1 Introduction

Discrete Family Symmetry from F-Theory GUTs

[12mm] Athanasios Karozas 111E-mail:akarozas@cc.uoi.gr, Stephen F. King 222E-mail: king@soton.ac.uk, George K. Leontaris 333E-mail: leonta@uoi.gr, Andrew K. Meadowcroft 444E-mail: am17g08@soton.ac.uk

[-2mm]

School of Physics and Astronomy, University of Southampton,

SO17 1BJ Southampton, United Kingdom

Physics Department, Theory Division, Ioannina University,

GR-45110 Ioannina, Greece

We consider realistic F-theory GUT models based on discrete family symmetries and , combined with GUT, comparing our results to existing field theory models based on these groups. We provide an explicit calculation to support the emergence of the family symmetry from the discrete monodromies arising in F-theory. We work within the spectral cover picture where in the present context the discrete symmetries are associated to monodromies among the roots of a five degree polynomial and hence constitute a subgroup of the permutation symmetry. We focus on the cases of and subgroups, motivated by successful phenomenological models interpreting the fermion mass hierarchy and in particular the neutrino data. More precisely, we study the implications on the effective field theories by analysing the relevant discriminants and the topological properties of the polynomial coefficients, while we propose a discrete version of the doublet-triplet splitting mechanism.

## 1 Introduction

F-theory is defined on an elliptically fibered Calabi-Yau four-fold over a threefold base [1]. In the elliptic fibration the singularities of the internal manifold are associated to the gauge symmetry. The basic objects in these constructions are the D7-branes which are located at the “points” where the fibre degenerates, while matter fields appear at their intersections. The interesting fact in this picture is that the topological properties of the internal space are converted to constraints on the effective field theory model in a direct manner. Moreover, in these constructions it is possible to implement a flux mechanism which breaks the symmetry and generates chirality in the spectrum.

F-theory Grand Unified Theories (F-GUTs) [2, 3, 4, 5, 6, 7, 8] represent a promising framework for addressing the flavour problem of quarks and leptons (for reviews see [9, 10, 11, 12, 13, 14]). F-GUTs are associated with D7-branes wrapping a complex surface in an elliptically fibered eight dimensional internal space. The precise gauge group is determined by the specific structure of the singular fibres over the compact surface , which is strongly constrained by the Kodaira conditions. The so-called “semi-local” approach imposes constraints from requiring that S is embedded into a local Calabi-Yau four-fold, which in practice leads to the presence of a local singularity [15], which is the highest non-Abelian symmetry allowed by the elliptic fibration.

In the convenient Higgs bundle picture and in particular the spectral cover approach, one may work locally by picking up a subgroup of as the gauge group of the four-dimensional effective model while the commutant of it with respect to is associated to the geometrical properities in the vicinity. Monodromy actions, which are always present in F-theory constructions, may reduce the rank of the latter, leaving intact only a subgroup of it. The remaining symmetries could be factors in the Cartan subalgebra or some discrete symmetry. Therefore, in these constructions GUTs are always accompanied by additional symmetries which play important role in low energy pheomenology through the restrictions they impose on superpotential couplings.

In the above approach, all Yukawa couplings originate from this single point of enhancement. As such, we can learn about the matter and couplings of the semi-local theory by decomposing the adjoint of in terms of representations of the GUT group and the perpendicular gauge group. In terms of the local picture considered so far, matter is localised on curves where the GUT brane intersects other 7-branes with extra symmetries associated to them, with this matter transforming in bi-fundamental representations of the GUT group and the . Yukawa couplings are then induced at points where three matter curves intersect, corresponding to a further enhancement of the gauge group.

Since is the highest symmetry of the elliptic fibration, the gauge symmetry of the effective model can in principle be any of the subgroups. The gauge symmetry can be broken by turning on appropriate fluxes [16] which at the same time generate chirality for matter fields. The minimal scenario of GUT has been extensively studied  [17, 18, 19]. Indeed only the simplest GUTs can in principle avoid exotic matter in the spectrum [4]. However, by considering different fluxes, other models have been constructed with different GUT groups, such as and  [20, 21, 22, 23, 24]. In particular, it is possible to achieve gauge coupling unification from in the presence of TeV scale exotics originating from both the matter curves and the bulk [25].

All of the approaches mentioned so far exploit the extra symmetries as family symmetries, in order to address the quark and lepton mass hierarchies. While it is gratifying that such symmetries can arise from a string derived model, where the parameter space is subject to constraints from the first principles of the theory, the possibility of having only continuous Abelian family symmetry in F-theory represents a very restrictive choice. By contrast, other string theories have a rich group structure embodying both continuous as well as discrete symmetries at the same time [26]-[31]. It may be regarded as something of a drawback of the F-theory approach that the family symmetry is constrained to be a product of symmetries. Indeed the results of the neutrino oscillation experiments are in agreement with an almost maximal atmospheric mixing angle , a large solar mixing , and a non-vanishing but smaller reactor angle , all of which could be explained by an underlying non-Abelian discrete family symmetry (for recent reviews see for example [32, 33, 34]).

Recently, discrete symmetries in F-theory have been considered [35] on an elliptically fibered space with an GUT singularity, where the effective theory is invariant under a more general non-Abelian finite group. They considered all possible monodromies which induce an additional discrete (family-type) symmetry on the model. For the GUT minimal unification scenario in particular, the accompanying discrete family group could be any subgroup of the permutation symmetry, and the spectral cover geometries with monodromies associated to the finite symmetries , and their transitive subgroups, including the dihedral group and , were discussed. However a detailed analysis was only presented for the case, while other cases such as were not fully developed into a realistic model.

In this paper we extend the analysis in [35] in order to construct realistic models based on the cases and , combined with GUT, comparing our results to existing field theory models based on these groups. We provide an explicit calculation to support the emergence of the family symmetry as from the discrete monodromies. In section 2 we start with a short description of the basic ingredients of F-theory model building and present the splitting of the spectral cover in the components associated to the and discrete group factors. In section 3 we discuss the conditions for the transition of to discrete family symmetry “escorting” the GUT and propose a discrete version of the doublet-triplet splitting mechanism for , before constructing a realistic model which is analysed in detail. In section 4 we then analyse in detail an model which was not considered at all in  [35] and in section 5 we present our conclusions. Additional computational details are left for the Appendices.

## 2 General Principles

F-theory is a non-perturbative formulation of type IIB superstring theory, emerging from compactifications on a Calabi-Yau fourfold which is an elliptically fibered space over a base of three complex dimensions. Our GUT symmetry in the present work is which is associated to a holomorphic divisor residing inside the threefold base, . If we designate with the ‘normal’ direction to this GUT surface, the divisor can be thought of as the zero limit of the holomorphic section in , i.e. at . The fibration is described by the Weierstrass equation

 y2=x3+f(z)x+g(z),

where are eighth and twelveth degree polynomials respectively. The singularities of the fiber are determined by the zeroes of the discriminant and are associated to non-Abelian gauge groups. For a smooth Weierstrass model they have been classified by Kodaira and in the case of F-theory these have been used to describe the non-Abelian gauge group.555For mathematical background see for example ref [43] Under these conditions, the highest symmetry in the elliptic fibration is and since the GUT symmetry in the present work is chosen to be , its commutant is . The physics of the latter is nicely captured by the spectral cover, described by a five-degree polynomial

 C5:5∑k=0bks5−k=0, (1)

where are holomorphic sections and is an affine parameter. Under the action of certain fluxes and possible monodromies, the polynomial could in principle be factorised to a number of irreducible components

 C5→Ca1×⋯×Can,1+⋯+n<5

provided that new coefficients preserve the holomorphicity. Given the rank of the associated group (), the simplest possibility is the decomposition into four factors, but this is one among many possibilities. As a matter of fact, in an F-theory context, the roots of the spectral cover equation are related by non-trivial monodromies. For the case at hand, under specific circumstances (related mainly to the properties of the internal manifold and flux data) these monodromies can be described by any possible subgroup of the Weyl group . This has tremendous implications in the effective field theory model, particularly in the superpotential couplings. The spectral cover equation (1) has roots , which correspond to the weights of , i.e. . The equation describes the matter curves of a particular theory, with roots being related by monodromies depending on the factorisation of this equation. Thus, we may choose to assume that the spectral cover can be factorised, with new coefficients that lie within the same field as . Depending on how we factorise, we will see different monodromy groups. Motivated by the peculiar properties of the neutrino sector, here we will attempt to explore the low energy implications of the following factorisations of the spectral cover equation

 i)C4×C1,ii)C3×C2,iii)C3×C1×C1. (2)

Case involves the transitive group and its subgroups while cases and incorporate the , which is isomorphic to . For later convenience these cases are depicted in figure 1.

In case for example, the polynomial in equation (1) should be separable in the following two factors

 C4×C1:(a1+a2s+a3s2+a4s3+a5s4)(a6+a7s)=0 (3)

which implies the ‘breaking’ of the to the monodromy group , (or one of its subgroups such as ), described by the fourth degree polynomial

 C4:5∑k=1aksk−1=0 (4)

and a associated with the linear part. New and old polynomial coefficients satisfy simple relations which can be easily extracted comparing same powers of (1) and (3) with respect to the parameter . Table 1 summarizes the relations between the coefficients of the unfactorised spectral cover and the coefficients for the cases under consideration in the present work.

The homologies of the coefficients are given in terms of the first Chern class of the tangent bundle () and of the normal bundle (),

 [bk]=η−kc1,whereη=6c1−t. (5)

We may use these to calculate the homologies of our coefficients, since if then , allowing us to rearrange for the required homologies. Note that since we have in general more coefficients than our fully determined coefficients, the homologies of the new coefficients cannot be fully determined. For example, if we factorise in a arrangement, we must have unknown parameters, which we call .

In the following sections we will examine in detail the predictions of the and models.

## 3 A4 models in F-theory

We assume that the spectral cover equation factorises to a quartic polynomial and a linear part, as shown in (3). T he homologies of the new coefficients may be derived from the original coefficients. Referring to Table 1, we can see that the homologies for this factorisation are easily calculable, up to some arbitrariness of one of the coefficients - we have seven and only six . We choose in order to make this tractable. It can then be shown that the homologies obey:

 [ai]=η−(6−i)c1−χ,Where i∈{1,…,5}[a7]=c1+χ[a6]=χ. (6)

This amounts to asserting that the five of ‘breaks’ to a discrete symmetry between four of its weights ( or one of its subgroups) and a .

The roots of the spectral cover equation must obey:

 b05∏i=1(s−ti)=0, (7)

where are the weights of the five representation of . When , this defines the tenplet matter curves of the [36], with the number of curves being determined by how the result factorises. In the case under consideration, when , . After referring to Table 1, we see that this implies that . Therefore there are two tenplet matter curves, whose homologies are given by those of and . We shall assume at this point that these are the only two distinct curves, though appears to be associated with (or a subgroup) and hence should be reducible to a triplet and singlet.

Similarly, for the fiveplets, we have

 b05∏i=1(s−ti−tj)=0fori≠j, (8)

which can be shown666See for example [36]. to give the defining condition for the fiveplets: . Again consulting the Table 1, we can write this in terms of the coefficients:

 P5=(a3a6+a2a7)2(a2a6a1a7)−(a4a6+a3a7)(a3a6+a2a7)(a1a6)+a21a5a26a7. (9)

Using the condition that must be traceless, and hence , we have that . An Ansatz solution of this condition is and , where is some appropriate scaling with homology , which is trivially derived from the homologies of and (or indeed and ) [35]. If we introduce this, then splits into two matter curves:

 P5=(a22a7+a2a3a6∓a0a1a26)(a3a26+(a2a6+a1a7)a7)=0. (10)

The homologies of these curves are calculated from those of the coefficients and are presented in Table 2. We may also impose flux restrictions if we define:

 FY⋅χ=N,FY⋅c1=FY⋅η=0, (11)

where and is the hypercharge flux.

Considering equation (7), we see that , so there are at most five ten-curves, one for each of the weights. Under and it’s subgroups, four of these are identified, which corroborates with the two matter curves seen in Table 1. As such we identify with this monodromy group and the coefficient and leave to be associated to .

Similarly, equation (8) shows that we have at most ten five-curves when , given in the form with . Examining the equations for the two five curves that are manifest in this model after application of our monodromy, the quadruplet involving forms the curve labeled , while the remaining sextet - with - sits on the curve.

### 3.1 The discriminant

The above considerations apply equally to both the as well as discrete groups. From the effective model point of view, all the useful information is encoded in the properties of the polynomial coefficients and if we wish to distinguish these two models further assumptions for the latter coefficients have to be made. Indeed, if we assume that in the above polynomial, the coefficients belong to a certain field , without imposing any additional specific restrictions on , the roots exhibit an symmetry. If, as desired, the symmetry acting on roots is the subgroup the coefficients must respect certain conditions. Such constraints emerge from the study of partially symmetric functions of roots. In the present case in particular, we recall that the discrete symmetry is associated only to even permutations of the four roots . Further, we note now that the partially symmetic function

 δ=(t1−t2)(t1−t3)(t1−t4)(t2−t3)(t2−t4)(t3−t4)

is invariant only under the even permutations of roots. The quantity is the square root of the discriminant,

 Δ=δ2 (12)

and as such should be written as a function of the polynomial coefficients so that too. The discriminant is computed by standard formulae and is found to be

 Δ(ak)=256a31a35−(27a42−144a1a3a22+192a21a4a2+128a21a23)a25−2(2(a22−4a1a3)a33−(9a22−40a1a3)a2a4a3+3(a22−24a1a3)a1a24)a5−a24(4a4a32+a23a22−18a1a3a4a2+(4a33+27a1a24)a1) (13)

In order to examine the implications of (12) we write the discriminant as a polynomial of the coefficient  [35]

 Δ≡g(a3)=4∑n=0cnan3 (14)

where the are functions of the remaining coefficients and can be easily computed by comparison with (13). We may equivalently demand that is a square of a second degree polynomial

 g(a3)=(κa23+λa3+μ)2

A necessary condition that the polynomial is a square, is its own discriminant to be zero. One finds

 Δg∝D21D32

where

 D1=a22a5−a1a24D2=(27a21a4−a32)a34−6a1a22a5a24+3a2(9a32−256a21a4)a25+4096a31a35 (15)

We observe that there are two ways to eliminate the discriminant of the polynomial, either putting or by demanding  [35].

In the first case, we can achieve if we solve the constraint as follows

 a22=2a1a3a24=2a3a5 (16)

Substituting the solutions (16) in the discriminant we find

 Δ=δ2=[a2a4(a23−2a2a4)(a23−a2a4)/a33]2 (17)

The above constitute the necessary conditions to obtain the reduction of the symmetry [35] down to the Klein group . On the other hand, the second condition , implies a non-trivial relation among the coefficients

 (a22a5−a24a1)2 = (a2a4−16a1a53)3 (18)

Plugging in the solution, the constraint (44) take the form

 (a22a7+a0a1a26)2 = a0(a2a6+16a1a73)3 (19)

which is just the condition on the polynomial coefficients to obtain the transition .

### 3.2 Towards an SU(5)×A4 model

Using the previous analysis, in this section we will present a specific example based on the symmetry. We will make specific choices of the flux parameters and derive the spectrum and its superpotential, focusing in particular on the neutrino sector.

It can be shown that if we assume an monodromy any quadruplet is reducible to a triplet and singlet representation, while the sextet of the fives reduces to two triplets (details can be found in the appendix).

#### 3.2.1 Singlet-Triplet Splitting Mechanism

It is known from group theory and a physical understanding of the group that the four roots forming the basis under may be reduced to a singlet and triplet. As such we might suppose intuitively that the quartic curve of decomposes into two curves - a singlet and a triplet of .

As a mechanism for this we consider an analogy to the breaking of the group by . We then postulate a mechanism to facilitate Singlet-Triplet splitting in a similar vein. Switching on a flux in some direction of the perpendicular group, we propose that the singlet and triplet of will split to form two curves. This flux should be proportional to one of the generators of , so that the broken group commutes with it. If we choose to switch on flux in the direction of the singlet of , then the discrete symmetry will remain unbroken by this choice.

Continuing our previous analogy, this would split the curve as follows:

 (10,4)={(10,1)=M+Ns(10,3)=M. (20)

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The homologies of the new curves are not immediately known. However, they can be constrained by the previously known homologies given in Table 2. The coefficient describing the curve should be expressed as the product of two coefficients, one describing each of the new curves - . As such, the homologies of the new curves will be determined by .

If we assign the flux parameters by hand, we can set the constraints on the homologies of our new curves. For example, for the curve given in Table 2 as would decompose into two curves - and , say. Assigning the flux parameter, , to the curve, we constrain the homologies of the two new curves as follows:

 [101]= aη+bc1 = cη+dc1−χ Where: a+c=1 and b+d=−5.

Similar constraints may also be placed on the five-curves after decomposition.

Using our procedure, we can postulate that the charge will be associated to the singlet curve by the mechanism of a flux in the singlet direction. This protects the overall charge of in the theory. With the fiveplet curves it is not immediately clear how to apply this since the sextet of can be shown to factorise into two triplets. Closer examination points to the necessity to cancel anomalies. As such the curves carrying and must both have the same charge under . This will insure that they cancel anomalies correctly. These motivating ideas have been applied in Table 3.

#### 3.2.2 GUT-group doublet-triplet splitting

Initially massless states residing on the matter curves comprise complete vector multiplets. Chirality is generated by switching on appropriate fluxes. At the level, we assume the existence of fiveplets and tenplets. The multiplicities are not entirely independent, since we require anomaly cancellation,777For a discussion in relaxing some of the anomaly cancellation conditions and related issues see [41]. which amounts to the requirement that . Next, turning on the hypercharge flux, under the symmetry breaking the and representations split into different numbers of Standard Model multiplets [55]. Assuming units of hyperflux piercing a given matter curve, the fiveplets split according to:

 n(3,1)−1/3−n(¯3,1)+1/3=M5,n(1,2)+1/2−n(1,2)−1/2=M5+N, (21)

Similarly, the tenplets decompose under the influence of hyperflux units to the following SM-representations:

 n(3,2)+1/6−n(¯3,2)−1/6=M10,n(¯3,1)−2/3−n(3,1)+2/3=M10−N,n(1,1)+1−n(1,1)−1=M10+N. (22)

Using the relations for the multiplicities of our matter states, we can construct a model with the spectrum parametrised in terms of a few integers in a manner presented in Table 3.

In order to curtail the number of possible couplings and suppress operators surplus to requirement, we also call on the services of an R-symmetry. This is commonly found in supersymmetric models, and requires that all couplings have a total R-symmetry of 2. Curves carrying SM-like fermions are taken to have , with all other curves .

### 3.3 A simple model: N=0

Any realistic model based on this table must contain at least 3 generations of quark matter (), 3 generations of leptonic matter (), and one each of and . We shall attempt to construct a model with these properties using simple choices for our free variables.

In order to build a simple model, let us first choose the simple case where N=0, then we make the following assignments:

 MT1=MF4=0MT2=1MT3=2MF1=MF2=−MF3=−1 (23)

Note that it does not immediately appear possible to select a matter arrangement that provides a renormalisable top-coupling, since we will be required to use our GUT-singlets to cancel residual charges in our couplings, at the cost of renormalisability.

### 3.4 Basis

The bases of the triplets are such that triplet products, , behave as:

 1=a1b2+a2b2+a3b31′=a1b2+ωa2b2+ω2a3b31′′=a1b2+ω2a2b2+ωa3b331=(a2b3,a3b1,a1b2)T32=(a3b2,a1b3,a2b1)T

where and . This has been demonstrated in the Appendix A, where we show that the quadruplet of weights decomposes to a singlet and triplet in this basis. Note that all couplings must of course produce singlets of by use of these triplet products where appropriate.

### 3.5 Top-type quarks

The Top-type quarks admit a total of six mass terms, as shown in Table 5. The third generation has only one valid Yukawa coupling - . Using the above algebra, we find that this coupling is:

 (1×1)×(3×3) →1×1 →1 (T3×T3)×Hu×θa →(T3×T3)viai i=1,2,3

With the choice of vacuum expectation values (VEVs):

 ⟨Hu⟩=(v,0,0)T⟨θa⟩=(a,0,0)T⟨θb⟩=b (24)

this will give the Top quark it’s mass, . The choice is partly motivated by algebra, as the VEV will preserve the S-generators. This choice of VEVs will also kill off the the operators and , which can be seen by applying the algebra above.

The full algebra of the contributions from the remaining operators is included in Appendix B. Under the already assigned VEVs, the remaining operators contribute to give the overall mass matrix for the Top-type quarks:

 mu,c,t=va⎛⎜⎝y3b2+y4a2y3b2+y4a2y2by3b2+y4a2y3b2+y4a2y2by2by2by1⎞⎟⎠ (25)

This matrix is clearly hierarchical with the third generation dominating the hierarchy, since the couplings should be suppressed by the higher order nature of the operators involved. Due to the rank theorem [37], the two lighter generations can only have one massive eigenvalue. However, corrections due to instantons and non-commutative fluxes are known as mechanisms to recover a light mass for the first generation [37][38].

### 3.6 Charged Leptons

The Charged Lepton and Bottom-type quark masses come from the same GUT operators. Unlike the Top-type quarks, these masses will involve SM-fermionic matter that lives on curves that are triplets under . It will be possible to avoid unwanted relations between these generations using the ten-curves, which are strictly singlets of the monodromy group. The operators, as per Table 5, are computed in full in Appendix B.

Since we wish to have a reasonably hierarchical structure, we shall require that the dominating terms be in the third generation. This is best served by selecting the VEV . Taking the lowest order of operator to dominate each element, since we have non-renormalisable operators, we see that we have then:

 me,μ,τ=v⎛⎜⎝y7d2b+y11d3ay7d2b+y11d3ay3d2y5ay5ay2d1y4by4by1⎞⎟⎠. (26)

We should again be able to use the Rank Theorem to argue that while the first generation should not get a mass by this mechanism, the mass may be generated by other effects [37][38]. We also expect there might be small corrections due to the higher order contributions, though we shall not consider these here.

The bottom-type quarks in have the same masses as the charged leptons, with the exact relation between the Yukawa matrices being due to a transpose. However this fact is known to be inconsistent with experiment. In general, when renormalization group running effects are taken into account, the problem can be evaded only for the third generation. Indeed, the mass relation at can be made consistent with the low energy measured ratio for suitable values of . In field theory GUTs the successful Georgi-Jarlskog GUT relation can be obtained from a term involving the representations but in the F-theory context this is not possible due to the absence of the 45 representation. Nevertheless, the order one Yukawa coefficients may be different because the intersection points need not be at the same enhanced symmetry point. The final structure of the mass matrices is revealed when flux and other threshold effects are taken into account. These issues will not be discussed further here and a more detailed exposition may be found in [49], with other useful discussion to be found in [58].

### 3.7 Neutrino sector

Neutrinos are unique in the realms of currently known matter in that they may have both Dirac and Majorana mass terms. The couplings for these must involve an singlet to account for the required right-handed neutrinos, which we might suppose is . It is evident from Table 5 that the Dirac mass is the formed of a handful of couplings at different orders in operators. We also have a Majorana operator for the right-handed neutrinos, which will be subject to corrections due to the singlet, which we assign the most general VEV, .

If we now analyze the operators for the neutrino sector in brief, the two leading order contribution are from the and operators. With the VEV alignments and , we have a total matrix for these contributions that displays strong mixing between the second and third generations:

 m=⎛⎜⎝y0va000y1vay9bv0y8bvy1va⎞⎟⎠, (27)

where . The higher order operators, and , will serve to add corrections to this matrix, which may be necessary to generate mixing outside the already evident large 2-3 mixing from the lowest order operators. If we consider the operator,

 θc⋅F⋅Hu⋅θd⋅θb→⎛⎜⎝0z3vd2bz2vd3bz1vd2b00z4vd3b00⎞⎟⎠ (28)

We use coefficients to denote the suppression expected to affect these couplings due to renormalisability requirements. We need only concern ourselves with the combinations that add contributions to the off-diagonal elements where the lower order operators have not given a contribution, as these lower orders should dominate the corrections. Hence, the remaining allowed combinations will not be considered for the sake of simplicity. If we do this we are left a matrix of the form:

 MD=⎛⎜⎝y0vaz3vd2bz2vd3bz1vd2by1vay9bvz4vd3by8bvy1va⎞⎟⎠ (29)

The right-handed neutrinos admit Majorana operators of the type , with . The operator will fill out the diagonal of the mass matrix, while the operator fills the off-diagonal. Higher order operators can again be taken as dominated by these first two, lower order operators. The Majorana mass matrix can then be used along with the Dirac mass matrix in order to generate light effective neutrino masses via a see-saw mechanism.

 MR=M⎛⎜⎝100010001⎞⎟⎠+y⎛⎜⎝0d3d2d30d1d2d10⎞⎟⎠ (30)

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The Dirac mass matrix can be summarised as in equation (29). This matrix is rank 3, with a clear large mixing between two generations that we expect to generate a large . In order to reduce the parameters involved in the effective mass matrix, we will simplify the problem by searching only for solutions where and , which significantly narrows the parameter space. We will then define some dimensionless parameters that will simplify the matrix:

 Y1= y1y0≤1 (31) Y2,3= y8,9by0a (32) Z1= z1d2by0a (33) Z2= z2d3by0a