Color Unifed Dynamical Axion

We consider an enlarged color sector which solves the strong CP problem via new massless fermions. The spontaneous breaking of a unified color group into QCD and another confining group provides a source of naturally large axion mass $m_a$ due to small size instantons. This extra source of axion mass respects automatically the alignment of the vacuum, ensuring a low-energy CP-conserving vacuum. The mechanism does not appeal to a $Z_2$ 'mirror' copy of the SM, nor does it require any fine-tuning of the axion-related couplings at the unification scale. There is no light axion and uncharacteristically the lighter spectrum contains instead sterile fermions. The axion scale $f_a$ can be naturally brought down to a few TeV, with an exotic spectrum of colored pseudoscalars lighter than this scale, observable at colliders exclusively via strong interactions. The $\{m_a, f_a\}$ parameter space which allows a solution of the strong CP problem is thus enlarged well beyond that of invisible axion models.


Introduction
Phenomenological analyses based on chiral perturbation theory and supported by lattice computations indicate that all Standard Model (SM) quarks have non-zero masses. This disfavors the solution to the strong CP problem via one massless SM quark, which automatically guarantees a U (1) axial invariance at the classical level. The interesting possibility of having a massless up quark in the microscopic theory which appears as massive at QCD scales due to non-perturbative instanton contributions [1] does not seem to be realized in nature, even if this option is not completely excluded [2][3][4][5][6][7].
It is still possible to solve the strong CP problem using massless fermions if the SM up quark is massive. The idea is to enlarge the SM gauge group with a new confining sector [8][9][10], whose scale is much larger than that of the QCD group, SU (3) c . Extra massless quarks charged under both QCD and the new confining sector may realistically solve the problem [11]. A new spectrum of confined states results.
The term "axion" denotes any (pseudo)Goldstone boson (pGB) of a global chiral U (1) symmetry which is exact at the classical level but has anomalous couplings to the field strength of a confining group. 1 Axions are characteristic of solutions to the strong CP problem based on an anomalous U (1) axial symmetry, usually called Peccei-Quinn (PQ) symmetry [12]. When the number of axions in a given theory outnumbers the total number of distinct instanton-induced scales other than QCD to which they couple, one (or more) light axions remain. Axions which are not elementary but composed of fermions are typical of theories with an enlarged confining sector and are often referred to as "dynamical" or composite axions. Very heavy dynamical axions made out of massless fermions will typically acquire the bulk of their mass from the largest instanton-induced scale Λ to which they exhibit anomalous couplings, For a very light axion coupled only to QCD, the mixing with the η pseudoscalar is relevant instead, and that axion obeys the usual relation [13,14] m 2 a f 2 a ∼ m 2 where m π , f π , m u , m d denote the pion mass and coupling constant, and the up and down quark masses, respectively. The first step in the direction of solving the strong CP problem with exotic massless quarks was the proposal by K. Choi and E. Kim [11] to enlarge the confining gauge sector of the SM to SU (3) c × SU (Ñ ), with the latter having a scale larger than that of QCD,Λ Λ QCD . Two confined charges would then exist in nature, color and axicolor respectively, and correspondingly two distinct sources of instanton potentials. A massless color-triplet quark Q, charged also under axicolor and singlet under the SM electroweak symmetry, would solve the QCD strong CP problem. The fact that the axicolor scale is very large would explain the non-observation of exotic bound states at low energies. An issue arises because there are now two potentially harmful vacuum angles to absorb: θ c of QCD andθ. Only one combination of them would be redefined away by a chiral rotation of Q. This was easily remedied by adding a second exotic quark χ charged only under SU (Ñ ). In the limit of vanishing QCD coupling, the SU (Ñ ) sector described four flavors which seed two singlet pseudoscalars with anomalous couplings: two dynamical axions. Finally, taking into account the SM quark sector and thus the SM η , three flavor-singlet pseudoscalars result in the low-energy spectrum for only two instanton sources of masses: the η , a very heavy axion with mass ∼Λ and a second axion almost massless and obeying Eq. (2). Because of this last axion, the axicolor construction can be seen as an ultraviolet dynamical completion of the invisible axion paradigm. As usual, a very large f a scale is required to be orders of magnitude larger than the electroweak (EW) one, albeit with the advantage of being free from scalar potential fine-tunings.
In a different and recent attempt [15] to solve the strong CP problem with extra massless quarks, the same SU (3) c × SU (Ñ ) confining sector is considered. No light axion remains in the low-energy spectrum, though, as only two pseudoscalar mesons are present which couple to the two anomalous currents: the customary η meson and one axion. This is achieved by assuming only one exotic massless quark Q instead of two. In the limit of vanishing QCD coupling, the SU (Ñ ) sector then describes only three flavors, resulting in only one gauge singlet pseudoscalar with anomalous SU (Ñ ) couplings: a dynamical axion. Both the η and the axion thus acquire a mass, and the axion mass is induced by SU (Ñ ) instantons, m a , f a ∼Λ. For such a heavy axion the f a scale can be as low as the TeV range without incurring unacceptable phenomenological consequences. The issue of the two θ parameters is solved in this proposal by imposing a discrete Z 2 symmetry relating the two sectors. Unfortunately, the practical implementation of this idea requires a complete Z 2 -"mirror" copy of the SM. The Z 2 symmetry is explicitly broken by a scalar potential which gives the second Higgs field a very large vacuum expectation value, e.g. 10 14 GeV, in order to sufficiently modify the running of the two confining scales. This is overall a quite complex, tuned, and large structure.
Even more recently, the SM massless quark avenue has been revisited in a theory that involves a product of SU (3) groups which break spontaneously to QCD [16]. A very interesting aspect developed by the same authors in a previous work [17] is the impact of small size instantons of the group which undergoes spontaneous symmetry breaking. These instantons are shown to provide a possible extra source of large masses for the putative axions of the model.
We will develop in what follows a new solution to the strong CP problem via massless fermions, in which the issue of the different θ parameters that arise in the presence of two or more confining groups is solved via color unification. Color unification with massless quarks is attempted here for the first time. This path is an alternative to the axicolor-type constructions and will lead to different phenomenology. QCD will be unified with another confining sector singlet under the electroweak gauge symmetry. The color unified theory (CUT) breaks spontaneously to QCD and another confining group. The small-size instantons of the unified color group provide an extra source of high masses for the axions of the theory, and it will be shown that no axion remains at low scales. The exotic low-energy spectrum is instead fermionic. Furthermore, it will be shown that interesting new phenomenological signals can be explored at colliders. The complete ultraviolet completion of this idea will be developed, implementing two different scenarios: in one of them the two resulting heavy axions are dynamical, while in the other one axion is elementary.
The structure of the paper can be easily inferred from the Table of Contents.

SU(6) Color Unification
We propose a scenario in which QCD is unified with another confining group into SU (6), and a single, strictly massless SU (6) fermion rotates away simultaneously all θ parameters. The unification path in the context of an extended strong sector to solve the strong CP problem was first proposed by Rubakov long ago, [18] in a Grand Unification construction that relied on traditional models à la DFSZ [19,20] with massive exotic fields, and required a Z 2 mirror copy of the complete SM field content. Another recent attempt [21] using unification ideas also relied on massive exotic fermions à la DFSZ. Here we instead consider color unification in the presence of massless fermions. The massless SU (6) fermion belongs to the 20 representation of the SU (6) CUT, having a definite chirality (e.g. left-handed) while being a singlet of the SM SU (2) L ×U (1) Y gauge symmetry: At a color unification scale Λ CUT much higher than the EW one, the SU (6) group breaks into The parameters θ c of SU (3) c andθ of SU (3) are necessarily equal and unphysical down to the unification scale, and will remain so even below the unification scale as long as Ψ remains massless, protected by chiral symmetry. Under spontaneous symmetry breaking of the CUT symmetry, Ψ decomposes as where the charges under the U (1) group in Eq. (3) are shown in parenthesis for completeness. If the components of the Ψ L field are to remain massless under the CUT scale, SU (3) must confine. The two confining scales Λ QCD andΛ need to be separated withΛ Λ QCD , as no bound states are observed other than those compatible with QCD. The 20-dimensional representation is thus advantageous because all its components charged under QCD are also charged under SU (3), and so will form bound states at the higher scaleΛ. This representation is also pseudo-real, and so the theory is anomaly free. The non-trivial issue of how to separateΛ and Λ QCD is discussed further below.
The colored-axicolored massless fermions in Eq. (4) will be denoted ψ L,R , see Tab. 2, while ψ ν will refer to the singlet massless fermions to convey that they act like sterile 2 neutrinos. The ψ ν fields only connect to the other fields through the unified strong forces, and thus their couplings to the visible universe will be safely suppressed by Λ CUT , provided the U (1) gauge group in (3) is also broken near that scale. 3 SU (6) color unification is thus a successful path to solve the strong CP problem, and this fact will remain at the heart of the developments in this paper. The remaining problem is to obtain a low-energy spectrum which is fully compatible with observations. Table 2: The massless fermion sector of the SU (6) construction below the unification scale. The notation is such that The SM fermions Because of color unification, the SM quarks must belong to SU (6) multiplets. The simplest option is to include them in six-dimensional fundamental representations. For each fermion generation, where q L , u R and d R denote the SM quarks, while their SU (6) partners are signaled by tildes. Theq L fields are necessarily electroweak doublets, and this character turns out to be the major practical issue of this model: -Leaving the tilde-quark sector massless but confined is unacceptable, as the condensateassuming chiral symmetry breaking of SU (3) -would typically break the SM EW symmetry at the largeΛ scale.
-Alternatively, giving much larger masses ( ≥Λ ) to the tilde quarks is not viable either in this SU (6) setup without spoiling SM quark masses, since they belong to the same multiplet. If a scalar field gave high masses to the tilde quarks 4 by obtaining a high vacuum expectation value (vev), that scalar field would have to be an SU (2) L doublet. Then its large vev would spontaneously break SM EW symmetry, giving gigantic masses to the W and Z boson.
The main problem of this model is then the unacceptably light tilde-fermion sector. We will develop next an extension whose only purpose is precisely to achieve high masses for the tildesector quarks, decoupling them from the low-energy spectrum. By the same token, the necessary separation of Λ QCD and a larger confining scale will naturally follow. 5 We will develop in detail two realistic ultraviolet (UV) completions.
3 The realistic Color Unified Theory: It is necessary to give large masses to the tilde-quark sector without giving masses to the SM quarks, a challenging enterprise as explained above due to the SU (6) unification. An external mechanism is ideal for this task. The color unified SU (6) group which contains QCD is enlarged via an external non-abelian SU (3 ) group with additional fermions charged only under the latter. In fact, all fermions in the theory will be charged under only one of the two groups, SU (6) or SU (3 ). Ψ L will be thus taken to be a singlet of SU (3 ) and the same applies to the multiplets in Eq. (5) which contain the SM quarks. The two sectors are connected exclusively via a new scalar ∆. QCD remains a subgroup of SU (6), whose θ-parameter is rotated away by the massless Ψ L fermion in Eq. (4). This type of auxiliary extension was suggested in Ref. [21] to give high masses to exotic fermions in a different context. The field content of our model is summarized in Tab. 3, in which all fermions except Ψ L will become massive. It is easy to see that the theory with this matter content is anomaly free. The scalar ∆ appearing in the table belongs to the bifundamental of SU (6) × SU (3 ), and its vev breaks color unification at a scale Λ CUT , taken to be much larger than all SM scales, The fermion quantum numbers under the two resulting groups are also shown in Tab. 3. A simple CUT-invariant Yukawa Lagrangian which connects the SU (6) and the auxiliary SU (3 ) extension reads The CUT symmetry is spontaneously broken upon ∆ taking a vev of the order of the CUT breaking scale Λ CUT This breaking generates a large mass for both the tilde-and prime-quark sectors, leaving massless only the SM fermion components of the original fermionic fields: 6 Unless otherwise stated, we will assume in what follows that all κ i Yukawa couplings are O(1), meaning all tilde and prime fermion masses are of order Λ CUT . Some tuning of the κ i values could be acceptable, though, as discussed further below.
The SM fermions get their masses through the usual SM Higgs doublet Φ, which in this model is a singlet of SU where Y SM i denote the SM Yukawa couplings. Analogously, the most general Lagrangian compatible with all symmetries discussed above allows us to write Yukawa couplings of the Higgs field to the prime-sector fermions, Eqs. (10) and (11) induce contributions to the tilde and fermion masses which are quantitatively irrelevant in comparison with those from Eq. (9). In addition, the couplings in Eq. (11) will be absent for symmetry reasons in one of the models to be developed in this paper (model II in Sect. 3.2). Both SU (3) c and SU (3) diag can now remain unbroken and confine at two different scales, Λ QCD and Λ diag , with Λ diag Λ QCD . The task of achieving different values for the two confining scales and getting rid of the tilde sector or any other dangerous exotic sector is thus accomplished.
Note that the Yukawa-type Lagrangian in Eq. (7) has an inherent global U (1) symmetry under which only the prime fermions and the ∆ field would transform -a generalized Baryon number symmetry in the prime sector, with charges This symmetry is not chiral, thus not anomalous under SU (3 ), and irrelevant to the strong CP problem. An associated pGB 8 results after spontaneous breaking, albeit with its couplings safely suppressed by the CUT scale and interacting only with the very heavy prime and tilde sectors. 9 Finally, the theory below Λ CUT contains phenomenologically interesting bound states formed from the massless ψ L,R fermions, to be studied below. The spectrum of free eigenstates below the EW scale contains the usual SM spectrum, plus a harmless pGB and sterile neutrinos. 6 Note that we take ∆ to be real. The phases of the nonvanishing entries in Eq. (8) can all be made equal by an SU (6) × SU (3 ) transformation; the remaining phase can be removed by a transformation under the U (1) defined in Eq. (12). 7 In this notation taken from unified models the contraction of the spinor indices is implicit, more precisely the first term would read Q T L CΦU c L , where C = iγ2γ0 is the charge conjugation matrix. 8 This symmetry is broken at loop level by SU (2)L sphalerons, in the same way that in the SM baryon number current is anomalous. For our purposes this effect is negligible. 9 As suggested in Ref. [17], this type of pGB could be entirely removed by gauging the U(1) group. There is no real need to implement this procedure in our case, though, given the strongly suppressed couplings of this pGB. θ issue The extension of the strong sector by the auxiliary external group SU (3 ) brings a new θ parameter into the game: where G i denote gauge field strengths with tensorial indices omitted. G c , G 6 , G and G diag correspond respectively to the SM QCD gauge group, SU (6), SU (3 ) and SU (3) diag . While the rotation of the massless field Ψ was designed to reabsorb θ 6 and ultimately θ c , θ may source back a SM strong CP problem through the contamination to the visible sector via the ∆ scalar. Indeed, at low energies the massless quark ψ transforms as a (3,3), therefore the phase θ 6 cannot be fully reabsorbed in the Lagrangian since the chiral rotation that removes the SU (3) c θ-term generates a new contribution to the SU (3) diag topological term. Ref. [21] acknowledges this issue (in the context of a different model which does not rely on massless fermions) and leaves it unsolved hoping that some UV completion solves it. In what follows, we will determine and exhaustively analyze two UV solutions, via the simple addition of either -An extra massless fermion transforming only under SU (3 ).
-A second bifundamental scalar field, which automatically endows PQ invariance to the above extension procedure.
The first solution is more in line with the spirit of the present paper, as all θ parameters inducing a strong CP problem are made unphysical via massless fermions, and it is developed next.

Model I: Adding a massless fermion charged under SU (3 ).
The θ parameter of the auxiliary SU (3 ) gauge group can be made unphysical by the addition of a massless fermion field χ that transforms as a fundamental of SU (3 ) and is an EW and SU (6) singlet. In other words, the field content for this solution is that previously shown in Tab. 3 plus the massless fermion χ with quantum numbers shown in Tab. 4. Additional composite bound states will result from χ, among them composite pseudoscalars with anomalous couplings -dynamical axions -whose masses are discussed further below.

Running of the coupling constants.
The CUT breaking pattern in Eq. (6) imposes the following relations among the gauge couplings with the constraint where α c , α diag , α and α 6 denote respectively the coupling strength of QCD, SU (3) diag , SU (3 ) and SU (6). As shown in Fig. 1, there is a discontinuity in the running of the coupling constants at the CUT-breaking scale that allows α to have large values while reproducing the known QCD running at low scales. Those α values will seed a source of large axion masses, as discussed in Sect. 3.1.3 further below. Although the relation in Eq. (14) imposes α diag (Λ CUT ) < α c (Λ CUT ), the presence of the SM q L , u R , and d R quarks at energies well below Λ CUT slows down the running of QCD with respect to that of SU (3) diag . In this regime ψ and χ are the only fields left charged under SU (3) diag (thẽ q sector generically decouples as their mass scale is set by Λ CUT , see Eq. (9)). As a consequence, α diag runs faster and thus the SU (3) diag group confines at a higher scale than Λ QCD , see Fig. 1. This mechanism easily achieves the separation of the two confining scales. We computed both the one-and two-loop running and the latter actually reinforces the pattern, as illustrated in the figure for the choice Λ diag = 4 TeV. Lower values of Λ diag are also phenomenologically acceptable, see Sect. 4.1 below.

Confinement of SU (3) diag and pseudoscalar anomalous couplings to the confining interactions
At the scale Λ diag , SU (3) diag confines and the remaining massless fermions (ψ and χ, see Tab. 5) will form massive QCD-colored bound states. In the limit in which α c is switched off, the SU (3) diag Lagrangian exhibits at the classical level a global flavor symmetry U (4) L × U (4) R −→ U (4) V . 10 The chiral symmetry is spontaneously broken by the quark condensates ψ L ψ R and χ L χ R . This decomposed here in terms of their QCD charges. There is a QCD octet plus a singlet with flavor contentψψ (3 c × 3 c = 8 c + 1 c ). The two QCD triplets, 3 c and3 c , correspond to the combinations ψχ andχψ. Finally, a color-singlet composite state is made out ofχχ. The fourteen colored mesons in Eq. (16) acquire large masses induced by gluon loops that are quadratically divergent and therefore sensitive to the cutoff scale Λ diag , 11 The remaining two QCD singlets will be denoted here by η ψ and η χ and are shown next to be dynamical axions. The associated currents are where f d denotes the SU (3) diag pGB scale, with Λ diag ≤ 4πf d . These classically conserved currents are broken at the quantum level by the SU (6) and SU (3 ) instantons, and so the currents are anomalous. The anomalous terms are These anomalous terms modify the classical equations of motion of the η ψ and η χ , and give rise to an effective Lagrangian, (24) η ψ and η χ are thus two dynamical axions. It is to be stressed that the PQ scale in this model is f d ∼ Λ diag for both axions and not the much larger Λ CUT scale. When the SM quarks are taken into account, the η QCD pseudoscalar meson is also present at energies below the QCD confinement scale, and the effective Lagrangian of anomalous couplings reads where Λ QCD ≤ 4πf π . As a consequence, the two instanton-induced scales Λ QCD and Λ diag provide a contribution to the masses of the pseudoscalars which have anomalous couplings; the corresponding effective potential is very well approximated by It follows that there are only two sources of mass (disregarding corrections from SM quark masses) for three states coupling to anomalous currents: η QCD , η ψ and η χ . In the absence of supplementary mass sources, one axion would get a mass of order Λ diag while another one would have remained almost massless, see Eqs.

Impact of small-size instantons of the spontaneously broken CUT
There is an additional and putatively large contribution to the axion mass(es) in the presence of a spontaneously broken theory: the small-size instantons (SSI) of the theory at the breaking scale, as pointed out long ago in Refs. [22][23][24] and very recently in Ref. [17]. SSI can induce a large mass even for perturbative theories if the breaking scale is large enough to overcome the exponential suppression of instanton effects. In our model, the instantons of the color-unified theory in Eq. (6) near the Λ CUT scale provide automatically this third source of axion mass. The SU (6) SSI can be neglected because of the smallness of α 6 at Λ CUT (e.g. see Fig. 1) and the analysis below will focus on the SU (3 ) SSI instantons.
It is well known [25][26][27][28] that, in the absence of fermions, the effective Lagrangian that describes instanton configurations for a pure Yang-Mills theory SU (N c ) induces a scale Λ inst in the instanton potential given by where ρ is the instanton size, D[α ] is the dimensionless instanton density, The constant C inst reads [29,30] and the function c(x) defined in Ref. [25] such that c( 1 /2) = 0.145873 and c(1) = 0.443307. For the SU (3 ) instantons of our model C inst = 0.0015. 12 In order to compute the integral in Eq. (27), the running of the coupling constant α (µ) must be included. At one loop this reads where α CUT ≡ α (Λ CUT ) and b is the one-loop β-function coefficient. For the spontaneously broken theory, only the SSI instantons with size ≤ 1/Λ CUT are relevant, This has the form where f (α , b) is given by For instance for the benchmark value α CUT = 0.3, the value of SU (3 ) SSI-induced scale in the absence of fermions (for which b = 10) is Nevertheless, the presence of fermions dramatically changes the value of this scale [27]. A suppression factor appears, which results from the interplay of the instantons of the theory and the fermionic spectrum. For the SU (3 ) theory under consideration, the prime-fermion Yukawa couplings are relevant. For generic values of all Yukawa couplings of order one, the dominant contribution stems from the y i couplings in Eq. (11). They are illustrated by the one-instanton "flower" contribution in Fig. 2a. Its impact is to suppress the pure gauge result in Eq. (27) by two factors: the χ chiral condensate and the y i Yukawas couplings of the primed sector, χχ is the order parameter controlling SU (3) diag chiral symmetry breaking and thus expected to be χχ −Λ 3 diag . For ρ ≤ 1/Λ CUT , the product ρ 3 χχ 1 reduces the SSI-induced scale by orders of magnitude, with where, in the presence of N f Dirac fermions, Figure  The integral in Eq. (36) can be computed exactly, although a good estimation follows from the approximation which leads to For the benchmark α CUT = 0.3, and substituting N f = 7 and b = 16/3, this gives The complete computation including fermions can be compared with the one in in Eq. (34), a strong suppression by a factor of order (Λ diag /Λ CUT ) 3 is in fact present.
There is in addition a subdominant contribution to the SSI scale, suppressed by the χ chiral condensate, the product of κ i Yukawa coupling of the prime-fermion sector in Eq. (7) and the product of SM Yukawa couplings. This contribution is illustrated by the instanton "double flower" in Fig. 2b, and given by where the power of the 4π factor results from the 6 SM Yukawa couplings and the 12 κ i couplings in the product. For ρ ≤ 1/Λ CUT , this contribution is well approximated by In summary, putting together the dominant and subdominant instanton contributions discussed, the SSI scale is given by Overall, the size of the new scale Λ SSI is quite sensitive to the value of the SU (3 ) coupling constant at the CUT-breaking scale. Fig. (3) illustrates the η χ axion mass induced by the small size instantons. For the benchmark examples studied, Λ SSI significantly affects the properties of the pseudoscalars. It provides a new contribution to the effective potential of the form A mass is thus generated for the η χ axion, given by where the replacement Λ diag 4πf d has been used. The other dynamical axion of the theory has been shown to acquire a mass of order Λ diag , see Eq. (48). Both dynamical axions have thus acquired masses of order TeV, as a direct and unavoidable consequence of the instanton potentials inherent to the theory.
How light can the axion that couples to SSI become?
The y i values are very relevant for the size of SSI scale and a priori provide the dominant contribution, as explained above. Nevertheless, the mass spectrum and thus the running of coupling constants is basically unaffected by them. Should the y i couplings be negligible, Λ SSI would be determined by the second term in Eq. (43). How small can this scale become, and thus how light can the axion coupled to it be? For vanishing y i values and generic κ i couplings of O(1), the product of Y SM i /4π factors in the second term in Eq. (43) would suppress the η χ axion mass values illustrated in Fig. 3 by a factor of O(10 −6 ). In other words, the η χ axion would then have a mass smaller than m η ψ ∼ Λ diag and under the TeV scale. Can this axion be lighter still by assuming both negligible y i couplings and small κ i couplings? The answer is yes although this possibility is limited by the fact that the κ i couplings determine essentially the spectrum of the tilde sector which, if lighter, may strongly impact the running of the coupling constants. To get an estimate neglecting y i couplings, with all κ i values of O(10 −1 ) there is little impact on the running while the Λ SSI scale is low enough to make the η χ axion accessible at colliders; κ i values of O(10 −2 ) also allow a realistic setup and would bring down the η χ mass even under the GeV regime and maybe it could be as light as an invisible axion. Other patterns of κ i couplings may also be possible. Nevertheless, in what follows we will not pursue this ad hoc avenue of fine-tuning the y i and κ i couplings to very small values. Unless stated otherwise, O(1) values will be assumed for all prime-fermion Yukawa couplings.

Solution to the strong CP problem.
It is pertinent to briefly re-check the status of the strong CP problem after taking into account the impact of the SSI of the spontaneously broken symmetry discussed above. Any new mass term for the axions breaks the PQ symmetry and therefore perturbs the axion potential; it is then important to verify that the vevs of the axions remain in the CP-conserving minimum, solving the strong CP problem. Indeed, this is the case with our color-unified proposal as, according to Eq. (13), the potential including θ i dependencies explicitly reads For this potential, the minimum is CP-conserving: since all θ i dependences cancel. A word of caution is pertinent as the exact dependence of the potential on the phases of the different couplings in the Lagrangian which participate in fermion mass generation (κ i , Y SM i , y i , . . . ) remains to be computed. Nevertheless, the two massless fermions Ψ and χ guarantee that at energies above CUT the two parametersθ 6 andθ are unphysical. Below CUT, the spectrum of the theory is exclusivley the SM one plus massless fermions, and the EW SM contributions are known to be negligible [31], even if a mismatch remained in spite of the low-energy presence of the massless quarks. An explicit computation of the threshold effects is elaborate and it is left for future work. Note that without the presence of the second PQ mechanism, that is, without the presence of the η χ field and its vev, all θ i in Eq. (13) would not have been reabsorbed, while Eq. (47) demonstrates that its inclusion does ensure a CP-conserving minimum.
It is very positive that in this model there is no contribution to the EW hierarchy problem coming from axion physics. No potential connects the EW and axion scales: the PQ scale f a 13 is set by Λ diag and not Λ CUT , and all axions are dynamically generated. This is a feature that our model I shares with the original axicolor model, and in general with models of composite dynamical axion(s). There remains instead the customary fine-tuning in spontaneously broken unified theories, as Λ CUT and the EW scale are connected via the scalar potential, but the latter does not communicate to our PQ mechanism.
3.1.5 Computation of the pseudoscalar mass matrix: η χ , η ψ , η QCD and light spectrum After the replacement of the pGBs with anomalous couplings by their physical excitations, η χ −→ η χ + η χ , η ψ −→ η ψ + η ψ , the effective low-energy Lagrangian for the axions and the SM η QCD field is given by (disregarding the effects of SM quark masses) Expanding to second order in the fields yields the following mass matrix: As advertised, only the usual QCD η QCD phys remains as a light eigenstate, while the two composite dynamical axions will be very heavy: one will have typically a mass of tens of TeV and the other will be orders of magnitude heavier. 14

Low energy spectrum and observable effects
The two dynamical axions η ψ, phys and η χ,phys are typically heavier than the TeV scale and thus not easy to directly observe at the Large Hadron Collider (LHC). Under a few TeV the spectrum of the theory contains: -The SM pseudoscalar meson η QCD phys , plus the rest of the SM hadronic spectrum.
-The exotic QCD-colored "pions" -color octets and color triplets -whose masses are given in Eq. (17) as m 2 ∼ α c Λ 2 diag . With masses naturally lighter than the TeV scale, these QCD-colored pions can be easily produced at the LHC.
-The two sterile fermions stemming from the 20-representation Ψ. They are basically invisible as their interactions with the visible world are suppressed by Λ CUT , which is much larger than Λ diag without any tunning.
-Possibly, a GB associated with generalized baryon number. This GB is harmless as its interactions are suppressed by Λ CUT . It can also easily be made arbitrarily heavy by gauging that global symmetry.
A further pertinent comment is that the massless χ fermions may themselves acquire an effective mass due to the instantons of SU (3) d and to SU (3 ) SSI instantons, similar to the effective mass in QCD for a hypothetically massless SM up quark. This is an interesting question which will not be further developed in this paper.
The very interesting phenomenological bounds and detection prospects for the exotic QCDcolored mesons will be quite similar to those applying to the next model. The ensemble will then be briefly developed in Sect. 4 further below. The same applies to the cosmological consequences of the two color-unified UV completions developed in this paper.

Model II: Addition of a second ∆ scalar.
This solution to the θ problem is an alternative to extending the spectrum by a massless fermion, discussed in the previous subsection. In this second model no extra fermion is added to the SU (6) × SU (3 ) Lagrangian, while a second ∆ field will be considered instead. The spectrum is that in Tab. 3 albeit with the scalar line duplicated, ∆ → {∆ 1 , ∆ 2 }. This simple extension allows the implementation of a PQ symmetry which reabsorbs the θ contribution to the strong CP problem. The corresponding PQ symmetry is automatic if the terms in Eq. (11) are omitted and Eq. (7) is replaced by This Lagrangian is invariant under two independent abelian global symmetries; one of them is anomalous with respect to SU (3 ) and corresponds to the PQ charge assignment 15 The vevs of ∆ 1 and ∆ 2 generalize the CUT spontaneous breaking in Eq. (8) and at the same time break spontaneously the PQ symmetry; therefore, this PQ scale coincides with the CUT scale. This distinguishes model 2 from model 1, as in the latter the PQ scale coincided with Λ diag . A pGB -an elementary axion-is generated at this stage. The corresponding PQ conserved current is given by The ∆ i fields are parameterized as where v ∆ 1 and v ∆ 2 denote respectively the ∆ 1 and ∆ 2 vevs, both of which we take to be real for simplicity. Decoupling the heavy radial modes, the PQ current reads where the elementary axion field a(x) corresponds to the GB combination with This classically exact PQ symmetry is broken at the quantum level by the SU (3 ) anomaly, which at lower energies translates into an anomalous current for the SU (3) diag gauge theory.
where N and N diag are the group factors, and where T a P Q corresponds to the PQ generator and t b = λ b 2 to the Gell-Mann matrices for the SU (3) generators. The anomalous term modifies the classical equations of motion of the axion, and gives rise to an effective Lagrangian, The impact of the SSI of the spontaneously broken theory will again add further contributions, inducing a putatively high mass for the elementary axion as discussed further below.

Running of the coupling constants
The matter content allows SU (3) diag to confine at higher scales than the QCD group SU (3) c as in model I. The separation of both scales is made even sharper in the model II because α diag runs faster. In model II, only one massless fermion charged under SU (3) diag is present under the CUT scale (compare Tabs. 5 and 6). We have estimated both the one and two-loop running, as illustrated in Fig. 4 for Λ diag = 4 TeV.

Confinement of SU (3) diag and pseudoscalar anomalous couplings to the confining interactions.
At the scale Λ diag the QCD coupling constant is small, and the SU (3) diag spectrum with only one massless fermion in Tab The gauge QCD group SU (3) c is again a subgroup of U (3) V which remains unbroken. The octet of pGBs colored under QCD will acquire large masses due to gluon loops, which at one-loop is given by The QCD singlet 1 c , denoted η ψ , is a dynamical axion. Note that it has the same quark composition as the η ψ meson in model I. The η ψ couples to both the SU (3) diag and SU (3) c anomalies, resulting in a low-energy effective Lagrangian for this axion given by In summary, this solution to the strong CP problem is a hybrid one with two axions: a heavy dynamical axion η ψ with mass of order Λ diag stemming from a PQ symmetry which reabsorbs the original θ SU (6) (and thus θ QCD ) parameter as in the previous section, and a second elementary Figure 5: Instanton contribution in model II. The long dashed lines connecting the SM Yukawa interactions correspond to φ propagators while the short dashed lines depict ∆ 1 or ∆ 2 propagators. axion a resulting from solving the external SU (3 ) sector à la PQWW [12,13,33]. Up to now, only two sources of masses have been identified for the ensemble of three pseudoscalars with anomalous couplings (η QCD , η ψ and a). We analyze next the SSI of this model which provide a large source of axion mass for the elementary axion a.

Impact of small-size instantons of the spontaneously broken CUT
The analysis of SSI for model II under discussion is simpler than that for model I developed in the previous subsection. No massless fermions charged under SU (3 ) are present in model II (in contrast with model I). Furthermore, PQ symmetry forbids here the y i Yukawa couplings which gave the dominant contribution in model I. In consequence, the terms proportional to κ i and mediated by the ∆ 1 and ∆ 2 scalars and the Higgs field will dominate Λ SSI . This is illustrated by the instanton "flower" in Fig. 5. It results in Λ SSI given by which can be written as (67) with b = 5 in this case. Here the approximation in Eq. (38) is no longer valid, and the result corresponds to that of the pure Yang-Mills case (Eq. (32)) with the extra suppression factor of the Yukawa couplings, where the function f (α CUT , b) is defined in Eq. (33). This result translates into a new contribution to the instanton-induced effective potential of the form δL ef f = Λ 4 SSI cos 12 Taking into account that in this model the elementary axion scale coincides with the CUT scale, it follows that for α CUT = 0.3, The linear dependence of m a on Λ CUT implies that this second axion is generically much heavier than the corresponding one in model I (e.g. Eq. (45)), see Fig. (6) for illustration. In summary, the dynamical axion has a mass of order Λ diag and thus of a few TeV or above, while the elementary axion is generically extremely heavy (with again the caveat in case of Yukawa fine tunings discussed in Sec. 3.1.3). As in model I, no light axion remains.

Solution to the strong CP problem.
The minimum of the axion potential can be easily shown to remain CP-conserving after including all contributions to the axion masses. Indeed, the θ i dependence of the Lagrangian can be again read off of Eq. (13), for which the following bosonic vevs lead to a CP-conserving minimum, 12 a f a +θ = 0 , We recall once again that computing the exact dependence of the potential on the phases of the Yukawa couplings is a task which remains for future work. After the replacement a −→ a + a, η ψ −→ η ψ + η ψ and introducing as well the QCD η QCD field, the effective low-energy mass Lagrangian for the physical mesons which couple to anomalous currents is given by where all the CP violating phases have been relaxed to zero.

3.2.5
Computation of the pseudoscalar mass matrix: a, η ψ , η QCD and light spectrum Taking into account all contributions except the SM quark masses, the following mass matrix results for the singlet pseudoscalars of the theory which couple to anomalous currents: (75) In this model both, the usual QCD η and the axion a phys remain as light eigenstates, while the other eigenstate η ψ, phys will have a mass generically above the TeV scale. Apart from the axion, the lowest set of exotic states is an octet of exotic "pions" whose masses are TeV, see Eq. This model II with an additional scalar may be less appealing than than model I with an extra massless fermions for two reasons: a) its axion sector contributes directly to the EW hierarchy problem, as its elementary axion results from a scalar potential which a priori communicates with the Higgs potential; b) it is a hybrid model with both one elementary and one dynamical axion, while model I is more aligned with the spirit of solving fully the strong CP problem via massless fermions.

Phenomenological and cosmological limits on the lightest exotic states
A common feature of both ultraviolet complete models constructed above is that the generic spectrum under the EW scale is the SM spectrum plus sterile fermions, in contrast with usual axion models.

Collider observable signals
The lowest set of observable exotic states are expected to be the exotic SU (3) c -colored "pions" whose masses may lie under the TeV scale. These resulted from the chiral symmetry breaking of the confining group SU (3) diag . In model I, QCD color-triplet and color-octet bound states are made out of ψ and χ massless fermions, shown in Tab 5 and Eq. (17). Model II gives only color octets composed of ψ fermions, as shown in Tab. 6 and Eq. (63).
In this color-unified axion solution, the exotic fundamental fermions have no SM SU (2) × U (1) charges. The heavy pions will be produced in colliders only via QCD interactions, e.g. gluon-gluon couplings, through which they also presumably decay before they can hadronize to make colour neutral states. As they are coloured, they do not mix with ordinary pions or other visible matter.

Color-octet pions from SU (3) diag confinement
The lightest scalar octets in Eqs. (17) and (63) are denoted by π d ≡ 8 c . Their effective coupling to QCD gluons can be written as where D µ denotes the SU (3) c covariant derivative and d abc is the corresponding symmetric group structure constant. The second term in Eq. (76) results at one-loop from the triangle diagram with the fermions χ running in the loop. The color-octet exotic pions can thus be either produced in pairs through the gluon-gluon-π d -π d coupling in the kinetic term, or singly produced through their anomalous coupling to gluons. The kinetic term dominates the scalar octet production channels, while the second term allows the π d decay, yielding a dijet final state. Experimental limits on scalar octet pair production via the gluonic interactions in the kinetic term can be inferred by recasting searches of sgluons. A recent search of sgluon pair production by ATLAS using 36.7 fb −1 of √ s = 13 TeV data [32], whose prediction was obtained from the NLO computation in Ref. [34] for √ s = 8 TeV (rescaled to 13 TeV according to Ref. [35]), sets a bound on the octet scalar exotic pions given by From Eqs. (17) and (63), this translates into For model I, an alternative bound may be inferred from the limits on color-triplet scalars, which can be produced via their color interactions. In the absence of couplings which mediate their decay (as the second type of coupling in Eq. (76) is not possible for scalar triplets), they will bind with SM quarks to form stable hadrons. This search is expected to result in a sensitivity similar to the one above [36].
It is very interesting to pursue the experimental search for colored pseudoscalars and stable exotic hadrons. Their detection would be a powerful indication of the dynamical solution to the strong CP problem proposed here.

Dynamical axion and exotic fermions
The dynamical axion denoted above by η ψ, phys with instanton-induced mass of order Λ diag , Eqs. (50) and (75), can a priori be either pair-produced through the kinetic coupling or singly produced through the dimension five anomalous operator. It would decay dominantly to two back-to-back jets and can be searched for in dijet resonance searches. Its production, however, may be suppressed by its high mass. For instance, for Λ diag 2.9 TeV, m η ψ, phys ∼ 90 TeV, which is beyond the reach of present collider searches. The other mesons and baryons resulting from the SU (3) diag confinement have masses in the TeV range and above; they would lead to collider signatures similar to those for the exotic pseudo-goldstone bosons.

The axion coupled to SSI
In model I with its additional massless fermion, this second axion is also dynamical and was denoted by η χ, phys . For the benchmark case with Yukawa couplings in the prime-fermion sector of order one, its mass is expected to lie above that of the η ψ, phys and out of reach, as shown in Eq. (50). As discussed in Sec. 3.1.3, it is possible for the prime-sector fermions to have smaller Yukawa couplings. The SSI-induced scale which sources this axion mass would then be substantially lowered, resulting in an η χ, phys mass similar to or much smaller than the η ψ, phys mass. In fact, the η χ, phys could even become the lightest exotic state observable at colliders, given that its couplings to the visible world are only suppressed by f d . A dedicated study would be required to examine how light this axion can become while maintaining this realistic ultraviolet model.
The second axion in model II, denoted a and of elementary nature, could analogously become very light by fine-tuning the prime-fermion Yukawa structure. Nevertheless, as its associated axion scale is the color-unified scale, f a ∼ Λ CUT , its couplings to the visible world are extremely suppressed. It follows that such light axion would not be accessible at present or foreseen colliders.

Cosmological and Gravitational aspects
We briefly discuss next the cosmological aspects of the models constructed above, as well as the putative instability threat from gravitational non-perturbative effects.

Stable particles and cosmological structures
Stable particles with masses higher than about 10 5 GeV may lead to cosmological problems, dominating the mass density and overclosing the universe [37]. This is often a problem in previous models of composite axions because exotic stable baryons bound by the extra confining force [11,38] are expected.
However, as pointed out in Refs. [38,39], if axicolor can be unified with a SM gauge group, then the unified forces could mediate the decay of axihadrons into lighter states, and the model would be cosmologically safe. The color unification of this work automatically employs this mechanism. The heavy exotic hadrons decay to the sterile fermions ψ ν , which are part of the CUT massless multiplet Ψ in Eq. (4) and Tab. 2. It remains to be determined whether the lifetime for these CUTinduced decay channels is too large to avoid problems from stable exotic hadrons. If the decays are made to be fast enough, the resulting sterile fermions may induce in turn other cosmological problems. This analysis is left to a future work.
A similar concern pertains to the domain walls which may arise due to the spontaneous breaking of the discrete shift symmetry in the instanton potential, Eqs. (48) and (73). The walls need to disappear before they dominate the matter density of the universe, or else other mechanisms must be applied to solve the domain wall problem [40][41][42][43][44].
In any case, if the universe went through an inflation phase any relic previously present will be wiped out. If the reheating temperature is lower than the PQ scales, meaning lower than Λ diag here, neither the heavy stable particles nor any putative domain walls are produced again after inflation, and the problems mentioned above are avoided altogether. We defer to a future work the in-depth study of the cosmological aspects of model I and model II, with either high-or low-scale inflation.

The question of gravitational quantum effects
Gravitational quantum corrections have been suggested to be relevant and dangerous for axion models in which the PQ scale is not far from the Planck scale. In model I, both PQ scales correspond to Λ diag , which is much lower than the Planck scale, and no instability resulting from gravitational quantum effects is at stake. In model II instead, while the dynamical PQ scale is analogously low, the second PQ symmetry is realized at the CUT scale and gravitational quantum effects could be relevant.
It has often been argued that all global symmetries may be violated by non-perturbative quantum gravitational effects, see for instance Refs. [41,[45][46][47][48]. For instance, a black hole can eat global charges and subsequently evaporate. Similar effects may exist with virtual black holes. Another indication that gravity might not respect global symmetries comes from wormhole physics [49][50][51][52]. The natural scale of violation in this case is the wormhole scale, usually thought to be very near (within an order of magnitude or so) the Planck mass M Planck .
For axion models with high PQ scales, such as the typical scale of invisible axion models f a ∼ 10 9 − 10 12 GeV, it has been argued that those non-perturbative quantum gravitational effects could lead to extreme fine-tunings. The authors of Ref. [41,[45][46][47] concentrated on the simplest (and the most dangerous) hypothetical dimension five effective operator where g 5 is a dimensionless coefficient and Φ would be a field whose vev breaks the PQ invariance. This term threatens the standard invisible axion solutions to the strong CP problem, as it would change the shape of the effective potential. The minimum moves unacceptably away from a CP-conserving solution unless the coefficient is strongly fine-tuned, for instance g 5 < 10 −54 for f a ∼ 10 12 GeV. These potentially dangerous terms can be avoided if the global PQ symmetry arises accidentaly as a consequence of other gauge groups [53,54]. Nevertheless, the idea that gravity breaks all global symmetries is indeed an assumption -and sometimes an incorrect one -at least at the Lagrangian level. 16 Furthermore, very recently the impact of non-perturbative effects has been clarified and quantified in Ref. [56]. The effects turn out to be extremely suppressed by an exponential dependence on the gravitational instanton action, and they are harmless even with high axion scales. The demonstration relied only on assuming that the spontaneous breaking of the PQ symmetry is implemented through the vev of a scalar field, and thus it directly applies to our model II. 17 In summary, both model I and model II are safe from instabilities induced by gravitational quantum corrections.

Conclusions
Color unification with massless quarks has been proposed and developed here for the first time. As a simple implementation of the idea, the SM color group has been embedded in SU (6), which is spontaneously broken to QCD and a second confining and unbroken gauge group. An exactly massless SU (6) fermion multiplet solves the strong CP problem. We have fully developed two ultraviolet completions of the mechanism.
In order to implement this idea successfully, it is necessary to give satisfactorily high masses to the SU (6) partners of the SM quarks to achieve a separation between the QCD scale and that of the second confining group. For this purpose, an auxiliary SU (3 ) gauge group is introduced under which the aforementioned massless fermion is a singlet. SU (6) × SU (3 ) −→ SU (3) c × SU (3) diag is a simple and realistic option. Both final groups remain unbroken and confine at two different scales, Λ QCD and Λ diag , with Λ diag ∼ O(# TeV) Λ QCD . The scale Λ diag then gives the order of magnitude of the mass of the dynamical composite axion inherent to the color-unified mechanism. Furthermore, massless (or almost massless) sterile fermions are a low-energy trademark remnant of the massless multiplet that solves the SM strong CP problem.
In order to avoid the SU (3 ) sector sourcing back an extra contribution to the strong CP problem, a minimal extension of its matter sector suffices. Two examples of ultraviolet complete models have been explored in this work: in model I an extra SU (3 ) massless fermion is added, while model II includes instead a second scalar with the same quantum numbers as the colorunification breaking scalar. From the point of view of the strong CP problem, those two models are very different. Model I features a second dynamical axion with a second PQ scale of order Λ diag . In model II, this second PQ scale coincides with the much larger color-unification scale, and the associated axion is elementary. We computed the two-loop running of all coupling constants involved, showing that the desired separation of all relevant scales is achieved naturally: a colorunification scale much larger than the two confining ones, Λ diag and Λ QCD , and the subsequent separation of the last two. This separation of scales is robust and stable over a wide range of parameter values.
We have found that regardless of the details of the ultraviolet implementation, generically there are the three sources of anomalous currents: the instantons of the confining SU (3) c , the instantons of the confining SU (3) diag , and finally the small-size instantons of the spontaneously broken colorunified theory. There are thus three diverse sources of mass for the three pseudoscalars in the theory which couple to anomalous currents: the QCD η , the dynamical axion inherent to color unification, and the second axion (either dynamical or elementary) associated to the solution of the θ problem. These three bosons then acquire masses of order Λ QCD , Λ diag and Λ SSI , respectively, and no standard invisible axion is left in the low-energy spectrum. This is generically a very interesting mechanism from the point of view of solving the strong CP problem with heavy axions and scales around the TeV. The mechanism allows a wide extension beyond the invisible axion range of axion parameter space which solves the strong CP-problem.
With axion masses and scales around the TeV, observable signals at colliders are expected, as well as other rich phenomenology. Generically, the lightest exotic bound states are colored pseudoscalars (QCD octets and triplets in model I, and only octets in model II). We have recast the results from present experimental searches of heavy colored mesons to infer a 2.9 TeV bound on the confinement scale of the second confining group, SU (3) diag , which is directly related in model I to the axion scale. It is possible, however, that the lightest hadron is one of the axions instead of a colored meson, although this possibility has been shown to require some fine-tuning of the Yukawa couplings of the theory and is consequently less appealing.
Overall, model I may be preferred as: i) it is exclusively based on solving the strong CP problem dynamically via massless quarks; ii) from the point of view of naturalness it does not require any fine-tuning to ensure the hierarchy between the PQ and the electroweak scales, as no PQ field is involved in the scalar potential. In model II instead, one PQ field participates in the color-unification scalar potential, and furthermore this model is a hybrid dynamical-elementary axion solution to the strong CP problem.
Model I is also unquestionably safe from the point of view of stability of the axion solution with respect to non-perturbative effects of quantum gravity, as its PQ scales are Λ diag ∼ TeV. Furthermore, recent advances suggest that the quantum gravity threat should not be considered a risk even for model II. The other issue of the cosmological impact of (quasi) stable heavy exotic hadrons and of the (almost) massless sterile fermion remnants can be simply avoided by introducing an inflation scale and reheating temperature lower than Λ diag . This last subject deserves future detailed attention in particular in view of the dark matter puzzle.