Flavored axion in the UV-complete Froggatt-Nielsen models
Leon M.G. de la Vega, Newton Nath, Stefan Nellen, Eduardo Peinado
FFlavored axion in the UV-complete Froggatt-Nielsen models
Leon M.G. de la Vega, ∗ Newton Nath, † Stefan Nellen, ‡ and Eduardo Peinado § Instituto de F´ısica, Universidad Nacional Aut´onoma deM´exico, A.P. 20-364, Ciudad de M´exico 01000, M´exico.
We propose UV-completions of Froggatt-Nielsen-Peccei-Quinn models of fermion masses andmixings with flavored axions, by incorporating heavy fields. Here, the U (1) Froggatt-Nielsensymmetry is identified with the Peccei-Quinn symmetry to solve the strong CP problem alongwith the mass hierarchies of the Standard Model fermions. We take into account leading or-der contributions to the fermion mass matrices giving rise to Nearest-Neighbour-Interactionstructure in the quark sector and A texture in the neutrino sector. A comprehensive numer-ical analysis has been performed for the fermion mass matrices. Subsequently, we investigatethe resulting axion flavor violating couplings and the axion-photon coupling arising from themodel.
1. INTRODUCTION
The axion [1–3] is one of the best motivated elementary particles beyond the Standard Model(BSM). It provides the most cogent solution to the “strong CP problem” of quantum chromody-namics (QCD), where the spontaneous breaking of a global U (1) symmetry, called the Peccei-Quinn(PQ) symmetry U (1) P Q , gives rise to a pseudo-Nambu-Goldstone (pNG) boson [2, 3]. It is also agood candidate for cold dark matter within a certain allowed parameter space of the axion scale [6–10]. In the literature, there exist two common approaches to realize the PQ symmetry that give riseto the axion, in one case the Higgs sector is extended (called the DFSZ model) [11, 12], whereas ina different scenario heavy quarks are introduced (called the KSVZ model) [13, 14]. As per the ex-perimental testability of the axion is concerned, it is the axion couplings to the photon g aγ and theelectron g ae that play very important roles. One can establish a relation between these couplingswith the PQ charges of fermions [15]. It is the constant E/N , i.e., the ratio of the electromagneticover the QCD anomaly coefficient which determines the strength of the photon-axion coupling,and it is a free parameter in generic axion models.Another puzzle in the Standard Model (SM) is the unexplained hierarchies in masses andmixings between elementary particles or the “hierarchical flavor structure” of the Yukawa couplings.Many flavor models have been studied to address the SM mass hierarchies involving additionalsymmetries. Among them, the Froggatt-Nielsen (FN) symmetry emerges as a leading candidateto account for the flavor hierarchies [16]. In this mechanism, one introduces a new complex scalar ∗ leonm@estudiantes.fisica.unam.mx † newton@fisica.unam.mx ‡ stefannellen@estudiantes.fisica.unam.mx § epeinado@fisica.unam.mx It is known that the theory of strong interactions i.e., QCD violates CP symmetry. However, the smallness ofCP violation in QCD has been a long-standing puzzle in particle physics, called as the “strong CP problem”. Atpresent, the most stringent bounds on the CP-violating term come from experimental limits on the neutron electricdipole moment and result in (cid:46) − [4, 5]. a r X i v : . [ h e p - ph ] F e b field called flavon, whose vacuum expectation value ( vev ) generates the hierarchical flavor structure,where a global flavor symmetry U (1) F N is imposed. Now, here one can ask, if it is possible touse the PQ symmetry as a flavor symmetry in order to explain the hierarchical flavor structurewhile solving the strong CP problem of the SM. Indeed, in [17] a connection between the PQsymmetry with flavor symmetries has been addressed. Recently, many attempts have been madewhere the PQ symmetry U (1) P Q is unified with the FN symmetry U (1) F N to address both issuesof the SM. The resultant axion that arises from this flavor-dependent framework has been called“flaxion” [18, 19], “axiflavon” [20–22], or “flavorful axion” [23–25].In this work, we study various low energy phenomenologies of the flavor-dependent axion thatarises from the PQ symmetry providing a solution to the strong CP problem. Here, only thethird family of quarks is generated at dimension four while other terms are introduced at higherdimensions. We also extended the flavor structure to the leptonic sector. In doing so, we identifythe U (1) P Q symmetry with the U (1) F N that gives rise to the fermion hierarchies, where the axionfield σ is treated like a flavon. We consider the DFSZ type axion models and show that one canconstruct a Nearest-Neighbour-Interaction (NNI) structure for both the up- and down-quark massmatrices M u/d = (cid:63) (cid:63) (cid:63) (cid:63) (cid:63) , (1)which was originally proposed in [26]. For the DFSZ models, we affix the PQ charges to all theSM fermions as well as to the Higgs sector in such a way that after the PQ and electroweak (EW)symmetry breaking, one ends up with the NNI structure of the quark mass matrices as givenby Eq. 1. Later, we present the UV-complete Froggatt-Nielsen-Peccei-Quinn (FNPQ) models byincorporating heavy fermionic fields . Two examples have been adopted within the frameworkof the type-I and -II Dirac seesaw mechanisms. In both the scenarios, it is shown that once weintegrate out heavy fields, one finds NNI-type quark mass matrices.Also, to explain the masses and mixings of the leptonic sector, we find the diagonal charged-lepton mass matrix in the FNPQ-framework. Moreover, within the framework of the type-I seesawmechanism [27–31], we obtain an A -type [32] neutrino mass matrix which is allowed by the latestglobal analysis of the neutrino oscillation data [33–35]. We also perform a numerical analysis of thequarks as well as leptonic mass matrices to find masses and mixings of fermions. Later, we exploreflavor violating decays with axions, where we estimate the branching ratios (BR) of K + → π + a and B + → K + a . These further help us to constrain the axion breaking scale. Furthermore, becauseof the presence of two Higgs doublets in the DFSZ axion model, we calculate the flavor violatingprocess t → hc . Finally, the cosmological consequences of the FNPQ axion are discussed.We plan the manuscript as below. In Sec. 2 we give a brief overview of the FN symmetry alongwith the NNI quark mass matrices. A detailed discussion of our theoretical framework that leads Notice that UV-completion of such flavored axion within the formalism of SUSY models has been discussed in [25],where authors have added two singlet superfields. Also, in their formalism they have considered the gauged FNsymmetry in absence of any heavy vector-like or chiral fields. to NNI quark textures together with the UV-complete mechanisms are presented in Sec. 3 and itssubsequent sub-sections, while the leptonic sector is described in Sec. 3 3.3. The scalar sector andaxion couplings are presented in Sec. 3 3.4, and Sec. 3 3.5, respectively. We analyze our numericalresults in Sec. 4, and conclusions are summarized in Sec. 5. U (1) F N
AND NEAREST-NEIGHBOUR-INTERACTION
Here, we start with a brief overview of the Froggatt-Nielsen symmetry and Nearest-Neighbour-Interaction structure for the fermion mass matrices. In order to address the hierarchical structureof the fermion masses Froggatt and Nielsen developed a symmetry called FN-symmetry [16], wherethey introduced a SM singlet “flavon” field σ . In their framework, one can express the necessaryYukawa terms for the quark sector as − L ⊃ y dij (cid:16) σ Λ (cid:17) n dij Q i Hd Rj + y uij (cid:16) σ Λ (cid:17) n uij Q i (cid:101) Hu Rj , (2)where n u/dij are complex numbers, σ is the flavon field and Λ is the scale of flavor dynamics. Also, Q i , u Ri , d Ri for i, j = 1 , , vev and breaks the symmetry, it leads to hierarchyin the masses of quarks and leptons. Thus, one can notice from Eq. 2 that the hierarchy of theYukawa couplings can be correlated to the smallness of (cid:104) σ (cid:105) / Λ, and hence to the fermion massmatrices m ij = y ij v ( (cid:104) σ (cid:105) / Λ) n , with v being the SM Higgs vacuum expectation value.Furthermore, this global U (1) symmetry can be used to obtain NNI-textures in the quarksector. The NNI-texture for a N × N mass matrix consists of a general matrix m with m ij (cid:54) = 0only for i = j ± i = N = j . Notice that one can connect the NNI-texture with the Fritzschtexture [36, 37] with an additional constraint of Hermiticity on both the quark mass matrices.However, the Fritzsch texture was ruled out as it failed to simultaneously generate the small valueof the CKM matrix element V cb together with the large value of top-quark mass m t [38, 39]. TheNNI texture is phenomenologically compatible with the quark sector, but contains more degreesof freedom in contrast to the experimental constraints, i.e. 6 masses, 3 angles and a CP-violatingphase of the quark sector. In the SM, the NNI texture may be obtained through Weak-Basis (WB)transformations from an arbitrary Yukawa matrix [26], and thus imposes no physical constraints.However, in models with more than one scalar a global symmetry that imposes the NNI on thequark mass matrix imposes a particular structure of scalar interactions [40], potentially leading tophysical observables. The proposed framework for the flavored PQ symmetry accommodates theDFSZ style of axion models, while leading to the NNI structure of the quark mass matrices andthe A structure of the neutrino mass matrix in the diagonal charged lepton basis.
3. FERMION MASSES IN FLAVORED DFSZ AXION MODELS
Now, we propose UV-complete models of quark masses, where a U (1) global symmetry enforcesNNI texture in the up and down quark mass matrices. The (3 ,
3) entry of the matrix is generatedby a dimension-4 operator, while the other entries are obtained by dimension-5 operators withthe addition of a symmetry-breaking flavon, thus obtaining mass hierarchies in a Froggatt-Nielsenstyle. Additionally, through the color anomalies of the global U (1) symmetry we obtain a QCDaxion from the flavon fields, which solves the strong CP problem and is a Dark Matter candidate.We present two realizations within the framework of the DFSZ style of QCD axions.In the DFSZ axion model, two Higgs doublets are needed in addition to the scalar singletwhich introduces the PQ symmetry breaking scale. We list the relevant field contents with theircorresponding SU (2) L × U (1) Y and U (1) P Q charges in Table I.
Fields/Symmetry Q iL u iR d iR H u H d σSU (2) L × U (1) Y (2, 1/6) (1, 2/3) (1, -1/3) (2, -1/2) (2, 1/2) (1, 0) U (1) PQ (9/2, -5/2, 1/2) (-9/2, 5/2, -1/2) (-9/2, 5/2, -1/2) 1 1 1 TABLE I.
Field content and transformation properties of the PQ-symmetry under the DFSZ type-I seesaw model,where i = 1 , , The effective Lagrangian that describes the up-quark sector in the model with the SM × U (1) P Q charges of Table I can be written as
L ⊃ C u Λ Q L H u u R σ + C u Λ Q L H u u R σ + C u Λ Q L H u u R σ + C u Λ Q L H u u R σ + C u Λ Q L (cid:101) H d u R σ ∗ + C u Λ Q L (cid:101) H d u R σ ∗ + C u Λ Q L H u u R σ + C u Λ Q L (cid:101) H d u R σ ∗ + y u Q L H u u R , (3)where C uij represents coupling constant and Λ is the cut-off scale of the model. Similarly, one canwrite the Lagrangian for the down-quark sector L ⊃ C d Λ Q L H d d R σ + C d Λ Q L H d d R σ + C d Λ Q L H d d R σ + C d Λ Q L H d d R σ + C d Λ Q L (cid:101) H u d R σ ∗ + C d Λ Q L (cid:101) H u d R σ ∗ + C d Λ Q L H d d R σ + C d Λ Q L (cid:101) H u d R σ ∗ + y d Q L H d d R , (4)with a similar meaning for C dij and Λ as the up-quark sector. Notice that these Higgs doublets coupleto both the quark sectors. After the symmetry breaking, and counting terms up to dimension-7,we find NNI-type quark-mass matrices as M u/d = εv u/d C u/d εv u/d C u/d εv d/u C u/d εv d/u C u/d y u/d v u/d , (5) Notice that the higher dimensional operators are either suppressed by a high enough energy scale compared todimension-4 or 5, or one can forbid them by using an additional symmetry, and hence one can safely deal with theNNI-type quark texture.
Fields/Symmetry F uC F uC F uC F uC F dC F dC F dC F dC U (1) Y U (1) PQ TABLE II.
Vector like fermions and their transformation properties of the PQ-symmetry under the DFSZ type-Iseesaw model, where C = L, R . where, ε = (cid:104) σ (cid:105) / Λ or (cid:104) σ (cid:105) ∗ / Λ. For typical values of ε ∼ .
2, one can safely neglect terms proportionalto ε and ε as has been pointed in Eqs. 3, 4. First, we consider a type-I Dirac seesaw model, where the heavy mediators are vector-like quarks.Furthermore, this restriction will exclude them from contributing to the anomalous couplings ofthe axion. The UV-complete operator for the up-quark sector, as given by Eq. 3, within the DFSZtype-I seesaw formalism can be achieved as follows L UVu ⊃ Y u Q L H u F uR + M u F uR F uL + Y u (cid:48) F uL σu R + Y u Q L H u F uR + M u F uR F uL + Y u (cid:48) F uL σu R + Y u Q L (cid:101) H d F uR + M u F uR F uL + Y u (cid:48) F uL σ ∗ u R + Y u Q L (cid:101) H d F uR + M u F uR F uL + Y u (cid:48) F uL σ ∗ u R , (6)where F ijqC are the vector like fields and their PQ-charges are given in Table II.Similarly, the UV-completion of the down-quark sector as given by Eq. 4 can be achieved byusing F ijqC (see Table II for their charges) and the corresponding Lagrangian can be written as L UVd ⊃ Y d Q L H d F dR + M d F dR F dL + Y d (cid:48) F dL σd R + Y d Q L H d F dR + M d F dR F dL + Y d (cid:48) F dL σd R + Y d Q L (cid:101) H u F dR + M d F dR F dL + Y d (cid:48) F dL σ ∗ d R + Y d Q L (cid:101) H u F dR + M d F dR F dL + Y d (cid:48) F dL σ ∗ d R . (7)In Fig. 1 we show the Feynman diagram corresponding to Eqs. 6 and 7.After the PQ and EW symmetry breaking by the vev s v σ , (cid:104) H u (cid:105) and (cid:104) H d (cid:105) we obtain the followingquark mass matrices in the ( Q L , F ( u/d ) L ) × ( q R , F ( u/d ) R ) basis as M u/d = (cid:32) M Q L q R M Q L F R M F L q R M F L F R (cid:33) × , (8)where the 4 submatrices of M u/d are given by Q iL H q q jR σF ijqR F ijqL FIG. 1.
UV complete diagram within the DFSZ type-I seesaw framework as apparent from Eqs. 6 and 7. M Q L q R = y u/d v u/d , (9) M Q L F R = Y u/d v u/d Y u/d v u/d Y u/d v d/u
00 0 0 Y u/d v d/u , (10) M F L q R = Y (cid:48) u/d v σ Y (cid:48) u/d v σ Y (cid:48) u/d v ∗ σ Y (cid:48) u/d v ∗ σ , (11) M F L F R = diag( M u/d , M u/d , M u/d , M u/d ) . (12)Now, the light quark mass matrix in this Dirac type-I seesaw scenario can be written as m u/d = M Q L q R − M Q L F R M − F L F R M F L q R , (13)at leading order. The resulting form of this matrix is given by m u/d = Y u/d Y (cid:48) u/d M v u/d v σ Y u/d Y (cid:48) u/d M v u/d v σ Y u/d Y (cid:48) u/d M v ∗ d/u v ∗ σ Y u/d Y (cid:48) u/d M v ∗ d/u v ∗ σ y u/d v u/d , = A u/d B u/d C u/d D u/d E u/d . (14)where A, B, C, D , and E are complex entries. As an alternative, we consider a type-II Dirac seesaw framework of a DFSZ model. Withinthis model, the UV-completion of the quark-sectors (see Eqs. 3, 4) can be achieved by introducing Q iL H q σ q jR Φ q FIG. 2.
UV complete diagram within the DFSZ type-II seesaw framework as apparent from Eqs. 15, and 16. two additional BSM doublets namely, Φ u (2 , − / , d (2 , / , SU (2) L × U (1) Y × U (1) P Q , respectively. Here, no additional heavy quark statesare added. We express the UV-complete Lagrangian for the up-quark sector as follows: L UVu ⊃ Y u Q L Φ u u R + Y u Q L Φ u u R + µ u H u Φ † u σ + Y u Q L (cid:101) Φ d u R + Y u Q L (cid:101) Φ d u R + µ d (cid:101) H d Φ d σ ∗ . (15)The SM × U (1) P Q charges for the remaining fields are given by Table I. Similarly, the down-quarksector can be written as L UVd ⊃ Y d Q L Φ d d R + Y d Q L Φ d d R + µ u H d Φ † d σ + Y d Q L (cid:101) Φ d d R + Y d Q L (cid:101) Φ d d R + µ d (cid:101) H u Φ d σ ∗ . (16)The resulting quark mass matrices after the PQ and EW symmetry breaking can be read as m u/d = Y u/d v Φ u/d Y u/d v Φ u/d Y u/d v Φ d/u Y u/d v Φ d/u y u/d v u/d , (17)where the vev s v Φ u,d of the additional doublets are determined by the scalar potential . Finally,we like to remind here that the quark mass matrices in Eqs. 14 and 17 have a NNI structure. In our formalism, the Yukawa Lagrangian invariant under SM × U (1) P Q for charged-leptonsand neutrinos is given by −L Y ⊃ y e L eL H d (cid:96) eR + y µ L µL H d (cid:96) µR + y τ L τL (cid:101) H u (cid:96) τR y ν L eL H u N + y ν L µL H u N + y ν L τ (cid:101) H d N . (18)The charge assignments for the leptonic fields are given in Table III. In [41], the fermion mass hierarchies and the strong CP problem with four Higgs doublets along with the PQsymmetry have been discussed.
Fields/Symmetry L iL (cid:96) iR N i σ (cid:48) SU (2) L × U (1) Y (2, -1/2) (1, -1) (1, 0) (1, 0) U (1) PQ (1, 0, -3) (0, -1, -2) (0, -1, -2) 2 TABLE III.
Field content and transformation properties of the leptonic fields and a scalar field σ (cid:48) under the DFSZtype-I seesaw model, where i = 1 , , This leads to diagonal charged-lepton as well as Dirac neutrino mass matrices, respectively. Onthe other hand, the Lagrangian involving right-handed neutrinos is given by −L Majorana ⊃ N c N + N c N σ + N c N σ (cid:48) + N c N σ (cid:48) . (19)Therefore, the Majorana neutrino mass matrix takes the form M R = × × ×× × × . (20)Now, in the type-I seesaw formalism, a light neutrino mass matrix is given by − m ν ≈ M TD M − R M D ,which can be read as m ν = × × × × × × , (21)and it corresponds to the type A neutrino mass matrix.It is to be noted here that the inclusion of σ (cid:48) allows a non-zero dimension-6 terms in the quarksector, which are zero at dimension-5 as given by Eq. 5. The Yukawa Lagrangian at dimension-6for the up-quark sector can be written as −L d =6 Y ⊃ y u Q L H u u R σ (cid:48) + y u Q L H u u R σ (cid:48) + y u Q L (cid:101) H d u R σ (cid:48)∗ . (22)However, we find that the UV-completion of the first term of Eq. 22 can only be achieved usingthe vector like fermions as given by Table II, whereas one needs new fermions to do the same forsecond and third terms, respectively. We explicitly show the UV-completion of the first term andthe corresponding Lagrangian can be written as L UVu ⊃ Y u Q L H u F uR + M u F uR F uL + Y u (cid:48) F uL σ (cid:48) F uR + M u F uR F uL + Y u (cid:48) F uL σ (cid:48) u R . (23)Similarly, the down-quark sector can be written as −L d =6 Y ⊃ y d Q L H d d R σ (cid:48) + y d Q L H d d R σ (cid:48) + y d Q L (cid:101) H u d R σ (cid:48)∗ . (24) Notice that the same leptonic Lagrangian was obtained in the context of a gauged U (1) (cid:48) symmetry in [42]. Q L H q σ q R σ F qR F qL F qR F qL FIG. 3.
UV complete diagram for a dimension-6 operator within the DFSZ type-I seesaw framework in presence offlavon field σ (cid:48) as follows from Eqs. 23, and 25. Here, we also find that one can achieve the UV-completion only for the first term of Eq. 24 andthe UV-complete Lagrangian can be written as L UVd ⊃ Y d Q L H d F dR + M d F dR F dL + Y d (cid:48) F dL σ (cid:48) F dR + M d F dR F dL + Y d (cid:48) F dL σ (cid:48) d R . (25)Therefore, in presence of dimension-6 terms, we generate a non-zero (1, 3) element of the quarkmass matrix as given by Eq. 5. However, in our analysis, we focus on the leading dimension-4and 5 contributions to the mass matrices, noting that given the matter content proposed, this isthe minimal set of operators needed to obtain the fermion masses and mixings compatible withobservations. We assume higher order operators are either suppressed by a high enough energyscale to be subdominant, or forbidden by an additional symmetry. In minimal DFSZ models the SU (2) L singlet scalar field couples with the doublets either throughthe cubic H u (cid:101) H † d σ term or through the quartic H u (cid:101) H † d σ term [11, 43]. In this work, the PQ chargesof the scalars are such that both the cubic and quartic couplings between the SU (2) L doublets andsinglets are present. We define the scalar fields present in the first DFSZ model as H u = (cid:32) h u + iA u h − u (cid:33) , H d = (cid:32) h + d h d + iA d (cid:33) , σ = S + iA , σ (cid:48) = S (cid:48) + iA (cid:48) , (26)whereas the second DFSZ model contains two additional SU (2) L doublets Φ u and Φ d , which wedefine as Φ u = (cid:32) φ u + iA u φ − u (cid:33) , Φ d = (cid:32) φ + d φ d + iA d (cid:33) . (27)The scalar potentials for both models have been provided in Appendix B. For each model we maywrite the Goldstone mode eaten by the Z-boson as A Z = (cid:80) i Y i v i A i (cid:113)(cid:80) i Y i v i , (28)0where Y i is the hypercharge of each scalar field, v i its vev , for the type-I seesaw model A i ∈{ A, A (cid:48) , A u , A d } and A i ∈ { A, A (cid:48) , A u , A d , A (cid:48) u , A (cid:48) d } for the type-II seesaw. The Goldstone bosonrelated to the PQ symmetry is likewise given by A P Q = (cid:80) i X i v i A i (cid:113)(cid:80) i X i v i . (29)The physical axion, however, is orthogonal to the Z-boson Goldstone, so to obtain it from thesetwo equations, we must perform the following subtraction [15] a = A P Q − (cid:80) i X i Y i v i (cid:113)(cid:80) i Y i v i (cid:113)(cid:80) i X i v i A Z . (30)The possible values of v i for the doublet fields are bounded by the SM, (cid:113)(cid:80) i Y i v i = 246 GeV. Onthe other hand, the possible values of (cid:113)(cid:80) i X i v i are bounded from below by its relationship tothe axion decay constant f a . This means that there must be a strong hierarchy between the vev s ofthe σ and σ (cid:48) fields and the vev s of the doublet fields. From Eqs. 28 - 30 we can see that this resultsin the physical axion being composed to a high degree of only the σ and σ (cid:48) fields. Additionally,we choose to adopt the hierarchy v (cid:48) σ << v σ as it leads to negligible mixing between the A and A (cid:48) fields, resulting in a stronger coupling of the axion to the quark fields, as in our framework quarkmasses are derived from the couplings of σ at leading order. With these considerations in mind weextract the axion-quark couplings by setting a ∼ A . We would also like to mention that the detailsabout the energy scale of f a and related phenomenology are analyzed in subsequent sections. The couplings of the axion to the gluon and the photon are determined by the electromagneticanomaly factor E and the QCD anomaly factor N . The effective Lagrangian that relates the axioncouplings with anomalies ( E and N ) can be read as L ⊃ α s π af a ( F (cid:101) F ) C + EN α EM π af a ( F (cid:101) F ) EM , (31)where ( F (cid:101) F ) C = ε µνρσ F bµν F bρσ , F bµν is the color field strength ( b = 1 , ..., F (cid:101) F ) EM = ε µνρσ F µν F ρσ , F µν is the electromagnetic field strength. The anomaly factors E and N are givenby the expressions [15] N = (cid:88) i X i T ( R i ) , E = (cid:88) i X i Q i D ( R i ) , (32)where the sums run over all fermions, X i is the PQ charge of fermion i , T ( R i ) is the index of the SU (3) c representation R i of fermion i (in particular T (3) = 1 / , T (6) = 5 / , T (8) = 3). Also, Q i represent the electric charge and D ( R i ) the dimension of R i . Moreover, the axion decay constant1is given by [44] f a = (cid:113)(cid:80) i X i v i N , (33)where this sum runs over all scalars with PQ charge X i and vev v i . For simplicity we will considerthe dominant contribution to f a be from σ , so we may write f a ∼ v σ /N . The axion mass in termsof the decay constant is [44] m a = 5 . µ eV (cid:18) GeV f a (cid:19) . (34)Finally, the coefficients of the anomalous axion-gluon coupling and axion-photon coupling are givenby [44] g aγ = α EM πf a (cid:18) EN − . (cid:19) , g ag = α s πf a . (35)In the Georgi-Glashow model [45] of gauge unification one can show that E/N = 8 /
3, using thisin Eq. 35, the axion-photon coupling can be expressed as g SU (5) aγ = 1 . GeV (cid:18) m a µ eV (cid:19) , (36)where the Fine-structure constant α EM = 1 /
137 has been used. Here, for the DFSZ models the SMfermions are the only contributors to the anomalies, resulting in N = 5 and E = 28 /
3. Therefore,compared to this benchmark the model presented here has the suppression | g SU (5) aγ || g DF SZaγ | = 14 . (37)For completeness, we briefly summarize here that the axion-photon interactions in the ( g aγ − m a )plane for m a ≥ m a ≤ O (10 − ) eV, as the reason of interest for the model, it is the AxionDark Matter eXperiment (ADMX) [48] searching for cold dark matter axions with a haloscopedetector, provides the most stringent bound. It can be seen from figure-5 of [48] that the ADMX canexplore 2 × − ≤ m a ≤ . × − eV for the coupling strength of O (10 − ) (GeV) − . On the otherhand, the ADMX Phase IIa/Gen-2 can improve their sensitivity of axion mass to (1 . , × − eV for | g aγ | one order smaller compared to the latest ADMX bound, i.e., O (10 − ) (GeV) − , fordetails see figure-9 of [48]. For definiteness, by choosing f a ∼ GeV, and m a ∼ − eV, wefind | g aγ | ∼ − (GeV) − using Eq. 35 and hence the suppression of this coupling places ourmodel beyond the reach of projected ADMX Phase IIa/Gen-2 sensitivity [48].We now proceed to discuss various low energy phenomenologies within the FNPQ-model insubsequent sections. As per the quark sector goes, we consider the quark mass matrix as given byEq. 14 with a vanishing (1, 3) element and we adopt this as it can be seen from Eq. 5 that the (1,3) element arises from dimension-6 which is 1 / Λ suppressed.2 Parameter Best fit | A u | / (10 − GeV) 1 . | B u | / (10 − GeV) − . | C u | / GeV − . | D u | / (10 GeV) 3 . | E u | / (10 GeV) 1 . | A d | / (10 − GeV) − . | B d | / (10 − GeV) 1 . | C d | / (10 − GeV) − . | D d | / (10 − GeV) − . | E d | / GeV − . α u / ◦ . β u / ◦ . σ range θ q / ◦ .
09 13 . → .
12 12 . θ q / ◦ .
207 0 . → .
213 0 . θ q / ◦ .
32 2 . → .
37 2 . δ q / ◦ .
53 66 . → .
10 68 . m u / (10 − GeV) 1 .
288 0 . → .
550 1 . m c / (10 − GeV) 6 .
268 6 . → .
459 6 . m t / GeV 171 .
68 170 . → .
18 171 . m d / (10 − GeV) 2 .
751 2 . → .
151 2 . m s / (10 − GeV) 5 .
432 5 . → .
728 5 . m b / GeV 2 .
854 2 . → .
880 2 . χ q TABLE IV.
Best-fit values of the model parameters in the quark sector are shown in the upper table. The globalbest-fit as well as their 1 σ error [5, 49] for the various observables are given in the second and third columns of thelower table. Also, the best-fit values of the various observables are listed in the last column of the lower table.
4. NUMERICAL RESULTS4.1. Masses and mixings of fermions
Here we perform a χ analysis to find masses and mixing parameters of both quarks and leptonssectors. First, a global χ fit is conducted to find the values of the parameters of the up and downquark matrices in our framework. The χ function takes the following form χ = (cid:88) ( µ exp − µ fit ) σ exp , (38)where the sum runs over all observables. Also, µ fit represent the masses and mixings calculatedusing fitting parameters, as given by Eq. 39, we list them in Table IV. Here µ exp and σ exp are theobservables and their standard deviation [5, 49] as given by Table IV.Note that by redefining the quark fields it can be shown that there exist only two non-zerophases. In the appendix A, we give a detailed analysis of the phase redefinition. Thus, the up- and3 Parameter Best fit a/ (10 − eV) 9 . b/ (10 − eV) 2 . c/ (10 − eV) 2 . d/ (10 − eV) 2 . φ a / ◦ . φ b / ◦ . φ c / ◦ . φ d / ◦ − . σ range θ l / ◦ . . → . . θ l / ◦ .
45 8 . → .
61 8 . θ l / ◦ . . → . . δ l / ◦
218 191 →
256 258 . α/ ◦ . β/ ◦ . m / (10 − eV ) 7 .
55 7 . → .
75 7 . m / (10 − eV ) 2 .
424 2 . → .
454 2 . (cid:80) m ν / (10 − eV) 6 . m e / MeV 0 . . → . m µ / GeV 0 . . → . m τ / GeV 1 . . → . χ l TABLE V.
Best-fit values of the model parameters in the lepton sector are shown in the upper table. The globalbest-fit as well as their 1 σ error [5, 49] for the various observables are given in the second and third columns of thelower table. Also, the best-fit values of the various observables are listed in the last column of the lower table. down-quark mass matrices that are used in the fit are given by m u/d = | A u/d | | B u/d | e − iα u/d | C u/d | e − iα u/d | D u/d | e − iβ u/d | E u/d | e − iβ u/d . (39)Also, as pointed out in appendix A, it is the difference in the up- and down- quark phasematrices that are relevant for the quark mixing matrix, hence in our numerical analysis, phases α d and β d have been fixed to zero. There are 12 parameters that need to be fitted, 10 amplitudes (5 foreach matrix) and 2 phases. We fit these 12 parameters to account for the 10 physical observablesrelated to them, the 6 quark masses, the 3 CKM angles and the CP-violating phase.The observables are obtained from these matrices using the MPT package [50]. The fit is doneat the energy scale M Z [49]. The initial values for the fitting procedure are randomized. Theresults from the best fit are given in Table IV, where χ q = 1 . A -texture and takes the form m ν = a e iφ a a e iφ a b e iφ b c e iφ c c e iφ c d e iφ d . (40)The χ function is identical as in Eq. 38, the only differences being the observables and theirstandard deviations. The leptonic masses and mixings obtained from the fit, which are compatiblewith the latest global fit data, can be seen in Table V at χ l = 2 . The models described here contain flavor violating Yukawa couplings. Of particular interest arethe axion flavor violating couplings to quarks as they lead to the decays with final state axions q i → q j a . These processes can be probed by meson decays with final state axions. These flavorviolating couplings are common to both models presented. From the effective Lagrangian of Eqs.3 and 4 we can write the Yukawa couplings to σ in the interaction basis as L σ = y uij u jL σu iR + y u (cid:48) ij u jL σ ∗ u iR + y dij d jL σd iR + y d (cid:48) ij d jL σ ∗ d iR , (41)where the Yukawa coupling matrices y u ( (cid:48) ) ij and y d ( (cid:48) ) ij are y u/d = 1 v σ A u/d B u/d , y u (cid:48) /d (cid:48) = 1 v σ C u/d D u/d . (42)In the quark mass basis the couplings to sigma are obtained by transforming the fermion fields tothe physical basis L σq (cid:48) = y uij U † ikL u (cid:48) kL σU jlR u (cid:48) lR + y u (cid:48) ij U † ikL u (cid:48) kL σ ∗ U jlR u (cid:48) lR + y dij V † ikL d (cid:48) kL σV jlR d (cid:48) lR + y d (cid:48) ij V † ikL d (cid:48) kL σ ∗ V jlR d (cid:48) lR , (43)where u (cid:48) /d (cid:48) denote quark mass eigenstates, U L and U R diagonalize the up quark mass matrix and V L and V R diagonalize the down quark mass matrix, respectively. By defining λ u ( (cid:48) ) = U † L y u ( (cid:48) ) U R and λ d ( (cid:48) ) = V † L y d ( (cid:48) ) V R we may write L σq (cid:48) = λ uij u (cid:48) iL σu (cid:48) jR + λ u (cid:48) ij u (cid:48) iL σ ∗ u (cid:48) jR + λ dij d (cid:48) iL σd (cid:48) jR + λ d (cid:48) ij d (cid:48) iL σ ∗ d (cid:48) jR . (44)Finally, by neglecting axion mixing with other scalars, we write the quark couplings to the axionas L aq (cid:48) = ia ( (cid:15) uij u (cid:48) i u (cid:48) j + (cid:15) dij d (cid:48) i d (cid:48) j + (cid:15) (cid:48) uij u (cid:48) i γ u (cid:48) j + (cid:15) (cid:48) dij d (cid:48) i γ d (cid:48) j ) , (45)where (cid:15) u,dij = ( λ ij − λ † ij ) / (cid:15) (cid:48) u,dij = ( λ ij + λ † ij ) / K + → π + a process. The decay ratio for the Kaon decayto axion and pion is given byΓ( K + → π + a ) ≈ m K π | (cid:15) d | B S (cid:18) − m π m K (cid:19) , (46)where B S is a non-perturbative parameter B S ∼ . | (cid:15) d | , this leads toΓ( K + → π + a ) ≈ . × − GeV v σ . (47)Using the latest constraint of the BR of Kaon decay from the E949 Collaboration [52] i.e.,BR( K + → π + a ) = Γ( K + → π + a )Γ T otal ( K + ) < . × − , (48)we may constrain v σ (cid:62) . × GeV. For the models described above we have v σ ≈ f a N with N = 5, and hence using this approximation one can translate bounds of v σ to the axion decayconstant f a , where f a (cid:62) × GeV . (49)Analogously, we may consider the B + → K + a decay, where the bottom to strange quark transitionis probed. The decay width of this process is given byΓ( B + → K + a ) ≈ m B π | (cid:15) d | ( f K (0)) (cid:18) m B m b − m s (cid:19) (cid:18) − m K m B (cid:19) , (50)with f K (0) ∼ . B + → K + a ) < − − − [20], which would lead to v σ (cid:62) (1 . × − GeV . (51)In our framework, v σ ≈ N f a , the bound above translates to f a (cid:62) × (10 − ) GeV . (52)The axion decay constant f a > × GeV bound translates in our models to the equivalentlimits m a < . × − eV, and | g aγ (GeV − ) | < . × − . These bounds are two orders ofmagnitude stronger than the limits from astrophysics, see figure-1 of [46], for completeness. Weend this section by noting that these bounds are obtained by neglecting σ − σ (cid:48) mixing. When thismixing is sizable these bounds on f a are relaxed. The presence of two Higgs doublets in the DFSZ models raises the possibility of flavor changingneutral currents (FCNC) mediated by scalar particles. Moreover, the identification of the U(1)6flavor symmetry with the Peccei-Quinn symmetry makes the implementation of natural flavorconservation difficult by requiring additional discrete symmetries and/or additional Higgs doublets.Therefore, in the DFSZ scheme of flavored axion models, scalar FCNCs are expected on minimalmodels. On the phenomenological side, these FCNCs are strongly constrained by processes, namely K L → µ − µ + or top decays such as t → hc, hu [54]. The masses of the additional scalar particles andthe non-diagonal couplings of the light Higgs may be constrained by the experimental limits on theseprocesses. The high scale of PQ breaking induces a decoupling of the components of H u and H d from the components of σ i . Hence, we assume one may write at leading order h ≈ h u cos α + h d sin α , H ≈ − h u sin α + h d cos α , where one identifies h as the 125 GeV boson observed at LHC, and H asan additional heavy scalar. The couplings of these two particles to SM fermions may be obtainedfrom the effective Lagrangian and read as follows L ⊃ C u v u u L u R h u + C u v d u L u R h d + C d v d d L d R h d + C d v u d L d R h u , (53)where the matrices C u/di are given by C u/d = A u/d B u/d E u/d , C u/d = C u/d D u/d . (54)In terms of the physical scalar and quark fields we have L ⊃ hu (cid:48) L u (cid:48) R (cid:18) C (cid:48) u v u cos α − C (cid:48) u v d sin α (cid:19) + Hu (cid:48) L u (cid:48) R (cid:18) C (cid:48) u v u sin α + C (cid:48) u v d cos α (cid:19) + ( u → d ) , (55)where the matrices C (cid:48) u/di are defined as C (cid:48) ui = U † L C ui U R , C (cid:48) di = V † L C di V R . (56)We see that in the limit v u /v d = cot α the Yukawa couplings of h are proportional to the upand down quark mass matrices, while the couplings of H are not. After the transformation of thefermion states to the mass basis the couplings of h are diagonal, while the couplings of H are not.This limit does not affect the couplings of the axion, which means that the processes discussed inthe previous section do not vanish. A deviation of α from the flavor conserving limit will introduceflavor violating couplings of the light Higgs, which can be probed with observables such as t → hc in the up-sector and h → bs in the down sector. The t → hc decay channel currently has an upperbound set by ATLAS [55] BR( t → hc ) LHC < . × − , (57)while future experiments HL-LHC [56], ILC and CLIC [57] project the following sensitivities tothis process BR( t → hc ) HL − LHC < × − , (58)BR( t → hc ) ILC / CLIC < − . (59)7 FIG. 4.
Exclusion region plot in the (tan α − tan β ) plane obtained from the non-observation of the t → hc flavorviolating decay. The gray colored region is excluded by ATLAS data [55], the purple colored region is expected tobe probed in the future by the HL-LHC experiment [56] and the orange region will be further probed by ILC orCLIC [57]. The uncolored region (see white thin band) predicts a branching ratio beyond the sensitivity of theseexperiments. The dashed line indicates limit of no flavor violation in light Higgs Yukawa couplings. Thus, the branching ratio can be calculated from Eq. 55 as [58]Γ t → hc = C tc m t π (cid:112) [1 − ( R c − R h ) ] [1 − ( R c + R h ) ] (cid:2) ( R c + 1) − R h (cid:3) , (60)where the coupling C tc is defined as C tc = [( C (cid:48) u ) + ( C (cid:48) u ) ] cos αv SM sin β − [( C (cid:48) u ) + ( C (cid:48) u ) ] sin αv SM cos β , (61) m t is the top quark mass, R h is the higgs to top mass ratio R h = m h /m t , and R c is the charm to topmass ratio R c = m c /m t . Using the experimental value for the total width of the top quark [5] wederive the constraints on the free parameters tan α and tan β as shown in Fig. 4. We use the bestfit point given in Table IV, to obtain a numerical value of Eq. 60 and we derive the approximateconstraint (cid:12)(cid:12)(cid:12)(cid:12) cos α sin β (1 − tan α tan β ) (cid:12)(cid:12)(cid:12)(cid:12) ≤
17 Γ
Expt → hc [ GeV ] , (62)for a given experimental input of the decay width Γ Expt → hc . We see from Fig. 4 that, as expectedfrom Eq. 62, small values for tan β allow only large values of tan α and vice versa. It can also beseen that ATLAS data has already ruled out a large portion of the parameter space (gray-region)8and HL-LHC (purple-region) and CLIC (orange-region) will leave only a small region around thetan β = cot α limit unprobed (see Fig. 4 caption for details). Finally, we would like to mentionthat the t → hu and h → cu decays can also place constraints on α and β , however we find theseto be numerically much weaker than the constraints from t → hc , given that the hierarchy in the C ui (see Eq. 54) matrices is preserved in the physical C (cid:48) ui matrix (see Eq. 56). Moreover, currentlythere are no experimental constraints on h → bs decays, although they are phenomenologicallyinteresting in the context of two Higgs Doublet models [59]. The axion can be a good dark matter candidate, provided a sufficient amount of them wasproduced in the early universe. There are several production mechanisms for axions. The relicdensity produced by the misalignment mechanism is [60, 61]Ω a h ≈ × (cid:18) f a GeV (cid:19) / (cid:104) θ a,i (cid:105) , (63)where θ a,i is the initial misalignment angle of the cosmological axion field and it lies in the range(0 , π ). Now, we notice that for the axion breaking scale 5 × < f a < × (GeV), onecan match the axion relic density to the observed dark matter relic abundance Ω DM h ∼ .
12 for0 < θ a,i < π . It is worthwhile to mention that the N >
5. CONCLUSION
In this work, to address the hierarchal flavor structure of fermion masses as well as the strongCP problem of QCD, we adopt a formalism based on the Froggatt-Nielsen symmetry. It is wellknown that the FN symmetry is one of the leading mechanisms to explain fermion masses andtheir mixings. On the other hand, the Peccei-Quinn symmetry provides an elegant solution to thestrong CP problem. Here, an attempt has been made by identifying the PQ symmetry U (1) P Q with the FN symmetry U (1) F N to address both the drawbacks of the SM. In doing so, we assignPQ charges to the SM and BSM fields (see Table I), within the framework of the DFSZ style axionmodels, in such a manner that the (3, 3) entry of the quark mass matrices are generated by adimension-4 operator, whereas remaining entries are obtained at the dimension-5 level. We endup with the Nearest-Neighbor-Interaction structure for the quark mass matrices in two differentUV-completions, namely the type-I, and -II Dirac seesaw mechanisms for quark masses. Moreover,9upon the breaking of the assigned symmetry, a flavored-axion from the flavon fields has beenobtained, which solves the strong CP problem and also is a Dark Matter candidate. For the leptonsector, we also extended the model by considering the type-I seesaw mechanism. We obtained the A -type Majorana neutrino mass matrix in the diagonal charged lepton mass basis.A comprehensive numerical analysis has been performed to find low energy fermion massesand mixings, which are summarized in Table IV, V and are consistent with the latest global-fitdata. Besides this, based on the ratio between the electromagnetic to the QCD anomaly factor,we have determined the axion-photon coupling, which is suppressed by a factor of 14 (as given byEq. 37) compared to the SU (5) GUT model. The models discussed here also contain axion flavorviolating couplings to quarks. Using these couplings, we have calculated the branching ratios of K + → π + a and B + → K + a processes that involve the flavored-axion. These indicate that theaxion decay constant is f a > × GeV, whereas the axion mass limit is m a < . × − eV,and the axion-photon coupling is | g aγ (GeV − ) | < . × − . We also have the possibility of flavorviolating neutral currents mediated by scalar particles due to the presence of two Higgs doublets.The branching ratio for the t → hc decay channel has been calculated for the model and the resultsare summarized in Fig. 4, where the latest bounds of ATLAS as well as the future experimentslimits of HL-LHC, ILC and CLIC have been incorporated. Finally, we have pointed out that whenthe axion breaking scale lies in the range f a ∈ (5 × , × ) GeV, then the axion relic densitycan be matched to the observed dark matter relic abundance. ACKNOWLEDGMENTS
This work is supported by the grants CONACYT CB-2017-2018/A1-S-13051 (M´exico), DGAPA-PAPIIT IN107118, DGAPA-PAPIIT IN107621 and SNI (M´exico). NN is supported by the post-doctoral fellowship program DGAPA-UNAM. LMGDLV is supported by CONACYT.
Appendix A: Phase redefinition of quark mass matrices
The up- and down-quark mass matrices as given by Eq. 14 can be written as m u/d = | A u/d | e iα u/d | B u/d | e iβ u/d | C u/d | e iγ u/d | D u/d | e iδ u/d | E u/d | e i(cid:15) u/d , (A1)with | A | , | B | , | C | , | D | , | E | , α and β as real parameters. The above mass matrix m u/d can bediagonalized by bi-unitary transformation of the form m diag = V † L mV R = O T P † L mP R O , (A2)where L and R depict the left- and right-chiral fields, respectively. Also, V L = P L O and V R = P R O are the unitary matrices that diagonalize m † m and mm † and P L = diag(1 , e iα , e iβ ), P R =0diag( e iρ , e iρ , e iρ ) are the diagonal phase matrices, respectively. Notice that we drop subscript( u/d ) to demonstrate phase redefinition as the following formalism is same for both the up anddown sector. Now, given the form of P L and P R , one can construct a real solution of the quarkmass matrix (Eq. A1) following the transformation P † L mP R as mentioned by Eq. A2. Thus, thequark mass matrix in terms of real parameters can be written as m = | A | | B | | C | | D | | E | . (A3)Notice that for most of the application V R is irrelevant, it is the V L that enters in the CKMparameterization. Therefore, one can write quark mixing matrix as V CKM = ( V † L ) u ( V L ) d = ( O T P † L ) u ( P L O ) d , = O Tu diag(1 , e − i ( α u − α d ) , e − i ( β u − β d ) ) O d . (A4)We see here that it is the difference in the diagonal-phase matrix that enters in the V CKM . Thus,in our numerical an analysis we have used the quark mass matrix of the form P † L m = | A | | B | e − iα | C | e − iα | D | e − iβ | E | e − iβ . (A5) Appendix B: Scalar potentials
Here, we give a full scalar potential for both the type-I and -II DFSZ seesaw models correspond-ing to their charge assignments as given by Tables I, III, respectively. For the type-I DFSZ seesawmodel the full scalar potential is given by V = µ u H † u H u + µ d H † d H d + µ σσ ∗ + µ σ (cid:48) σ (cid:48)∗ + λ u ( H † u H u ) + λ d ( H † d H d ) + λ ( σσ ∗ ) + λ (cid:48) ( σ (cid:48) σ (cid:48)∗ ) + λ ( H † u H d )( H † d H u ) + λ ( H † u H u )( H † d H d ) + λ ( (cid:101) H † u H d )( (cid:101) H † d H u ) + λ ( σσ ∗ )( H † u H u )+ λ ( σσ ∗ )( H † d H d ) + λ ( (cid:101) H † u H d ) σ + λ ( σ (cid:48) σ (cid:48)∗ )( H † u H u ) + λ ( σ (cid:48) σ (cid:48)∗ )( H † d H d ) + λ ( σ (cid:48) σ (cid:48)∗ )( σσ )+ κ ( σ σ (cid:48)∗ + σ ∗ σ (cid:48) ) + λ ( (cid:101) H † u H d ) σ (cid:48) . (B1)For the type-II DFSZ seesaw model we enlarge the previous model by two additional Higgs Doubletstransforming as Φ u ∼ ( − / ,
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