A 4 modular ﬂavour model of quark mass hierarchies close to the ﬁxed point τ = ω

We investigate the possibility to describe the quark mass hierarchies as well as the CKM quark mixing matrix without ﬁne-tuning in a quark ﬂavour model with modular A 4 symmetry. The quark mass hierarchies are con-sideredinthevicinityoftheﬁxedpoint τ = ω ≡ exp


Introduction
In spite of the remarkable success of the standard model (SM), the flavour problem of quarks and leptons is still a challenging issue.In order to solve the flavour problem, a remarkable step was made in Ref. [1], where the idea of using modular invariance as a flavour symmetry was put forward.This new original approach based on modular invariance opened up a new promising direction in the studies of the flavour physics and correspondingly in flavour model building.
The main feature of the approach proposed in Ref. [1] 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.In addition, both the couplings and the matter fields (supermultiplets) are assumed to transform in representations of an inhomogeneous (homogeneous) finite modular group Γ (′) N .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 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].When the flavour symmetry is broken, the elements of the Yukawa coupling and fermion mass matrices get fixed, and a certain flavour structure arises.As a consequence of the modular symmetry, in the lepton sector, for example, the charged-lepton and neutrino masses, neutrino mixing and the leptonic CPV phases are simultaneously determined in terms of a limited number of coupling constant parameters.This together with the fact that they are also functions of a single complex VEV -that of the modulus τ -leads to experimentally testable correlations between, e.g., the neutrino mass and mixing observables.Models of flavour based on modular invariance have then an increased predictive power.
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 fine-tuning some of the constant parameters present in the models. 2Perhaps, the only notable exceptions are Refs.[116][117][118], in which modular weights are used as Froggatt-Nielsen charges [119], and additional scalar fields of non-zero modular weights play the role of flavons.The recent work in Ref. [120] has proposed the formalism 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 fields.Indeed, authors have succeeded to reproduce the charged lepton mass hierarchy without fine-tuning keeping the observed lepton mixing angles.On the other hand, it is still challenging to reproduce quark masses and the Cabibbo, Kobayashi, Maskawa (CKM) quark mixing matrix in quark flavour models with modular symmetry.
It was noticed in [8] and further exploited in [15,33,43,62] that for the three fixed points of the VEV of τ in the modular group fundamental domain, τ sym = i, τ sym = ω ≡ exp(i 2π/3) = − 1/2 + i √ 3/2 (the 'left cusp'), and τ sym = i∞, the theories based on the Γ N invariance have 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 [108] for further details).
The fermion mass matrices are strongly constrained in the points of residual symmetries [8,15,33,43,62,[120][121][122].This suggests that fine-tuning could be avoided in the vicinity of these points if the charged-lepton and quark mass hierarchies 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.Relatively small deviations of the modulus VEV from the symmetric point might also be needed to ensure the breaking of the CP symmetry [3].
In this work, we study the possibility of obtaining the quark mass hierarchies as well as the CKM matrix without fine-tuning along the lines proposed in Ref. [120] in a model with A 4 modular quark flavour symmetry.Since A 4 symmetry is rather simple, it can be used to clearly understand the problems facing the construction of no-fine-tuned modular invariant flavour models of quark mass hierarchies and CKM mixing.After introducing the necessary tools in Section 2, we present the A 4 modular invariant model in Section 3. In Section 3, we describe how one can naturally generate hierarchical mass patterns in the vicinity of symmetric points, and then, investigate the flavour structure of the quark mass matrices.In Section 4, quark masses and CKM parameters are discussed numerically.In Section 5, the CP problem is discussed.We summarize our results in Section 6.In Appendix A, the decomposition of tensor products are presented.In Appendix B, the relevant modular forms with higher weights are listed.In Appendix C, the modular forms are presented at close to τ = ω.In Appendix D, the measure of goodness of numerical fitting is presented.

Modular symmetry of flavours and residual symmetries
We start by briefly reviewing the modular invariance approach to flavour.In this supersymmetric (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 [1,106,123], 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 [108].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 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.
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, namely [8]: • 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 Quark mass hierarchy in A 4 modular invariant model In theories where modular invariance is broken only by the VEV of modulus, the flavour structure of mass matrices in the limit of unbroken supersymmetry is determined by the value of τ and by the couplings in the superpotential.At a symmetric point τ = τ sym , flavour textures can be severely constrained by the residual symmetry group, which may enforce the presence of multiple zero entries in the mass matrices.As τ moves away from its symmetric value, these entries will generically become non-zero.The magnitudes of such (residual-)symmetry-breaking entries will be controlled by the size of the departure ǫ from τ sym and by the field transformation properties under the residual symmetry group (which may depend on the modular weights).We present below a more detailed discussion of this approach to the fermion (charged lepton and quark) mass hierarchies following [120].Consider a 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 K ≡ k + k c .Modular invariance requires M(τ ) to transform as 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 ).
Let us discuss the case, where τ is in the vicinity of τ sym = ω.We consider the basis where the product ST is represented by a diagonal matrix.In this ST -diagonal basis, we define which are representations under the residual symmetry group.By setting γ = ST in Eq. (12), one finds It is now convenient to treat the M ij as functions of [120] u so that, in this context, |u| denotes the deviation of τ from the symmetric point.Note that the entries M ij (u) depend analytically on u and that u ST −→ ω 2 u.Thus, in terms of u, Eq. ( 14) reads where Mij (u) ≡ (1 − u) −K M ij (u).Expanding both sides in powers of u, one obtains where M(n) ij denotes the n-th derivative of Mij with respect to u.It follows that for τ ≃ ω the mass matrix entry M ij ∼ Mij is only allowed to be O(1) when ρc i ρj = 1.More generally, if ρc i ρj = ω ℓ with ℓ = 0, 1, 2, then the entry M ij ∼ Mij is expected to be O(|u| ℓ ) in the vicinity of τ = ω.The factors ρ(c) i depend on the weights k (c) , see Eq. ( 13).Thus, the leading terms of the components of the mass matrix is any of O(1), O(|u|) and O(|u| 2 ) in the vicinity of τ = ω.This result allows to obtain fermion mass hierarchies without fine-tuning [120].
3.2 Flavour structure of the A 4 modular invariant quark model

The model of quarks
We present next a simple model of quark mass matrices with modular A 4 flavour symmetry, which we consider in the vicinity of the fixed point τ = ω.We assign the A 4 representation and the weights for the relevant chiral superfields of quarks as • The Higgs fields coupled to up and down sectors H u , H d are A 4 singlet 1 with weight 0.
These are summarized in Table 1 5 .The superpotential terms giving rise to quark mass matrices are written by using modular forms with weights 2, 4 and 6 as follows: Table 1: Assignments of A 4 representations and weights for relevant chiral super-fields.
We take the modular invariant kinetic terms simply by where ψ (I) denotes a chiral superfield with weight k I , and τ is the anti-holomorphic modulus.
After taking VEV of modulus, one can set τ = τ * .It is important to address the transformation needed to get the kinetic terms of matter superfields in the canonical form because the terms in Eq. ( 19) are not canonical.Therefore, we normalize the superfields as: The canonical form is obtained by an overall normalization, which shifts our parameters such as where τ = ω + ǫ (|ǫ| ≪ 1).We have: By using the tensor product decomposition rules given in Appendix A, we obtain the following expressions for the mass matrices M d and M u of down-type and up-type quarks 6 : 6 We note that the mass matrices are written in RL convention. where i , Y We note that the CKM quark mixing matrix U CKM is given by the product of the unitary matrices U uL and U dL , which diagonalise respectively It is clear from the expressions of M u and M d in Eq. ( 23) that only the absolute values squared of the constants α q ( αq ), β q ( βq ) and γ q (γ q ), q = d, u, enter into the expressions for M † u M u and M † d M d .Thus, these constants cannot be a source of CP violation and without loss of generality can be taken to be real.In contrast, the constants g q , q = d, u, if complex, may cause violation of the CP symmetry and therefore we will consider them, in general, as complex parameters.

Quark mass matrices at τ = ω
Consider the quark mass matrices in Eq. ( 23) at the fixed point τ = ω.In the symmetric basis of S and T generators given in Appendix A (Eq. ( 62)), in which the mass matrices in Eq. ( 23) are obtained, the modular forms Y It is easily checked that the 1st, 2nd and 3rd rows of M q are proportional to each other.That is, the mass matrix of Eq. ( 25) is of rank one.It proofs convenient to analyse the quark mass matrices M q in the diagonal basis of the ST generator for the A 4 triplet, in which the flavour structure of M q becomes explicit.The STtransformation of the A 4 triplet of the left-handed quarks where the representations of S and T for the triplet are given explicitly in Appendix A. The STeigenstate Q ST is obtained with the help of a unitary transformation.We use the unitary matrix V ST , which leads to the diagonal basis of the ST generator of interest: Then, the ST -eigenstate is The right-handed quarks q c i (q = d, u), which are singlets (1, 1 ′′ , 1 ′ ) with weights (4, 2, 0), are eigenstates of ST : where we have used ST charges of (1, 1 ′′ , 1 ′ ) which read (1, ω 2 , ω) (see Eq. ( 65)).Finally, we have It can be shown using, in particular, the preceding results that the Dirac mass matrix in the ST diagonal basis, M q , is related to the mass matrix in the initial S and T symmetric basis as follows: Thus, we get: which is a rank one matrix.We have two massless up-quarks and two massless down-quarks at the fixed point τ = ω.The matrix M † q M q is transformed as: We see that only the third generation down-types quarks and up-type ones get non-zero masses.

Quark mass matrices in the vicinity of τ = ω
The quark mass matrices in Eq. ( 32) are corrected due to the small deviation of τ from the fixed point of τ = ω.By the Taylor expansion of the modular forms in the vicinity of τ = ω as seen in Appendix C, we estimate the off-diagonal elements of M 2 q in Eq. ( 33).In the ST diagonal basis, the correction is parametrised by a relatively small variable ǫ, where The parameter ǫ describing the deviation of τ from ω is related to the "deviation" parameter u introduced in [120] (see Eq. ( 15)): Up to 2nd order approximation in ǫ, the quark mass matrix M q is given by: where ǫ 1 , ǫ 2 , k 2 and k 3 are given in Appendix C : with Using Eq. ( 35) we obtain the elements of (M q ≡ M 2 q in leading order in ǫ 1 : where the factors and γq Y 1 are absorbed in αq , βq and γq , respectively.We note that in the case of |ǫ| ≪ 1 of interest the factor Y 1 is close to 1 7 .The flavour structure of (M q is given in terms of powers of ǫ as: We can obtain the mass eigenvalues m q1 , m q2 and m q3 approximately as follows.The determinant of M 2 q is given as which is independent of g q .We also have and It is easy to find that in the case of |g q | ∼ 1 and αq ∼ βa ∼ γq the mass ratios satisfy: On the other hand, if |g q | ≫ 1 and αq ∼ βa ∼ γq , we have Then, the quark mass ratios are approximately given by Namely, in the case of |g q | ≫ 1 and αq ∼ βa ∼ γq the quark mass hierarchies are given effectively in terms of |ǫ 1 /g q | ∼ |ǫ/g q |.Indeed, we have succeeded in explaining both down-quark and up-quark mass hierarchies numerically for |g u | ∼ 15 and |g d | ∼ 1.Thus, we see the scaling of quark masses with ǫ.The CKM elements are also scale roughly with ǫ.However, they depend also on the constants and phases of both mass matrices because both the up and down quark mass matrices contribute to them.

Quark masses and CKM mixing without fine-tuning
In this Section, we discuss the possibility of reproducing the observed quark masses and CKM quark mixing parameters without fine-tuning in the vicinity of τ = ω, i.e., without strong dependence of the results on the constants present in the model.We investigate first whether it is possible to describe the up-quark and down-quark mass hierarchies in terms of powers of the small parameter ǫ ≡ τ − ω avoiding fine-tuning of the constants present in the model.Correspondingly, we suppose that the constants α q , α ′ q , β q and γ q in Eq. ( 18) are real and are of the same order, i.e., g q ≡ α ′ q /α q ≃ β q /α q ≃ γ q /α q ∼ O(1), so that their influence on the strong quark mass hierarchies of interest is insignificant [120].The reality of the constants can be ensured by imposing the condition of exact gCP symmetry in the considered model [3].The gCP symmetry will be broken by the complex value of ǫ = τ − ω = 08 .It can be broken also by some (or all) constants being complex.
In the modular invariance approach to the flavour problem the modulus τ obtains a VEV, which breaks the modular flavour symmetry, at some high scale.Thus, the quark mass matrices, and correspondingly the quark masses, mixing angles and CP violating phase, are derived theoretically in the model at this high scale.The values of these observables at the high scale are obtained from the values measured at the electroweak scale by the use of the renormalization group (RG) equations.In the framework of the minimal SUSY scenario the RG running effects depend, in particular, on the chosen high scale and tan β.In the analysis which follows we use the GUT scale of 2 × 10 16 GeV and tan β = 5 as reference values.The numerical values of the quark Yukawa couplings at the GUT scale for tan β = 5 are given by [127,128]: The quark masses are obtained from the relation 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| (or |ǫ 1 | = 2.235|ǫ|), we have where we have given also the values of |ǫ 1 | suggested by fitting the down-type and up-type quark mass ratios given in Eq. (46).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 (corresponding to |ǫ 1 | = 0.045 ∼ 0.067).
In the considered case we have eight real parameters in the down-type and up-type quark mass matrices, α q , β q , γ q , g q ≡ α ′ q /α q , q = d, u, and one complex parameter τ = ω + ǫ.Taking 10), we can reproduce the observed quark mass values.A sample set of values of these parameters for which the quark mass hierarchies are described correctly is given in Tables 2 and 3.   Table 3: Results on the quark mass ratios compared with those at the GUT scale including 1σ error, given in Eq. (46).
A quantitative criterion of fine-tuning, i.e., of high sensitivity of observables to model parameters, was proposed by R. Barbieri and G. Giudice in [135] in a different context, but is applicable also in the case of quark mass hierarchies studied by us.The Barbieri-Giudice measure of fine-tuning [135] in the quark sector, max(BG), corresponds to the largest of quantities |∂ ln(mass ratio)/∂ ln α( ′ ) q |, |∂ ln(mass ratio)/∂ ln βq | and |∂ ln(mass ratio)/∂ ln γq |, equivalently, to the largest of |∂ ln(mass ratio)/∂ ln α( ′ ) q |, |∂ ln(mass ratio)/∂ ln β q | and |∂ ln(mass ratio)/∂ ln γ q |.An observable O is typically considered fine-tuned with respect to some parameter p if BG ≡ |∂ ln O/∂ ln p| 10 [135].The criterion is satisfied by the quark mass ratios in the models considered in our work.This can be easily checked using the analytic expressions for the quark masses in terms of the constant parameters, Eqs. ( 40) -( 42) and (44).We should add that, as was shown in [120], when applied to mixing angles the Barbieri-Giudice criterion leads to incorrect results.At present there does not exist a reliable formal no-fine-tuning criterion for the mixing angles and the CP violating phase.So, we and other authors use the simple criterion that the constant parameters present in the quark mass matrices be of the same order of magnitude, the rational being that these parameters are introduced on equal footing and there is no a priori reason why they should have vastly different values.

Reproducing the CKM mixing angles
As discussed in the previous sections, the quark mass hierarchies are reproduced due to ǫ, which denotes the deviation from the fixed point τ = ω, and the help of |g u | ∼ O (10).Next, we study the CKM mixing angles by taking the values of β q /α q ,γ q /α q and g d to be of order 1.The present data on the CKM mixing angles are given in Particle Data Group (PDG) edition of Review of Particle Physics [129] 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 [127,128]: 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 [129] 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 [130] is also given in [129]: 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) We will discuss the CP violation in our model in the next Section.We try to reproduce approximately the observed CKM mixing angles with real g q , q = d, u.In our scheme, the CKM mixing angles are given roughly in terms of powers of ǫ 1 , as seen in Eq. (38).In order to reproduce the observed ones precisely, the numerical values of the order one ratios of the parameters β q /α q ,γ q /α q as well as of g d "help" somewhat (no fine-tuning) since both up and down quark mass matrices contribute to them.We will show those numerical values in Tables.
We scan parameters with the constraint of reproducing the observed values of the quark masses, Cabibbo angle and |V ub | including the 3σ uncertainties.A sample set for the fitting and the results are presented in Tables 4 and 5.     46), ( 51) and ( 54) and obtained from the measured ones.
As seen in Table 5, the values of the CKM elements |V us | and |V ub | found in the fit are consistent with the observed values.On the other hand, the magnitude of V cb is large, almost twice as large as the observed one.This result can be understood using the results in Eq. (38).For the values of the parameters in Table 4 both down-type quark sector and up-type quark one contribute to |V cb | additively in O(ǫ 1 ), each contribution being close to the observed one.
The CP violating measure J CP is much smaller than the observed one.As g q is real, the CP violating phase is generated by Im ǫ 1 , which corresponds to Re ǫ.This contribution is strongly suppressed, as discussed in Section 5.

Reproducing CP violation
As we have seen, the CP violating measure J CP is suppressed in the case of real parameters of g d and g u due to the extremely small value of the CPV phase δ CP .In order to try to reproduce the observed value of δ CP we take either of the two parameters (or both) to be complex.In the case of complex g d or g u , the gCP symmetry is broken explicitly by the complex constant.Having both g d and g u complex is effectively equivalent to not imposing the gCP symmetry requirement at all.As in the preceding subsection, we scan the parameters with the constraint of reproducing the observed values of the quark masses, Cabibbo angle and |V ub | including 3σ uncertainties.

Complex g d
First, we take g d to be complex but real g u .We present a sample set of the results of the fitting in Tables 6 and 7. We obtain a large value of the CPV phase δ CP around 80  46), ( 51), ( 52) and ( 54) and obtained from the measured ones.

Complex g u
We consider next the case of complex g u and real g d .A sample set for a fitting is in Tables 8 and  9.The magnitude of V cb is also larger than the observed one by a factor ∼ 2. In this case, the magnitude of the CPV phase δ CP is close to 180  46), ( 51), ( 52) and ( 54) and obtained from the measured ones.

4.3.3
The case of complex g d and g u Finally, we consider the case of complex g d and g u .A sample set of the results of the fitting is given Tables 10 and 11.We obtain a value of the CPV phase δ CP consistent with measured one.On the other hand, the magnitude of V cb is still larger than that at the GUT scale given in Eq. ( 51).Therefore, |J CP | is also larger approximately by a factor of 2 than its value at the GUT scale (see Eq. ( 54 Table 11: Results of the fit of the quark mass ratios, CKM mixing angles, δ CP and J CP with complex ǫ, g d and g 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.

CKM mixing angle and CPV phase with two moduli τ d and τ u
In the previous Subsections, we have considered a common modulus τ in the mass matrices M d and M u .We have seen that the mass hierarchies of both down-type and up-type quarks can be reproduced with the real constant g u having a value |g u | of O( 10) and all other constants being O(1).We can consider phenomenologically also the possibility of having two different moduli in up-and down-quark sectors, τ q , q = d, u (see Section 5 for further discussion of this possibility).The d-and u-type quark mass hierarchies in this case are given respectively by 1 : In this case the mass hierarchies in Eqs. ( 47) and ( 48) can be easily reproduced with |g u | ∼ 1. Table 12: 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 q = ω + ǫ q with complex ǫ q , q = d, u, Table 13: 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.
We present a sample set of results of the fitting with complex g d and g u in Tables 12 and  13.The magnitude of V cb is almost consistent with the observed one at the GUT scale.In this case, the up-type quark sector contribution to |V cb | is of O(|ǫ u |), which is much smaller than the down-type quark sector one that is of O(|ǫ d |).Although we did not search for the χ 2 minimum, we show the magnitude of the measure of goodness of the fitting Nσ, which is defined in Appendix D, as a reference value.In this numerical result, we obtain Nσ = 6.8.The fit does not look so good.It is a consequence of the extremely high precision of the data which we are fitting using simple method without making efforts to improve the quality of the fit by varying the value of tan β and/or the threshold effects in the RG running.

Improved 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 order 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 the weights (4, 2 , 0) of the right-handed quarks (d c , s c , b c ) and (u c , c c , t c ) with weights (6, 4 , 2), respectively, in Table 1.Then, the quark mass matrices are given as follows: where The additional parameters f d and f u of the model play an important role in reproducing the observed CKM parameters.Indeed, we have obtained a good fit of CKM matrix with and one τ .We show the numerical result with complex ǫ and f d , while real g d , g u and f u in order to reduce free parameters.The numerical results are presented in Tables 14 and 15.The measure of goodness of fit is considerably improved as Nσ = 1.6 in this case.We can get better fit with Nσ < 1 in the case that g d , g u and f u also complex.The goodness of the fit might be also improved by using a different value of tan β and/or different set of threshold corrections.46), ( 51), ( 52) and ( 54) and obtained from the measured ones.
we will refer in what follows in this Section to both hierarchies in Eqs. ( 57) and ( 58) as being of the type 1 : |ǫ| : |ǫ| 2 .Then from the point of view of CP violation the quark mass matrices M d,u in Eq. ( 35) have the following generic structure: where we have used ǫ 1 = i 2.235ǫ and kept only the leading order terms in ǫ in the first, second and third columns of M q .The different real coefficients in the elements of M q shown in the second matrix in Eq. ( 35), including the factors (2.235) 2 and 2.235 in the first and second column, as well as the common factors of the 1st, 2nd and 3rd rows of M q , namely, αq ωY 3 1 , βq ω 2 Y 2 1 and γq Y 1 , which we have not included in the expression Eq. ( 59) of M q , are not relevant for the present general discussion of CP violation.It is not difficult to show that (M gen q ) † M gen q of interest can be cast in the form: where we have used ǫ q = |ǫ q |e i κq taking into account the possibility of two different deviations from τ = ω in the down-quark and up-quark sectors 11 .The matrix in Eq. ( 60) is diagonalised by U gen qL = P(κ q )O q , where P(κ q ) = diag(e −i (κq+π/2) , 1, e i (κq+π/2) ) and O q is a real orthogonal matrix.The CKM matrix in this schematic analysis of "leading order" CP violation is given by: In the "minimal" case of one and the same deviation of τ from ω we have ǫ 1d = ǫ 1u and, consequently, κ d = κ u .It follows from Eq. ( 61) that in this case U gen CKM is real and thus CP conserving.This implies that to leading order in ǫ in the elements of the quark mass matrices M q , there will be no CP violation in the quark sector in the considered model: the CP violating phase in U CKM , δ th CP = 0, π, while it follows from the data that δ CP ≃ 66.2 • .The CP violation arises as a higher order due to the corrections to the leading terms in the elements of M q .Since it is possible to describe correctly the quark mass hierarchies only if we have |ǫ| ≪ 1, the CPV phase δ th CP , which is generated by the higher order corrections in ǫ in the elements of M q , is generically much smaller than the measured value of δ CP , i.e., δ th CP ≪ 66.2 • , which is incompatible with the data.This conclusion is confirmed by our numerical analysis in Section 4.2.
Thus, we arrive at the conclusion that in the considered quark flavour model with A 4 modular symmetry, supplemented by the gCP symmetry, and one modulus τ having a VEV in the vicinity of the left cusp, τ = ω + ǫ, the description of the quark mass hierarchies in terms of ǫ, which has the generic structure 1 : |ǫ| : |ǫ| 2 and thus implies |ǫ| ≪ 1, is incompatible with the description of CP violation in the quark sector.On the basis of the general results presented in [120] we suppose that the problem of incompatibility between the "no-fine-tuned" description of the quark mass hierarchies in the vicinity of the left cusp τ = ω + ǫ with |ǫ| ≪ 1, and the description of CP violation in the quark sector, will be present in any quark flavour model based on the finite modular groups S 3 , A (′) , S (′) 4 and A (′) 5 and gCP symmetry.In the modular A 4 model studied by us we have considered phenomenologically also the possibility of having two different moduli in down-and up-type quark sectors, τ q , q = d, u, acquiring VEVs in the vicinity of the left cusp, τ q = ω + ǫ q , q = d, u, with ǫ d = ǫ u12 .The d-and u-quark mass hierarchies in this case are given respectively by 1 : Since ǫ d = ǫ u , we have also κ d = κ u and thus P * (κ u )P(κ d ) = diag(e i (κu−κ d ) , 1, e −i (κu−κ d ) ) = 1 in Eq. ( 61), where 1 is the unit matrix.The factor P * (κ u )P(κ d ) may, in principle, be a source of the requisite CP violation provided κ u − κ d is sufficiently large.However, performing a numerical analysis we find that the CPV phase δ CP thus generated is too small to be compatible with the measured value.Thus, the problem of correct description of the CP violation in the quark sector in the model considered by us persist also in this case as long as the gCP symmetry constraint is imposed.

Summary
We have investigated the possibility to describe the quark mass hierarchies as well as the CKM quark mixing matrix 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 = ω ≡ exp(i 2π/3) (the left cusp of the fundamental domain of the modular group), τ being the VEV of the modulus.In the considered A 4 model the three left-handed (LH) quark doublets Q = (Q 1 , Q 2 , Q 3 ) are assumed to furnish a triplet irreducible representation of A 4 and to carry weight 2, while the three righthanded (RH) up-quark and down-quark singlets are supposed to be the A 4 singlets (1, 1 ′′ , 1 ′ ) carrying weights (4,2,0), respectively.The model involves modular forms of level 3 and weights 6, 4 and 2, and contains eight constants, only two of which, g d and g u , can be a source of CP violation in addition to the VEV of the modulus, τ = ω + ǫ, (ǫ) * = ǫ, |ǫ| ≪ 1.
We find that in the case of real (CP-conserving) g d and g u and common τ (ǫ) for the downtype quark and up-type quark sectors, the down-type quark mass hierarchies can be reproduced without fine tuning with |ǫ| ∼ = 0.03, all other constants being of the same order in magnitude, and correspond approximately to 1 : |ǫ| : |ǫ| 2 .The description of the up-type quark mass hierarchies requires a ten times smaller value of |ǫ|.It can be achieved with the same |ǫ| ∼ = 0.03 allowing the constant g u to be larger in magnitude than the other constants of the model, |g u | ∼ O (10), and corresponds to 1 : |ǫ|/|g u | : |ǫ| 2 /|g u | 2 .In this setting the description of the CKM element |V cb | is problematic.We have shown that a much more severe problem is the correct description of the CP violation in the quark sector since it arises as a higher order correction in ǫ, which has to be sufficiently small in order to reproduce the quark mass hierarchies.This problem may be generic to modular invariant quark flavour models with one modulus, in which the gCP symmetry is imposed and the quark mass hierarchies are obtained in the vicinity of fixed point τ = ω.In the considered model the CP violation problem is not alleviated in the case of complex g u and real g d , while in the cases of i) real g u and complex g d , and ii) complex g d and g u , the rephasing invariant J CP has a value which is larger by a factor of ∼ 1.8 than the correct value due to a larger than observed value of |V cb |.We show also that an essentially correct description of the quark mass hierarchies and the CKM mixing matrix, including the CP violation in the quark sector, is possible in the vicinity of the left cusp with all constants being of ∼ O(1) in magnitude and complex g d and g u , if there are two different moduli τ d = ω + ǫ d and τ u = ω + ǫ u in the down-quark and up-quark sectors, with down-type and up-quark mass hierarchies given by 1 : A correct description is also possible in a modification of the considered model which involves level 3 modular forms of weights 8, 6 and 4. Thus, ten observables of quark sector can be reproduced quite well.On the other hand, there is no room to predict somethings in the quark masses and mixings.However, the model has predictive power for the flavor phenomena in SMEFT, such as in B meson decays as well as the flavor violation of the charged lepton decays [39].
The results of our study show that describing correctly without sever fine-tuning the quark mass hierarchies, the quark mixing and the CP violation in the quark sector is remarkably challenging within the modular invariance approach to the quark flavour problem.

Appendix A Tensor product of A 4 group
We take the generators of A 4 group for the triplet as follows: where ω = e i 2 3 π .In this basis, the multiplication rules are: where Further details can be found in the reviews [132][133][134].

C Modular forms at close to τ = ω
In what follows we present the behavior of modular forms in the vicinity of τ = ω.We perform Taylor expansion of modular forms Y 1 (τ ), Y 2 (τ ) and Y 3 (τ ) around τ = ω.We parametrize τ as: where |ǫ| ≪ 1.Then, the modular forms are expanded in terms of ǫ.In order to get up to 2nd order expansions of ǫ, we parametrize the modular forms as: These parameters are determined numerically in Taylor expansions as: The values are obtained by using the first and second derivarives.The constraint Y 2 2 + 2Y 1 Y 3 = 0 in Eq. ( 67) gives ǫ 2 = 2ǫ 1 and 4k 3 = 1 + 2k 2 , which are satisfied also numerically.
We expres also the higher weight triplet modular forms Y (k) i , k = 4, 6, in terms of ǫ 1 , k 2 and k 3 using ǫ 2 = 2ǫ 1 .For the weight 4 modular form we get: In a similar way we get the expressions for the two relevant weight 6 modular forms:

3. 1
Fermion mass hierarchy without fine-tuning close to τ = ω 1,2,3), are the components of the weight 2, 4 and 6 modular forms furnishing triplet representations of A 4 .Explicit expressions for the the modular forms of interest are presented in Appendix B.

3 ′
take simple forms at τ = ω as shown in Appendix B. Correspondingly, at τ = ω the quark mass matrices can written as:

4. 1
Quark mass hierarchies with common τ in M d and M u ǫ

Table 4 :
Values of the constant parameters obtained in the fit of the quark mass ratios and of CKM mixing angles.See text for details.

Table 5 :
Results of the fit of the quark mass ratios, CKM mixing angles and J CP factor.'Exp' denotes the respective values at the GUT scale, including 1σ errors, quoted in Eqs. (

Table 6 :
• .The magnitude of V cb is still larger than the observed one by a factor ∼ 2. 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 ǫ and g d .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 ǫ and g d .'Exp' denotes the values of the observables at the GUT scale, including 1σ errors, quoted in Eqs. (

Table 8 :
• .Correspondingly, the rephasing invariant |J CP | is smaller approximately by a factor of 20 than that at the GUT scale, Eq. (54).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 ǫ and g u .

Table 9 :
Results of the fit of the quark mass ratios, CKM mixing angles, δ CP and J CP with complex ǫ and g u .'Exp' denotes the values of the observables at the GUT scale, including 1σ error, quoted in Eqs. (

Table 10 :
Values of the constant parameters obtained in the fit of the quark mass ratios, CKM mixing angles, the CPV phase δ CP and the J CP factor with complex ǫ, g d and g u .
)).It is possible to improve the result for |V cb |, e.g., by modification of the model, as is discussed in the next Subsections.
arg [g d ]

Table 14 :
Values of the constant parameters obtained in the fit of the quark mass ratios, CKM mixing angles, the CPV phase δ CP and J CP with complex ǫ and f d , while real g d , g u and f u .

Table 15 :
Results of the fit of the quark mass ratios, CKM mixing angles, δ CP and J CP with complex ǫ and f d , while real g d , g u and f u .'Exp' denotes the values of the observables at the GUT scale, including 1σ error, quoted in Eqs. ( As a measure of goodness of fit, we use the sum of one-dimensional ∆χ 2 for eight observable quantitiesq 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 .(86)