A 4 Modular Flavour Model of Quark Mass Hierarchies close to the Fixed Point τ = i ∞

We study the possibility to generate the quark mass hierarchies as well as the CKM quark mixing and CP violation without ﬁne-tuning in a quark ﬂavour model with modular A 4 symmetry. The quark mass hierarchies are considered in the vicinity of the ﬁxed point τ = i ∞ , τ being the vacuum expectation value of the modulus. We consider ﬁrst a model in which the up-type and down-type quark mass matrices M u and M d involve modular forms of level 3 and weights 6, 4 and 2 and each depends on four constant parameters. Two ratios of these parameters, g u and g d , can be sources of the CP violation. If M u and M d depend on the same τ , it is possible to reproduce the up-type and down-type quark mass hierarchies in the considered model for | g u | ∼ O (10) with all other constants being in magnitude of the same order. However, reproducing the CP violation in the quark sector is problematic. A correct description of the quark mass hierarchies, the quark mixing and CP violation is possible close to τ = i ∞ with all constant being in magnitude of the same order and complex g u and g d , if there are two diﬀerent moduli τ u and τ d in the up-type and down-type quark sectors. We also consider the case of M u and M d depending on the same τ and involving modular forms of weights 8, 4, 2 and 6


Introduction
The idea of using modular invariance opened up a new promising direction to challenge the flavour problem of quarks and leptons [1].The main feature of the approach is that the elements of the Yukawa coupling and fermion mass matrices in the Lagrangian of the theory are modular forms of a certain level N which are functions of a single complex scalar field τ -the modulus -and have specific transformation properties under the action of the modular group.The matter fields (supermultiplets) are assumed to transform in representations of an inhomogeneous (homogeneous) finite modular group Γ (′) N , while the modular forms furnish irreducible representations of the same group.For N ≤ 5, the finite modular groups Γ N are isomorphic to the permutation groups S 3 , A 4 , S 4 and A 5 (see, e.g., [2]), while the groups Γ ′ N are isomorphic to the double covers of the indicated permutation groups, S ′ 3 ≡ S 3 , A ′ 4 ≡ T ′ , S ′ 4 and A ′ 5 .These discrete groups are widely used in flavour model building.The theory is assumed to possess the modular symmetry described by the finite modular group Γ (′) N , which plays the role of a flavour symmetry.In the simplest class of such models, the vacuum expectation value (VEV) of modulus τ is the only source of flavour symmetry breaking, such that no flavons are needed.Another very appealing feature of the proposed framework is that the VEV of τ can also be the only source of breaking of the CP symmetry [3].
There is no VEV of τ which preserves the full modular symmetry.However, as was noticed in [4] and further exploited in [5][6][7][8], there exists three fixed points of the VEV of τ in the modular group fundamental domain, which do not break the modular symmetry completely.These "symmetric" points are τ sym = i, τ sym = ω ≡ exp(i 2π/3) = − 1/2 + i √ 3/2 (the 'left cusp'), τ sym = i∞, and for the theories based on the Γ N invariance, preserve, respectively, Z S 2 , Z ST 3 , and Z T N residual symmetries.In the case of the double cover groups Γ ′ N , the Z S 2 residual symmetry is replaced by the Z S 4 and there is an additional Z R 2 symmetry that is unbroken for any value of τ (see [9] for further details).When the flavour symmetry is fully or partially broken, the elements of the Yukawa coupling and fermion mass matrices get fixed and a certain symmetry-constrained flavour structure arises.
The approach to the flavour problem based on modular invariance has been widely explored so far primarily in the framework of supersymmetric (SUSY) theories.Within the framework of rigid (N = 1) SUSY, modular invariance is assumed to be a property of the superpotential, whose holomorphicity restricts the number of allowed terms.
Following a bottom-up approach, phenomenologically viable "minimal" lepton flavour models based on modular symmetry, which do not include flavons, have been constructed first using the group Γ 4 ≃ S 4 [10] and Γ 3 ≃ A 4 [1] (shown to be describing correctly the available data in Ref. [11])."Non-minimal" models with flavons based on Γ 2 ≃ S 3 and Γ 3 ≃ A 4 have been proposed respectively in [12] and [13].
Reproducing correctly the mass hierarchies of quarks and charged leptons is one of the major difficulties in the flavour models.In almost all phenomenologically viable flavour models based on modular invariance constructed so far the hierarchy of the charged lepton and quark masses is obtained by severe fine-tuning of some of the constant parameters present in the models 2 .Perhaps, the only notable exceptions are the non-minimal models proposed in Refs.[92][93][94], in which modular weights are used as Froggatt-Nielsen charges [95], and additional scalar fields of non-zero modular weights play the role of flavons.
In Ref. [96] a formalism was developed that allows to construct models in which the fermion (e.g.charged-lepton and quark) mass hierarchies follow solely from the properties of the modular forms, thus avoiding the fine-tuning without the need to introduce extra scalar fields.Indeed, in [96] the authors have succeeded in constructing a viable lepton flavour model in which the charged lepton mass hierarchy is reproduced without fine-tuning of the constant parameters present in the charged lepton mass matrix.
The fermion mass matrices are strongly constrained in each of the three symmetric points (i.e., the points of residual symmetries) of the VEV of the modulus τ in the fundamental domain of the modular group discussed above [4][5][6][7][8][96][97][98][99].This fact was exploited in [96] where it was shown that fine-tuning can be avoided in the vicinity of the symmetric points τ = ω and τ = i∞ where the charged-lepton and quark mass hierarchies can follow from the properties of the modular forms present in the corresponding fermion mass matrices rather than being determined by the values of the accompanying constants also present in the matrices.
Following the approach in [96] quark flavour models based on modular invariance have been proposed in [14,[100][101][102][103].These studies showed, in particular, that although it is possible to reproduce without fine-tuning the up-type quark or the down-type quark mass hierarchies, reproducing both the up-type quark or down-type quark masses as well as the Cabibbo, Kobayashi, Maskawa (CKM) quark mixing and CP violation in the quark sector without fine-tuning is a highly challenging problem in flavour models with modular symmetry.
More specifically, in [14] we have investigated the possibility to describe the quark mass hierarchies as well as the CKM mixing and CP violation close to symmetric point τ sym = ω in a model without flavons with modular A 4 symmetry.In this model the quark doublets furnish a triplet irreducible representation of A 4 , while the right-handed quark fields are (different) singlets of A 4 carrying different modular weights.According to the general results in [96], each of the up-type and down-type quark mass hierarchies can be obtained in the model without fine-tuning due to the Z ST 3 residual symmetry.The model contains eight constants, only two ratios of which, g u and g d , can be a source of the CP violation in addition to the VEV of the modulus, τ = ω + ǫ, (ǫ) * = ǫ, |ǫ| ≪ 1.This is the minimal number of parameters ensuring non-zero values of all quark masses.We have shown in [14], in particular, that in the case of real (CP-conserving) g u and g d and common τ (ǫ) in the up-type and down-type quark sectors, the down-type quark mass hierarchies can be reproduced without fine tuning with |ǫ| ∼ = 0.03 and correspond approximately to 1 : |ǫ| : |ǫ| 2 , all other constants being of the same order in magnitude.The up-type quark mass hierarchies, which are smaller by a factor ∼ 10 than the down-type ones, can be obtained with the same |ǫ| ∼ = 0.03 but allowing |g u | ∼ O (10) and correspond to 1 : |ǫ|/|g u | : |ǫ| 2 /|g u | 2 .In this setting the correct description of the CP violation in the quark sector represents a severe problem since it arises as a higher order correction in ǫ with |ǫ| ≪ 1.This problem can be alleviated in the case of complex (CP-violating) g u and g d with |g u | ∼ O (10), but the model failed to reproduce the value of the V cb element of the CKM matrix.A correct non-fine-tuned description of the quark mass hierarchies, the quark mixing and CP violation is possible with all constants being of the same order in magnitude and complex g u and g d , if one allows different values of ǫ in the up-type and down-type quark sectors, or in a non-minimal modification of the considered model which involves one modulus and five constants in each of the two quark sectors one of which is CP violating.
The results obtained in [14] encourage us to search for alternative modular invariant models describing correctly without sever fine-tuning the quark mass hierarchies, the quark mixing and CP violation.
In this work we study the quark mass hierarchies, the CKM mixing angles and the CP violating (CPV) phase close to the fixed point τ sym = i∞ at which the Z T N residual symmetry holds.We investigate models based on modular A 4 symmetry (N = 3) since it is rather simple and the modular forms are restricted only to triplets and singlets.At τ sym = i∞ we have Z 3 residual symmetry as in the case of τ sym = ω.However, the description of the quark mass hierarchies in the vicinity of τ sym = i∞ differs significantly from that in the vicinity of τ sym = ω due, in particular, to the significantly larger value of Im τ .
After introducing the modular symmetry briefly in Section 2, we present the general framework of generating the mass hierarchy close to τ = i∞ without fine-tuning in Section 3. In Section 4, we construct a "minimal" A 4 modular invariant model.We then describe how one can naturally generate hierarchical quark mass patterns in the vicinity of τ sym = i∞ and investigate the flavour structure of the quark mass matrices in the model.After summarizing the input quark data in Section 5, we perform numerical analysis of the description of the quark mass hierarchies and CKM parameters by the "minimal" model.In Section 6.2 we consider phenomenologically the case of existence of two different moduli τ u and τ d in the up-type and down-type quark sectors and perform a numerical analysis of this case.We also present in Section 6.3 an alternative model in which the quark mass matrices depend on a common modulus τ , involve altogether eight constant parameters (two of which can break the CP symmetry) and the lightest u-quark mass is generated either by a tiny SUSY breaking contribution or by dimension six operators and depends on one additional real parameter.We perform also a numerical analysis of the compatibility of the model with the existing data.We summarize our results in Section 7. In Appendix A, the decompositions of the tensor products of the irreducible representations of A 4 are given.In Appendix B, the measure of goodness of numerical fitting is presented.Appendix C contains a proof that the determinant of the up-type quark mass matrix considered in Section 6.3 is zero.

Modular symmetry and its Residual Symmetries
We start by briefly reviewing the modular invariance approach to flavour.In the SUSY framework, one introduces a chiral superfield, the modulus τ , transforming non-trivially under the modular group Γ ≡ SL(2, Z).The group Γ is generated by the matrices obeying S 2 = R, (ST )3 = R 2 = 1, and RT = T R. The elements γ of the modular group act on τ via fractional linear transformations, while matter superfields transform as "weighted" multiplets [1,104,105], where k ∈ Z is the so-called modular weight 3 and ρ(γ) is a unitary representation of Γ.In using modular symmetry as a flavour symmetry, an integer level N ≥ 2 is fixed and one assumes that ρ(γ) = 1 for elements γ of the principal congruence subgroup Hence, ρ is effectively a representation of the (homogeneous) finite modular group Γ ′ N ≡ Γ Γ(N) ≃ SL(2, Z N ).For N ≤ 5, this group admits the presentation The modulus τ acquires a VEV which is restricted to the upper half-plane and plays the role of a spurion, parameterising the breaking of modular invariance.Additional flavon fields are not required, and we do not consider them here.Since τ does not transform under the R generator, a Z R 2 symmetry is preserved in such scenarios [9].If also matter fields transform trivially under R, one may identify the matrices γ and −γ, thereby restricting oneself to the inhomogeneous modular group Γ ≡ P SL(2, Z) ≡ SL(2, Z) / Z R 2 .In such a case, ρ is effectively a representation of a smaller (inhomogeneous) finite modular group Γ N ≡ Γ Γ(N) ∪ Z R 2 .For N ≤ 5, this group admits the presentation In general, however, R-odd fields may be present in the theory and Γ and Γ ′ N are then the relevant symmetry groups.For N = 2, 3, 4, 5, Γ N , as is well known, is isomorphic to the non-Abelian discrete symmetry groups S 3 , A 4 , S 4 , A 5 and Γ ′ N is isomorphic to their respective double covers.Finally, to understand how modular symmetry may constrain the Yukawa couplings and mass structures of a model in a predictive way, we turn to the Lagrangian -which for an N = 1 global supersymmetric theory is given by Here K and W are the Kähler potential and the superpotential, respectively.The superpotential W can be expanded in powers of matter superfields ψ I , where one has summed over all possible field combinations and independent singlets of the finite modular group.By requiring the invariance of the superpotential under modular transformations, one finds that the field couplings Y I 1 ...In (τ ) have to be modular forms of level N.These are severely constrained holomorphic functions of τ , which under modular transformations obey N .The breakdown of modular symmetry is parameterised by the VEV of the modulus and there is no value of τ which preserves the full symmetry.Nevertheless, at certain so-called symmetric points τ = τ sym the modular group is only partially broken, with the unbroken generators giving rise to residual symmetries.In addition, as we have noticed, the R generator is unbroken for any value of τ , so that a Z R 2 symmetry is always preserved.There are only three inequivalent symmetric points (in the fundamental domain of the modular group), namely [4]: • τ sym = i, invariant under S, preserving Z S 4 (recall that S 2 = R) , • τ sym = ω ≡ exp(2πi/3), 'the left cusp', invariant under ST , preserving 3 Mass hierarchy without fine-tuning close to τ = i∞ In theories where modular invariance is broken only by the VEV of modulus τ , the fermion mass matrices (in the limit of unbroken SUSY) are expressed in terms of modular forms of a given level N and a limited number of couplings in the superpotential.The flavour structure of the mass matrices is determined by the properties of the respective modular forms at the value of the VEV of τ .It can be severely constrained at the points of residual symmetries τ = τ sym which typically enforce the presence of multiple zero entries in the mass matrices due to zero values of the corresponding modular form components.As τ moves away from its symmetric value, these entries will generically become non-zero but are suppressed and thus a flavour structure arises.This approach to fermion (charged lepton and quark) mass hierarchies was explored in, e.g., [14,96,97].We present below a more detailed discussion of the approach following [96].If ǫ parameterises the deviation of τ from a given symmetric point τ sym , |ǫ| ≪ 1, the degree of suppression of the (residual-)symmetry-breaking entries is determined by |ǫ| l , where l > 0 is an integer.The integer constant l is not another free parameter of the theory.The values l can take depend on the symmetric point, i.e., on the residual symmetry.As was shown in [96], i) for τ sym = i, l = 0, 1, ii) for τ sym = ω, l = 0, 1, 2, and iii) for τ sym = i∞ of interest and Γ N (Γ ′ N ) finite modular group, N ≤ 5, l can take values l = 0, 1, ..., N − 1.Thus, for Γ 3 ≃ A 4 and τ sym = i∞, l = 0, 1, 2. For a specific would-be-zero element of the fermion mass matrix the value of l is uniquely determined by the residual symmetry and by the transformation properties of the fermion fields, associated with the entry, under the residual symmetry group [96].
Indeed, consider the modular-invariant bilinear where the superfields ψ and ψ c transform under the modular group as4 so that each M(τ ) ij is a modular form of level N and weight Taking τ to be close to the symmetric point, and setting γ to the residual symmetry generator, one can use this transformation rule to constrain the form of the mass matrix M(τ ) [96].
At τ sym = i∞ of interest we have Z T N residual symmetry group gnerated by the T generator of Γ N (Γ ′ N ).Consider the T -diagonal representation basis for group generators in which ρ (c) (T ) = diag(ρ (c) i ), and assume that τ is "close" to τ sym = i∞, i.e., that Im τ is sufficiently large.By setting γ = T in Eq. ( 12), one finds It is now convenient to treat the M ij as functions of q ≡ exp(i2πτ /N), so that parametrises the deviation of τ from the symmetric point [96].Note that the entries M ij (q) depend analytically on q and that q T − → ξ q, with ξ = exp(i 2π/N).Thus, in terms of q, Eq. ( 13) reads: Expanding both sides in powers of q, one finds: where denotes the n-th derivative of M ij with respect to q.It follows that M (n) ij (0) can only be non-zero for values of n such that (ρ c i ρ j ) * = ξ n .In particular, the entry in the vicinity of the symmetric point.It crucially follows that the entry M ij is expected to be O(ǫ ℓ ) whenever Im τ is sufficiently large 5 .The power ℓ is uniquely determined by how the Γ N (Γ ′ N ) representations of ψ c i and ψ j decompose under the residual symmetry group Z T N [96].In the case of A 4 , as we have already indicated, ℓ can take values ℓ = 0, 1, 2.
The discussed result allows, in principle, to obtain fermion mass hierarchies without fine-tuning.
4 A 4 modular invariant flavor model

Models of quarks
We consider next simple models of quark mass matrices in models with the level N = 3 modular symmetry (A 4 modular flavor symmetry).We assign the A 4 representation and the weights for the relevant chiral superfields as • quark singlets (righ-handed) (d c , s c , b c ) and (u c , c c , t c ) : A 4 singlets (1 ′ , 1 ′ , 1 ′ ) with weight (4, 2, 0), respectively, • up and down sector Higgs fields H u,d = H u,d : A 4 singlets 1 with weight 0, which are summarized in Table 1.Then, the superpotential terms of the down-type and up-type Table 1: Assignments of A 4 representations and weights in our model.
quark masses are written for the case in Table 1: where the subscripts of 1 ′ , 1 ′′ denote the A 4 representations.The decompositions of the tensor products in A 4 are given in Appendix A. We will also investigate the possibility of right-handed quark assignments being A 4 singlets (1 ′ , 1 ′ , 1 ′ ) with weights (4, 2, 0) for (d c , s c , b c ) and (6, 2, 0) for (u c , c c , t c ) in Subsection 6.3.In this case modular forms of weight 8 are also involved.

Modular forms of A 4
The weight 2 triplet modular forms are given as: where q = exp (2πiτ ).They satisfy also the constraint [1]: The weight 4, 6 and 8 modular forms of interest can be expressed in terms of the weight 2 modular forms.For the weight 4 modular forms we have: where Y 1 ′′ vanishes due to the constraint of Eq. (20).The expressions for the seven modular forms of weigh 6 read: We give the expressions also for the nine modular forms of weight 8: We note that Y 3 .At τ = i∞, the modular forms of interest take very simple forms: where we can take a normalization Y 0 = 1.

Modular forms "close" to τ = i∞
We present next the behavior of modular forms "close" to τ = i∞.By "close to", or "in the vicinity" of, τ = i∞ we mean values of Im [τ ] which are sufficiently large, e.g., Im [τ ] ∼ 2.5 − 3.0.As we will see, values of Im [τ ] ∼ 2.5 are also typically required to fit the quark mass and mixing data.We find first the expressions of the modular forms Y 1 (τ ), Y 2 (τ ) and Y 3 (τ ) in Eq. ( 19) close to τ = i∞.We use the small real parameter ǫ as given in Eq. ( 14), and the phase p in terms of τ as: where ǫ ≪ 1.At Im [τ ] = 2.5, for example, ǫ ∼ = 5.322 × 10 −3 , ǫ 2 ∼ = 2.832 × 10 −5 , ǫ 3 ∼ = 1.507 × 10 −7 , ǫ 4 ∼ = 8.020 × 10 −10 , ǫ 5 ∼ = 4.268 × 10 −12 and ǫ 6 ∼ = 2.271 × 10 −14 .The expansion parameter q in Eq. ( 19) is written in terms of ǫ and p as: For the components of triplet weight 2 modular form Y (2) 3 we obtain in terms of the small parameter (p ǫ): It follows from Eq. ( 18) that only triplet modular forms will enter into the expressions of the down-type and up-type quark mass matrices.We use the results in Eq. ( 27) and Eqs. ( 21), ( 22) and ( 23) to express the higher weight triplet modular forms Y (k) i in terms of (p ǫ).For the weight 4 triplet modular form we get: In the case of the two weight 6 triplet modular forms the expressions read: Finally we give the expansions for the two triplet weight 8 modular forms: In Eqs. ( 28) -( 30) the numerical estimates of the order of the corrections are for Im [τ ] = 2.5 and we have kept the terms in the expansions which are not smaller than ∼ 10 −5 .

The Quark mass matrices
We obtain the quark mass matrices from the superpotential in Eq. ( 18).In the right-left (RL) convention they have the form: (2) i ≡ Y i and v d (u) denotes the vacuum expectation value of H d (u) , and We take the modular invariant kinetic terms simply by6 where ψ (I) denotes a chiral superfield with weight k I , and τ is the anti-holomorphic modulus.
Since the (anti-)holomorphic modulus is a dynamical field, it becomes a complex number after the modulus τ takes a VEV.Then, one can set τ = τ * .
It is important to address the transformation needed to get the kinetic terms of matter superfields in canonical form because the terms in Eq. ( 33) are not canonical.Therefore, we normalize the superfields as: The canonical form is obtained by an overall normalization, which shifts our parameters such as:

Quark mass matrices at τ = i∞
Since ǫ = 0 at τ = i∞, the modular forms and Y (k = 2, 4, 6) vanish.Then, the quark mass matrices are given by: These are rank one matrices.Thus we have two massless down-type quarks and two massless up-type quarks at the fixed point τ = i∞.The matrices q , q = d, u, whose eigenvalues are the squared quark masses, are given by: 4.6 Quark mass matrices close to τ = i∞ and mass hierarchy The quark mass matrices in Eq. ( 36) are corrected due to the deviation from τ = i∞.By using modular forms of weight 2, 4 and 6 in Eqs. ( 27), ( 28), (29), we obtain the deviation from M (0) q given in Eq. (36).The correction is expressed in terms of the small variable ǫ and the phase p defined in Eq. (25) In the leading order approximation in ǫ, M q is given by: Correspondingly, M 2 q ≡ M † q M q has the following structure: Using Eq. ( 38), we obtain the elements of M 2 q in leading order in ǫ: where the elements are given in units of v 2 q .For the determinant of M 2 q we find: with q ] is g q independent.The heaviest quark mass m q 3 can be obtained to a good approximation from m 2 q 3 ∼ = Tr(M 2 q ).We find: It is also not difficult to obtain an expression for m 2 q 2 m 2 q 3 , which to a good approximation is given by the determninant of the 2-3 sector: Suppose that α ′ q , β q and γ q , which appear in the superpotential in Eq. ( 18) are in magnitude of the same order.As follows from Eq. ( 35), α′ q , βq and γq are enhanced by the factors associated with the renormalisation of the matter fields (see Eqs. (33) and (34)).These factors are powers of 2Im τ , which for τ → i∞ has a arelatively large value (in the case of interest, Im τ ∼ 2.5).After taking into account that for |α ′ q | ∼ |β q | ∼ |γ q |, as it follows from Eq. ( 35), we have , we get from Eqs. ( 41) -( 43): m q 3 ≃ α′ q g q = α ′ q I 3 τ g q , m q 2 ≃ g q + 1 g q βq (12 ǫ) = g q + 1 g q β q I 2 τ (12 ǫ) , where I τ ≡ 2Im τ and we have assumed for simplicity that the constant g q is real and that g q ∼ > 1.
Thus, for the mass ratios we obtain: This result implies that in the case of |α ′ q | ∼ |β q | ∼ |γ q | and |g q | ∼ 1 in the considered model, the quark mass hierarchies are determined by Im τ (we recall that ǫ = exp(− 2πIm τ /3)).The constant |g q | can play a role in obtaining the correct quark mass ratios if, e.g., |g q | >> 1.
Indeed, as we show below, both down-type and up-type quark mass hierarchies can be explained with a common τ in the down-type and up-type quark matrices with Imτ ∼ 2.5, by taking |g d | ∼ 1 and |g u | ∼ 10.We will explore also the alternative possibility of having two different moduli in the down-type and up-type quark mass matrices τ d and τ u and thus two different small parameters ǫ d and ǫ u , ǫ d = ǫ u .
We note finally that the unitary matrices U q (q = d, u), which diagonalise the matrix M † q M q as M † q M q = U q diag(m 2 q 1 , m 2 q 2 , m 2 q 3 ) U † q , with u 1 ≡ u, u 2 ≡ c, u 3 ≡ t, d 1 ≡ d, d 2 ≡ s and d 3 ≡ b, form the CKM quark mixing matrix: We use the parametrisation of U CKM , and thus the convention for three quark mixing angles and CPV phase present in U CKM , θ 12 , θ 23 , θ 13 and δ, proposed in [110].

Input data of quark masses, CKM elements
The modulus τ breaks the modular invariance by obtaining a VEV at some high mass scale.We assume this to be the GUT scale.Correspondingly, the values of the quark masses and CKM parameters at the GUT scale play the role of the observables that have to be reproduced by the considered quark flavour model.They are obtained using the renormalisation group (RG) equations which describe the "running" of the observables of interest from the electroweak scale, where they are measured, to the GUT scale.In the analyses which follow we adopt the numerical values of the quark Yukawa couplings at the GUT scale 2 × 10 16 GeV derived in the framework of the minimal SUSY breaking scenarios with tan β = 5 [111]: The quark masses are given as m q = y q v H with v H = 174 GeV.The choice of relatively small value of tan β allows us to avoid relatively large tan β-enhanced threshold corrections in the RG running of the Yukawa couplings.We set these corrections to zero.
Assuming that both the ratios of the down-type and up-type quark masses, m b , m s , m d and m t , m c , m u , are determined in the model by the small parameter ǫ, which is defined in the comparison of the following equations with in Eq. ( 45), which is defined in the following equations, to be compared with Eq. (45), we have: Thus, the required ǫ for the description of the down-type and up-type quark mass hierarchies differ approximately by one order of magnitude.As indicated by Eq. ( 45), this inconsistency can be "rescued" by relaxing the requirement on the constant |g u | in the up-quark sector, such as with ǫ = 0.02 ∼ 0.03.The quark flavor mixing is given by the CKM matrix, which has three independent mixing angles and one CP violating phase.These mixing angles are given by the absolute values of the following three CKM elements.We take the present data on the three CKM elements in Particle Data Group (PDG) edition of Review of Particle Physics [112] as: By using these values as input and tan β = 5 we obtain the CKM mixing angles at the GUT scale of 2 × 10 16 GeV [111]: The tree-level decays of B → D ( * ) K ( * ) are used as the standard candle of the CP violation.The CP violating phase of latest world average is given in PDG2022 [112] as: Since the phase is almost independent of the evolution of RGE's, we refer to this value in the numerical discussions.The rephasing invariant CP violating measure J CP [113] is also given in [112]: Taking into account the RG effects on the mixing angles for tan β = 5, we have at the GUT scale 2 × 10 16 GeV: J CP = 2.80 +0.14 −0.12 × 10 −5 . (54)

Numerical Analyses
In present Section we show results of the numerical analyses of the considered A 4 quark flavour model.We perform a fit of the quark masses, CKM mixing angles and CP violating phase δ CP at the GUT scale, whose values are given in Section 5.
More specifically, we perform a systematic scan restricting model parameters in the relevant ranges to reproduce the values of the observables by using a measure of goodness of the fit defined in Appendix B, as shown in the following subsections.Several phenomenological possibilities are investigated, as specified below.
In the considered model we have four constant parameters in each of the two quark sectors α q , α ′ q , β q and γ q , q = d, u.We are interested first of all whether it is possible to describe the down-type and up-type quark mass hierarchies in terms of powers of the small parameter ǫ = exp(−2πIm τ /3) avoiding fine-tuning of the constants present in the model.Thus, except possibly for |g u | = |α u /α ′ u |, all other constants should be of the same order in magnitude, i.e., . In this case their influence on the strong quark mass hierarchies of interest is insignificant [96].The minimal number of parameters in the model corresponds the case of real constants α q , α ′ q , β q and γ q , q = d, u.The reality of the constants can be ensured by imposing the condition of exact gCP symmetry in the model [3].The gCP symmetry will be broken by the complex value of τ7 .It can be broken also by some, or all, constants being complex and we will analyse also these cases.

Quark mass hierarchies with common τ in M d and M u
In the present subsection we investigate the case that the down-type and up-type quark mass matrices depend on the same modulus τ .In what concerns the violation of the CP symmetry, there are four phenomenological possibilities, which we consider below.

Real g d and g u
In this case the CP symmetry is violated only by the VEV of the modulus τ .A tentative sample set for the fitting of the values of the observables at the GUT scale given in Section 5 and the results obtained are presented in Tables 2 and 3.For the expansion parameter we get ǫ ≃ 5.8 × 10 −3 .The CP violating measure J CP is much smaller than the observed one.As g q (q = d, u), are real, the CP violating phase is generated by Re τ .This contribution is strongly suppressed for the same reason it is suppressed in the quark flavour model with A 4 modular symmetry with real g d and g u , considered in the vicinity of the symmetric point τ = ω in [14].The cause of the suppression is analysed in detail in Section 5 of ref. [14] and we will not repeat the analysis here.We only note that, as it follows from the discussion in [14], the suppression of interest might not take place if some of the constants present in the down-type and up-type quark mass matrices M d and M u are complex and CP-violating and/or if M d and M u depend on two different CP-violating moduli τ d and τ u , respectively, with τ d = τ u .In what follows we explore phenomenologically these possibilities.

6.1.2
Complex g d (g u ) We assume first that g d is complex, while g u is real.We present a sample set of the results of the fitting of the input data in Tables 4 and 5.
The results collected in Table 5 show that in this case it is possible to reproduce the observed value of the CPV phase δ CP .However, owing to the fact that the |V cb | and |V ub | elements of the  46), ( 51), ( 52) and ( 54) and obtained from the measured ones.
CKM matrix are larger respectively by the factors of 1.9 and 1.2 than the central values of these observables at the GUT scale, the J CP factor is also larger by more than a factor of 2 than the value that should be reproduced.
We have investigated also the case of complex g d and real g u .We find that in this case it is impossible to reproduce the observed value of the CPV phase δ CP and of the J CP factor: the values we obtain for these observables are too small.The problem with reproducing the value of |V cb | also persists.

6.1.3
Complex g d and g u We consider further the case of both g d and g u being complex.We present a sample set of the results of the fitting in Tables 6 and 7. We find that, as in the case of complex g d and real g u , it is possible to reproduce the observed value of the CPV phase δ CP but the value of the J CP factor is larger by more than a factor of 2 than the correct value.(see Table 7).This can be traced again to the larger than the GUT  46), ( 51), ( 52) and ( 54) and obtained from the measured ones.
scale values of |V cb | and |V ub | obtained in the fit.We did not attempt to improve the description of |V cb | and |V ub | by varying tan β and taking into account the threshold correction effetcs in the RG evolution of these two observables.Accounting for these effects warrants an independent comprehensive study.
It follows from the results reported in the preceding subsections that it is possible to reproduce the down-type and up-type quark mass hierarchies in the considered model with 10) when the down-type and up-type quark mass matrices depend on the same modulus τ .However, reproducing the CP violation in the quark sector is problematic.In certain cases this is due to the fact that the values of |V ub | and/or |V cb | resulting from the fit are larger than the GUT scale values of these observables.
Thus, in the next subsection we investigate phenomenologically also the possibility of having two different moduli in the down-type and up-type quark mass matrices τ d and τ u and thus two different small parameters ǫ d and ǫ u , ǫ d = ǫ u8 .We consider the cases of real CP-conserving g d and g u constants as well as complex CP-violating g d and g u constants.In both cases the two moduli τ d and τ u have CP violating VEVs.
6.2 Two moduli τ d and τ u

Real g d and g u
We present a sample set of the results of the fitting in this case in Tables 8 and 9.
4234 + i 2.4672 −0.4528 + i 3.5211 3.57 1.32 −0.703 1.70 0.82 −0.774Table 8: Values of the constant parameters obtained in the fit of the quark mass ratios, CKM mixing angles and of the CPV phase δ CP in the case of two moduli τ d and τ u .9: Results of the fits of the quark mass ratios, CKM mixing angles, J CP and δ CP in the vicinity of two different moduli in the down-quark and up-quark sectors, τ d and τ u , τ d = τ u .'Exp' denotes the values of the observables at the GUT scale, including 1σ error, quoted in Eqs. ( 46), ( 51), ( 52) and ( 54) and obtained from the measured ones.
It follows from Table 9 that the CP violation in the quark sector cannot be reproduced: the value of the CPV phase δ CP is approximately by a factor of 5 smaller than the observed value at the GUT scale.Correspondingly, the J CP factor is also smaller than the observed value.

Complex g d and g u
We consider next the case of both constants g d and g u being complex.A sample set of the results of the fitting in this case is presented in Tables 10 and 11.
As follows from Tables 10 and 11, all quark observables can be successfully reproduced with all constant being in magnitude of the same order.The magnitude of the measure of goodness of the fitting Nσ, which is defined in Appendix B, is Nσ = 2.8.The number of parameters is rather large -this successful model has altogether 14 real parameters (10 real constants and 4 phases).
In the next subsection we consider alternative models with a common modulus τ in the down-type and up-type quark mass matrices, with smaller number of parameters.46), ( 51), ( 52) and ( 54) and obtained from the measured ones.

Alternative model with a common modulus τ
Finally, we discuss an alternative model.In this model we introduce weight 8 modular forms in addition to the weights 4 and 6 ones in an attempt to get a correct description of the observed three CKM mixing angles and CP violating phase with one modulus τ .The model is obtained from the considered one by replacing in Table 1 the weights (6, 2 , 0) of the right-handed quarks (u c , c c , t c ).
The weights are same ones (4, 2 , 0) for right-handed quarks (d c , s c , b c ).These assigments are summarized in Table 12.
Table 12: Assignments of A 4 representations and weights in our model.
Then, the quark mass matrices are easily obtained as: where In order to get the canonical form of the kinetic term, as discussed in Eq. ( 34), the couplings are The u-quark mass m u = 0 could be generated by supersymmetry breaking [97].If supersymmetry is broken by some F-term, the Yukawa couplings are corrected by terms of the order of F/Λ 2 , where F is the supersymmetry breaking expectation value with dimension of mass square and Λ is the SUSY breaking messenger scale [13,97] As a second possibility of generating m u = 0, one can consider the dimension six operators [97]: whose Wilson coefficient should be a modular form of the appropriate weight.In order for this mechanism to work, the weight assignments in Table 12 should be modified.The following conditions have to be fulfilled: with the additional constraint where k Hd and k Hu denote weights of H d and H u , respectively.The conditions in Eq. ( 63) ensure that the superpotentials discussed in the previous subsections have weight zero, and the one in Eq. ( 64) implies that the operator of Eq. ( 62) has different weight from the corresponding renormalisable Yukawa term, so that it couples to an independent modular form multiplet with weight 8 + k Hd + k Hu , 4 + k Hd + k Hu , 2 + k Hd + k Hu .These terms make the resulting up-type quark mass matrix of rank three.Such a mechanism generates tiny quark mass m u ∼ v d v 2 u /Λ 2 , where Λ is the scale at which the operator in Eq. ( 62) is generated.
Adding the tiny corrections due to the SUSY breaking or the dimension six operator discussed above, the up-type quark mass matrix M u is modified as follows: (2) where the mass correction terms appear in the first column.Their contribution are negligibly small in the second and third columns for τ close to τ = i∞ (e.g., Imτ ∼ 2.5).The contribution to the down-type quark mass matrix is negligible as well.Unless C u1 = C u2 = C u3 , the determinant of M 2 u is non-vanishing in this case: where Thus, Det [M 2 q ] depends on |C u | 2 .Indeed, as shown numerically below, the observed u-quark mass is reproduced for |C u | ∼ 0.1 and |f u | = O(1).The contributions of |C ui | ∼ 0.1 (i = 1, 2, 3) to the quark mixing angles and the CPV phase are negligible because they are much smaller than the magnitudes of modular forms Ỹ (k) 3 .In the considered model the up-type quark mass hierarchy differs from that discussed after Eq. ( 43 where as before I τ = 2Im τ .Thus, in the case of |α ′ u | ∼ |β u | ∼ |γ u | for the mass ratios we get: The observed up-type quark mass hierarchies are reproduced for Im τ ∼ 2.5 and small |C u | with f u = O(1).For example, we get m u /m t = 5 × 10 −6 if Im τ = 2.5 and |C u /f u | = 0.1 in Eq. ( 69).
The ratios of the up-type quark masses should be compared with the down-type quark mass ratios: We note that the relatively large value of Im τ together with |C u | ∼ 0.1 gives the requisite stronger hierarchies of up-type quarks masses as compared to the down-type quark mass hierarchies even if f u = O (1).
In what follows we show samples of results of the numerical analysis of the model.We take C u1 to be real and neglect C u2 and C u3 for simplicity.

6.3.1
Real g d and f u In the case of real g d and f u the model contains 11 real parameters.A tentative sample set of fitting of the values of the observables at the GUT scale, given in Section 5, is shown in Tables 13  and 14.Since g d and f u are real and thus CP-conserving, the value of the CP violating measure J CP is much smaller than the value of observed one at the GUT scale.The reason is practically identical to that discussed in [14].By introducing an additional source of the CP violation in the form of complex g d or/and f u , the quality of the fit is improved considerably, as is shown below.This is the case with minimum number of parameters in which we can possibly have sufficiently large violation of the CP symmetry in the quark sector.The CP violation is generated by the complex parameter f u and the modulus τ .The total number of parameters is 12 (10 real constants and 2 phases).The numerical results of the fitting are presented in Tables 15 and 16.  46), ( 51), ( 52) and ( 54) and obtained from the measured ones.
As follows from Tables 15 and 16 all quark observables can be successfully reproduced with all constant being in magnitude of the same order.The magnitude of the measure of goodness of the fit Nσ is Nσ = 2.0.We emphasise that both the mass hierarchies of down-type and up-type quarks are reproduced by the common modulus τ with one complex parameter f u , which is of order one in magnitude.

6.3.3
Complex g d and f u The case of the complex g d with real f u does not give us a fitting result with Nσ smaller than Nσ = 5, so, we omit the discussion of this case.In the case with both complex g d and f u the number of parameters is altogether 13.The sources of the CP violation in this case are the phases of the modulus τ and of the constants g d and f u .The numerical results are presented in Tables 17 and 18.  46), ( 51), ( 52) and ( 54) and obtained from the measured ones.
As it follows from Tables 17 and 18 all quark observables can be successfully reproduced with all constant being in magnitude of the same order.The magnitude of the measure of goodness of the fitting Nσ is Nσ = 1.58.

Summary
We have studied the quark mass hierarchies as well as the CKM quark mixing and CP violation without fine-tuning in a quark flavour model with modular A 4 symmetry.The quark mass hierarchies are considered close to the fixed point τ = i∞, τ being the VEV of the modulus.In the considered A 4 model the three left-handed quark doublets Q = ((u, d), (c, s), (t, b)) are assumed to furnish a triplet irreducible representation of A 4 and to carry weight 2, while the three righthanded down-type quark and up-type quark singlets are supposed to be the A 4 singlets (1 ′ , 1 ′ , 1 ′ ) carrying weights (4, 2, 0), respectively.The down-type and up-type quark mass matrices in the model, M d and M u , involve modular forms of level 3 and weights 6, 4 and 2, and each contains four constants, only two ratios of which, g d and g u , can be a source of the CP violation in addition to the VEV of the modulus, τ .If M d and M u depend on the same τ , it is possible to reproduce the down-type and up-type quark mass hierarchies in the considered model for |g u | ∼ O (10) with all other constants being in magnitude of the same order.However, reproducing the CP violation in the quark sector is problematic.This is due to the fact that the values of |V ub | and/or |V cb | elements of the CKM matrix resulting from the fit are larger than the values of these observables.We have shown that a correct description of the quark mass hierarchies, the quark mixing and CP violation is possible close to τ = i∞ with all constant being in magnitude of the same order and complex g d and g u , if there are two different moduli τ d and τ u in the down-type and up-type quark sectors.
We have considered also the case when M d and M u depend on the same τ and involving modular forms of weights 6, 4, 2 and 8, 4, 2, respectively, with M u receiving a tiny SUSY breaking or higher dimensional operator contribution as well.Both the mass hierarchies of down-type and up-type quarks as well and the CKM mixing angles and the CP violating phase are reproduced successfully with one (or two) complex parameter(s) and all other parameters being in magnitude of the same order.The relatively large value of Im τ , needed for describing the down-quark mass hierarchies, is crucial for obtaining the correct up-type quark mass hierarchies.
We would like to note that if all theoretical uncertainties (due to the scale of SUSY breaking, threshold corrections etc.), which are not included in our analyses, are taken into account, the numerical conclusions may change.Then, it is not excluded that the models we have found to fail to describe correctly the data on certain observable (|V cb |, for example) might actually be successful in describing the corresponding observable.
Finally we comment on the number of parameters of our models.The minimum number of parameters is 12 (10 real constants and 2 phases) as shown in subsection 6.3.1.Although it is larger than the 10 quark observables, the model is valuable in revealing the possible origin of the down-type and up-type quark mass hierarchies.
The results of our study show that describing correctly without severe fine-tuning the quark mass hierarchies, the quark mixing and CP violation in the quark sector is remarkably challenging within the modular invariance approach to the quark flavour problem.
where ω = e i 2 3 π .In this basis, the multiplication rules are: where Further details can be found in the reviews [114][115][116].

B A measure of fit
As a measure of goodness of fit, we use the sum of one-dimensional ∆χ 2 for eight observable quantities q j = (m d /m b , m s /m b , m u /m t , m c /m t , |V us |, |V cb |, |V ub |, δ CP ).By employing the Gaussian approximation, we difine Nσ ≡ ∆χ 2 , where ∆χ 2 = j q j − q j,best fit σ j 2 .
C Massless quark in the mass matrix with weight (6, 2, 0) Consider the mass matrix by only replacing in Table 1 the weights (6, 2 , 0) of the right-handed quarks.Then, the mass matrix is obtained easily as: where
) in subsection 4.6.Suppose that |α ′ u |, |β u | and |γ u | are of the same order.Bringing the kinetic terms to their canonical forms leads to normalisation factors of α ′ u , β u and γ u which differ from those in Eq. (35) since the weights of the right-handed up-type quarks are different in the considered alternative model (k u c = 6, k c c = 2, k y c = 0).After taking into account that due to the renormalisation factors | α′ u | ≫ | βu | ≫ |γ u | and assuming for simplicity that f u is real and f u 1 we have:

Table 2 :
459 3.33 0.98 −0.70 2.43 1.07 −11.8Values of the constant parameters obtained in the fit of the quark mass ratios and of CKM mixing angles.See text for details.

Table 3 :
Results of the fit of the quark mass ratios and CKM mixing angles.J CP factor is output.
'Exp' denotes the respective values at the GUT scale, including 1σ errors .

Table 4 :
394 3.47 1.27 0.88 161 • 1.85 1.15 −10.54Values of the constant parameters obtained in the fit of the quark mass ratios, CKM mixing angles and of the CPV phase δ CP with complex g d .See text for details.

Table 5 :
Results of the fit of the quark mass ratios, CKM mixing angles, δ CP and J CP with complex ǫ and g d .'Exp' denotes the values of the observables at the GUT scale, including 1σ errors, quoted in Eqs. (

Table 6 :
3862 + i 2.4132 3.60 1.04 0.86 161.2 • 2.02 1.34 10.4 205.6 • Values of the constant parameters obtained in the fit of the quark mass ratios, CKM mixing angles and of the CPV phase δ CP with complex g d and g u .See text for details.

Table 7 :
Results of the fit of the quark mass ratios, CKM mixing angles, δ CP and J CP with complex g d and g u .'Exp' denotes the values of the observables at the GUT scale, including 1σ errors, quoted in Eqs. (

Table 10 :
3087 + i 2.4012 0.1231 + i 3.8110 4.60 1.33 0.80 198.5 • 3.00 2.10 1.22 218.3 • Values of the constant parameters obtained in the fit of the quark mass ratios, CKM mixing angles and of the CPV phase δ CP in the case of two moduli τ d and τ u .

Table 11 :
Results of the fits of the quark mass ratios, CKM mixing angles, J CP and δ CP in the vicinity of two different moduli in the down-quark and up-quark sectors, τ d and τ u , τ d = τ u .'Exp' denotes the values of the observables at the GUT scale, including 1σ error, quoted in Eqs. (

Table 13 :
3416 + i 2.3818 3.89 1.12 −0.66 1.92 3.45 −1.78 −0.069Values of the constant parameters obtained in the fit of the quark mass ratios and of CKM mixing angles.See text for details.

Table 14 :
Results of the fit of the quark mass ratios and CKM mixing angles.J CP factor is output.'Exp' denotes the respective values at the GUT scale, including 1σ errors.

Table 15 :
3952 + i 2.4039 3.82 1.17 −0.677 1.72 3.21 1.68 127.3 • -0.07147Values of the constant parameters obtained in the fit of the quark mass ratios, CKM mixing angles and of the CPV phase δ CP with real C u1 , real g d and complex f u .See text for details.

Table 16 :
Results of the fit of the quark mass ratios, CKM mixing angles, δ CP and J CP with real g d and complex f u .'Exp' denotes the values of the observables at the GUT scale, including 1σ errors, quoted in Eqs. (

Table 17 :
3629 + i 2.4090 3.82 1.06 0.68 177.7 • 1.60 3.28 1.61 122.5 • -0.0707Values of the constant parameters obtained in the fit of the quark mass ratios, CKM mixing angles and of the CPV phase δ CP with real C u1 , and complex g d and f u .See text for details.

Table 18 :
Results of the fit of the quark mass ratios, CKM mixing angles, δ CP and J CP with complex g d and f u .'Exp' denotes the values of the observables at the GUT scale, including 1σ errors, quoted in Eqs. (