Fermion Mass Hierarchies from Modular Symmetry

We show how quark and lepton mass hierarchies can be reproduced in the framework of modular symmetry. The mechanism is analogous to the Froggatt-Nielsen (FN) mechanism, but without requiring any Abelian symmetry to be introduced, nor any Standard Model (SM) singlet flavon to break it. The modular weights of fermion fields play the role of FN charges, and SM singlet fields with non-zero modular weight called weightons play the role of flavons. We illustrate the mechanism by analysing $A_4$ (modular level 3) models of quark and lepton (including neutrino) masses and mixing, with a single modulus field. We discuss two examples in some detail, both numerically and analytically, showing how both fermion mass and mixing hierarchies emerge from different aspects of the modular symmetry.


Introduction
The origin of the three families of quarks and leptons and their extreme range of masses remains a mystery of particle physics. According to the Standard Model (SM), quarks and leptons come in complete families that interact identically with the gauge forces, leading to a remarkably successful quantitative theory describing practically all data at the quantum level. The various quark and lepton masses are described by having different interaction strengths with the Higgs doublet, also leading to quark mixing and chargeparity (CP) violating transitions involving strange, bottom and charm quarks. However, the SM provides no understanding of the pattern of quark and lepton masses, quark mixing or CP violation.
The discovery of neutrino mass and mixing makes the flavour puzzle hard to ignore, with the fermion mass hierarchy now spanning at least 12 orders of magnitude, from the neutrino to the top quark. However, it is not only the fermion mass hierarchy that is unsettling. There are now 28 free parameters in a Majorana-extended SM, of which 22 are associated with flavour, surely too many for a fundamental theory of nature. While the quark mixing angles are small, the lepton sector has two large mixing angles θ 12 , θ 23 and one small mixing angle θ 13 which is of the same order of magnitude as the quark Cabibbo mixing angle [1].
One early attempt to understand the quark and lepton mass hierarchies is the Froggatt-Nielsen (FN) mechanism [2]. This approach assumes an additional U (1) f l symmetry under which the quarks and leptons carry various charges and a cut-off scale M f l is associated with the breaking of the U (1) f l symmetry. In the SM the top quark mass of 173 GeV is given by a Yukawa coupling times the Higgs vacuum expectation value of 246 GeV divided by the square root of two. This implies a top quark Yukawa coupling close to unity. From this point of view, the top quark mass is not at all puzzling -it is the other fermion masses associated with much smaller Yukawa couplings that require explanation. According to FN, the fermions are assigned various U (1) f l charges and small Yukawa couplings are forbidden at the renormalisable level due to the U (1) f l symmetry. The symmetry is broken by the vacuum expectation value of a new "flavon" field θ, where θ is a neutral scalar under the SM but carries one unit of U (1) f l charge. Small effective Yukawa couplings then originate from non-renormalisable contact operators where the fermion charges are compensated by powers of θ, leading to suppression by powers of the small ratio θ /M f l (where M f l acts as a cut-off scale of the contact interaction).
To account for family replication and to address the question of large lepton mixing, theorists have explored a larger non-Abelian family symmetry, SU (3) f l [3], where the three families are analogous to the three quark colours in quantum chromodynamics (QCD).
Many other examples have been proposed based on subgroups of SU (3) f l , including nonabelian discrete flavour symmetry (for reviews see e.g. [4][5][6][7][8][9][10]). Moreover, the leptonic CP violation phases can be predicted and the precisely measured quark CKM mixing matrix can be accommodated if the discrete flavour symmetry is combined with generalized CP symmetry [11][12][13][14]. However the main drawback of all such approaches that the flavour symmetry must be broken down to different subgroups in the neutrino and charged lep-ton sectors at low energy and this requires flavon fields to obtain vacuum expectation values (VEVs) along specific directions in order to reproduce phenomenologically viable lepton mixing angles. As a consequence, the scalar potential of discrete flavour symmetry models is rather elaborate, and auxiliary abelian symmetries are usually needed to forbid dangerous operators.
Recently, modular symmetry has been suggested as the origin of flavour symmetry, with neutrino masses as complex analytic functions called modular forms [15]. The starting point of this novel idea is that non-Abelian discrete family symmetries may arise from superstring theory in compactified extra dimensions, as a finite subgroup of the modular symmetry of such theories (i.e. the symmetry associated with the non-unique choice of basis vectors spanning a given extra-dimensional lattice). It follows that the 4D effective Lagrangian must respect modular symmetry. This implies that Yukawa couplings may be modular forms. So if the leptons transform as triplets under some finite modular symmetry, then the Yukawa couplings must transform nontrivially under the modular symmetry and they are modular forms which are holomorphic functions of τ [15]. At a stroke, this removes the need for flavon fields and ad hoc vacuum alignments to break the family symmetry, and potentially greatly simplifies the particle content of the theory. Moreover, all higher-dimensional operators in the superpotential are completely determined by modular invariance if supersymmetry is exact. Models with modular flavour symmetry can be highly predictive; the neutrino masses and mixing parameters can be predicted in terms of few input parameters, although the predictive power of this framework may be reduced by the Kähler potential which is less constrained by modular symmetry [16].
It has been realised that, if the VEV of the modulus τ takes some special value, a residual subgroup of the finite modular symmetry group Γ N would be preserved. The phenomenological implications of the residual modular symmetry have been discussed in the context of modular A 4 [24,27], S 4 [27,29] and A 5 [32] symmetries. If the modular symmetry is broken down to a residual Z 3 (or Z 5 ) subgroup in charged lepton sector and to a Z 2 subgroup in the neutrino sector, the trimaximal TM1 and TM2 mixing patterns can be obtained [24,29].
In this paper, we show how fermion mass hierarchies can be reproduced in the framework of modular symmetry. The mechanism is analogous to the FN mechanism, but without requiring any Abelian symmetry to be introduced, nor any SM singlet flavon to break it. The modular weights of fermion fields play the role of FN charges, and a SM singlet field φ with non-zero modular weight (called a weighton) plays the role of a flavon θ. We illustrate the mechanism with modular level 3 (A 4 ) models of quark and lepton (including neutrino) masses and mixing, using a single modulus field τ and where the charged fermion mass hierarchies originate from a single weighton φ. We discuss two such viable models in some detail, both numerically and analytically, showing how both fermion mass and mixing hierarchies emerge from the modular symmetry. The class of modular level 3 (with even weight modular forms) examples of the mechanism we present here is by no means exhaustive; the new mechanism may be be applied to other levels and choices of weights, and to models with any number of moduli fields and weightons.
We highlight the two of the new features of this work compared to previous analyses.
• We introduce a new mechanism for charged fermion mass hierarchies analogous to the FN mechanism, but without any flavons, using instead modular weights and a complete singlet "weighton" field φ. The only other paper to address the charged lepton mass hierarchy with modular symmetry is [31] which obtained suppression for charged lepton masses in a level 4 (S 4 ) model using a triplet flavon, whereas here our weighton is a complete singlet.
• We discuss quark mass hierarchies using our mechanism. There are only two papers which address the quark masses and mixings using modular symmetry [34,36], and neither paper has attempted to account for the quark mass hierarchies. In [36], texture zeroes in the quark Yukawa matrices were enforced using odd weight modular forms, in particular the double cover of A 4 , namely Γ 3 ∼ = T , but no attempt was made to account for the quark or charged lepton mass hierarchies.
We also remark that the approach here differs from an early work based on U (1) F N broken by a flavon θ, where all fields carried both FN charge and modular weight [47]. In our approach, the Yukawa couplings are modular forms, which means that the modular weights do not have to sum to zero, and are triplets under the A 4 modular symmetry, which constrains the rows of the Yukawa matrices. We also emphasise that we do not have any U (1) F N symmetry, nor any flavon θ to break such a symmetry. Our weighton φ is an A 4 singlet which does not break any flavour symmetry and is therefore not a flavon.
The layout of the remainder of the paper is as follows. In section 2 we briefly review modular symmetry. In section 3 we give the necessary results for level 3 modular forms. In section 4 we show how a previously proposed A 4 modular model of leptons can be recast in natural form by introducing a single weighton, then apply similar ideas to 27 possible models in the quark sector. In section 5 we analyse all the quark models, combined with the natural lepton model, and identify two viable combinations, which can successfully describe all quark and lepton (including neutrino) masses and mixing, using a single modulus field τ , and in which all charged fermion mass hierarchies originate from a single weighton. We discuss these two viable models in some detail, both numerically and analytically, showing how all fermion mass and mixing hierarchies emerge from the modular symmetry. Section 6 concludes the paper.

Modular symmetry
The modular group Γ is the group of linear fraction transformations which acts on the complex modulus τ in the upper half complex plane as follow, We note that the map is an isomorphism from the modular group to the projective matrix group P SL(2, Z) ∼ = SL(2, Z)/{±I}, where SL(2, Z) is the group of two-by-two matrices with integer entries and determinant equal to one.
The modular group Γ can be generated by two generators S and T which are represented by the following two matrices of P SL(2, Z), We can check that the generators S and T obey the relations, The principal congruence subgroup of level N is the subgroup which is an infinite normal subgroup of SL(2, Z). It is easy to see that T N is an element of Γ ( A crucial element of the modular invariance approach is the modular form f (τ ) of weight k and level N . The modular form f (τ ) is a holomorphic function of the complex modulus τ and it is required to transform under the action of Γ(N ) as follows, The modular forms of weight k and level N span a linear space of finite dimension. It is always possible to choose a basis in this linear space such that the modular forms can be arranged into some modular multiplets f r ≡ (f 1 (τ ), f 2 (τ ), ...) T which transform as irreducible representation r of the finite modular group Γ N for even k [15,35], i.e.
where γ is the representative element of the coset γΓ(N ) in Γ N , and ρ r (γ) is the representation matrix of the element γ in the irreducible representation r.
The superpotential W (Φ I , τ ) can be expanded in power series of the supermultiplets Φ I , where Y I 1 ...In is a modular multiplet of weight k Y and it transforms in the representation The requirement of modular invariance of the superpotential implies where the supermultiplet Φ I 1 is assumed to transform in a representation ρ I 1 of Γ N , with a modular weight −k I 1 , and so on for the other supermultiplets.
In the present work we shall adopt the same convention as [15,26,27]. The finite modular group Γ 3 is isomorphic to A 4 which is the symmetry group of the tetrahedron. It contains twelve elements and it is the smallest non-abelian finite group which admits an irreducible three-dimensional representation. The A 4 group has three one-dimensional representations 1, 1 , 1 and a three-dimensional representation 3. In the singlet representations, we have 1 : S = 1, For the representation 3, we will choose a basis in which the generator T is diagonal. The explicit forms of S and T are with ω = e 2πi/3 = −1/2 + i √ 3/2. The basic multiplication rule is where the subscripts S and A denotes symmetric and antisymmetric combinations respectively. If we have two triplets α = (α 1 , α 2 , α 3 ) ∼ 3 and β = (β 1 , β 2 , β 3 ) ∼ 3, we can obtain the following irreducible representations from their product, The linear space of the modular forms of integral weight k and level N = 3 has dimension k + 1 [15]. The modular space M 2k (Γ(3)) can be constructed from the Dedekind etafunction η(τ ) which is defined as The Dedekind eta-function η(τ ) satisfies the following identities There are only three linearly independent modular forms of weight 2 and level 3, which are denoted as Y i (τ ) with i = 1, 2, 3. We can arrange the three modular functions into a vector Y The modular forms Y i can be expressed in terms of η(τ ) and its derivative as follow [15]: The q-expansions of the triplet modular forms Y They satisfy the constraint [15] (Y L e c 3 e c 2 e c 1 N c H u,d A 4 3 1 1 1 3 1 k I 1 1 1 1 1 0 Multiplets of higher weight modular forms can be constructed from the tensor products of Y (2) 3 .

Using the
we can obtain five independent weight 4 modular forms, Similarly there are seven modular forms of weight 6, which can be decomposed as 1⊕3⊕3 under A 4 [15], Y It has been realised that, if the VEV of the modulus τ takes some special value, a residual subgroup of the finite modular symmetry group Γ 3 would be preserved. Thus, the fixed points  [27]. For example,

The Feruglio model of leptons
In this subsection we review an example of a model of lepton masses and mixing based on A 4 modular symmetry, first introduced as example 3 in [15] and later reanalysed in the light of current data in [26]. In this example, there is no flavon field other than the modulus τ . The Higgs doublets H u and H d are assumed to transform as 1 under A 4 and their modular weights k Hu,H d are vanishing. The neutrino masses are assumed arise from the type I seesaw mechanism. In this example [15], the three generations of left-handed lepton doublets L ≡ (L 1 , L 2 , L 3 ) T and of the CP conjugated right-handed T are organised into two triplets 3 of A 4 with modular weights denoted as k L and k N , which will be fixed to take the values of unity shown in Table 1.
When the three CP conjugated right-handed charged leptons e c 3,2,1 are assigned to three different singlets 1 , 1 and 1 of A 4 as in previous works [15,17,18,[21][22][23][24], their modular weights could be identical, which will be fixed to take the values of unity as shown in Table 1, and only the lowest weight modular form Y is necessary in the minimal model.
Then the superpotential for the charged lepton masses takes the form The invariance of W e under modular transformations implies the following relations for the weights, which implies where all values are fixed to be unity as shown in Table 1. This is exactly the case considered in the literature [15,17,18,[21][22][23][24]. We can straightforwardly read out the charged lepton Yukawa matrix For example, τ T = i∞ implies Y If neutrino masses are generated through the type-I seesaw mechanism, for the triplet assignments of both right-handed neutrinos N c and left-handed lepton doublets L, the most general form of the superpotential in the neutrino sector is where f N (Y ) and f M (Y ) are generic functions of the modular forms Y (τ ). Motivated by the principle of minimality, we consider the following example: 3 , which implies, The modular weights of N c and L correspond to k L = k N = 1 as shown in Table 1.
We find M D and M N take the following form The light neutrino mass matrix is given by the seesaw formula, This is the original Feruglio model introduced as example 3 in [15], corresponding to the case of D 10 in [26], giving an excellent fit to current experimental data. The best fit (allowed range) of the modulus for D 10 in [26] is: Re τ = 0.0386(0.0307 ∼ 0.1175), Im τ = 2.230(1.996 ∼ 2.50), which approximates the fixed point case τ T = i∞, since the real part is much less than the imaginary part.

A natural model of charged leptons
In this subsection we show how Feruglio's A 4 modular model of charged leptons can be recast in natural form by introducing a single weighton. The neutrino sector will remain unchanged to leading order. The resulting model of leptons shown in Table 2 now involves a single "weighton" φ which is defined to be a SM and A 4 singlet field with k φ = 1 (i.e. weight −1). We show how such a model can generate a natural charged lepton mass hierarchy. In the next subsection we extend the idea to the quark sector, thereby explaining all charged fermion masses naturally. The three right-handed charged leptons e c 3,2,1 are assigned to three different singlets 1 , 1 and 1 of A 4 as before but now their their modular weights are not identical, and correspond to k e c 3,2,1 = 0, −1, −3 (i.e. weights 0, 1, 3) such that powers of φ with k φ = 1 are required compensate the terms in the previous model, with the combinations e c 3 φ, e c 2 φ 2 , e c 1 φ 4 each having combined weights of unity as before. The weighton φ is assumed to develop a vacuum expectation value (vev) so that the corresponding terms are suppressed by powers ofφ where M f l is a dimensionful cut-off flavour scale. This generates the charged lepton mass hierarchy naturally, with m τ,µ,e ∝φ,φ 2 ,φ 4 , with only the lowest weight modular form Y being necessary as before.
After the weighton develops its vev, the superpotential for the charged lepton masses takes the form which gives a charged lepton Yukawa matrix similar to Eq.26, except that it involves powers ofφ controlling the hierarchies, For example, τ T = i∞ implies Y ∝ (1, 0, 0) T , leading to a diagonal and naturally hierarchical charged lepton Yukawa matrix with m e : m µ : m τ = α eφ 4 : β eφ 2 : γ eφ . The empirically observed charged lepton mass ratios m e /m µ = 1/207 and m µ /m τ = 1/17 suggest that we fixφ ≈ 1/15 to account for the charged lepton mass hierarchy, with the mass ratios m e /m µ ∼φ 2 and m µ /m τ ∼φ, assuming order one coefficients α e , β e , γ e ∼ 1. The small parameterφ ≈ 1/15 defined to to be the ratio of scales in Eq.31 now provides an explanation for the charged lepton mass hierarchies.
However now there will be additional terms corresponding to higher weight modular forms, Y 3 , compensated by extra powers of weighton fields φ, which will give corrections to the charged lepton superpotential, ∆W e = α e e c 1φ 6 (LY (4) where from Eq.21 the weight 4 Yukawa couplings are given in terms of the weight 2 Yukawa couplings, This yields the additive correction to the charged lepton mass matrix in Eq.33, Since the modular weights of L and N c are unchanged, and their representations are the same, we expect the seesaw neutrino matrices to be the same as in the original model at lowest order, where no weighton field φ appears and as in Eq.28. Thus the seesaw matrices in this model are exactly the same as in Eq.29. However now there will higher order corrections involving weightons, the leading correction being suppressed byφ 2 ,  which is of the same form as in Eq.28, yielding additive corrections to the seesaw matrices of the same form as in Eq.29 but suppressed byφ 2 and with the primed Yukawa couplings given by Eq.35. As before, sinceφ ≈ 1/15, these corrections are expected to be about 0.5%, so in the neutrino sector we can safely ignore these corrections and use the same results as before. Thus we expect that the modulus best fit to point to be the same value quoted as before, approximating the fixed point case τ T = i∞.

Natural models of quarks
Quarks have been considered with A 4 modular symmetry in [34]. However there has been no attempt to explain the quark mass hierarchy. Using similar ideas developed in the previous section for the charged leptons, we now consider models for the down type quark Yukawa matrix with m d : m s : m b ∼φ 4 :φ 3 :φ, which turns out to be a good description of the down quark mass hierarchies as we shall see. As in the charged lepton sector, the weighton is assumed to develop a vacuum expectation value (vev) so that the corresponding terms are suppressed by powers ofφ = φ /M f l , where M f l is a dimensionful cut-off flavour scale, which we assume to be the same scale as for the charged leptons.
We introduce the quark modular weights in Table 3 which can achieve this, using the same weighton φ as in the charged lepton sector. We assign the quark doublets Q to a triplet of A 4 with k Q = 1 analogous to the lepton doublets. The three right-handed down type quarks d c 3,2,1 are assigned to three different singlets 1 , 1 and 1 of A 4 , analogous to how the charged lepton Yukawa matrix was constructed.
Unlike in the charged lepton sector, here we allow higher weight modular forms in the quark sector, which will prove necessary to describe quark mixing. We therefore have more freedom in assigning various modular weights to d c 3,2,1 such that powers of φ with k φ = 1 are required compensate the terms, with the combinations d c 3 φ, d c 2 φ 3 , d c 1 φ 4 appearing, analogous to the charged lepton assignments. This generates the down type quark mass hierarchy naturally, with m b,s,d ∝φ,φ 3 ,φ 4 .
After the weighton develops its vev, the superpotential for the down type quark masses with k d c 3,2,1 = 0, −2, −3 takes the form which gives a similar form of Yukawa matrix for the down type quarks as for the charged leptons in Eq.33, albeit the second row being more suppressed than before, where without loss of generality we may take α d , β d , γ u to be real. However now there will be additional terms corresponding to higher weight modular forms, Y 3 , compensated by extra powers of weighton fields φ, which will give corrections to the down type quark superpotential, analogous to the higher order corrections to the charged lepton superpotential in Eq.34. Since these corrections will yield a matrix with a similar structure to the lowest order matrix but with each element having an additional correction be suppressed by a relative power ofφ 2 . This yields the additive correction to the down type quark mass matrix in Eq.39, where α d , β d , γ d are new free complex coefficients (also assumed to be of order unity) while the weight 4 Yukawa couplings are given in Eq.35.
Other alternatives include k d c 3,2,1 = 2, −2, −3: Also we consider k d c 3,2,1 = 4, −2, −3: It is worth noting that we have achieved a single power of suppressionφ for the third down type family in several ways, by choosing an even weight for k d c 3 = 0, 2, 4, . . . so that d c 3φ Q is also even and may be compensated by a Yukawa coupling modular form of weight 2, 4, 6, . . ., leading to the three possibilities for the third row of the down type Yukawa matrix as above (with more possibilities at even higher weight). On the other hand higher powers of suppression such asφ 3 ,φ 4 for the first two families may only be achieved at lowest order by a Yukawa coupling modular form of weight 2.
Turning to the up type quark sector, we first consider m u : m c : m t ∼φ 4 :φ 2 : 1, assuming φ ≈ 1/15 as before. In order to achieve this, the up type quarks are assigned the modular weights as shown in Table 3. The three right-handed up type quarks u c 3,2,1 are assigned to three different singlets 1 , 1 and 1 of A 4 . In this case the up type quark mass hierarchy is much stronger and the top quark Yukawa coupling is of order unity, which suggests that it should be unsuppressed without any weighton field being involved. Moreover, as shown in [34], the lowest weight modular forms Y are not sufficient to describe quark mixing so here we shall utilise weight 6 modular form Y for all three families then the up quark Yukawa matrix would have rows proportional to that of the down quark Yukawa matrix, leading to zero quark mixing angles, so we need to use higher weight modular forms for the up Yukawa matrix, at least for the second or third families, and here we use weight 6 only for the third family. This motivates the assignments k u c 3,2,1 = 5, −1, −3 such that the combinations Qu c 3 , Qu c 2 φ 2 , Qu c 1 φ 4 imply the modular forms Y 3 , respectively, where powers of φ with k φ = 1 are required. Actually there are two independent weight 6 modular forms Y (6) 3,I and Y (6) 3,II and both must be considered as contributing independently.
Although the above assignments satisfies our requirements, we need to check that these are indeed the leading order terms. Firstly Qu c 3 has weight −6 so the leading term is Y  are forbidden at all orders). Secondly, although Qu c 2 has weight zero, this term is forbidden since it is an A 4 triplet and Y (0) 3 does not exist. Therefore the leading allowed term is Qu c 2 φ 2 with weight −2, compensated by Y After the weighton develops its vev, the leading order superpotential for the up type quark masses takes the form 3,II )H u , Thus k u c 3,2,1 = 5, −1, −3 leads to the up quark Yukawa matrix, where the weight 6 Yukawa couplings are given in Eq.22. This is consistent with a diagonal and naturally hierarchical up type quark Yukawa matrix with m u : m c : m t ∼φ 4 :φ 2 : 1, where without loss of generality we may take α u , β u , γ I u to be real, while in general γ II u can be complex.
Before performing a numerical study of this case, we recall that, 3,II ∝ (0, 0, 0) T so near this limit Y (6) 3,II will not contribute. However we need to go away from this limit to explain quark mixing angles. There is a potential problem with the Yukawa structures in Eqs.39,43 since analytically (ignoring third family mixing angles) we expect θ d 12 ∼ θ u 12 ∼ Y 2 /Y 1 , so the physical Cabibbo angle θ 12 ∼ θ d 12 − θ d 12 ∼ 0 due to cancellation.
To avoid this problem we also consider an alternative model with the assignments k u c 3,2,1 = 5, 2, −3 (i.e. only differing by the assignment k u c 2 = 2) such that the combinations Qu c 3 , Qu c 2 φ, Qu c 1 φ 4 imply the modular forms Y 3 , respectively, where powers of φ with k φ = 1 are required. This may avoid the cancellation problem of the Cabibbo angle, since now θ u 12 ∼ Y is different from θ d 12 ∼ Y 2 /Y 1 , but is slightly less natural, being consistent with a diagonal and naturally hierarchical up type quark Yukawa matrix with m u : m c : m t ∼φ 4 :φ : 1. Thus k u c 3,2,1 = 5, 2, −3 leads to: The analysis of the alternative model using the up quark Yukawa matrix in Eq.44 is very similar to that using the up quark Yukawa matrix in Eq.43, but we expect that the Cabibbo angle will be reproduced more easily, with the down quark Yukawa matrix in Eq.39 being the same in both cases.
We also consider a third model with the assignments k u c 3,2,1 = 5, 4, −3 (i.e. differing by the assignment k u c 2 = 4) such that the combinations Qu c 3 , Qu c 2 φ, Qu c 1 φ 4 imply the modular forms Y 3 , respectively, where powers of φ with k φ = 1 are required. Thus with k u c 3,2,1 = 5, 4, −3 we have: All the above three possibilities for the up quark Yukawa matrices have the third family controlled by a weight 6 modular form, resulting from the choice k u c 3 = 5. We now consider third family modular forms of weight 4 corresponding to the choice k u c 3 = 3. This would lead to three more possibilities as shown below.
In the the next section we perform a numerical analysis of our models. First we check the lepton sector results, based on the matrices in Eqs.29,33, then go on to the quark sector using one of the Yukawa matrices in Eq.39,41 or 42 combined with one of Eq.43-51. Without loss of generality we take α e,d,u , β e,d,u , γ e,d,u to be real, with β I u , γ I u real while β II u , γ II u are complex. We allowφ to be free but find that the numerical fits prefer φ ≈ 1/15, as expected.

Input data and global analysis
The charged fermion mass matrices are given by The scale of Yukawa couplings in this model is given by the string compactification scale, and hence we use couplings calculated at the GUT scale from a minimal SUSY breaking scenario, with tan β = 5, as done in [34,48,49]. Similarly, we use the CKM parameters also at this scale as derived by the same authors. For the charged lepton and down type Yukawa masses, the physical particle masses are given by m For the neutrino parameters, we use the data from NuFit 4.1 (2019) [50], without SK atmospheric data, which we summarise below for Normal Ordering (NO), where we write For our numerical study, we follow a procedure similar to that described in [31], but here generalised to the quark sector, to find the minimum χ 2 min,Q contribution from the CKM and quark Yukawa pulls. We consider all 27 combinations of Y I d Y I u , . . . , Y III d Y IX u , restricting τ to be within a range which is acceptable to the lepton sector, based on the matrices in Eqs.29,33, which is approximately the same as found in [26], model D 10 in their notation.
We display our best fit points in and Y II d Y III u , for which we found an arbitrarily good χ 2 min < 1. For the remainder of this paper, we will focus on these two successful models, and do not list the benchmark points for the other models which do not well reproduce data.    In the left panel are the lepton observables and pulls (in fractions of 1σ), the χ 2 min,L contribution from the lepton sector, as well as predictions for neutrino masses, phases, neutrinoless double beta decay and mass ordering. The inputs for the lepton sector are displayed at the bottom. In the right panel we have the quark observables and pulls, the χ 2 min,Q quark contribution, and quark inputs. At the bottom right we list the τ and φ inputs which are common to both sectors. We note thatφ = 1/15 = 0.06667 for example may be fixed exactly to find an equivalently good benchmark point.

Model
fermion Yukawa couplings (at GUT scale), we also find very strong agreement with the CKM angles and phase.
To explain why this is the case, we first look at numerical motivations and then go on to study the analytic properties of this point. Firstly, we list the two numerical mixing matrices which produce the CKM for our best fit point found in Tab. 5 are as follows.
, and similarly for the down sector we find the following two diagonalising matrices, We can see that the Cabbibo angle, θ 12 is mostly generated by the mixing in the up sector, since U u L 1,2 U d L 1,2 , however both sectors will play a similar role in generating the other two angles, with the imaginary part playing a significant role for θ 23 . To examine this further, we turn to an analytic study of the structure of our up and down Yukawa matrices.

Analytic results
We may approximate the analytic forms of our successful model of Y V I u , Y III d rewriting the weighton and weight two modular forms as follows, using Eq.19, and writing the higher weight forms directly in terms of these weight two approximations as in Eqs. 21, 22,φ 0.057 ≡ 1 , where q = e 2πiτ . We find numerically that 2 1 is a similar order to 2 2 , and to 3 . Consequently, we may take the first non-trivial term at the order 3 ), dropping higher corrections in each entry of our successful model. We find the following results for the up and down quark Yukawa matrices, respectively, making a leading order approximation for each element of the matrix, Since the matrices are hierarchical, we can make an estimate for the three mixing angles as follows, which accurately reproduces the fully calculated CKM angles. We then express

Numerical study
We now study a second successful model, comprised of Y III u , Y II d , with input and output parameters found in Tab. 6. This section will proceed analagously to the previous one.
We first present the two numerical diagonalising matrices as before. The numerical values of this model are similar to the previous scenario, where θ 12 is dominated by the contribution from the up quark sector.

Analytic results
We again proceed with the same analytic approach as before, and will find very similar analytic approximations as with the previous scenario. For the new scenario (with slightly different input values of (ϕ, τ )), we again see the relation 3 ), and take the lowest non trivial order in each entry in the two Yukawa matrices.   Table 6: Results of the fit to lepton and quark data for model combining M ν , Y e , Y III u , Y II d . In the left panel are the lepton observables and pulls (in fractions of 1σ), the χ 2 min,L contribution from the lepton sector, as well as predictions for neutrino masses, phases, neutrinoless double beta decay and mass ordering. The inputs for the lepton sector are displayed at the bottom. In the right panel we have the quark observables and pulls, the χ 2 min,Q quark contribution, and quark inputs. At the bottom right we list the τ and φ inputs which are common to both sectors. We note thatφ = 1/15 = 0.06667 for example may be fixed exactly to find an equivalently good benchmark point.
The parameters g 1 , g 2 and 2 , 3 are determined by the fit to the neutrino mass squared differences and PMNS mixing parameters, which arise predominantly from the neutrino sector, due to the very small charged lepton mixing corrections. The large elements in the neutrino mass matrix occurring in the (1, 1) and (2, 3) positions, controlled by g 1 , g 2 , are responsible for the quasi-degenerate neutrino masses m 1 ∼ m 2 ∼ 0.11 eV, and m 3 ∼ 0.12 eV, with the neutrinoless double beta becay parameter m ee ∼ 0.11 eV in the sensitivity region of current experiments, and the cosmological sum of neutrino masses m i ∼ 0.34 eV being in the disfavoured region. Either this lepton model will be discovered soon or it will be excluded in the near future. In any case we remark that the neutrino sector considered here is identical to that of the Feruglio model, being independent of the weighton φ, and hence 1 .

Conclusion
In this paper we have shown how quark and lepton mass hierarchies can be reproduced in the framework of modular symmetry. The mechanism we have proposed is analogous to the Froggatt-Nielsen (FN) mechanism, but without requiring any Abelian symmetry to be introduced, nor any Standard Model (SM) singlet flavon to break it. The modular weights of fermion fields play the role of FN charges, and SM singlet fields with non-zero modular weight called weightons play the role of flavons.
We have illustrated the mechanism by analysing A 4 (modular level 3) models of quark and lepton (including neutrino) masses and mixing, with a single modulus field. We showed how a previously proposed A 4 modular model of leptons can be recast in natural form by introducing a single weighton, then applied similar ideas to 27 possible models in the quark sector. We analysed all the quark models, combined with the natural lepton model, and identified two viable combinations, which can successfully describe all quark and lepton (including neutrino) masses and mixing, using a single modulus field τ , and in which all charged fermion mass hierarchies originate from a single weighton.
We have discussed these two particular examples in some detail, both numerically and analytically, showing how both fermion mass and mixing hierarchies emerge from the modular symmetry. The analytic results clearly show how the fermion mass hierarchies are controlled by the powers of the weighton field which multiply a particular row of the Yukawa matrix, while the smallness of the quark mixing angles arises because of the proximity of the modulus field to the fixed point case τ T = i∞, which results in exponentially suppressed entries within a particular row of the Yukawa matrix. This leads to a simple analytic understanding of the smallness of quark mixing angles.
We emphasise that the mechanism introduced in this paper is quite unlike the traditional FN mechanism, based on an Abelian symmetry, in which the suppression of both rows and columns of the Yukawa matrices arises from FN charges. In the present approach, fermion mass hierarchies and small quark mixing angles emerge from different aspects of the modular symmetry, without having to introduce an extra Abelian symmetry and an additional flavon to break it. The A 4 flavour symmetry arises as a finite subgroup of the underlying modular symmetry, and the weightons responsible for the charged fermion mass hierarchies are A 4 singlets which do not break the flavour symmetry.
Finally we note that the class of modular level 3 (with even weight modular forms) examples of the mechanism we present here are by no means exhaustive. The new mechanism may be be applied to other levels and choices of weights, and to models with any number of moduli fields and weightons.