Fermion masses and flavor mixings and strong CP problem

For all the success of the Standard Model (SM), it is on the verge of being surpassed. In this regard we argue, by showing a minimal flavor-structured model based on the non-Abelian discrete $SL_2(F_3)$ symmetry, that $U(1)$ mixed-gravitational anomaly cancellation could be of central importance in constraining the fermion contents of a new chiral gauge theory. Such anomaly-free condition together with the SM flavor structure demands a condition $k_1\,X_1/2=k_2\,X_2$ with $X_i$ being a charge of $U(1)_{X_i}$ and $k_i$ being an integer, both of which are flavor dependent. We show that axionic domain-wall condition $N_{\rm DW}$ with the anomaly free-condition depends on both $U(1)_X$ charged quark and lepton flavors; the seesaw scale congruent to the scale of Peccei-Quinn symmetry breakdown can be constrained through constraints coming from astrophysics and particle physics. Then the model extended by $SL_2(F_3)\times U(1)_X$ symmetry can well be flavor-structured in a unique way that $N_{\rm DW}=1$ with the $U(1)_X$ mixed-gravitational anomaly-free condition demands additional Majorana fermion and the flavor puzzles of SM are well delineated by new expansion parameters expressed in terms of $U(1)_X$ charges and $U(1)_X$-$[SU(3)_C]^2$ anomaly coefficients. And the model provides remarkable results on neutrino (hierarchical mass spectra and unmeasurable neutrinoless-double-beta decay rate together with the predictions on atmospheric mixing angle and leptonic Dirac CP phase favored by the recent long-baseline neutrino experiments), QCD axion, and flavored-axion.


I. INTRODUCTION
Symmetries play an important role in physics in general and in quantum field theory in particular. The standard model (SM) as a low-energy effective theory has been very predictive and well tested, due to the symmetries satisfied by the theory -Lorentz invariance plus the SU(3) C × SU(2) L × U(1) Y gauge symmetry in addition to the discrete space-time symmetries like P and CP. However, it leaves many open questions for theoretical and cosmological issues that have not been solved yet. These include the following: inclusion of gravity in gauge theory, instability of the Higgs potential, cosmological puzzles of matterantimatter asymmetry, dark matter, dark energy, and inflation, and flavor puzzle associated with the SM fermion mass hierarchies, their mixing patterns with the CP violating phases, and the strong CP problem. Moreover, there is no answer to the question: why there are three generations in the SM. The SM, therefore, cannot be the final answer. So it is widely believed that the SM should be extended a more fundamental underlying theory.
Neutrino mass and mixing is the first new physics beyond SM and adds impetus to solving the open questions in particle physics and cosmology. Moreover, a solution to the strong CP problem of QCD through Peccei-Quinn (PQ) [1] mechanism 1 may hint a new extension of gauge theory realized in gauge/gravity duality [2]. If nature is stringy, string theory, the only framework we have for a consistent theory with both quantum mechanics and gravity, should give insight into all such fundamental issues. String theory when compactified to four dimensions can generically contain G F = anomalous gauged U(1) plus non-Abelian finite symmetries. In this regard, in order to construct a model with the open questions one needs more types of gauge symmetry beside the SM gauge theory. One of simple approaches to a neat solution for those could be accommodated by a type of symmetry based on seesaw [4] and Froggatt-Nielsen (FN) [5] frameworks, since it is widely believed that non-renormalizable operators in the effective theory should come from a more fundamental underlying renormalizable theory by integrating out the heavy degrees of freedom. Therefore, one can anticipate that there may exist some correlations between low energy and high energy physics; e.g. the flavored-axion [2] can easily fit into a string theoretic framework, and appear cosmologically as a form of cold dark matter. Even gravity (which is well- 1 See, its related reports [3]. described by Einstein's general theory of relativity) lies outside the purview of the SM, once the gauged U(1)s are introduced in an extended theory, its mixed gravitational-anomaly should be free. And we assume that the heavy gauge bosons associated with the gauged U(1)s are decoupled, and thus in the model we consider the gauged U(1)s will be treated as the global U(1)s symmetries at low energy. As shown in Ref. [2], the FN mechanism formulated with global U(1) flavor symmetry could be promoted from the string-inspired gauged U(1) symmetry. Such flavored-PQ global symmetry U(1) acts as a bridge for the flavor physics and string theory [2,6]. Flavor modeling on the non-Abelian finite group has been recently singled out as a good candidate to depict the flavor mixing patterns, e.g., Ref. [2,7,8], since it is preferred by vacuum configuration for flavor structure. Hence, flavored-PQ symmetry modeling extended to G F could be a powerful tool to resolve the open questions for particle physics and cosmology.
In this paper we present, by showing an extended flavored-PQ model which extend to a compact symmetry 2 G F for new physics beyond SM, that the U(1) mixed-gravitational anomaly cancellation is of central importance in constraining the fermion contents of a new chiral gauge theory, and the flavor structure of G F is 3 strongly correlated with physical observables. So, finding the SM fermion mass spectra and their peculiar flavor mixing patterns in modeling is very important, since it is the first step toward establishing an effective low-energy Lagrangian of an extended theory. Unlike the A 4 symmetry containing one-and three-dimensional representations used in Refs. [2,8] the non-abelian discrete SL 2 (F 3 ) symmetry [7,9,10] contains two-dimensional representation in addition to one-and three-dimensional representations, in which the three dimensional representation is mainly responsible for the large leptonic mixing angles while the two dimensional representation is mainly to fit the quark masses and small mixing angles (especially the Cabbibo angle).
Moreover, depending on the quantum number of flavored U(1) X the group G F can give different structures of quark and lepton mass texture. Together with U(1) X symmetry, such SL 2 (F 3 ) could make the model compact providing an economic mass texture (see Eq. (21)) 2 Here the meaning of a 'compact' symmetry is a symmetry that provides only requisite parameters it is not hard to disprove; for example, see the quark and lepton mass textures in Eqs. (21) and (64) provided by the well-sewed supepotentials (18) and (45) under the SL 2 (F 3 ) × U (1) X symmetry. 3 Here we assume that, below the scale associated with U (1) Xi gauge bosons, the gauged U (1) Xi leaves behind low-energy symmetries which are QCD anomalous global U (1) Xi , see Eq. (1).
for the quark mass spectra and mixings, especially, the Cabbibo angle. On the other hand, if one uses A 4 symmetry in the same framework, it is expected that there are uncontrollable redundant parameters in the quark mass textures which should be fine-tuned by hand to realize the quark mass spectra and mixings. So taking G F = SL 2 (F 3 ) × U(1) X may have a good advantage to compactly describe the peculiar mixing patterns of quarks and leptons including their masses. Contrary to Ref. [11], the present model provides another possibility of flavor modeling in virtue of the quantum number of U(1) X , leading to completely different mass textures of quark and lepton. And in turn its results give an upper bound on QCD axion mass with different values of tan β in Eq. (32) and g Aee in Eq. (58), since axion to leptons and quarks couplings depend on structure of the quark and lepton sector. In this sense, if the astronomical constraint of star cooling [12] favored by the model in Ref. [11] is really responsible for the QCD axion, the present model will be ruled out.
And it is expected that the upcoming NA62 experiment expected to reach the sensitivity of Br(K + → π + +A i ) < 1.0×10 −12 [13] will soon rule out or favor the scenario in Ref. [11], while for the present model just gives an upper bound on the scale of PQ symmetry breakdown.
The rest of this paper is organized as follows. In Sec. II we set up a minimalistic SUSY model for quarks, leptons, and flavored-axions (and its combination QCD axion), which contains a G F = SL 2 (F 3 ) × U(1) X symmetry for a compact description of new physics beyond SM. In Sec. III the SL 2 (F 3 ) × U(1) X symmetry-invariant superpotential for vacuum configurations is constructed and its vacuum structure is analyzed. In Sec. IV we describe the Yukawa superpotential for quarks and flavored-axions and show that the SM quark masses and mixings could well be described by new expansion parameters defined under the U(1) X × [gravity] 2 anomaly-free condition. In turn, in order to show that the quark sector works well we perform a numerical simulation. And we show that the constraint coming from the particle physics on rare decay K + → π + + A i [11,14,15] on the U(1) X symmetry breaking scale is much stronger than that from the astroparticle physics on QCD axion cooling of stars. Along the line of quark sector, in Sec. V we show that the Yukawa superpotential for leptons and flavored-axions could well be flavor-structured, which gives testable predictions on the neutrino mass ordering, δ CP and θ 23 . And we show that the U(1) X symmetry breaking scale can also be constrained via the astrophysical constraint on flavored-axion cooling of stars, but its constraint is smaller than that from K + → π + + A i .
What we have done is summarized in Sec. VI, and we provide our conclusions. In appendix we consider possible next-to-leading order corrections.

II. THE MODEL SETUP
Assume we have a SM gauge theory based on the G SM = SU(3) C × SU(2) L × U(1) Y gauge group, and that the theory has in addition a G F = SL 2 (F 3 ) × U(1) X for a compact description of new physics beyond SM. Here the symmetry group of the double tetrahedron [7,9,10] 4 is mainly for the peculiar flavor mixing patterns. Here we assume that the non-Abelian finite group SL 2 (F 3 ) could be realized in field theories on orbifolds and it is a subgroup of a gauge symmetry that can be protected from quantum-gravitational effects. Since chiral fermions are certainly a main ingredient of the SM, the gauge-and gravitational-anomalies of the gauged U(1) X are 5 generically present, making the theory inconsistent, where Some requirements and constraints needed for the extended theory are: cancelled by the Green-Schwarz (GS) mechanism [18]. Hereafter the gauged U(1) will be treated as the global U(1) symmetry. Note that the global symmetry U(1) X we consider is the remnant of the U(1) X gauge symmetry broken by the GS mechanism. Hence, the spontaneous breaking of U(1) X realizes the existence of the Nambu-Goldstone (NG) modes (called axions) and provides an elegant solution to the strong CP problem.
(ii) The non-vanishing anomaly coefficient of the quark sector constrains the quantity N f j X ψ j in the gravitational instanton backgrounds (with N f generations well defined in the non-Abelian discrete group), and in turn whose 4 The details of the SL 2 (F 3 ) group are shown in Appendix A. 5 As shown in Refs. [2,6] with the well-defined Kahler potential based on type-IIB string theory, the author demonstrated that, while the two massive gauge bosons associated with the gauged U (1) Xi eat two degree of freedom, the other two axionic directions survive to low energies as the flavored-PQ axions, leaving behind low energy symmetries which are the QCD anomalous global U (1) Xi .
in the QCD instanton backgrounds, where the t a are the generators of the representation of SU(3) to which Dirac fermion ψ i belongs with X-charge. Thanks to the two QCD anomalous U(1) we have a relation [8] |δ indicating that the ratio of QCD anomaly coefficients is fixed by that of the decay constants f a i of the flavored-axions A i . Here f a i set the flavor symmetry breaking scales, and their ratios appear in expansion parameters of the quark and lepton mass spectra (see Eqs. (24) and (25)). As studied in Refs. [2,8], in the so-called flavored-PQ models the scale of PQ symmetry breakdown is congruent to the seesaw scale via Eq. (3), which could well be fixed 6 and/or constrained through the constraints and/or hints coming from astroparticle physics on axion cooling of stars with the fine-structure of axion to electron α Aee < 6 × 10 −27 [17], 4.1 × 10 −28 α Aee 3.7 × 10 −27 [17], and the coupling of axion to neutron g Ann < 8 × 10 −10 [21] etc. as well as the constraints coming from particle physics on rare flavor violating decay processes induced by the flavored-axions Br(K + → π + A i ) < 7.3×10 −11 [16] and Br(µ → e γ A i ) 1.1×10 −9 [22] etc..
where 7 k i (i = 1, 2) are nonzero integers, which is a conjectured relationship between two anomalous U(1)s. The U(1) X i is broken down to its discrete subgroup Z N i in the backgrounds of QCD instanton, and the quantities N i (nonzero integers) associated to the axionic domain-wall are given by Then, from Eqs. (4) and (5) one obtains |δ G 1 | = N 1 and |δ G 2 | = N 2 . Clearly, in the QCD instanton backgrounds if N 1 and N 2 are relative prime, there is no Z N DW discrete symmetry and therefore no domain wall problem 8 . Now, we will see that the domainwall condition with the U(1) X × [gravity] 2 anomaly free-condition is dependent on the U(1) X charged quark and lepton flavors. Eq. (2) can be expressed δ G 1 = α X 1 and δ G 2 = ω X 2 , where α and ω are some integer numbers. To make sure that no axionic domain-wall problem occurs, the following two conditions are required: (i) The numbers α and ω coming from U(1) X charged quark flavors should be 'relative prime'.
If the quantum numbers X 1 and X 2 are given by −2p and −q, respectively, from Eq. (4) one obtains k 1 p = k 2 q. So the number k i coming from the U(1) X × [gravity] 2 anomaly-free condition depends on both the U(1) X charged quark and lepton flavors.
Then, Eq. (5) is expressed as (ii) Hence, the number k 2 should be relative prime with |ω| and k 1 , as well as the number k 1 should not be a multiple of 2 and should be relative prime with |α|.
Consequently 9 , under the U(1) X × [gravity] 2 anomaly-free condition, to make sure that no axionic domain-wall problem occurs in a theory one could introduce additional U(1) X charged Majorana fermions and/or could assign well flavor-structured U(1) X quantum numbers to fermion contents that can protect k 1 to be a multiple of 2.
7 For −k 1 = k 2 = 1 in Ref. [2], additional Majorana fermions are introduced to satisfy the U (1) X ×[gravity] 2 anomaly free-condition. Note that, however, in general, k 2 /k 1 = integer. 8 Note that, in the present model, since the non-Abelian finite symmetry SL 2 (F 3 ) is broken completely by higher order effects, there is no residual symmetry; so, there is no room for a spontaneously broken discrete symmetry to lead to domain-wall problem. 9 Of course, one can consider the cases of the domain-wall number N DW > 1 if the PQ phase transition occurred during (or before) inflation.
As we shall see later, even though the integer k i depends on both the U(1) X charged quark and lepton flavors, it does not play the role of constraining the QCD axion decay constant

III. VACUUM CONFIGURATION
In this section, the SL 2 (F 3 ) × U(1) X symmetry-invariant superpotential for vacuum configurations is constructed and its vacuum structure is analyzed. First we present the representations of the field contents responsible for vacuum configuration. Apart from the usual two Higgs doublets H u,d responsible for electroweak symmetry breaking, which are invariant under SL 2 (F 3 ) (i.e. flavor singlets 1), the scalar sector is extended via two types of new scalar multiplets, flavon fields responsible for the spontaneous breaking of the flavor symmetry Φ T , Φ S , Θ,Θ, η, Ψ,Ψ that are G SM -singlets and driving fields Φ T 0 , Φ S 0 , η 0 , Θ 0 , Ψ 0 that are to break the flavor group along required vacuum expectation value (VEV) directions and to allow the flavons to get VEVs, which couple only to the flavons: we take the flavon fields Φ T , Φ S to be SL 2 (F 3 ) triplets, η to be a SL 2 (F 3 ) doublet (2 ′ representation), and Θ,Θ, Ψ,Ψ to be SL 2 (F 3 ) singlets (1 representation), respectively, that are G SM -singlets, and driving fields Φ T 0 , Φ S 0 to be SL 2 (F 3 ) triplets, η 0 to be a SL 2 (F 3 ) doublet (2 ′′ representation) and Θ 0 , Ψ 0 to be SL 2 (F 3 ) singlets. The flavored-PQ symmetry U(1) X is composed of two anomalous symmetries U(1) X 1 × U(1) X 2 generated by the charges X 1 ≡ −2p and X 2 ≡ −q. with W → e iξ W , whereas flavon and Higgs fields remain invariant under an U(1) R symmetry. As a consequence of the R symmetry, the other superpotential term κ α L α H u and the terms violating the lepton and baryon number symmetries are not allowed. In addition, dimension 6 supersymmetric operators like Q i Q j Q k L l (i, j, k must not all be the same) are not allowed either, and stabilizing proton. Here the global U(1) symmetry is a remnant of the broken U(1) gauge symmetry which can connect string theory with flavor physics [2,6] (see also [23]).
representations of the driving, flavon, and Higgs fields are summarized as in Table I. The superpotential depending on the driving fields, invariant Here U (1) X ≡ U (1) X 1 × U (1) X 2 symmetries which are generated by the charges X 1 = −2p and where higher dimensional operators are neglected, and µ i=T,Ψ,η are dimensional parameters and g T,η , g 1,...,8 are dimensionless coupling constants. Note here that the model implicitly has two U(1) X ≡ U(1) X 1 × U(1) X 2 symmetries which are generated by the charges X 1 = −2p and X 2 = −q. The fields Ψ andΨ charged by −q, q, respectively, are ensured by the U(1) X symmetry extended to a complex U(1) due to the holomorphy of the supepotential. So, the PQ scale µ Ψ = v Ψ vΨ/2 corresponds to the scale of spontaneous symmetry breaking of the U(1) X 2 symmetry. Since there is no fundamental distinction between the singlets Θ andΘ as indicated in Table I, we are free to defineΘ as the combination that couples to Φ S 0 Φ S in the superpotential W v [24]. At the leading order the usual superpotential term µH u H d is not allowed, while at the leading order the operator driven by Ψ 0 and at the next leading order the operators driven by Φ T 0 and η 0 are allowed which is to promote the effective µ-term Actually, in the model once the scale of breakdown of U(1) X symmetry is fixed by the constraints coming from astrophysics and particle physics, the other scales are automatically fixed by the flavored model structure. And it is clear that at the leading order the scalar supersymmetric W (Φ T Φ S ) terms are absent due to different U(1) X quantum numbers, which is crucial for relevant vacuum configuration in the model to produce compactly the present lepton and quark mixing angles. Now we consider how a desired vacuum configuration for compact description of quark and lepton mixings could be derived. In SUSY limit, the vacuum configuration is obtained by the F -terms of all fields being required to vanish. The vacuum alignments of the flavons Φ T and η are determined by From this set of five equations, we can obtain the supersymmetric vacua for Φ T and η where g T and g η are dimensionless couplings, and v T and v η are not determined. The minimization equations for the vacuum configuration of Φ S and (Θ,Θ) are given by And from Eq. (12), we can get the supersymmetric vacua for the fields Φ S , Θ,Θ where v Θ is undetermined. As can be seen in Eq. (13), the VEVs v Θ and v S are naturally of the same order of magnitude (here the dimensionless parameters g 3 and g 4 are the same order of magnitude). Finally, the minimization equation for the vacuum configuration of Ψ is given by where µ Ψ is the U(1) X breaking scale and g 7 is a dimensionless coupling. From the above equation we can get the supersymmetric vacua for the fields Ψ,Ψ Note that, once the scale of breakdown of U(1) X symmetry is fixed, all the other scales of VEVs are determined by the present flavor structured model. As can be seen in Eqs. (13) and (15), in the SUSY limit there exist flat directions along which the scalar fields Φ S , Θ and Ψ,Ψ do not feel the potential. The SUSY-breaking effect lifts up the flat directions and corrects the VEV of the driving fields, leading to soft SUSY-breaking mass terms (here we do not specify a SUSY breaking mechanism in this work).
The flavon field F charged under U(1) X is a scalar field which acquires a VEV and breaks spontaneously the flavored-PQ symmetry U(1) X . In order to extract NG modes resulting from spontaneous breaking of U(1) X symmetry, we set the decomposition of complex scalar fields as follows 10 in which we have set Φ S1 = Φ S2 = Φ S3 ≡ Φ Si in the supersymmetric limit, and v g = v 2 Ψ + v 2 Ψ . And the NG modes A 1 and A 2 are expressed as [2] with the angular fields φ S , φ θ and φ Ψ .

IV. QUARKS AND FLAVORED-AXIONS
Let us impose SL 2 (F 3 ) × U(1) X quantum numbers on SM quarks in a way that quark masses and mixings are well described as well as no axionic domain-wall problem occurs 11 .
, the quantum numbers of the SM quark fields are summarized as in Table II. The U(1) X invari- ance forbids renormalizable Yukawa couplings for the light families, but would allow them through effective nonrenormalizable couplings suppressed by (F /Λ) n with some positive integer n. Here Λ, above which there exists unknown physics, is the scale of flavor dynamics, and is associated with heavy states which are integrated out. The Yukawa superpotential where the hat Yukawa coupling denotes order of unity i.e., 1/ √ 10 |ŷ| √ 10, and Higher dimensional operators driven by Φ T and η fields, Hu Λ 2 with y c =ŷ c (Ψ/Λ) is neglected here, but will be included in numerical calculation.
Once the scalar fields Φ S , Θ,Θ, Ψ andΨ get VEVs, the flavored U(1) X symmetry is spontaneously broken 12 . And at energies below the electroweak scale, all quarks and leptons obtain masses. The relevant quark interaction terms with chiral fermions is given by where q u = (u, c, t), q d = (d, s, b), and g is the SU (2) 12 If the symmetry U (1) X is broken spontaneously, the massless modes A 1 of the scalar Φ S (and/or Θ) and A 2 of the scalar Ψ(Ψ) appear as phases. 13 Here we took η = vη √ 2 (+1, 0). 14 Even there seem to have vacuum corrections to the leading order picture in Eq. (21), e.g. where Here One of the most interesting features observed by experiments on the quarks is that the mass spectrum of the up-type quarks exhibits a much stronger hierarchical pattern to that of the down-type quarks, which may indicate that the CKM matrix [26] is mainly generated by the mixing matrix of the down-type quark sector. So the following new expansion parameters could be defined in a way that the diagonalizing matrices V d L and V u L satisfy the CKM matrix in the Wolfenstein parametrization where arg(ŷ i ) ≡ φ i and B = A ρ 2 + η 2 with the Wolfenstein parametrization 15 (λ, ρ, η, A) [25]. Note that the expansion parameters ∇ Ψ and ∇ Θ (∇ S ) associated with the U(1) X charged fields are defined by the relation Eq. (3) associated with the two QCD anomalous U(1), containing the model dependent parameter From the empirical down-type quark mass ratios calculated from the measured values , we can obtain roughly the down-type quark mixing angles in the standard parametrization [27] And their corresponding down-type quark masses are roughly given by 15 We take λ = 0.22509 +0.00091 Note that the parametrization of Eq. (23) is very crucial to reproduce the d-and s-quark mass and the mixing angle θ d 12 . From the mass ratio of t-and b-quark (m b /m t ) PDG 2.41 +0.03 −0.03 × 10 −2 in PDG [26] the value of tan β ≡ v u /v d can be obtained in a good approximation: The top Yukawa couplingŷ t can be directly obtained from the top quark mass m t = |ŷ t |v u = 173.1 ± 0.6 GeV [26]. From the hierarchical mass ration between u-and c-quark and its corresponding mixing angle In turn, the expansion parameter ∇ η is defined by using (m c /m t ) PDG 7.39 +0.20 −0.20 × 10 −3 : As designed, with the fields redefinition the CKM matrix with J quark Hence it is very crucial for obtaining the right values of the new expansion parameters to reproduce the empirical results of the CKM mixing angles and quark masses. In addition, such right values are needed to reproduce the empirical results of the charged leptons and the light active neutrino masses in our model. In the following subsequent section we will perform a numerical simulation.

A. Numerical analysis for Quark sector
We perform a numerical simulation 16 using the linear algebra tools of Ref. [29]. With the inputs tan β = 7.40 , and |ŷ d | = 0.9200 (φ d = 6.2100 rad), |ŷ d | = 3.1400, |ŷ s | = 0.3300 (φ s = 2.9300 rad), we obtain the mixing angles and Dirac CP phase θ q Below the scale of spontaneous SU(2) L × U(1) Y gauge symmetry breaking, the running mass includes corrections from QCD and QED loops [26]. In order to explain the experimental data on quark and lepton masses 17 we have used, it is meaningful to use the masses at a common momentum scale µ which is heavier than the QCD scale of about 1 GeV. Hence, in the MS scheme for the light quark (u-, d-, and s-quark) the renomalization scale has been chosen to be a common scale µ ≈ 2 GeV and their masses are current quark masses at µ ≈ 2 GeV, and for heavy quarks (b-and c-quark) the renormalization scale equal to the quark mass are chosen to bem Q (µ) at µ =m Q . For top quark (t-quark), the t-quark mass at scales below the pole mass is unphysical since the t-quark decouples at its scale, hence its mass is more directly determined by experiments, see Ref. [26], leading to the value we have used. 16 Here, in numerical calculation, we have only considered the mass matrices in Eq. (21) since it is expected that the corrections to the VEVs due to dimensional operators contributing to Eq. (7) could be small enough below a few percent level, see Appendix B. 17 For charged leptons (e, µ, τ ) we have used the experimental data [26] in this work since the difference between pole mass and running mass are less significant.
(i) Below the chiral symmetry breaking scale, the axion-hadron interactions are meaningful for the axion production rate in the core of a star where the temperature is not as high as 1 GeV, which is given by where the QCD axion decay constant is given by and ψ N is the nucleon doublet (p, n) T (here p and n correspond to the proton field and neutron field, respectively). The couplings of the axion to the nucleon can be rewritten whereX q = δ G 2 X 1q + δ G 1 X 2q with q = u, d, s and X 1u = 8, X 1d = 8, X 1s = 0, X 2u = 3, X 2d = 1, X 2s = −1. From Eqs. (34)(35) the QCD axion coupling to the neutron can be obtained as where the neutron mass m n = 939.6 MeV, and the axion-neutron coupling, X n , related to axial-vector current matrix elements by Goldberger-Treiman relations [26] is obtained as where η = (1 + z + ω) −1 with z = m u /m d and ω = m u /m s ≪ z, and the ∆q are given by the axial vector current matrix element ∆q S µ = p|qγ µ γ 5 q|p . Now, for numerical estimations on Eq. (36) we adopt the central values of ∆u = 0.84 ± 0.02, ∆d = −0.43 ± 0.02 and ∆s = −0.09 ± 0.02, and take the Weinberg value for 0.38 < z < 0.58 [26] and ω = 0.315 z.
Then, the value of the axion-neutron coupling lies in ranges 0.007 X n 0.111. There is a hint for extra cooling from the neutron star in the supernova remnant "Cassiopeia A" by axion neutron bremsstrahlung, requiring a coupling to the neutron of size g Ann = (3.8 ± 3) × 10 −10 [31], which is translated into 9.94 × 10 6 F A /GeV 1.31 × 10 9 . However, since the cooling of the superfluid core in the neutron star can also be explained by neutrino emission in pair formation in a multicomponent superfluid state 3 P 2 (m j = 0, ±1, ±2) [30], one may not take it seriously. The range quoted is compatible with the state-of-the-art upper limit on the coupling from neutron star cooling g Ann < 8 × 10 −10 [21], whose upper bound is interpreted as the lower bound of the QCD axion decay constant: (ii) Since a direct interaction of the SM gauge singlet flavon fields charged under U(1) X with the SM quarks charged under U(1) X can arise through Yukawa interaction, the flavoredaxion interactions with the flavor violating coupling to the s-and d-quark is given by where 18 V d † L ≈ V CKM , f a 1 = |X 1 |v F , and f a 2 = |X 2 |v g are used. Then the decay width of K + → π + + A i is given by [11,14,15] where m K ± = 493.677 ± 0.013 MeV, m π ± = 139.57018(35) MeV [26], and From the present experimental upper bound Br(K + → π + A i ) < 7.3 × 10 −11 [16] with Br(K + → π + νν) = 1.73 +1.15 −1.05 × 10 −10 [33], we obtain the lower limits of flavored-axion decay constants and their corresponding QCD axion decay constant 18 Actually, in the standard parametrization the mixing elements of V d R are given by θ R 23 ≃ Aλ 2 ∇ η |ŷ s /ŷ b |, θ R 13 ≃ √ 2 Bλ 3 ∇ η ∇ S , and θ R 12 ≃ √ 2|ŷ d /ŷ s | 2 cos φd ∇ 2 S . Its effect to the flavor violating coupling to the sand d-quark is negligible: where is used. Note that the lower bounds of flavored-axion decay constants f a i are dependent on the values of k i , while the QCD axion decay constant F A does depend on the properties (2α and ω in Eq. (6)) from the QCD instanton background instead of the k i . Clearly, from Eqs. (38) and (42) the most stringent constraint on the QCD axion decay constant comes from the present experimental upper bound In the near future the NA62 experiment will be expected to reach the sensitivity of Br(K + → π + + A i ) < 1.0 × 10 −12 [13], which is interpreted as the flavored-axion decay constant and its corresponding QCD axion decay constant

V. LEPTONS AND FLAVORED-AXIONS
Next, we assign the left-handed charged lepton SU(2) L doublets denoted as L e , L µ , L τ to the (1, −p − Q y ν 1 ), (1 ′ , −p − Q y ν 1 ), and (1 ′′ , −p − Q y ν 1 ), respectively, while the right-handed charged leptons denoted as e c , µ c and τ c , the electron flavor to the (1, p + Q y ν 1 + 6q), the muon flavor to the (1 ′′ , p + Q y ν 1 − 3q), and the tau flavor to the (1 ′ , p + Q y ν 1 − q). And we assign the right-handed neutrinos SU(2) L singlets denoted as N c to the (3, p). Note that Q y ν 1 = Q y ν 2 = Q y ν 3 is assigned to give a tribimaximal (TBM)-like mixing pattern. In addition, additional Majorana fermions are introduced to have no axionic domain-wall problem, which link low energy neutrino oscillations to astronomical-scale baseline neutrino oscillations.
As mentioned before, with the conditions (4) Table III. The lepton Yukawa superpotential, similar Remark that, as in the SM quark fields since the U(1) X quantum numbers are arranged to lepton fields as in Table III with the conditions (4) and (D4) satisfied, it is expected that the SM gauge singlet flavon fields derive higher-dimensional operators, which are eventually visualized into the Yukawa couplings of leptons as a function of flavon fields Ψ(Ψ).
For pseudo-Dirac neutrino as the active neutrino to be realized in a way that the neutrino oscillations at low energies could have a direct connection to new neutrino oscillations available on high-energy neutrinos [2], two requirements are needed since the quantum numbers L e ,µ ,τ (or equivalently Q y ν i ) are not uniquely determined: (i) the quantum numbers Q y ν i and Q y s i should have opposite sign due to Q y ss 1 = 2(Q y s 1 − Q y ν 1 ) and Q y ss 2 = Q y ss 3 = Q y s 2 + Q y s 3 − 2Q y ν 1 , (ii) especially, the quantum numbers Q y s 2 and Q y s 3 should have the same sign for normal neutrino mass ordering, and (iii) As we shall see later, it could make a connection between the neutrino oscillation at low energies and new oscillations available on high-energy neutrinos through astronomical-scale baseline. Then, the quantum numbers Q y s i can be uniquely determined by taking into account both the U(1) X × [gravity] 2 anomaly-free condition in Eq. (D5) and the hat Yukawa coupling of order unity, 1/ √ 10 |ŷ s i | √ 10, we obtain (i) |Q y s 3 | ≫ |Q y s 1 | ≥ |Q y s 2 | for inverted mass ordering (IO), and (ii) |Q y s 1 | ≫ |Q y s 2 | ≥ |Q y s 3 | for normal mass ordering (NO). In such case, considering the observed neutrino mass hierarchy ∆m 2 sol ≡ m 2 ν 2 −m 2 ν 1 ≃ 7.50×10 −5 eV 2 and ∆m 2 atm ≃ 2.52 × 10 −3 eV 2 where ∆m 2 atm ≡ m 2 ν 3 − m 2 ν 1 for NO; |m 2 ν 2 − m 2 ν 3 | for IO, we have the followings: For the case-I with E/N = 3.83 in Eq. (D3) the Yukawa couplings of charged-leptons are represented with Q yτ = −q, Q yµ = 3q, Q ye = −6q as the U(1) X quantum numbers of Yukawa couplings of pseudo-Dirac neutrinos are given for k 1 = +k 2 = 1 in Eq. (D6) as Here for NO the quantum numbers Q y s 2 and Q y s 3 should have the same sign, while for IO Q y s 1 and Q y s 2 should have the opposite sign.
For the case-III with E/N = 1.83 in Eq. (D3) the Yukawa couplings of charged-leptons are represented with Q yτ = −q, Q yµ = −3q, Q ye = 6q as the U(1) X quantum numbers of Yukawa couplings of pseudo-Dirac neutrinos are given for k 1 = +k 2 = 1 in Eq. (D6) as The hat Yukawa couplingsŷ e,µ,τ are fixed by the numerical values in Eq.
for IO (Q y s 1 = ∓17q, Q y s 2 = ±17q, Q y s 3 = 28q): and within the 3σ constraints of the low energy neutrino oscillations [35] by using the value However, there still remain two physical parameters undetermined, the scale of U(1) X symmetry breakdown and Q y ν i , which correspond to the physical observables, the QCD axion mass and mass splittings ∆m 2 k for new neutrino oscillations through astronomical-scale baseline. Note that the neutrino mixing angles can be determined through the lepton Yukawa superpotential in Eq. (45) structured by the SL 2 (F 3 ) symmetry together with the desired VEV directions in Eqs. (11,13,15), as will be seen later. As seen in superpotential (45) since the SM charged-lepton fields (which are nontrivially X-charged Dirac fermions) have U(1) EM charges, the axion A 2 coupling to electrons are added to the Lagrangian through a chiral rotation. And the axion A 2 couples directly to electrons, thereby the axion can be emitted by Compton scattering, atomic axio-recombination and axio-deexcitation, and axio-bremsstrahlung in electron-ion or electron-electron collisions [38]. The axion A 2 coupling to electron in the model reads where m e = 0.511 MeV. Such weakly coupled flavored-axion A 2 has a wealth of interesting phenomenological implications in the context of astrophysics 19 , like the formation of a cosmic diffuse background of axions from core collapse supernova explosions [36] or neutron star 19 From the cooling of white-dwarfs with the fine-structure of axion to electron, which is recently improved 4.1×10 −28 α Aee 3.7×10 −27 in Ref. [17], implying axion decay constant f a2 = (1.42−4.27)×10 10 GeV and its corresponding QCD axion decay constant F A = (0.34 − 1.01) × 10 10 GeV. See also the most recent analysis α Aee = 2.04 +0.81 −0.77 × 10 −27 at 1σ [19] leading to f a2 = 1.92 +0.52 −0.29 × 10 10 GeV which is interpreted as F A = 4.52 +1. 22 −0.69 × 10 9 GeV. These hints including Ref. [20] seem incompatible with the bound in Eq. (42) from the decay process K + → π + A i . However, if one relinquishes N DW = 1 by considering N DW > 1 in the case that the PQ phase transition happened during (or before) inflation, one can easily construct a model for accommodating the debating constraints under the present flavored-PQ scenario.
Here the fine-structure constant, α Aee = g 2 Aee /4π, is related to the axion-electron coupling constant g Aee . Then, the astrophysical lower bound of the PQ breaking scale f a 2 and its corresponding QCD axion decay constant F A is derived from the above mentioned upper limits f a2 > (3.98 × 10 8 − 1.23 × 10 10 ) GeV ⇔ F A > (9.38 × 10 7 − 2.90 × 10 9 ) GeV . (59) Since this limit for the QCD axion decay constant is much lower than the bound from its axion photon coupling expressed in terms of the axion mass, pion mass, pion decay constant, z and w, The axion coupling to photon g aγγ divided by the axion mass m a is dependent on E/N. Left plot in Fig. 1 shows the E/N dependence of (g aγγ /m a ) 2 so that the experimental limit is independent of the axion mass m a [8]: the values of (g aγγ /m a ) 2 of our model are located lower than that of the experimentally excluded bound (g aγγ /m a ) 2 ≤ 1.44×10 −19 GeV −2 eV −2 from ADMX [43]. for E/N = 11/6 (case-III), respectively.

B. Neutrinos
Even in the present model the quantum numbers Q y ν i (or equivalently Q Le,µ,τ ) are not uniquely determined through the model setup, together with the conditions above Eq. (46) their quantum numbers can be assigned by their corresponding physical observables which are the pseudo-Dirac mass splittings ∆m 2 k responsible for new oscillations available on highenergy neutrinos through astronomical-scale baseline [2,45,46].
fermions is given by And in the above Lagrangian (64) the charged-lepton and heavy Majorana neutrino mass terms read whereκ For NO, the Dirac and Majorana mass terms read where y 2 ≡ŷ ν 2 /ŷ ν 1 and y 3 ≡ŷ ν 3 /ŷ ν 1 . For IO, the Dirac and Majorana mass terms read Reminding that the hat Yukawa couplings in Eqs. (66-73) are all of order unity and complex numbers. From Eq. (64), by redefining the light neutrino field ν L as P ν ν L and transforming which will be used in numerical analysis, later.
After seesawing [2] due to the scale in Eq. (60) (or see Eqs. (43) and (59)) much larger than the electroweak scale, in a basis where charged lepton and heavy neutrino masses are real and diagonal, we obtain an effective light neutrino mass matrix in the basis (ν L , S c R ) Under the given quantum numbers the active neutrinos appear as pseudo-Dirac neutrinos.
And the pseudo-Dirac mass splittings in k-th pair ∆m 2 k ≡ m 2 ν k − m 2 S k are expressed as where the leptonic PMNS matrix U PMNS [26] is given by Eq. (C1), and In the limit of y 2 , y 3 → 1 the above mass matrix reflects exact TBM mixing [47] and its corresponding mass eigenvalues |δ ν 1 | = 3m 0 |F |, |δ ν 2 | = 3m 0 , |δ ν 3 | = 3m 0 |G|. Since in general it is expected deviations of y 2,3 from unity, Eq. (77) directly indicates that there could be deviations from the exact TBM, leading to a possibility to search for CP violation in neutrino oscillation experiments. In addition, due to the small value of θ 13 it is expected To obtain the pseudo-Dirac mass splittings, taking the scale of heavy neutrino M =ŷ Θ f a 1 /(|X 1 | 2(1 + κ 2 )) in Eq. (67) from the QCD axion decay constant in Eq. (60) and using the best-fit values of the low energy neutrino oscillations [35], we can obtain the pseudo-Dirac mass splittings in a good approximation: for NO withŷ s 1 = 1;

C. Numerical analysis for neutrino mixing parameters
In order to show model predictions on the leptonic Dirac CP phase δ CP incident to the atmospheric mixing angle θ 23 , we perform a numerical simulation by using the linear algebra tools of Ref. [29] with the 3σ constraints of the low energy neutrino oscillations [35].
Here both the lightest active neutrino mass and the Majorana CP phases contributing to the effective active neutrino masses are negligibly small enough in the model. Therefore, we can have reasonable model predictions on the Dirac CP phase δ CP incident to behavior of the large uncertainty on θ 23 .
The recent analysis based on global fits [35,53,54] of the neutrino oscillations enters into a new phase of precise determination of mixing angles and mass squared differences: we take the global fits at 3σ [35], shown in Table IV     And we have showed that the constraint on the U(1) X symmetry breaking scale coming from the particle physics on the rare decay K + → π + +A i is much stronger than that from the astroparticle physics on QCD axion and flavored-axion cooling of stars. So, in order to fix the scale of PQ phase transition we take a testable QCD axion decay constant, F A = 1.29 × 10 11 GeV, from the current bound and the future expected sensitivity on Br(K + → π + + A i ), The SL 2 (F 3 ) is the double covering of the tetrahedral group A 4 [7,9,10]. It contains 24 elements and has three kinds of representations: one triplet 3 and three singlets 1, where we have used the matrices The following multiplication rules between the various representations are calculated in Ref. [10], where α i indicate the elements of the first representation of the product and β i indicate those of the second representation. Moreover a, b = 0, ±1 and we denote 1 0 ≡ 1, 1 1 ≡ 1 ′ , 1 −1 ≡ 1 ′′ and similarly for the doublet representations. On the right-hand side the sum a + b is modulo 3.
The multiplication rule with the 3-dimensional representations is

Appendix B: Higher order corrections
We consider possible next-to-leading order corrections. Higher-dimensional operators invariant under SL 2 (F 3 ) × U(1) X symmetry, suppressed by additional powers of the cutoff scale Λ, could be added to the leading order terms in the superpotential. Then the mass and mixing matrices for fermions can be corrected by both a shift of the vacuum configuration and nontrivial next-to-leading operators contributing to the Yukawa superpotential.
For example, we show that next leading corrections to the renormalizable Majorana neutrino sector can well be under control. In addition to the leading order Yukawa superpotential W ℓν , we should also consider those higher dimensional operators that could be induced by the flavon fields Φ T and η which are not charged under the U(1) X . At the next leading order in the Majorana neutrino sector those operators triggered by the field Φ T are written as (N c N c ΘΦ T ) 1 /Λ and (N c N c Φ S Φ T ) 1 /Λ. Here the first term, after symmetry breaking, is absorbed into the leading order terms in the renormalizable superpotential and the corresponding Yukawa couplings are redefined. On the other hand, the second term could be non-trivial and it can be clearly expressed as Indeed at order 1/Λ, after symmetry breaking, there is a new structure contributing to M R , whose contribution is written as whereκ with i = 1, 2, 3, s, a. Even though these corrections to the leading order picture seem to nontrivial, these can be kept small, below few percent level due to ∇ T in Eq. (33) by keeping |ŷ R | |ŷ R i |, i.e.κ κ i with Eq. (67). Then, eventually, after seesawing in Eq. (77) the active neutrino mixing matrix at leading order could not be crucially changed.
Next, considering higher dimensional operators induced by Φ T , Φ S , Θ, Ψ, η invariant under SL 2 (F 3 ) × U(1) X in the driving superpotential W v , which are suppressed by additional powers of the cut-off scale Λ, they can lead to small deviations from the leading order vacuum configurations. The next leading order superpotential δW v , which is linear in the driving fields and invariant under SL 2 (F 3 ) × U(1) X × U(1) R , is given by By keeping only the first order in the expansion, one can obtain the minimization equations. The corrections to the VEVs, Eqs. (11,13,15), are of relative order 1/Λ and affect the flavon fields Φ S , Φ T , Θ,Θ, η and Ψ, and the vacuum configuration can be modified with relations among the dimensionless parameters (a 1 ...a 7 , b 1 ...b 12 , c 1 , c 2 , d 1 , f 1 ...f 4 ).
This vanishing anomaly, however, does not restrict Q y ν i (or equivalently Q y ss i ), whose quantum numbers can be constrained by the new neutrino oscillations of astronomical-scale baseline, which will be shown later. With the given above U(1) X quantum numbers, such U(1) X × [gravity] 2 anomaly is free for 1 +Q y s 2 +Q y s 3 − 13; case-Ĩ Q y s 1 +Q y s 2 +Q y s 3 − 11; case-IĨ Q y s 1 +Q y s 2 +Q y s 3 − 7; case-III whereQ y s i = Q y s 1 /X 2 . We take k 2 = ±15 for the U(1) X i charges to be smallest making no axionic domain-wall problem. Hence, forQ y s 1 +Q y s 2 +Q y s 3 = 28 (−2) for the case-I; 26 (−4) for the case-II; 22 (−8) for the case-III, the values of k i are rescaled as with p = k 2 and q = k 1 by k 1 p = k 2 q = k 1 k 2 . In the present model the color anomaly coefficients are given by δ G 1 = 2X 1 and δ G 2 = 3X 2 . Then, the axionic domain-wall condition in Eq. (5) is rewritten as ensuring that no axionic domain-wall problem occurs.