Flipped SU(5) with modular $A_4$ symmetry

We study Flipped $SU(5)\times U(1)$ Grand Unified Theories (GUTs) with $\Gamma_3\simeq A_4$ modular symmetry. We propose two models with different modular weights assignments, where the fermion mass hierarchy can arise from weighton fields. In order to relax the constraint on the Dirac neutrino Yukawa matrix we appeal to mechanisms which allow incomplete GUT representations, allowing a good fit to quark and charged lepton masses and quark mixing for a single modulus field $\tau$, with the neutrino masses and lepton mixing well determined by the type I seesaw mechanism, at the expense of some tuning. We also discuss the double seesaw possibility allowed by the extra singlets generically predicted in such string inspired theories.


Introduction
The existence of three fermion families and the origin of flavour mixing are long standing questions in particle physics. Regarding the first issue, there is no theoretical explanation why there are three families for the otherwise successful standard model and its field theory extensions such as Grand Unified Theories (GUT) and their supersymmetric analogues. A possible interpretation gaining attention these days lies in effective models derived within the context of string theory. In a wide class of such constructions the number of fermion generations is attributed to the topological properties -and in particular the Euler characteristic-of the compactification manifold. The origin of flavour mixing among the three families, however, is still unclear. Various mechanisms have been implemented, including those based on the geometry of the compactification manifold, abelian and discrete symmetries, fluxes and non-perturbative effects, but they are still debatable.
Over the last few decades abelian and non-abelian discrete symmetries have gained an increasing interest in the particle physics literature, with special focus on their rôle in model building. They have been introduced to predict viable fermion and particularly neutrino mass textures as well as to suppress processes leading to baryon and lepton number violating interactions. More recently, modular invariance has been suggested as a novel candidate for predicting viable fermion mass textures [1] 5 . This is an intrinsic symmetry in theories with ultra-violet completion, such as string theory, and can exist in parallel with some discrete group of a different origin. Indeed, Modular Invariance is a fundamental concept in string theory and is naturally expected to leave its trace in the effective field theory model (possibly based on some GUT). Among other implications, it governs the structure of the potential and particularly the Yukawa couplings. For example, in orientifold compactifications of Type II strings the Yukawa couplings are functions with specific modular properties and in orbifold compactifications of heterotic strings Yukawa couplings between twisted states are subject to restrictions from modular invariance and recent attempts to exploit these properties have already appeared (see for example [5,6]). In general, depending on the details of the derivation of the superstring model, Yukawa couplings are expected to be expressed in terms of certain modular forms exhibiting certain transformation properties under the modular group.
Grand unified theories (GUTs) are well motivated theories which reduce the three gauge interactions of the SM group into a simpler structure such as SU (5) [52]. In GUTs, the quark and lepton fields are embedded into fewer gauge multiplets, resulting in relations between quark and lepton mass matrices. There is good motivation for including also family symmetry together with GUTs in order to account for large lepton mixing [53]. The discrete group A 4 is the minimal choice which admits triplet representations [54]. Without modular symmetry, combining A 4 family symmetry with SU (5) GUTs [55] requires vacuum alignment of the flavons in order to break the A 4 , and this provides a further motivation for including modular symmetry. Modular symmetry was first combined with SU (5) GUTs in an (Γ 3 A 4 ) × SU (5) model in [14,56], and subsequently modular SU (5) GUT models have been constructed based on (Γ 2 S 3 ) × SU (5) [10,57], and (Γ 4 S 4 ) × SU (5) [58][59][60]. Recently SO(10) GUTs have been studied based on Γ 3 A 4 modular symmetry [61].
The above studied GUTs depend on a Higgs in an adjoint representation in order to break the gauge symmetry, unless an extra-dimensional mechanism is invoked such as Wilson lines or F-theory flux breaking. On the other hand, in some string theories the adjoint representation is not available to break GUT symmetry, for example the viable GUT models in the context of heterotic compactifications, are only those which do not rely on such Higgs fields. The most popular ones are the Flipped SU (5) [62][63][64][65][66] and the Pati-Salam models [67,68] In this paper, motivated by the above considerations, we study Flipped SU (5) × U (1) GUTs with Γ 3 A 4 modular symmetry. To illustrate the approach, we propose two models with different modular weights assignments, where the fermion mass hierarchy can arise from weighton fields, where one of the models is studied in detail using a numerical χ 2 analysis. In such models the neutrino sector can be tightly constrained by the up type quark mass matrix, in particular the up-quarks and Dirac neutrino mass matrices satisfy the relation m T D = m u at the GUT scale. In order to avoid this constraint we appeal to F-theory constructions where the components of the GUT multiplets may lie on different matter curves. With this constraint relaxed, we can fit the quark and lepton mass matrices and quark mixing for a single modulus field τ , with the neutrino masses and lepton mixing determined by the type I seesaw mechanism. We also discuss the double seesaw possibility allowed by the extra singlets possible in Flipped SU (5).
The layout of the remainder of the paper is as follows. In section 2 we present a short introduction to modular transformations, mainly focusing on the modular symmetry A 4 . In section 3 we start a brief account of the field theory version of the Flipped SU (5) model. Next, we proceed with a modular invariant version proposed in the present work. Considering that Yukawa couplings are certain modular forms, we derive the modular invariant superpotential and the induced mass matrices for up and down quarks, charged leptons and neutrinos. We perform an detailed numerical investigation and show that the proposed construction is in agreement with all low energy data regarding the fermion masses and their mixing. In section 4 we discuss the predictions of the suggested models and summarise the main results. We study the contribution of singlet neutrinos to the flavour structure in Appendix A. A variant of this model, based on a different choice of the modular properties is presented in the Appendix B.

Modular symmetries
In this section, we give a brief review of the modular symmetry and the tetrahedral group A 4 as a finite modular group.

The infinite modular symmetry
A modular transformation γ is defined as a linear fractional transformation on the complex modulus τ varying in the upper-half plane H = Im(τ ) > 0, where a, b, c, d are integers and ad − bc = 1. Each modular transformation can be represented by a 2 × 2 matrix with integer entries and the determinant equal to one, i.e., The modular groupΓ is defined as a group of these transformations, i.e., It includes infinite elements. All elements can be generated by S and T , given by They are represented by 2 × 2 matrices as The actions of S and T in H are given by: One can prove that the identities S 2 = (ST ) 3 = I are satisfied, namely, A subgroup of Γ is obtained by restricting a, d = 1 (mod N ) and b, c = 0 (mod N ), where N is a positive integer. Γ(N ) is also an infinite group. The quotient group Γ/Γ(N ) is finite and labelled as Γ N . It is equivalently obtained by requiring a, b, c, d ∈ Z N , namely For N = 2, 3, 4, 5, 7, Γ 2 S 3 , Γ 3 A 4 , Γ 4 S 4 , Γ 5 A 5 and Γ 7 Σ(168).
In N = 1 supersymmetric theories, a modular-invariant superpertential is expanded as series of polynomials in powers of supermultiplets φ I , Here, Y I 1 I 2 ···In (τ ) are called modular forms keeping the superpotential term invariant under any modular transformation. A modular form Y i (τ ) of level N and weight 2k is defined as a holomorphic function of the modulus τ with the modular transformations under Γ(N ), Under the quotient group Γ N , they are transformed not as holomorphic functions but linear superposition of a series of modular forms {Y 1 (τ ), Y 2 (τ ), · · · } which take the same weight and level, where j runs as an index of the series {Y 1 (τ ), Y 2 (τ ), · · · }, ρ(γ) is a unitary representation matrix of γ ∈ Γ N . For a given finite Γ N , one can choose a basis, where the representation ρ is decomposed to a few irreducible representations. In this basis, modular forms and fields appear as a series of irreducible representions of Γ N . Assigning this basis as the flavour basis of matter fields, the restriction of the modular invariance could strongly constrain the flavour structure. In this work, we will take Γ 3 A 4 as an example to discuss the flavour mixing in Flipped SU (5) framework.

The finite modular symmetry A 4
Γ 3 is a finite subgroup ofΓ, referring to N = 3. Due to the requirement a, b, c, d ∈ Z 3 , the generator T satisfies one more condition T 3 = I, leading to its isomorphism to the tetrahedral group A 4 .
It has three singlet (1, 1 and 1 ) and one triplet (3) irreducible representations. The generators S and T in these representations are given by where ω = e 2iπ/3 . Tensor products of two irreducible representations are decomposed as, where r = 1, 1 , 1 , 3, and "S" and "A" in the subscript denote symmetric and antisymmetric combinations, respectively. In particular, the decomposition of the tensor product of two triplets a = (a 1 , a 2 , a 3 ) T and b = (b 1 , b 2 , b 3 ) T is explicitly written as: Modular forms of level N = 3 and weight 2k form a linear space of dimension 2k + 1. All of them have been explicitly obtained in terms of the Dedekind eta-function η(τ ) [1]: • For 2k = 2, there are 3 linearly independent modular forms, transforming as a triplet of A 4 . Given the triplet basis in Eq. (13), this triplet modular form is written as Y Modular forms of higher weights are derived from products of those of lower weights.
• For 2k = 4, there are 5 linearly independent modular forms, derived from products of two modular forms of weight 2 and written as • For 2k = 6, the 7 linearly independent modular forms are given by We further list some singlet modular forms of higher weights ( 2k = 10 and 12), which may be useful for the rest of the work, 3 Model Building The Flipped SU (5) model has been proposed long time ago [64,65] as an alternative symmetry breaking pattern of the SO(10) gauge group. It is based on the SU (5) × U (1) χ gauge symmetry and has been reconsidered as a possible superstring alternative to Georgi-Glashow SU (5) due to the fact that its spontaneous breaking to SM symmetry requires only a pair of 10 + 10 Higgs representations and does not need any adjoint Higgs representation. In fact, this is a welcome property since in many string derived effective models the Higgs adjoint representation does not appear in the massless spectrum. Among other virtues the model admits a doublet-triplet mass splitting for the color triplets, and in the presence of additional neutral singlets, an extended seesaw mechanism for neutrino masses is naturally realised. The hypercharge generator is a linear combination of the U (1) inside SU (5) and the external abelian factor U (1) χ and it is no longer fully embedded in SU (5). This way Flipped SU (5) representations accommodate the SM matter fields differently.
To start with, the following quantum numbers follow from SO(10) The Flipped gauge symmetry SU (5) × U (1) χ , can be broken to the SM gauge symmetry via a two-step symmetry breaking, In the first step of symmetry breaking, representation decompositions follow as, , In the second step of symmetry breaking, the two U (1) symmetries are broken to U (1) Y with the hypercharge defined by Each representations then gain hypercharges as where Q = (u, d) and L = (ν, e). After the symmetry breaking, SM matter fields Q, L, u c , d c , e c , as well as the right-handed neutrino ν c and MSSM Higgses h u and h d , are seperated as shown on the right-hand side of the above formula. The definition of the hypercharge includes a component of the external U (1) χ in such a way that flips the positions of u c ↔ d c and e c ↔ ν c within these representations, while leaves the remaining unaltered.
In summary, we obtain the following 'Flipped' embedding of the SM representations. The chiral matter fields are The Higgs fields breaking GUT and SM symmetries reside in the following Flipped SU (5) representations A remarkable fact in the Flipped model, is that the5 matter field is completely distinguished from the5 Higgs field. Indeed, due to their different U (1) χ charge which is involved in the hypercharge definition, their SM components do not contain exactly the same type of SM-fields (the5 matter field contains u c , while the5 Higgs field contains the down-type D h ). Several R-parity violating terms will not be allowed because of this distinction.
The fermion masses arise from the following SU (5) × U (1) χ invariant couplings Also, a higher order term providing Majorana masses for the right-handed neutrinos can be written If additional singlet fields ν S , Φ i are present (which is the usual case in string derived models), then -depending on their specific properties-the following couplings could be generated In addition to SM representations, the Higgs sector contains dangerous colour triplets D h , D H + c.c.. They become massive through the following terms Note that, if no other symmetry exists the terms such as HF i h, Hf jh could be possible. Such terms would generate dangerous mixing bewteen Higgs and Matter: Remarkably, a Z 2 symmetry [66] which is odd only for H → −H excludes all these couplings from the lagrangian, while all the previous (useful) terms are left intact. Similar symmetries have been discussed in [69] For rank one mass textures the couplings in Eq. (28) predict m t = m ντ at the GUT scale. However, in contrast to the standard SU (5) model, down quark and lepton mass matrices are not related, since at the SU (5) × U (1) χ level they originate from different Yukawa couplings. This is an important difference with the ordinary SU (5). We know that in order to obtain the observed lepton and down quark mass spectrum at low energies, at the GUT scale the following relations should hold [70] In the ordinary SU (5), the masses are related and the relations can be attributed to the Higgs adjoint which couples differently. This mechanism though is not operative in Flipped SU (5) due to the absence of the adjoint, as noticed above, thus the mass matrices are not related and Yukawas can be adjusted accordingly.
We further review the derivation of the matching condition between gauge couplings of U (1) Y and those of SU (5) × U (1) χ . Computing the traces and finding normalisation constants so that final trace is 2 give In terms of normalised generators,Ỹ = 1 5Cy (ỹ + κχ) where the ratio is κ ≡ 2 Cy Cχ = 2 and for the U (1) Y gauge coupling or equivalently, For initial values a χ = a 5 , we obtain the standard relation of SU (5). In general, however, a χ = a 5 , and there is more flexibility. Fields

Modular-invariant Flipped SU (5)
Working in the Flipped SU (5) framework, we introduce modular invariance in the flavour space with chiral matter fields arranged as multiplets of A 4 . Assignments of matter and Higgs fields in SU (5) × U (1) χ and A 4 are shown in Table 1. In addition, a weighton ξ, which is a singlet scalar with non-trivial modular weight, is introduced to generate fermion mass hierarchies [30,59].
We discuss the constraint of the modular symmetry to the flavour structure. For the up quarks, the Yukawa superpotential, as shown in Eq. (28), takes the form FfΦ. Now including the flavour indices and constraining the superpotential by the modular invariance, we obtain the most general modular-invariant superpotential terms generating quark masses where λ u ij and λ d ij = λ d ji are free parameters andξ ≡ ξ/Λ. Subdominant terms such asξ 5 Y 3 F 3fΦ are possible in superpotential. However, these terms do not lead to significant deviations and thus we did not write them explicitly. These terms generate hierarchical Yukawa structures for quarks after the weighton ξ acquires the VEV v ξ . We write out Y d and Y u up to 6 (with ≡ v ξ /Λ) as where Y 3 2 of weight 6. Here, we have written the Yukawa matrices in the leftright notation, where " * " and " † " represent the complex and Hermitian conjugations,

as a complex and symmetric matrix, can be diagonalised via
The quark mixing matrix is given by V CKM = V † u V d . The complexity of Y u can be further addressed in the case of small . Then, Y u is approximatively written to be Eigenvalues of Y u , i.e., Yukawa couplings of u, c and t, can be analytically derived accordingly,ỹ where Y (2) = (Y 1 , Y 2 , Y 3 ) T , and the dot between two vectors a and b denotes the product i a i b * i . This expression shows that Yukawa couplings of the first, second and third generation up quarks are determined by modular forms of weights 2k = 6, 4, 2, respectively.
In the charged lepton sector, the superpotential terms to generate charged lepton masses are given by Here, λ e , λ µ and λ τ are dimensionless coefficients which can always be kept real by rotating phases of e c , µ c and τ c . The Yukawa matrix is given by Approximatively, the three eigenvalues are given bỹ where Y (2) = (Y 2 , Y 3 , Y 1 ) T . Different from the up quarks, Yukawa couplings of charged leptons are determined by only the modular forms of weight 2k = 2.
In the neutrino sector, the Yukawa matrix Y D between ν and ν c is obtained from the terms W u , which lead to the Yukawa matrix relation Y D = Y T u . Majorana masses for ν c are generated via where the dots represent negligible terms such asξ 10−i−j F i F jHH , which are also allowed by modular invariance but further suppressed by at least 4 .
The light neutrino masses, after integrating out ν c , are generated via the type-I seesaw formula, i.e., Note that non-renormalisable superpotential terms as cannot be forbidden by the modular symmetry. These terms, they lead to Higgs mass µ ∼ξ 3 Λ if Y (4) 1 = 0 and at some stabilisers [25,71], apply, Y = 0 and may be satisfied, µ ∼ξ 5 Λ. These suppressions are not enough to restrict µ ∼ TeV scale. It is possible to introduce an additional Z 2 symmetry (withΦ,f and e c i be Z 2 -odd and the other particles Z 2 -even) beyond the modular symmetry which can forbid these terms and does not lead to additional corrections to the flavour structure.

A mechanism for incomplete representations
As already noted, spontaneous symmetry breaking of the plain field theory Flipped SU (5) model down to the SM symmetry, entails certain fermion mass relations emanating from their common origin in the original SU (5) × U (1)-invariant Yukawa lagrangian. Indeed, recall that in the present model, the up-quark and Dirac neutrinos satisfy the relation m T D = m u at the GUT scale. When seeking an ultra-violet completion of the model, however, this is not always true. In string theory constructions, such as the heterotic and F-theory models, quite often the various GUT representations accommodating the MSSM fields are truncated by stringy projection mechanisms, magnetic fluxes etc 6 , and as a result, such strict relations among the mass matrices are no longer true. In the case of F-theory constructions, in particular, this observation can be illustrated with the following example. Within a generic context of F-theory constructions, matter representations are trapped on the various intersections of the GUT 'surface' wrapped by the appropriate number of 7-branes, with other 7-branes perpendicular to the GUT divisor. As a consequence, an equal number of two-dimensional Riemann surfaces usually called 'matter curves' is formed where the GUT symmetry is further enhanced. In the simplest scenario, each one of the matter curves and the GUT representation residing on them, are characterised by distinct U (1) 'charges' associated with the Cartan algebra of some covering group. For the sake of the argument, therefore, let us assume now that there are M 10 copies of chiral ten-plets on an appropriate matter curve, i.e., #(10 − ) with all of them accommodating fermion generations. Turing on a hypercharge flux of N, N 1 , N 2 units respectively, we obtain: Thus, this way the Dirac neutrino Yukawa coupling is splitted from the up quark Yukawa coupling.
Furthermore, it is worth to note that including multiplicities of matter field representations F ,f and e c can be embedded into different 16's of SO (10), and thus the modularinvariant Flipped SU (5) model can be embedded into a modular-invariant SO(10).
In any case, we can write out the superpotential terms W D with coefficients independent of from those in W u , Then, we arrive at a more generalised matrix where the dimensionless coefficients could 6 See for example [72].
be different from those in Y u , i.e., where Y D(6) i represent the linear combination of modular forms λ D 11 Y (6) 3 2 of weight 6. We consider an ecomonical case that the first two generations of u c and L are splitted but the third generation does not. Thus, λ D ij are independent of λ u ij except λ D 33 . Although Y D is still hierarchical, the relaxing of the coefficients makes the model easier to fit the numerical data, as will be discussed in the next subsection. M ν in this case is estimated to be This kind of mass structure predicts masses m 1 : m 2 : m 3 ∼ 4 : 2 : 1 and therefore, normal ordering (m 1 < m 2 < m 3 ) for neutrino masses. However, in order to generate the correct mass difference ratio ∆m 2 21 /|∆m 2 31 | ∼ 0.03, a fine-tuning of order 10 −2 is required.

Numerical analysis
We perform a χ 2 analysis at the GUT scale to show how well the model fits the data.
These data were derived from a minimal SUSY breaking scenario, with tan β = 5 [18,30,55,73]. They are insensitive to the exact values of tan β unless a very large tan β is taken. Three mixing angles and one CP-violating phase in the CKM mixing matrix are applied from the same literature, For neutrino masses and lepton mixing, we take global best-fit values (without including SK atmospheric data) from NuFIT 5.0 [74,75] and average the positive and negative 1σ errors.
for the normal ordering (NO, i.e., m 1 < m 2 < m 3 ) of neutrino masses. For tan β 10, the RG running effect mainly leads to an small overall enhancement to the neutrino mass scale but has negligible correction to the flavour structure due to the suppression of the charged lepton Yukawa couplings [76]. Thus, in this paper, we will directly use the measured values of lepton mixing angles and mass squared differences in the numerical fit.   We define the following two χ 2 functions in the quark sector and lepton sector, respectively, where p i are the model predictions, b i the current best-fit values and the errors σ i corre-   spond here to the average of the 1σ ranges for each observable.
The relevant free parameters and observables are listed in sets respectively. Here, we ignore the contribution of ν S to the neutrino masses. All coefficients in P q and P l , i.e., λ u ij , λ d ij , λ D ij , λ c ij , and λ e,µ,τ , are scanned with absolute values in the range (0, 1) with arbitrary phases in the range (0, 2π). In the set P q and P l , same values of and τ are respectively used. is scanned in (0, 0.2) and τ is scanned in the fundamental domain ofΓ (3). The latter is achieved by firstly scanned in the fundamental domain of Γ and then shifted to the rest region following modular transformations of the finite modular group Γ 3 , i.e., τ → γτ for all γ in Γ 3 . Furthermore, λ D 33 ≡ λ u 33 is always fixed in the scan as discussed before.
Due to the large pararemter space, a full scan in all parameter space is hard to be performed. Instead, we listed two benchmark points in Tables 2 and 3, respectively, as representatives. Both benchmarks fit the numerical values very well χ 2 q + χ 2 l < 10. In both cases, the second octant of θ 23 , i.e., θ 23 > 45 • is predicted and the leptonic CPviolation is small |δ| < 10 • . In the second benchmark, hierarchical right-handed neutrinos masses with the lightest one M 1 ∼ 2 × 10 10 GeV is predicted. It can be applied for baryogenesis via thermal leptogenesis. 7 This benchmark has a relatively small coefficient |λ u 33 | ∼ 0.1 as an input to predict the Top Yukawa couplingỹ t ∼ 0.5. It is achieved due to the enhancement of a large value of the modular form Y (2) = (−0.0405235 − 0.0260606i, 0.205766 + 0.397288i, 0.180747 − 4.15085i) T at τ = −1.48709 + 0.310071i (remind thatỹ t ≈ |λ u 33 | √ Y (2) · Y (2) ). We further make a scan around the first benchmark with all free parameters deviated by less than 1%. Predictions of mixing angles and δ are shown in Fig. 1.

Conclusion
In this work, the long-standing problem of the origin of flavour mixing among the fermion families has been investigated within the framework of modular symmetries. Inspired by a wide class of string theory effective models endowed with modular invariance and simplicity of the Higgs sector to implement the symmetry breaking, we have constructed a Flipped SU (5) model with A 4 modular symmetry, assigning specific modular weights to the fermion and Higgs fields. In this context, Yukawa couplings are modular forms, with the three families (anti-)five-plets of SU (5) transforming as a triplet under the A 4 modular symmetry, while the fermion ten-plets transform as singlet A 4 representations, distinguished by different modular weights, and the charged lepton electroweak singlets transform as non-trivial one-dimensional A 4 representations with different modular weights. The hierarchy of charged fermion masses is then achieved via higher dimensional operators coupled to a singlet weighton field which carries unit modular weight.
Flipped SU (5) models exhibit many interesting features which make them attractive string motivated candidates as compared to standard SU (5) GUTs. Among other merits, only a pair of 10 + 10 Higgs representations suffices to break the GUT symmetry. At the same time the down-type colour triplets of this Higgs pair combine with those of 5 +5 Higgs representations to realise the doublet-triplet splitting in an elegant manner. As for the mass matrix textures, because charged right-handed lepton fields are SU (5) singlets, the charged lepton mass matrix is unrelated to that of the down quarks. This way the restrictive mass constraints of the ordinary SU (5) are avoided and modular invariance is the main symmetry left over to organise the charged fermion mass matrices. Regarding the neutral fermion sector, a notable property, not shared by the standard SU (5), is that the right-handed neutrinos are contained in the ten-plet representation together with the quark doublets and the down-type colour triplets. This implies the relation Y D = Y T u between Dirac neutrino and up quark Yukawa matrices and, in the simplest version of the model, this restrictive relation makes it difficult to fit the neutrino oscillation data.
In order to overcome this problem and avoid fine tuning issues, we appeal to a stringinspired mechanism according to which magnetic fluxes turned on along the Abelian subgroup of SU (5) split the SU (5)-representations and disentangle the neutrino and up quark Yukawa couplings. Notwithstanding this mechanism, the Dirac neutrino mass matrix has a hierarchical structure, which can be partially canceled by the hierarchical Majorana mass matrix for right-handed neutrinos, resulting in a normally ordered and hierarchical pattern of light neutrino masses, with m 1 an order of magnitude smaller than m 2 . This way, we are able to fit the neutrino oscillation data, only with a slight fine tuning of parameters in the neutrino sector. For the considered model, we find a good fit to charged quark and lepton masses, with a relatively low value of χ 2 . The leptonic CP-violating oscillation phase is predicted to be δ = −8 • ± 8 • . A by-product of our approach is the prediction of hierarchical heavy neutrino masses. The lightest one may around 10 10 GeV, which is around the correct value for standard thermal leptogenesis, which however we do not pursue further here.

A Contribution of singlet neutrinos
At this point, we will study the case where, we take into account the presence of additional singlet neutrino superfields ν S , as predicted string derived Flipped SU (5) models. In this case, more neutrinos mass terms exist, These terms, together with those in W u superpotential, generate a general 9 × 9 mass matrix for neutrinos. In the basis (ν, ν c , ν S ), this mass matrix is written as where all sub-blocks are 3 × 3 matrices. In particular, In our numerical analysis as discussed in section 3.4, we have focused only in the scenario m S Λ c ∼ 10 16 GeV. Below, we give a brief discussion on the opposite scenario m S Λ c . Namely, the light neutrino masses is given by M ν = M D M D M −1 S M T D −1 M T D , which is also called the double seesaw formula. At leading order of , one can check that M ν approximates to up to an overall factor. It partially determines the flavour structure, but has only one non-zero eigenstate ∼ −4 m S v 2 u νc H 2 . Including the next-to-leading correction, one obtain the second lightest neutrino has mass m S v 2 u νc H 2 . Their hierarchy is too large to explain the ratio |r| ≡ ∆m 2 21 /|∆m 2 31 | ∼ 0.03 unless fine tuning between coefficients are considered.

B The second model
In this appendix we present a second Flipped SU (5) model which differs from the first with respect to the representations of the A 4 symmetry and the modular weights assigned to the fields. The transformation properties of the spectrum are shown in Table 4. The superpotential Yukawa couplings for the up and down quark are, where Y (2k)H r with r = 1, 2, 3 and 2k = 4, 6 represent three components of the linear combination of modular forms Y In this model, we consider case that, only the second generation of u c and L is splitted but the first and third generations do not. So, as we see in Eq. (70) the free parameters λ u 1 and λ u 3 are the same as those for the domain of up quarks, while only the second parameter is different.
Also, Majorana masses for ν c are generated via where λ ν ij are free parameters, with i, j = 1, 2, 3. Furthermore, Y 2k r are modular forms with r = 1, 1 , 1 representations of A 4 symmetry and 2k are modular weights. We consider the limit m S Λ c . In this case, light neutrino masses are given by the double seesaw formula, Note that non-renormalisable superpotential terms as, for 2k = 2, 4, ..., are suppressed due to the large power ofξ. This could also be forbidden by introducing additional Z 2 as discussed in the end of section 3.2.