$T'$ Models with High Quality Flaxions

The"gauged'' Peccei-Quinn (PQ) mechanism of Fukuda, Ibe, Suzuki and Yanagida is implemented in the flavorful axion model of Carone and Merchand. This model of flavor is similar to other successful ones based on the double tetrahedral group, but the flavor symmetry includes a global U(1) factor that leads to the presence of a flavorful axion. Here we gauge that U(1) symmetry and introduce a heavy sector that includes (1) the fermions necessary to cancel anomalies and (2) a second scalar flavon field that spontaneously breaks the U(1) symmetry. The full theory has an accidental U(1)$\times$U(1)$'$ global symmetry, anomalous with respect to QCD; U(1)$_\text{PQ}$ emerges as a linear combination. The gauged flavor symmetry restricts the possible PQ symmetry-breaking higher-dimension operators so that sufficient axion quality is preserved. We provide a model of the quark sector, as a proof of principle, and then a model which incorporates the standard model charged leptons as well. In both cases, the charge assignments that lead to acceptable axion quality also lead to a multiplicity of some of the heavy sector states; we check that the Landau pole for hypercharge remains above the cut off of the effective theory. We consider relevant phenomenological constraints on these models including those on the predicted axion couplings.


I. INTRODUCTION
The absence of a CP-violating term quadratic in the gluon field-strength tensor, i.e. one proportional to G µν G µν , remains one of the puzzles of the standard model. The Peccei-Quinn mechanism [1,2] posits the existence of a spontaneously broken U(1) global symmetry, one that is anomalous with respect to QCD; the goldstone boson of this symmetry, the axion, couples to G µν G µν so that this term vanishes when the axion sits at the minimum of its non-perturbatively generated potential. Aside from providing a dynamical mechanism for solving the strong-CP problem, the axion is also a plausible dark matter candidate [3].
Interest in axions has been heightened by the absence of compelling evidence for TeV-scale beyond-the-standard-model physics at the LHC, as well as the null results from dark matter experiments that search directly for weakly interacting massive particles at or around the electroweak scale.
Following earlier work [4], one possibility that has reemerged recently is that the anomalous global symmetry of the Peccei-Quinn mechanism may play a role in explaining the flavor structure of the standard model [5][6][7][8][9][10][11]. In the simplest models, a U(1) F flavor symmetry, spontaneously broken by a "flavon" field ϕ, provides an origin for both the axion field and Yukawa coupling hierarchies [9]. The Yukawa couplings (aside from that of the top quark) arise via higher-dimension operators of the form where we have used a charge −1/3 quark as an example, and where p depends on the U(1) F charge assignments of the fields. By judicious choices of these assignments, Yukawa matrix entries can arise in a hierarchical pattern, as determined by the powers p that control the various entries, as well as the ratio ϕ /M F which is taken to be a small parameter. The axion a can be identified using the nonlinear representation where ϕ = f / √ 2, and where σ is a heavy field, with mass of O(f ), that we will ignore.
Non-linear redefinitions of the fermion fields can remove a from the Yukawa couplings and shift it to the fermion kinetic terms, where it will appear as a field that is derivatively coupled to the U(1) F Noether current. Since the axion has couplings that are flavor-dependent, it has been called a "flavorful axion," [5] "axiflavon," [6][7][8] or "flaxion," [9] depending on the tastes of the authors. We will refer to this type of axion as a flaxion in the present work.
Interesting models of flavor that involve non-Abelian groups may include discrete and/or continuous Abelian factors. Non-Abelian groups often lead to more predictive models than purely Abelian ones, since the fermions can be embedded in representations with dimension greater than 1. (By contrast, there is significantly more freedom when one can assign a U(1) charge to each fermion field independently.) Nevertheless, Abelian factors are often necessary in these models, as is the case in a number of elegant models based on the double tetrahedral group T ′ . For example, the supersymmetric models of Ref. [12,13] based on T ′ × Z 3 require the Z 3 factor so that a subgroup exists that rotates the standard model fermion fields of the first generation (which reside within a T ′ doublet) by a phase. The subsequent breaking of this subgroup at a lower energy scale accounts for the smallness of the Yukawa couplings of the first generation fermions relative to the other generations. Differences between the up-and down-quark Yukawa matrices require additional symmetries, for example promoting Z 3 to Z 3 × Z 2 in one of the models of Ref. [13]; the non-supersymmetric T ′ flavor models studied in Ref. [14], on the other hand, utilized T ′ × Z 3 × Z 3 . In Ref. [10], similar nonsupersymmetric models were studied in which the second Z 3 factor was promoted to U(1), endowing the model with a flaxion to address the strong CP problem. (For supersymmetric flaxion models based on T ′ symmetry, see Ref. [11].) However, no origin was provided for this U(1) symmetry, which was assumed to arise as an artifact of some unspecified theory in the ultraviolet. One nontrivial feature of such a completion is that it would have to solve the axion quality problem, i.e., the problem that the Peccei-Quinn mechanism is easily rendered ineffective by quantum gravitational effects [15]. It is believed that quantum gravitational effects generally break all global symmetries [16], in this case through operators that would re-introduce the strong CP problem by triggering a non-zero value of the θ parameter. We will review this more explicitly later. Possible mechanisms of producing "high quality" axions have been proposed [17][18][19][20][21][22][23], but little discussion exists (as far as we are aware) in the context of flavored axion models.
In this work, we consider flaxion quality in the context of the T ′ ×Z 3 ×U(1) F flaxion model discussed in Ref. [10]. We go beyond that work by building variant models with heavy sectors that make explicit the physics that protects the flaxion quality. In particular, we promote the U(1) F factor of the original model to a gauge symmetry, and show that the extended model has an approximate U(1)×U(1) ′ global symmetry in which the U(1) F gauge group is embedded. One of the two goldstone bosons that arise from the spontaneous breaking of the global symmetry becomes the longitudinal component of the U(1) F gauge boson, while the other remains as a flaxion. The gauged flavor symmetry restricts the possible higherdimension operators that can break the U(1)×U(1) ′ global symmetry so that flaxion quality is sufficiently preserved. Thus, we present models that show how to successfully implement the "gauged Peccei-Quinn" approach proposed by Fukuda, et al. [17] to the T ′ × Z 3 ×U(1) F flaxion model of Ref. [10]. This places the results of that work on sounder theoretical footing.
Our paper is organized as follows. In Sec. II, we give a brief summary of the Yukawa textures that emerge in the T ′ × Z 3 ×U(1) F model of Ref. [10]. We will not need to review how the breaking of the T ′ × Z 3 symmetry leads to most of the features of these matrices, since only the factors associated with the breaking of the U(1) F factor will be relevant to our later discussion. In Sec. III, we present a model in which the T ′ × Z 3 ×U(1) F symmetry is applied only to the quark sector; this model is consistent with a wide range of other possible flavor groups that might be relevant in the lepton sector. In Sec. IV, we consider a flaxion model in which the same flavor symmetry is relevant to both the quark and lepton sectors.
In Sec. V, we summarize our conclusions.

II. TEXTURES
The models of interest are based on the flavor group T ′ ×Z 3 ×U(1) F . The flavor symmetry breaking fields fall in T ′ singlet and doublet representations; using the T ′ × Z 3 notation of Ref. [13], Details of T ′ group theory, including an explanation of this notation and the Clebsch-Gordan matrices necessary for constructing invariant Lagrangian terms, can be found in Ref. [13].
However, this will not be relevant to our subsequent discussion. One only needs to know the Yukawa textures generated via the breaking of the flavor symmetry and the role played by the s flavon, which is the only one that is charged under U(1) F ; we choose our normalization so that this charge is +1. More specifically, the symmetry breaking of the discrete factors is given by where Z D 3 refers to a diagonal subgroup of a Z 3 subgroup of T ′ and the additional Z 3 factor [12,13]. The dimensionless parameters ǫ and ǫ ′ are defined in terms of the symmetrybreaking vacuum expectation values (vevs) and the flavor scale M F , the cut off of the effective theory: 3) The additional dimensionless parameter ρ is determined by the U(1) F breaking scale. This leads to the leading-order Yukawa textures where the u i , d i and l i are (in general complex) O(1) parameters. These Yukawa matrices were shown to be phenomenologically viable in Ref. [10] via a global fit to the quark and lepton masses and the Cabibbo-Kobayashi-Maskawa (CKM) mixing angles; in this fit, the values of the free parameters were consistent with the expectations of naive dimensional analysis [24].

III. QUARK SECTOR MODEL
We focus in this section on a T ′ × Z 3 ×U(1) F model of quark flavor, corresponding to the quark sector of the model of Ref. [10]. An extension to the lepton sector that assumes the same flavor group is presented in Sec. IV. The quark-sector model presented in this section exemplifies our approach more directly, and is compatible with models of lepton flavor that may assume a different lepton flavor group structure.
The U(1) F in Ref. [10] was a global flavor symmetry whose spontaneous breaking at the flavor scale provided an origin for a flavored axion. This breaking was accomplished by a single flavon field s, whose flavor charge was normalized to +1. Of the quark fields, only the right-handed bottom quark carried a flavor charge, −1, so that Yukawa matrix entries that multiply d 3 R acquire a suppression factor given by s /M F ≡ ρ, where M F was the flavor scale. This factor, taken in addition to those related to the breaking of the T ′ symmetry, provides for the successful Yukawa textures that were summarized in the previous section.
Since the U(1) F symmetry is anomalous with respect to color, the flavored goldstone boson that emerges from spontaneous symmetry breaking serves as a viable flavored axion.
To implement the "gauged" Peccei-Quinn idea of Fukuda, et al. [17], we introduce another flavon field s ′ , with U(1) F charge −1/N, with N an integer to be determined later. This field will couple to N heavy colored states D j R and D j L , for j = 1 . . . N. We promote this symmetry to a gauged flavor symmetry. We will see that at leading order in a 1/M F expansion, the theory including the heavy sector fields has an enlarged global symmetry, U(1)× U(1) ′ , corresponding to separate phase rotations on the s and s ′ fields. Gauging the U(1) F flavor symmetry leaves the full theory with a residual U(1) global symmetry that is both anomalous and spontaneously broken, assuring the presence of a flavorful axion.
However, a consequence of the gauged flavor symmetry is that the set of operators that break the residual global symmetry explicitly occur only at very high order, so that the flavored axion evades the axion quality problem. In this section we assume the simplest possibility, that the flavor scale is identified with the reduced Planck scale, M F = M * , which provides the cut off for the low-energy effective theory. We will see that heavy particles needed to cancel anomalies associated with the gauged U(1) F flavor symmetry appear at two intermediate scales associated with the expectation values of the s and s ′ fields.
The gauge quantum numbers of the relevant fields are shown in the first two rows of Table I. Aside from the two scalars, s and s ′ , and the right-handed bottom quark,    Let V 0 (s, s ′ ) represent the scalar potential including only the renormalizable terms. For N > 3, V 0 is only a function of s * s and s ′ * s ′ , leading to an accidental U(1)×U(1) ′ global symmetry corresponding to separate phase rotations on the two flavon fields. We will normalize the global charges to be (1, 0) and (0, 1) for the s and s ′ fields, respectively. Using notation similar to Ref. [17], we adopt the nonlinear representation where s = f a / √ 2, and where g F is the flavor gauge coupling, the gauge charges of the s and s ′ fields are q and q ′ , respectively, with q = +1 and q ′ = −1/N for the model defined in Table I. We immediately identify the eaten linear combination b = 1 The orthogonal linear combination is the physical massless degree of freedom, the flavored axion a = 1 Under a U(1) F gauge transformation, the exponentiated fields shift a/f a → a/f a + q α and It is shown in Ref. [17] that a shift of the axion field a/F by 2π connects two gauge equivalent points in a-b space provided that We omit a repetition of that discussion here, but use the quantity F in our discussion below.
We next consider constraints on the parameters N and x. The coupling of the a and b fields to gluons and photons is determined by the U(1)×U(1) ′ color and electromagnetic anomalies, respectively, where G µν and F µν are the gluon and photon field strength tensors. The color anomaly factors are given by  We fix f a to this value with M F = M * , so that f a ≈ 2 × 10 16 GeV; one then finds that f A < 10 12 GeV implies, for example that f b < 10 13 GeV when N = 10. Note that for f b at this upper limit, we can compute the location of the Landau pole for hypercharge, which we expect to be drastically reduced by the multiplicity of heavy charged particles; we find this scale Λ LP ≈ 3 × 10 18 GeV, which nonetheless remains above the cutoff of our effective theory. We discuss this computation more explicitly below.
We next turn to the issue of axion quality. The accidental global symmetry of the potential is broken by terms that are not functions of s * s and s ′ * s ′ alone. The lowest order U(1) F gauge-invariant term of this form is where ξ is an order-one coupling that is generally complex. This contributes both to the axion mass as well as to a linear term in the axion potential: where m 0 is the standard QCD contribution to the axion mass, and (3.14) The linear term will shift the minimum of the axion potential away from the origin, reintroducing a non-vanishing value of the θ parameter, θ = a /f A = Im ξ Re ξ ∆m 2 m 2 0 + ∆m 2 .

(3.15)
Applying the phenomenological bound θ < 10 −10 [25], and assuming that the real and imaginary parts of ξ are of order unity, one concludes that ∆m 2 /m 2 0 < 10 −10 . Using the following estimate for the QCD contribution [25] m 0 = 5.691 as well as our previous choice of f a = 10 −2 M * , we find that this bound implies (3.17) If we saturate this bound with f a fixed as previously noted, the mass scales of the heavy particles that carry standard model charges are fixed, since these are determined via the with the sum on i = 1 . . . N implied. These will contribute significantly to the running of hypercharge so we must check that the associated Landau pole remains above the cut off of our effective theory. To do so, we evaluate the one-loop renormalization group equations between each threshold where we define the location of the Landau pole by α −1 Y (Λ LP ) = 0 using the standard model normalization of hypercharge 1 , and where the particle content of Table I    Taking the heavy particle thresholds to be m a ≈ f a and m b ≈ f b and α −1 Y (m Z ) = 98.43, we find the Landau pole locations shown in Table II. We see that the Landau pole remains above the cut off of our effective theory, M * , for a wide range in N; the value for this scale remains roughly constant, with the accelerated running caused by the greater particle multiplicity compensated by the heavier particle thresholds, which also increase with N, as given by the axion quality bound in Eq. (3.17). We don't have similar worries for the U(1) H gauge coupling since its value at low energies is not fixed phenomenologically and can be taken small enough to keep its Landau pole safely above the cut off.
Bounds on the axion-photon coupling, defined by g aγγ ≡ − αem Using the values of f A shown in Table II, as well as the estimate for the axion mass in Eq. (3.16), these bounds eliminate N ≤ 8, so that N ≥ 9 is necessary for a viable model.  also be arranged. We will not consider the issue of the stability of the heavy states further, since even in the case where they are exactly stable, it is possible that their abundance might be so low after re-heating [23] that there would be no negative consequences as far as direct searches or cosmology is concerned.

IV. EXTENSION WITH LEPTONS
The model presented in our previous work, Ref. Operator coefficients were found via this fit to be consistent with the expectations of naive dimensional analysis.
In this section, we present a model that is a closer match to the one of Ref. [10] in that both quarks and leptons are subject to the T ′ × Z 3 ×U(1) F flavor symmetry and M F is again fixed to 4 × 10 16 GeV, with f a = 10 −2 M F as suggested by the fit results. In this way, all the numerical results of Ref. [10] are unchanged. We will assume the most general set of M Fsuppressed higher-dimension operators, including those that could spoil the axion quality.
Despite the fact that the ultraviolet (UV) cut off M F is smaller than M * , the flavor-scale assumed in our quark-sector model, we will find that axion quality is sufficiently preserved.
With f a and M F fixed, there are two remaining free parameters, f b and N, which will be constrained by (a) the axion quality bound that we have previously derived, which is now written as However other production mechanisms can also be implemented that allow for a lower axion decay constant, see for example Refs. [29][30][31][32][33][34]. Thus we only impose the upper bound f A ≤ 10 13 GeV. It is also possible that dark matter has multiple components so that the relic density need not be saturated by the axion's contribution.
(c) the requirement that the Landau pole of the hypercharge gauge coupling remain above our UV cutoff, the flavor scale M F . This constraint is relevant given the multiplicity of states with non-zero hypercharge in our extended heavy sector.
Besides the above constraints, there are also constraints from the flavor-changing couplings of the axion. It was shown in Ref. [10] that the most stringent limit comes from the meson decay K + → π + a (see Eq. (3.19) in that reference). Since the most relevant limit concern quarks lets focus on that sector for now. Derivatively coupled, flavor-changing axion couplings were obtained in Ref. [10] by applying the nonlinear field redefinition where a was the axion field in that model, and then rotating to the quark mass eigenstate basis. In the scenario we consider here, however, the analogous redefinition will involve the ã field instead, which is not the axion field. Re-expressing the derivative interaction in terms of the linear combination of theã andb fields identified with the axion (c.f., Eqs. (3.4) and (3.5)), then the bound on f a given in (3.19) of Ref. [10] is modified to: which is trivially satisfied for our choice f a = 4 × 10 14 GeV. Therefore we will not be concerned by the flavor-changing neutral current constraints henceforth. We will show how other relevant constraints can be satisfied below.

A. The Model
The scalar fields and colored fermions charged under the gauged U(1) F of our quark-sector model remain unchanged while new color singlets are introduced to cancel gauge anomalies.
The charge assignments of this model are presented in Table III.  Mass terms for the exotics are given by (4.4) where the κ's are Yukawa couplings and the flavor indices on the heavy fields are omitted.
From this expression, we see how the accidental U(1) and U(1) ′ symmetries of the scalar potential may be extended to the Yukawa couplings, with the global charges identified in the third row of Table III.
The induced axion coupling to the G G term is given by the same formulas presented in the last section since the charges of the colored fermions under the accidental U(1) × U(1) ′ group are the same. However the axion coupling to photon pairs will be modified by the differences in the heavy particle spectra, including the presence of the new heavy leptons that are doublets under SU(2) W in the present theory. For each of the U(1) global symmetries there is an F F interaction corresponding to the associated anomaly. These are given by [10] L ⊇ α em 8π 5) where N B and N W are the anomaly coefficients for hypercharge and isospin respectively.
The value of these coefficients is completely determined once the charges of the scalar fields are fixed. Using the values presented in Table III one obtains leading to the axion-photon coupling Note that the numerical coefficient is the same as what one would find in the simplest DFSZ axion models [35].

B. Model Constraints
Since the exotic fermions with non-zero hypercharge, in this case D i and λ i , for i = 1 . . . N, obtain their masses from the same scalar, the running of the hypercharge gauge coupling will be modified above the threshold given approximately by the scalar s ′ vev. This is different from the model introduced in the last section where the heavy particles with hypercharge appeared at two distinct energy thresholds. At 1-loop order, the location of the Landau pole is determined here by where the contribution to the beta function is Analogous to Table II, we present the location of the Landau pole for different heavy particle multiplicities N, assuming that the scale f b saturates the axion quality condition, Eq. (4.1).
We also show the predicted value of the axion decay constant f A .    Table IV with N ≤ 9 are ruled out. We thus find that N ≥ 10 is necessary, similar to our quark sector-model.

V. DISCUSSION AND CONCLUSIONS
It has been long argued that the Peccei-Quinn (PQ) solution to the strong CP problem could be spoiled by the presence of higher-dimension operators that violate the PQ symmetry explicitly unless their accompanying dimensionless coefficients are unnaturally small, or if the operators arise at sufficiently high order [15]. In this paper, we have extended the flavorful axion model presented in our previous work [10] to address this problem by implementing a general strategy for preserving axion quality proposed in Ref. [17], the "gauged Peccei-Quinn" mechanism. The basic structure of our extended flavor sector was illustrated in a model of quark flavor presented in Sec. III. In that model, we made the theoretically economical choice of identifying the flavor scale M F (the cut off of our effective theory) with the reduced Planck scale M * . We then considered a more comprehensive model that included the charged leptons, with a flavor scale below the Planck scale; our choice of M F = 4×10 16 GeV as well as the various scales of flavor symmetry breaking were selected to match those of the model in Ref. [10], so that the results of the global fit to fermion masses and mixing angles presented in that work would trivially carry over to the present case.
However, since the gauged Peccei-Quinn mechanism renders the axion a linear combination of two scalar fields, some parametric differences in the flavor-changing axion couplings arise relative to our earlier results [10]; taking these into account, we showed that the most stringent bound from strange meson decays was trivially satisfied. We also showed that the ratio of the electromagnetic to color anomalies in this model was the same as in the original model proposed in Ref. [10], (i.e., 8/3), the same as the prediction of the simplest DFSZ axion models [35]. We confirmed that both models we presented were consistent with the relevant low-energy constraints on the flaxion couplings for a range of model parameters. Since our model involved a relatively large sector of heavy fermions, some charged under U(1) Y and all vector-like under the standard model gauge group, we considered the accelerated running of the hypercharge gauge coupling at higher energy scales, confirming that its Landau pole remains above the ultraviolet cut off of our effective theory. It is likely that flavor models exist that allow a simpler adaptation of the mechanism of Ref. [17] to address the problem of flaxion quality. Finding the models that allow the simplest implementation may provide a clue as to which flaxion models are more likely to be realized in nature.