have discarded large mixings of electron neutrino to mediummass sterile neutrinos [45,46]. We will show the constraints from kink searches to one of our benchmark points in Fig. 1. Since the sterile neutrino masses can be at O(10) TeV, there may be a chance to observe them at future colliders.
Our model predicts neutrinos in three different mass scales: the three sub-eV neutrinos, three heavy, mostly right-handed neutrinos, and three neutrinos with masses between the sub-eV and SU (3) L × U (1) X scales. The masses of the medium-scale neutrinos is determined by the SU (3) L × U (1) X -breaking scale and in the case of v heavy ∼ 50 TeV the medium-scale neutrino masses are around keV scale. The sub-eV neutrinos are constrained by their mass squared differences and mixings. The keV neutrinos that our model predicts are constrained by the LEP bound on the number of light neutrinos, "light" here meaning neutrinos with their masses smaller than m Z /2 [1,2,3,4,5,6,7]. The coupling of medium mass neutrinos to Z-bosons is heavily suppressed by the ratio between SU (2) L × U (1) Y -and SU (3) L × U (1) Xbreaking scales and they will pass the LEP limits on the number of light neutrinos. This becomes evident in our benchmark points.
Our model possesses the extended particle content of the 331-model. The additional gauge bosons and scalars of our model could potentially mediate the non-standard neutrino interactions. For example the additional charged gauge boson, V ± µ , mediates the CC-NSI given by, The V ± µ mediated contribution to the NSI will be heavily suppressed by its mass: making it negligible as m 2 V ± 5 TeV. Indeed, for our numerical benchmarks, the NSI parameters have magnitude O(10 −13 ). All the new gauge bosons and scalars of our model have their masses proportional to the SU (3) L × U (1) X -breaking scale. The non-standard interactions mediated by charged scalars will therefore also be suppressed due to their heavy masses.

The FN-charges for the numerical example
For the numerical example we take the SU (3) L × U (1) X -breaking scale v heavy to be around 5 to 50 TeV, as for this scale the quark sector was studied in [24,25]. We choose the values for the leptonic FN-charges so that the SU (3) L × U (1) X -breaking scale is fixed.
The FN-charge of the left-handed lepton triplet L L,i is determined by the light-neutrino masses. According to Eq. (51) all the light-neutrino masses m i will be: where the only free parameter is the FN-charge of the lepton-triplet. Experimentally the light-neutrino masses are constrained by [42]: where the neutrino mass squared differences are: ∆m 2 ij = m 2 i − m 2 j . By setting v light to electroweak scale and L ∼ 8 or 9, one obtains light-neutrino masses from the correct ballpark. We will choose these values for our numerical example.
When the FN-charge of the lepton triplet is fixed, the charged lepton mass hierarchy depends only on the FN-charges q(e R,i ) of the right-handed charged leptons as is evident from Eq. (24). The FN-charges q(e R,i ) are the sole source of charged lepton mass hierarchy, as all the left-handed lepton triplet FN-charges are identical. We choose the right-handed charged lepton charges so that their mass matrix texture becomes: As a summary the chosen lepton FN-charges are presented in Table 3. The FN-charges of the scalar triplets were presented in Table 1.

Numerical values for leptons
We have chosen three benchmark points BP1, BP2 and BP3, presented in Table 3. The order-one coefficients introduced in Eq. (24) and Eq. (29) are in the interval |c| ∈ [0.5, 5] to retain naturalness of the parameters. We choose different values for SU (3) L ×U (1) X -breaking VEVs u and v 2 , the SU (2) L ×U (1) Y -breaking VEVs v and v 1 , and the FN charges for charged leptons. Below we list the explicit values for the coupling matrices we used.   The computed values of neutrino masses, effective neutrino interaction strength and V ± µ -W ± µ mixing for our benchmarks.
From the Figures 1 and 2 it is apparent that next-generation neutrino oscillation experiments measuring ν µ disappearance or neutrinoless double beta decay (0νββ) experiments have a moderate possibility of supporting our model at BP3, since the present experimental limits are only approximately one degree of magnitude weaker. Of our three benchmark points, BP3 has the greatest prospect of being detected in future, since the sterile component of ν e has a disapprearance effect 6 j=4 |U ej | 2 ∼ 10 −3 and similarly the expected ν µ disappearance should be 6 j=4 |U µj | 2 ∼ 10 −4 . For BP1 and BP2 the disappearance effect is smaller by a factor of O(100) and O(10), respectively. The medium-mass neutrinos lie on the eV-scale. Nonetheless, they are too heavy to account for the MiniBooNe anomaly [43]. We calculated the active-medium neutrino mixing matrices and have illustrated them at constraint plots. Figure 1 shows the constraints from 0νββ experiments [44,45] and kink searches in single beta decay energy spectra of various unstable radioactive isotopes [45] for BP3. We have also included the expected sensitivity of KATRIN experiment after three-year run. Figure  2 shows the constraints from muon neutrino disappearance experiments [46] and the Mini-BooNe anomally for BP3.

Conclusion
The FN331-model is based on SU (3) c × SU (3) L × U (1) X gauge symmetry and economically incorporates the Froggatt-Nielsen mechanism into it, thus simultaneously explaining the number of fermion families and the mass hierarchy of charged fermions. In this work we extended the FN331-model with three right-handed neutrino singlets. This allowed for tree-level masses for all of the neutrinos, which the original FN331-model was lacking. The neutrino masses and mixings in this model are naturally explained by utilizing a combination of the seesaw and FN mechanisms. The light-neutrino masses acquire additional suppression due to the FN mechanism, allowing the Majorana mass scale to be quite low, around few TeV. This allows for the possible collider searches of the heavy neutrinos in the future colliders. The mixing of the neutrinos, represented by the PMNS-matrix, is also explained without fine-tuning since the FN mechanism can enforce the correct texture for the lepton mass matrices. As a summary the model presented here offers an explanation for the whole fermion sector: it explains the number of fermion families and the mass hierarchy of all of the fermions, thus solving the flavour problem while fulfilling all the experimental constraints.