Compact model for Quarks and Leptons via flavored-Axions

We show how the scales responsible for Peccei-Quinn (PQ), seesaw, and Froggatt and Nielsen (FN) mechanisms can be fixed, by constructing a compact model for resolving rather recent, but fast-growing issues in astro-particle physics, including quark and leptonic mixings and CP violations, high-energy neutrinos, QCD axion, and axion cooling of stars. The model is motivated by the flavored PQ symmetry for unifying the flavor physics and string theory. The QCD axion decay constant congruent to the seesaw scale, through its connection to the astro-particle constraints of both the stellar evolution induced by the flavored-axion bremsstrahlung off electrons $e+Ze\rightarrow Ze+e+A_i$ and the rare flavor-changing decay process induced by the flavored-axion $K^+\rightarrow\pi^++A_i$, is shown to be fixed at $F_A=3.56^{+0.84}_{-0.84}\times10^{10}$ GeV (consequently, the QCD axion mass $m_a=1.54^{+0.48}_{-0.29}\times10^{-4}$ eV, Compton wavelength of its oscillation $\lambda_a=8.04^{+1.90}_{-1.90}\,{\rm mm}$, and axion to neutron coupling $g_{Ann}=2.14^{+0.66}_{-0.41}\times10^{-12}$, etc.). Subsequently, the scale associated to FN mechanism is dynamically fixed through its connection to the standard model fermion masses and mixings, $\Lambda=2.04^{\,+0.48}_{\,-0.48}\times10^{11}\,{\rm GeV}$, and such fundamental scale might give a hint where some string moduli are stabilized in type-IIB string vacua. In the near future, the NA62 experiment expected to reach the sensitivity of ${\rm Br}(K^+\rightarrow\pi^++A_i)<1.0\times10^{-12}$ will probe the flavored-axions or exclude the model, if the astrophysical constraint of star cooling is really responsible for the flavored-axion.


I. INTRODUCTION
Many of the outstanding mysteries of astrophysics may be hidden from our sight at all wavelengths of the electromagnetic spectrum because of absorption by matter and radiation between us and the source. So, data from a variety of observational windows, especially, through direct observations with neutrinos and axions, may be crucial. Hence, axions and neutrinos in astro-particle physics and cosmology could be powerful sources for a new extension of SM particle physics [1,2], in that they stand out as their convincing physics and the variety of experimental probes. Fortunately, most recent analyses on the knowledges of neutrino (low-energy neutrino oscillations [9] and high-energy neutrino [10]) and axion (QCD axion [11,12] and axion-like-particle(ALP) [13,14]) enter into a new phase of model construction for quarks and leptons. In light of finding the fundamental scales, interestingly enough, there are two astro-particle constraints coming from the star cooling induced by the flavored-axion bremsstrahlung off electrons e + Ze → Ze + e + A i [13] and the rare flavor-chanting decay process induced by the flavored-axion K + → π + + A i [15], respectlely, 6.7 × 10 −29 α Aee 5.6 × 10 −27 at 3σ , where α Aee is the fine-structure of axion to electron.
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 for the aforementioned fundamental issues one needs more types of gauge symmetry beside the SM gauge theory. One of simple approaches to a neat solution for those issues could be accommodated by introducing a type of symmetry based on seesaw [3] and Froggatt-Nielsen (FN) [8] 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 heavy degrees of freedom. If so, one can anticipate that there may exist some correlations between low energy and high energy physics.
As shown in Ref. [1], the FN mechanism formulated with global U(1) flavor symmetry could be promoted from the string-inspired gauged U(1) symmetry. Such flavored-PQ symmetry U(1) acts as a bridge for the flavor physics and string theory [1,16]. Even gravity (which is well-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 gravitationalanomaly should be free. 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. [1,17,18], since it is preferred by vacuum configuration and string theory for flavor physics. In the socalled flavored PQ symmetry model where the SM fermion fields as well as SM gauge singlet fields carry PQ charges but electroweak Higgs doublet fields do not [1,7,17], the flavoredaxions (one linear combination QCD axion and its orthogonal ALP) couple to hadrons, photons and leptons, and its PQ symmetry breaking scale is congruent to the seesaw scale.
Hence, flavored-PQ symmetry modeling extended to G F could be a powerful tool to resolve the open questions for astro-particle physics and cosmology.
Since astro-particle physics observations have increasingly placed tight constraints on parameters for flavored-axions, it is in time for a compact model for quarks and leptons to mount an interesting challenge on fixing the fundamental scales such as the scales of seesaw, PQ, and FN mechanisms. The purpose of the present paper is to construct a flavored-PQ model along the lines of the challenge, which naturally extends to a compact symmetry G F for new physics beyond SM. Remark that [7]  The rest of this paper is organized as follows. In Sec. II we construct a compact model based on SL 2 (F 3 ) × U(1) X in a supersymmetric framework. Subsequently, we show that the model works well with the SM fermion mass spectra and their peculiar flavor mixing patterns. In Sec. III we show that the QCD decay constant (congruent to the seesaw scale) is well fixed through constraints coming from astro-particle physics, and in turn the FN scale is dynamically determined via its connection to the SM fermion masses and mixings.
And we show several properties of the flavored-axions. What we have done is summarized in Sec. V, and we provide our conclusions. In Appendix we consider possible next-to-leading order corrections to the vacuum expectation value (VEV). 3 Recently, studies on flavored-axion are gradually becoming amplified [19].
As mentioned in the Introduction, finding the scales responsible for seesaw [3], PQ [5], and FN [8] mechanisms, as a theoretical guideline to the aforementioned fundamental issues, could be one of big challenges. To resolve such interesting challenge, we construct a neat and economical model based on the flavored-PQ symmetry U(1) X embedded in the non-Abelian finite group, which may provide a hint and/or framework to accommodate all the fundamental issues on astro-particle physics and cosmology. Along this line, the G F quantum number of the field contents is assigned in a way that (a) the G F requires a desired vacuum configuration to compactly describe the quark and lepton masses and mixings, (b) the G F fits in well with the astro-particle constraints induced by the flavored-axions, and (c) the U(1) X mixed-gravitational anomaly-free condition with the SM flavor structure demands additional Majorana fermions as well as no axionic domain-wall problem.
Similar to Ref. [7] it is followed by 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 we assume that the symmetry group of the double tetrahedron SL 2 (F 3 ) [18,20,21] 4 is realized in field theories on orbifolds and 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 generically present, making the theory inconsistent. Hence some requirements needed for the extended theory are: anomalies should be cancelled by the Green-Schwarz (GS) mechanism [22] (see Ref. [1]).
(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 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 [17] |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. (38), (39), and (40)).
where 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 (iv) The U(1) X invariance forbids renormalizable Yukawa couplings for the light families, but would allow them through effective non-renormalizable couplings suppressed by (F /Λ) n with a flavon field F and positive integer n. Then the SM gauge singlet flavon field F is activated to dimension-four(three) operators with different orders [1,8,17,23] where OP 4(3) is a dimension-4(3) operator, and all the coefficients c n and c ′ 1 are complex numbers with absolute value of order unity. Here the flavon field F is a scalar field which acquires a VEV and breaks spontaneously the flavored-PQ symmetry U(1) X .
And the scale Λ, above which there exists unknown physics, is the scale of flavor dynamics, and is associated with heavy states which are integrated out. Such fundamental scale may come from where some string moduli are stabilized.
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. Here the global U(1) symmetry 5 including U(1) R is remnants of the broken U(1) gauge symmetries which can connect string theory with flavor physics [1,16]. Hence, the spontaneous breaking of U(1) X realizes the existence of the Nambu-Goldstone (NG) mode (called axion) and provides an elegant solution to the strong CP problem.

A. Vacuum configuration
In this section we will review the fields contents responsible for the vacuum configuration since the scalar potential of the model is the same as in Ref. [7]. Apart from the usual two Higgs doublets H u,d responsible for electroweak symmetry breaking, which transform as (1, 0) under SL 2 (F 3 ) × U(1) X symmetry, the scalar sector is extended via two types of new scalar multiplets that are G SM -singlets: flavon fields Φ T , Φ S , Θ,Θ, η, Ψ,Ψ responsible for the spontaneous breaking of the flavor symmetry, and driving fields Φ T 0 , Φ S 0 , η 0 , Θ 0 , Ψ 0 that are to break the flavor group along required VEV directions and to allow the flavons to In addition, the superpotential W in the theory is uniquely determined by the U(1) R symmetry, containing the usual R-parity as a subgroup: {matter f ields → e iξ/2 matter f ields} and {driving f ields → e iξ driving f ields}, with W → e iξ W , whereas flavon and Higgs fields 5 It is likely that an exact continuous global symmetry is violated by quantum gravitational effects [24].
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.
The superpotential dependent on the driving fields having U(1) R charge 2, which is , is given at leading order by [7] W 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. The fields Ψ andΨ charged by X 2 , 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 [25]. 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.
The vacuum configuration of the flavon fields, Φ T , Φ S , η,Θ, Ψ, andΨ, is obtained from the minimization conditions of the F -term scalar potential 6 . At the leading order the global minima of the potential are given [7] by where v Ψ = vΨ and κ = v S /v Θ in SUSY limit.

B. Quarks, Leptons, and flavored-Axions
Under SL 2 (F 3 ) × U(1) X with U(1) R = +1, the SM quark matter fields are sewed by the five (among seven) in-equivalent representations 1, 1 ′ , 1 ′′ , 2 ′ and 3 of SL 2 (F 3 ), and assigned as in Table II and III. Because of the chiral structure of weak interactions, bare fermion masses are not allowed in the SM. Fermion masses arise through Yukawa interactions 7 . Through 6 The vacuum configuration of the driving fields is not relevant in this work. And we will not consider seriously the corrections to the VEVs due to higher dimensional operators contributing to Eq. (7) since their effects are expected to be only few percents level, see Appendix B. 7 Since the right-handed neutrinos N c (S c ) having a mass scale much above (below) the weak interaction scale are complete singlets of the SM gauge symmetry, they can possess bare SM invariant mass terms. However, the flavored-PQ symmetry U (1) X guarantees the absence of bare mass terms M N c N c and M S S c S c .
As discussed in Refs. [1,7,17] , the quantum numbers of the lepton fields are summarized as in Table III. The lepton Yukawa superpotential, similar to the quark sector, invariant under G SM ×G F ×U(1) R reads at leading order In the above charged-lepton Yukawa superpotential, W ℓ , it has three independent Yukawa terms at the leading: apart from the Yukawa couplings, each term involves flavon field y β which appears in the superpotentials (13) and (14).
In the neutrino Yukawa superpotential 8 , W ν , two right-handed Majorana neutrinos S c and N c are introduced to make light neutrinos pseudo-Dirac particles and to realize an exact tri-bimaximal mixing (TBM) [26] at leading order, respectively. Such additional Majorana fermion S c plays a role of making no axionic domain-wall problem, which links low energy neutrino oscillations to astronomical-scale baseline neutrino oscillations. The different assignments of SL 2 (F 3 ) × U(1) X quantum number to Majorana neutrinos guarantee 8 We will study on neutrino in detail including numerical analysis in the next project.
the absence of the Yukawa terms S c N c × F . Consequently, two Dirac neutrino mass terms are generated; one is associated with S c , and the other is N c . The right-handed neutrino SU(2) L singlet denoted as N c transforms as the (3, p) and additional Majorana neutrinos denoted as S c e , S c µ , and S c τ transform as (1, r − Q y s 1 ), (1 ′′ , r − Q y s 2 ) and (1 ′ , r − Q y s 3 ), respectively. Below the cutoff scale Λ, the mass term of the Majorana neutrinos N c comprises an exact TBM pattern. Imposing the U(1) X symmetry explains the absence of the Yukawa terms LN c Φ S and N c N c Φ T as well as does not allow the interchange between Φ T and Φ S , both of which transform differently under U(1) X , so that the exact TBM is obtained at leading order. With the desired VEV alignment in Eq. (9) it is expected that the leptonic Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix at the leading order is exactly compatible with a TBM In order to explain the present terrestrial neutrino oscillation data, non-trivial next leading order corrections should be taken into account: for example, considering next leading order Yukawa superpotential in the Majorana neutrino sector triggered by the field Φ T are written (For neutrino phenomenology we will consider in detail in the next project. See also an interesting paper [27].) After including the higher dimensional operators there remain no residual symmetries 9 .
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 condition (4) (or Eq. (19)) satisfied, it is expected that the SM gauge singlet flavon fields derive higher-dimensional operators, which are eventually where Similarly, the U(1) X quantum numbers associated to the neutrinos can be assigned by the anomaly-free condition of U(1) X -[gravity] 2 together with the measured active neutrino observables: This vanishing anomaly, however, does not restrict Q yν (or equivalently Q y ss i ), whose quantum numbers can be constrained by the new neutrino oscillations of astronomical-scale baseline, as shown in Refs. [1,7,28]. With the given above U(1) X quantum numbers, such whereQ y s i = Q y s 1 /X 2 . We choose k 2 = ±21 for the U(1) X i charges to be smallest making no axionic domain-wall problem, as in Ref. [1,7]. Hence, for the case-IQ y s 1 +Q y s 2 +Q y s 3 = −10 (32); for the case-II −20 (22); for the case-III −14 (28), respectively, for k 2 = 21(−21). Then, the color anomaly coefficients are given by δ G 1 = 2X 1 and δ G 2 = −3X 2 , and subsequently from Eq. (5) the axionic domain-wall condition as in Ref. [7] is expressed with the reduced Clearly, in the QCD instanton backgrounds since the N 1 and N 2 are relative prime there is no Z N DW discrete symmetry, and therefore no axionic domain-wall problem occurs.
The model incorporates the SM gauge singlet flavon fields F A = Φ S , Θ, Ψ,Ψ with the following interactions invariant under the U(1) X ×SL 2 (F 3 ) and the resulting chiral symmetry, i.e., the kinetic and Yukawa terms, and the scalar potential V SUSY in SUSY limit 11 are of the form Here the V SUSY term is replaced by V total including soft SUSY breaking term when SUSY breaking effects are considered, and ψ stands for all Dirac fermions. The kinetic terms +higher order terms for canonically normalized fields are written as The scalar fields Φ S , Θ and Ψ(Ψ) have X-charges X 1 and X 2 (−X 2 ), respectively, that is where ξ k (k = 1, 2) are constants. So, the potential V SUSY has U(1) X global symmetry. 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 12 in which we have set Φ S1 = Φ S2 = Φ S3 ≡ Φ Si and h Ψ = hΨ in the SUSY limit, and v g = v 2 Ψ + v 2 Ψ . And the NG modes A 1 and A 2 are expressed as with the angular fields φ S , φ θ and φ Ψ . With Eqs. (23) and (25), the derivative couplings of A k arise from the kinetic terms 11 In our superpotential, the superfields Φ S , Θ and Ψ(Ψ) are the SM gauge singlets and have −2p and −q(q) X-charges, respectively. Given soft SUSY-breaking potential, the radial components of the X-fields |Φ S |, |Θ| |Ψ| and |Ψ| are stabilized. The X-fields contain the axion, saxion (the scalar partner of the axion), and axino (the fermionic superpartner of the axion). 12 Note that the massless modes are not contained in the fieldsΘ, where v F = v Θ (1 + κ 2 ) 1/2 and h F = (κh S + h Θ )/(1 + κ 2 ) 1/2 , and the dots stand for the orthogonal components h ⊥ F and A ⊥ 1 . Recalling that κ ≡ v S /v Θ . Clearly, the derivative interactions of A k (k = 1, 2) are suppressed by the VEVs v F and v Ψ . From Eq. (27), performing v F , v Ψ → ∞, the NG modes A 1,2 , whose interactions are determined by symmetry, are invariant under the symmetry and distinguished from the radial modes, h F and h Ψ .

Quarks and CKM mixings, and flavored-Axions
Now, let us move to discussion on the realization of quark masses and mixings, in which the physical mass hierarchies are directly responsible for the assignment of U(1) X quantum numbers. The axion coupling matrices to the up-and down-type quarks, respectively, are diagonalized through bi-unitary transformations: , and the mass eigenstates ψ ′ R = V ψ R ψ R and ψ ′ L = V ψ L ψ L . These transformation include, in particular, the chiral transformation necessary to make M u and M d real and positive. This induces a contribution to the QCD vacuum angle. Note here that under the chiral rotation of the quark fields the effective QCD vacuum angle is invariant, see Refs. [1,17]. With the desired VEV directions in Eq. (9), in the above Lagrangian (28) the mass matrices M u and M d for up-and down-type quarks, respectively, are expressed as where In the above mass matrices the corresponding Yukawa terms for up-and down-type quarks are given by 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 [29] is mainly generated by the mixing matrix of the down-type quark sector. Moreover, due to the diagonal form of the up-type quark mass matrix in Eq. (48) the CKM mixing matrix V CKM ≡ V u L V d † L coming from the charged quark-current term in Eq. (28) is generated from the down-type quark matrix in Eq. (30): in the Wolfenstein parametrization [30] and at higher precision [31], where λ = where P u and Q u are diagonal phase matrices, and V d L and V d R can be determined by diagonalizing the matrices for M † d M d and M d M † d , respectively. The physical structure of the upand down-type quark Lagrangian should match up with the empirical up-and down-type quark masses and their ratios calculated from the measured PDG values [29]: Then, the mixing matrix V d † L = V CKM is obtained by diagonalizing the Hermitian matrix The CKM mixing angles in the standard parametrization [33] can be roughly described as And with the quark fields redefinition the CKM CP phase is given as Subsequently, the up-and down-type quark masses are obtained as And the parameter of tan β ≡ v u /v d is given in terms of the PDG value in Eq. (36) by Since all the parameters in the quark sector are correlated with one another, it is very crucial for obtaining the values of the new expansion parameters to reproduce the empirical results of the CKM mixing angles and quark masses. Moreover, since such parameters are also closely correlated with those in the lepton sector, finding the value of parameters is crucial to produce the empirical results of the charged leptons (see below Eq. (48)) and the light active neutrino masses in our model. In the following subsequent subsection we will perform a numerical simulation for quark sector.

Numerical analysis for Quark masses and CKM mixing angles
We perform a numerical simulation 14 using the linear algebra tools of Ref. [34]. With the inputs tan β = 4.7 , κ = 0.33 ,  14 Here, in numerical calculation, we only have considered the mass matrices in Eqs. (29) and (30) 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.

III. SCALE OF PQ PHASE TRANSITION AND QCD AXION PROPERTIES
The couplings of the flavored-axions and the mass of the QCD axion are inversely proportional to the PQ symmetry breaking scale. In a theoretical view of Refs. [1,7,17], the scale of PQ symmetry breakdown congruent to that of the seesaw mechanism can push the scale much beyond the electroweak scale, rendering the flavored-axions very weakly interacting particles. Since the weakly coupled flavored-axions (one linear combination QCD axion and its orthogonal ALP) could carry away a large amount of energy from the interior of stars, according to the standard stellar evolution scenario their couplings should be bounded with electrons 16 , photons, and nucleons. Hence, such weakly coupled flavoredaxions have a wealth of interesting phenomenological implications in the context of astroparticle physics [1,7], like the formation of a cosmic diffuse background of axions from the Sun [35,36]; from evolved low-mass stars, such as red-giants and horizontal-branch stars in globular clusters [37,38], or white dwarfs [39,40]; from neutron stars [41]; and from the duration of the neutrino burst of the core-collapse supernova SN1987A [42] as well as the rare flavor changing decay processes induced by the flavored-axions K + → π + + A i [15,43] and µ → e + γ + A i [43,45] etc..
Such flavored-axions could be produced in hot astrophysical plasmas, thus transporting energy out of stars and other astrophysical objects, and they could also be produced by the rare flavor changing decay processes. Actually, the coupling strength of these particles with normal matter and radiation is bounded by the constraint that stellar lifetimes and energyloss rates [46] as well as the branching ratios for the µ and K flavor changing decays [15,45] should not be counter to observations. Interestingly enough, the recent observations also show a preference for extra energy losses in stars at different evolutionary stages -red giants, supergiants, helium core burning stars, white dwarfs, and neutron stars (see Ref. [13] for the summary of extra cooling observations and Ref. [1] on the interpretation to a bound of the QCD axion decay constant); the present experimental limit, Br(K + → π + A i ) < 7.3 × 10 −11 [15], puts a lower bound on the axion decay constant, and in the near future the NA62 experiment expected to reach the sensitivity Br(K + → π + A i ) < 1.0 × 10 −12 [47] will probe the flavored-axions or put a severe bound on the QCD axion decay constant F A (or flavored-axion decay constants F a i = f a i /δ G i ). According to the recent investigation in Ref. [1,7], the flavored-axions (QCD axion and its orthogonal ALP) would provide very good hints for a new physics model for quarks and leptons. Fortunately, in a framework of the flavored-PQ symmetry the cooling anomalies hint at an axion coupling to electrons, photons, and neutrons, which should not conflict with the current upper bound on the rare K + → π + A i decay. Remark that once a scale of PQ symmetry breakdown is fixed the other is automatic including the QCD axion decay constant and the mass scale of heavy neutrino 16 The second (µ) and third (τ ) generation particles are absent in almost all astrophysical objects. associated to the seesaw mechanism.
In order to fix the QCD axion decay constant F A (or flavored-axion decay constants F a i = f a i /δ G i ), we will consider two tight constraints coming from astro-particle physics: axion cooling of stars via bremsstrahlung off electrons and flavor-violating processes induced by the flavored-axions.

A. Flavored-Axion cooling of stars via bremsstrahlung off electrons
In the so-called flavored-axion framework, generically, the SM charged lepton fields are nontrivially U(1) X -charged Dirac fermions, and thereby the flavored-axion coupling to electrons are added to the Lagrangian through a chiral rotation.
In the present model since the flavored-axion A 2 couples directly to electrons, the axion can be emitted by Compton scattering, atomic axio-recombination and axio-deexcitation, and axio-bremsstrahlung in electron-ion or electron-electron collision [37]. The flavoredaxion A 2 coupling to electrons in the model reads where m e = 0.511 MeV, 2 and X e = −11X 2 . Indeed, the longstanding anomaly in the cooling of WDs (white dwarfs) and RGB (red giants branch) stars in globular clusters where bremsstrahlung off electrons is mainly efficient [39] could be explained by axions with the fine-structure constant of axion to electrons α Aee = (0.29 − 2.30) × 10 −27 [48] and α Aee = (0.41 − 3.70) × 10 −27 [40,49], indicating the clear systematic tendency of stars to cool faster than predicted. It is recently reexamined in Ref. [13] as Eq. (1) where α Aee = g 2 Aee /4π, which is interpreted in terms of the QCD axion decay constant in the present model as B. Flavor-Changing process K + → π + + A i induced by the flavored-axions Below the QCD scale (1 GeV≈ 4πf π ), the chiral symmetry is broken and π and K, and η are produced as pseudo-Goldstone bosons. 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 flavor-changing process K + → π + + A i is induced by the flavored-axions A i . Then, the flavored-axion interactions with the flavor violating coupling to the s-and d-quark is given by where 17 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 [43,44] where m K ± = 493.677 ± 0.013 MeV, m π ± = 139.57018(35) MeV [50], and where is used. From the present experimental upper bound in Eq. (1), , we obtain the lower limit on the QCD axion decay constant Hence, from Eqs. (51) and (55) we can obtain a strongest bound on the QCD axion decay constant F A = 3.56 +0.84 −0.84 × 10 10 GeV .
Interestingly enough, from Eqs. (47) and (56) In the near future the NA62 experiment will be expected to reach the sensitivity of [47], which is interpreted as the flavored-axion decay constant and its corresponding QCD axion decay constant f a i > 9.86 × 10 11 GeV ⇔ F A > 2.32 × 10 11 GeV .
Clearly, the NA62 experiment will probe the flavored-axions or exclude the present model. 17 In the standard parametrization the mixing elements of V d R are given by θ R , and θ R 12 ≃ 2 √ 2| sin φ d |λ 2 . Its effect to the flavor violating coupling to the s-and d-quark is negligible: 12 = 0 at leading order.

C. QCD axion interactions with nucleons
Below the chiral symmetry breaking scale, the axion-hadron interactions are meaningful (rather than the axion-quark interactions) 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 [17] where a is the QCD axion, its 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). Recently, the couplings of the axion to the nucleon are very precisely extracted as [52] where N = 2δ G 1 δ G 2 with δ G 1 = 2X 1 and δ G 2 = −3X 2 , andX q = δ G 2 X 1q + δ G 1 X 2q with q = u, d, s and X 1u = X 1 , X 1d = X 1 , X 1s = 0, X 1c = 0, X 1b = 0, X 1t = 0, X 2u = −4X 2 , X 2d = −X 2 , X 2s = X 2 , X 2c = −2X 2 , X 2b = 3X 2 , X 2t = 0. And the QCD axion coupling to the neutron is written as where the neutron mass m n = 939.6 MeV. The state-of-the-art upper limit on this coupling, g Ann < 8 × 10 −10 [53], from the neutron star cooling is interpreted as the lower bound of the QCD axion decay constant Clearly, the strongest bound on the QCD axion decay constant comes from the flavored-axion cooling of stars via bremsstrahlung off electrons in Eq. (51) as well as the flavor-changing process K + → π + + A i induced by the flavored-axions in Eq. (55).
Using the state-of-the-art calculation in Eq. (61) and the QCD axion decay constant in Eq. (56), we can obtain g Ann = 2.14 +0.66 −0.41 × 10 −12 , which is incompatible with the hint for extra cooling from the neutron star in the supernova remnant "Cassiopeia A" by axion neutron bremsstrahlung, g Ann = 3.74 +0.62 −0.74 × 10 −10 [54]. This huge discrepancy may be explained by considering other means in the cooling of the superfluid core in the neutron star, for example, by neutrino emission in pair formation in a multicomponent superfluid state 3 P 2 (m j = 0, ±1, ±2) [55].

D. QCD axion mass and its interactions with photons
With the well constrained QCD axion decay constant in Eq. (56) congruent to the seesaw scale we can predict the QCD axion mass and its corresponding axion-photon coupling.
As in Refs. [1,17], the axion mass in terms of the pion mass and pion decay constant is obtained as where 18 f π = 92.21 (14) MeV [29] and (3) and ω = 0.315 z . (66) Note that the Weinberg value lies in 0.38 < z < 0.58 [29,56]. After integrating out the heavy π 0 and η at low energies, there is an effective low energy Lagrangian with an axion-photon coupling g aγγ : where E and B are the electromagnetic field components. And the axion-photon coupling can be expressed in terms of the QCD axion mass, pion mass, pion decay constant, z and w: 18 Here F (z, ω) can be replaced in high accuracy as in Ref. [52] by The upper bound on the axion-photon coupling is derived from the recent analysis of the horizontal branch (HB) stars in galactic globular clusters (GCs) [57], which translates into the lower bound of decay constant through Eq. (65), as The bounds of Eqs. (69) and (70) are much lower than that of Eq. (56) coming from the present experimental upper bound Br(K + → π + A i ) < 7.3 × 10 −11 [15] as well as the axion to electron coupling 6.7 × 10 −29 α Aee 5.6 × 10 −27 at 3σ [13].
Hence, from Eqs. (56) and (65) The QCD axion coupling to photon g aγγ divided by the QCD axion mass m a is dependent on E/N. 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 [17]: for 0.38 < z < 0.58, the value of (g aγγ /m a ) 2 for the case-II and -III are located lower than that of the ADMX (Axion Dark Matter eXperiment) bound [12], while for the case-I is marginally 19 lower than that of the ADMX bound, where  Fig. 1, the uncertainties of (g aγγ /m a ) 2 for the case-II and -III are larger than that of case-I for 0.38 < z < 0.58. Fig. 2 shows the plot for the axion-photon coupling |g aγγ | as a function of the axion mass 19 In fact, this is the case for 0.54 z < 0.58. , which corresponds to the case-I, -II, and -III, respectively. As the upper bound on Br(K + → π + + A i ) gets tighter, the range of the QCD axion mass gets more and more narrow, and consequently the corresponding band width on |g aγγ | in Fig. 2 is getting narrower. In Fig. 2 the top edge of the bands comes from the upper bound on Br(K + → π + + A i ), while the bottom of the bands is from the astrophysical constraints of star cooling induced by the flavored-axion bremsstrahlung off electrons e + Ze → Ze + e + A i .
The model will be tested in the very near future through the experiment such as CAPP (Center for Axion and Precision Physics research) [61] as well as the NA62 experiment expected to reach the sensitivity of Br(K + → π + + A i ) < 1.0 × 10 −12 [47].

IV. SUMMARY AND CONCLUSION
Motivated by the flavored PQ symmetry for unifying the flavor physics and string theory [1,16], we have constructed a compact model based on SL 2 (F 3 ) × U(1) X symmetry for resolving rather recent, but fast-growing issues in astro-particle physics, including quark and leptonic mixings and CP violations, high-energy neutrinos, QCD axion, and axion cooling of stars. Since astro-particle physics observations have increasingly placed tight constraints on parameters for flavored-axions, we have showed how the scale responsible for PQ mechanism (congruent to that of seesaw mechanism) could be fixed, and in turn the scale responsible for FN mechanism through flavor physics. Along the lines of finding the fundamental scales, In the concrete, the QCD axion decay constant congruent to the seesaw scale, through its connection to the astro-particle constraints of stellar evolution induced by the flavored-axion bremsstrahlung off electrons e+Ze → Ze+e+A i and the rare flavor-changing decay process induced by the flavored-axion K + → π + + A i , is shown to be fixed at F A = 3.56 +0.84 −0.84 × 10 10 GeV (consequently, the QCD axion mass m a = 1.54 +0. 48 −0.29 × 10 −4 eV, wavelength of its oscillation λ a = 8.04 +1.90 −1.90 mm, axion to neutron coupling g Ann = 2.14 +0.66 −0.41 × 10 −12 , and axion to photon coupling |g aγγ | = 5.99 +1.85 −1.14 × 10 −14 GeV −1 for E/N = 23/6 (case-I), 4.89 +1.51 −0.93 × 10 −14 GeV −1 for E/N = 1/2 (case-II), 1.64 +0. 51 −0.31 × 10 −14 GeV −1 for E/N = 5/2 (case-III), respectively, in the case z = 4.8.). Subsequently, the scale associated to FN mechanism is automatically fixed through its connection to the SM fermion masses and mixings, Λ = 2.04 +0. 48 −0.48 ×10 11 GeV, and such fundamental scale might give a hint where some string moduli are stabilized in type-IIB string vacua.
We may conclude that in an extended SM framework by a compact symmetry G F = SL 2 (F 3 ) × U(1) X , if the scale responsible for the FN mechanism (whose scale is associated to some string moduli stabilization) is fixed, the scales responsible for seesaw and PQ mechanisms are dynamically determined in way that the SM fermion (including neutrino) masses and mixings are well delineated, which in turn provides predictions on several properties of the flavored-axions.
In the very near future, the NA62 experiment expected to reach the sensitivity of Br(K + → π + + A i ) < 1.0 × 10 −12 will probe the flavored-axions or exclude the model.
The SL 2 (F 3 ) is the double covering of the tetrahedral group A 4 [18,20,21]. It contains 24 elements and has three kinds of representations: one triplet 3 and three singlets 1, 1 ′ and 1 ′′ , and three doublets 2, 2 ′ and 2 ′′ . The representations 1 ′ , 1 ′′ and 2 ′ , 2 ′′ are complex conjugated to each other. Note that A 4 is not a subgroup of SL 2 (F 3 ), since the two-dimensional representations cannot be decomposed into representations of A 4 . The generators S and T satisfy the required conditions S 2 = R, T 3 = 1, (ST ) 3 = 1, and R 2 = 1, where R = 1 in case of the odd-dimensional representation and R = −1 for 2, 2 ′ and 2 ′′ such that R commutes with all elements of the group. The matrices S and T representing the generators depend on the representations of the group [21]: where we have used the matrices The following multiplication rules between the various representations are calculated in Ref. [21], 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