Color unified dynamical axion
Abstract
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 lowenergy CPconserving vacuum. The mechanism does not appeal to a \(Z_2\) “mirror” copy of the SM, nor does it require any finetuning of the axionrelated couplings at the unification scale. There is no very 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.
1 Introduction
Phenomenological analyses based on chiral perturbation theory and supported by lattice computations indicate that all Standard Model (SM) quarks have nonzero 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 nonperturbative 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 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 \(SU(3)_c \times SU({\tilde{N}})\), with the latter having a scale larger than that of QCD, \({\tilde{\Lambda }} \gg \Lambda _{\mathrm{QCD}}\). Two confined charges would then exist in nature, color and axicolor respectively, and correspondingly two distinct sources of instanton potentials. A massless colortriplet 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 nonobservation of exotic bound states at low energies. An issue arises because there are now two potentially harmful vacuum angles to absorb: \(\theta _c\) of QCD and \({\tilde{\theta }}\). 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 \(\chi \) charged only under \(SU({\tilde{N}})\). In the limit of vanishing QCD coupling, the \(SU({\tilde{N}})\) 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 \(\eta '\), three flavorsinglet pseudoscalars result in the lowenergy spectrum for only two instanton sources of masses: the \(\eta '\), a very heavy axion with mass \(\sim {\tilde{\Lambda }}\) 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 finetunings.
In a different and recent attempt [15] to solve the strong CP problem with extra massless quarks, the same \(SU(3)_c\times SU({\tilde{N}})\) confining sector is considered. No light axion remains in the lowenergy spectrum, though, as only two pseudoscalar mesons are present which couple to the two anomalous currents: the customary \(\eta '\) 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 \(SU({\tilde{N}})\) sector then describes only three flavors, resulting in only one gauge singlet pseudoscalar with anomalous \(SU({\tilde{N}})\) couplings: a dynamical axion. Both the \(\eta '\) and the axion thus acquire a mass, and the axion mass is induced by \(SU({\tilde{N}})\) instantons, \(m_a, f_a \sim {\tilde{\Lambda }}\). For such a heavy axion the \(f_a\) scale can be as low as the TeV range without incurring unacceptable phenomenological consequences [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27]. The issue of the two \(\theta \) 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. This \(Z_2\) 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 [22]. A very interesting aspect developed by the same authors in a previous work [23] 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 \(\theta \) 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 axicolortype 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 smallsize 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 typically no axion remains at low scales. The exotic lowenergy 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.
2 SU(6) color unification
The massless fermion sector of the SU(6) construction above the unification scale
SU(6)  \(SU(2)_L\)  U(1)  

\(\Psi _L\)  20  1  0 
The massless fermion sector of the SU(6) construction below the unification scale. The notation is such that \(\psi _L^c\equiv (\psi ^c)_L=(\psi _R)^c\)
\(SU(3)_c\)  \(SU({\tilde{3}})\)  

\(\psi _{L}\)  \(\Box \)  \(\bar{\Box }\) 
\(\psi _{L}^c\)  \(\bar{\Box }\)  \(\Box \) 
\(2\times \psi _\nu \)  1  1 
The coloredaxicolored massless fermions in Eq. (4) will be denoted \(\psi _{L,R}\), see Table 2, while \(\psi _\nu \) will refer to the singlet massless fermions to convey that they act like sterile^{2} neutrinos. The \(\psi _\nu \) 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 \(\Lambda _\text {CUT}\), 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 lowenergy spectrum which is fully compatible with observations.
The SM fermions

Leaving the tildequark sector massless but confined is unacceptable, as the condensate – assuming chiral symmetry breaking of \(SU({\tilde{3}})\) – would typically break the SM EW symmetry at the large \({\tilde{\Lambda }}\) scale.

Alternatively, giving much larger masses (\(\ge {\tilde{\Lambda }}\)) 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.
3 The realistic color unified theory: \(SU(6)\times SU(3')\)
The matter content. The table on the left describes matter above the CUT scale, while the one on the right gives the transformation properties under the gauge groups remaining after CUT spontaneous breaking. The fermions in bold have masses comparable to \(\Lambda _{\mathrm{CUT}}\) and are integrated out around the CUT scale. The quantum numbers under the EW gauge group correspond both to the highenergy and lowenergy fields
SU(6)  \(SU(3')\)  \(SU(3)_c\)  \(SU(3)_{\mathrm{diag}}\)  \(SU(2)_L\)  \(U(1)_Y\)  

\(Q_L\)  \( \Box \)  1  \(q_L\)  \(\Box \)  1  \(\Box \)  \(\frac{1}{6}\) 
\({\tilde{{\mathbf {q}}}_{\mathbf {L}}}\)  1  \(\Box \)  \(\Box \)  \(\frac{1}{6}\)  
\(U_L^c\)  \(\bar{\Box }\)  1  \(u_L^c\)  \(\bar{\Box }\)  1  1  \(\frac{2}{3}\) 
\({\tilde{{\mathbf {u}}}}_{\mathbf {L}}^{\mathbf {c}}\)  1  \(\bar{\Box }\)  1  \(\frac{2}{3}\)  
\(D_L^c\)  \(\bar{\Box }\)  1  \(d_L^c\)  \(\bar{\Box }\)  1  1  \(\frac{1}{3}\) 
\({\tilde{{\mathbf {d}}}}_{\mathbf {L}}^{\mathbf {c}}\)  1  \(\bar{\Box }\)  1  \(\frac{1}{3}\)  
\(\Psi _L\)  20  1  \(\psi _{L}\)  \(\Box \)  \(\bar{\Box }\)  1  0 
\(\psi ^c_{L}\)  \(\bar{\Box }\)  \(\Box \)  1  0  
\(2\times \psi _\nu \)  1  1  1  1  
\(q_L'\)  1  \(\bar{\Box }\)  \({\mathbf {q}}_{\mathbf {L}}^{\prime }\)  1  \(\bar{\Box }\)  \(\Box \)  \(\frac{1}{6}\) 
\({u_L'^c}\)  1  \(\Box \)  \({\mathbf {u}}_{\mathbf {L}}^{\prime {\mathbf {c}}}\)  1  \(\Box \)  1  \(\frac{2}{3}\) 
\({d_L'^c}\)  1  \(\Box \)  \({\mathbf {d}}_{\mathbf {L}}^{\prime {\mathbf {c}}}\)  1  \(\Box \)  1  \(\frac{1}{3}\) 
\(\Delta \)  \(\Box \)  \(\bar{\Box }\)  –  –  –  1  0 
Both \(SU(3)_c\) and \(SU(3)_{\mathrm{diag}}\) can now remain unbroken and confine at two different scales, \(\Lambda _{\mathrm{QCD}}\) and \(\Lambda _{\mathrm{diag}}\), with \(\Lambda _{\mathrm{diag}}\gg \Lambda _{\mathrm{QCD}}\). 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.
Finally, the theory below \(\Lambda _{\text {CUT}}\) contains phenomenologically interesting bound states formed from the massless \(\psi _{L,R}\) 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.
3.1 \({\varvec{\theta '}}\) issue

An extra massless fermion transforming only under \(SU(3')\).

A second bifundamental scalar field, which automatically endows PQ invariance to the above extension procedure.
3.2 Model I: adding a massless fermion charged under \(SU(3')\).
The table shows on the left (right) the quantum numbers above (under) the CUT scale for the massless \(\chi \) quarks which absorbs \(\theta '\) in Model I
SU(6)  \(SU(3)^{\prime }\)  \(SU(2)_L\)  \(U(1)_Y\)  \(SU(3)_c\)  \(SU(3)_\text {diag}\)  

\(\chi \)  1  \(\Box \)  1  0  1  \(\Box \) 
3.2.1 Running of the coupling constants
The massless quark sector charged under \(SU(3)_{\mathrm{diag}}\) remaining below the confining scale \(\Lambda _{\mathrm{diag}}\)
\(SU(3)_c\)  \(SU(3)_{\mathrm{diag}}\)  

\(\psi _{L}\)  \(\Box \)  \(\bar{\Box }\) 
\(\psi _{L}^c\)  \(\bar{\Box }\)  \(\Box \) 
\(\chi _{L}\)  1  \(\Box \) 
\(\chi _{L}^c\)  1  \(\bar{\Box }\) 
3.2.2 Confinement of \(SU(3)_{\mathrm{diag}}\) and pseudoscalar anomalous couplings to the confining interactions
At the scale \(\Lambda _{\mathrm{diag}}\), \(SU(3)_{\mathrm{diag}}\) confines and the remaining massless fermions will form massive QCDcolored bound states.
It follows that there are only two sources of mass (disregarding corrections from SM quark masses) for three states coupling to anomalous currents: \(\eta _{\mathrm{QCD}}'\), \(\eta '_\psi \) and \(\eta '_\chi \). In the absence of supplementary mass sources, one axion would get a mass of order \(\Lambda _{\mathrm{diag}}\) while another one would have remained almost massless, see Eqs. (1) and (2), as often happens in models with dynamical axions. The model would be simply an ultraviolet implementation of the invisible axion paradigm.
3.2.3 The impact of smallsize instantons on the dynamical axion mass
An additional and putatively large contribution to the axion mass(es) applies in the presence of a spontaneously broken theory: the smallsize instantons (SSI) of the theory at the breaking scale, as pointed out long ago in Refs. [31, 32, 33] and very recently in Ref. [23]. 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 colorunified theory in Eq. (6) near the \(\Lambda _{\mathrm{CUT}}\) scale provide automatically this third source of axion mass. The SU(6) SSI can be neglected because of the smallness of \(\alpha _6\) at \(\Lambda _{\mathrm{CUT}}\) (e.g. see Fig. 1) and the analysis below will focus on the \(SU(3')\) SSI contribution.
At the scales where we will compute the SSI effects, the \(SU(3')\) gauge coupling is perturbative. Therefore, for these instantons, the dilute gas approximation [34] gives a reliable estimate of the effective potential for the pseudoscalars. This was not the case for the previous instanton effects which, corresponding to the confinement scales, required the use of the effective chiral Lagrangian to obtain the potential from QCD observables.
3.2.4 Smallsize instantons with small Yukawa couplings
3.2.5 How light can the axion that couples to SSI become?
The \(y_i'\) values are very relevant for the size of SSI scale. Nevertheless, the mass spectrum and thus the running of coupling constants is basically unaffected by them. Should the \(y_i'\) couplings be negligible, \(\Lambda _{\mathrm{SSI}}\) would be determined by the second term in Eq. (46). For vanishing \(y_i'\) values and generic \(\kappa _i\) couplings of \({\mathcal {O}}(1)\), the product of \(Y^{SM}_i/4\pi \) factors in the second term in Eq. (46) would suppress the \(\eta '_\chi \) axion mass to values of order keV. Such low masses are excluded up to the GeV range [41] for axion scales not far from a TeV, as is the case here. The allowed range is illustrated in Fig. 3.^{15}
3.2.6 Solution to the strong CP problem
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 \(f_a\)^{16} is set by \(\Lambda _{\mathrm{diag}}\) and not \(\Lambda _{\mathrm{CUT}}\), 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 finetuning in spontaneously broken unified theories, as \(\Lambda _{\mathrm{CUT}}\) and the EW scale are connected via the scalar potential, but the latter does not communicate to our PQ mechanism.
The demonstrated possibility of lowering the PQ scale towards the electroweak one raises the question of the compatibility of the setup presented here with attempts to solve the EW hierarchy problem, e.g. via compositeness or supersymmetry. This is an interesting question which deserves future work. One could probably build supersymmetric or techincolor version of the models presented here. Nevertheless, the need to strongly separate the \(\Lambda _\mathrm{CUT}\) scale from the EW scale is a nontrivial source of instability in the scalar potential.
3.2.7 Computation of the pseudoscalar mass matrix: \(\eta '_\chi \), \(\eta '_\psi \), \(\eta '_{\mathrm{QCD}}\) and light spectrum
3.2.8 Low energy spectrum and observable effects

The SM pseudoscalar meson \(\eta '_{QCD\,phys}\), plus the rest of the SM hadronic spectrum.

The exotic QCDcolored “pions” — color octets and color triplets — whose masses are given in Eq. (17) as \(m^2\sim \alpha _c \Lambda ^2_\text {diag}\). With masses naturally lighter than the TeV scale, these QCDcolored pions can be easily produced at the LHC.

The two sterile fermions stemming from the 20 representation \(\Psi \). They are basically invisible as their interactions with the visible world are suppressed by \(\Lambda _{\mathrm{CUT}}\), which is much larger than \(\Lambda _{\mathrm{diag}}\) without any tuning.

Possibly, a GB associated with generalized baryon number. This GB is harmless as its interactions are suppressed by \(\Lambda _{\mathrm{CUT}}\). It can also easily be made arbitrarily heavy by gauging that global symmetry.
3.2.9 Smallsize instantons with \({\mathcal {O}}(1)\) Yukawa couplings
3.2.10 Solution to the strong CP problem
Upon \(SU(3)_{\text {diag}}\) confinement, only one massless fermion remains in the spectrum, \(\psi \). Therefore, the potential contains only one pseudoscalar meson that inherits a shift symmetry: \(\eta '_\psi \). This does not invalidate the solution to the strong CP problem, since the other phase \({{\bar{\theta }}}'\) in Eq. (49) is reabsorbed when the \(\chi \) is integrated out of the spectrum. This is a consequence of the mass of the \(\chi \) being generated by instanton effects: the phase of this mass term will be exactly that of the topological term, and it will be completely removed after integrating out the fermion.
3.2.11 Low energy spectrum and observable effects
For most of the parameter space, the two dynamical axions \({\eta '_{\psi ,\,phys}}\) and \(\eta '_{\chi ,phys}\) are typically heavier than the TeV scale and thus very difficult to observe at LHC. Otherwise, under a few TeV the spectrum is the same as that itemized in Sect. 3.1.4, except that there are no color triplet “axipions” because of the \(\chi \) absence at the relevant energies, see Eq. (29). Therefore only the color octet “axipion” can be searched for at the LHC.
3.3 Model II: Addition of a second \(\Delta \) scalar.
The massless quark sector charged under \(SU(3)_{\mathrm{diag}}\) below \(\Lambda _{\mathrm{CUT}}\) in the model with an extra scalar
\(SU(3)_c\)  \(SU(3)_{\mathrm{diag}}\)  

\(\psi _{L}\)  \(\Box \)  \(\bar{\Box }\) 
\(\psi _{L}\,^c\)  \(\bar{\Box }\)  \(\Box \) 
3.3.1 Running of the coupling constants
3.3.2 Confinement of \(SU(3)_{\mathrm{diag}}\) and pseudoscalar anomalous couplings to the confining interactions.
3.3.3 Impact of smallsize instantons on the dynamical axion mass
3.3.4 Smallsize instantons of the spontaneously broken CUT
3.3.5 Solution to the strong CP problem
3.3.6 Computation of the pseudoscalar mass matrix: a, \(\eta '_\psi \), \(\eta '_{\mathrm{QCD}}\) and light spectrum
Apart from the axion, the lowest set of exotic states is an octet of exotic “pions” whose masses are \(\lesssim \, \hbox {TeV}\), see Eq. (70). A similar colored octet is present in Model I discussed in Sect. 3.1, although Model I contains an additional set of colortriplet pseudoscalars in the case of small \(y_i'\) Yukawa couplings. In contrast, no colortriplet is ever expected here as the exotic classical flavor symmetry is \(U(3)_L\times U(3)_R\), see Eq. (69), instead of the \(U(4)_L\times U(4)_R\) symmetry of Model I.
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.
4 Phenomenological and cosmological limits on the lightest exotic states
A common feature of the ultraviolet complete models constructed above is that the generic spectrum under the EW scale includes, in addition to the SM spectrum, sterile fermions, in contrast with usual axion models. An axion may also be present in this range depending on the model parameters.
4.1 Collider observable signals
A 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 \(SU(3)_{\mathrm{diag}}\). All models exhibit as a common characteristic QCD coloroctet meson bound states made out of their massless fermions, shown in Table 5 and Eq. (17) for Model I and Table 6 and Eq. (70) for Model II. In addition, QCD colortriplet meson bound states may be observable for Model I with small Yukawa couplings in the primed sector.
In this colorunified axion solution, the exotic fundamental fermions have no SM \(SU(2)\times U(1)\) charges. The heavy pions will be produced in colliders only via QCD interactions, e.g. gluongluon couplings, through which they also presumably decay before they can hadronize to make color neutral states. As they are colored, they do not mix with ordinary pions or other visible matter.
4.1.1 Coloroctet pions from \(SU(3)_{\mathrm{diag}}\) confinement
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.
4.1.2 Dynamical axion and exotic fermions
The dynamical axion denoted above by \(\eta '_{\psi ,\,phys}\) with instantoninduced mass of order \(\Lambda _{\mathrm{diag}}\), Eqs. (53) and (82), can a priori be either pairproduced through the kinetic coupling or singly produced through the dimension five anomalous operator. It would decay dominantly to two backtoback jets and can be searched for in dijet resonance searches. Its production, however, may be suppressed by its high mass. For instance, for \(\Lambda _{\mathrm{diag}} \simeq 2.9 \, \hbox {TeV}\), \(m_{\eta '_{\psi ,\,phys}} \sim 90 \, \hbox {TeV}\), which is beyond the reach of present collider searches. The other mesons and baryons resulting from the \(SU(3)_{\mathrm{diag}}\) confinement have masses in the TeV range and above; they would lead to collider signatures similar to those for the exotic pseudogoldstone bosons.
4.1.3 The axion coupled to SSI

For \(y_i'\) couplings of \({\mathcal {O}}(1)\), this second axion becomes typically heavier than the TeV scale and is thus out of LHC reach.

For small \(y_i'\) couplings (\({\mathcal {O}}(0.2)\) have been illustrated), an observable axion with mass in the GeVtens of TeV regime is realistic. Given the low PQ scale (\(f_d\), close to a TeV) lower masses are excluded.
4.2 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 nonperturbative effects.
4.2.1 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 [50]. This is often a problem in previous models of composite axions because exotic stable baryons bound by the extra confining force [11, 51] are expected.
However, as pointed out in Refs. [51, 52], 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 \(\psi _\nu \), which are part of the CUT massless multiplet \(\Psi \) in Eq. (4) and Table 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. (51) and (80). 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 [53, 54, 55, 56, 57].
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 \(\Lambda _{\mathrm{diag}}\) 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 indepth study of the cosmological aspects of Model I and Model II, with either high or lowscale inflation.
4.2.2 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 \(\Lambda _{\mathrm{diag}}\), 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 nonperturbative quantum gravitational effects, see for instance Refs. [54, 58, 59, 60, 61]. 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 [62, 63, 64, 65]. 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 \(M_{\mathrm{Planck}}\).
5 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. \(SU(6)\times SU(3')\longrightarrow SU(3)_c\times SU(3)_{\mathrm{diag}}\) is a simple and realistic option. Both final groups remain unbroken and confine at two different scales, \(\Lambda _{\mathrm{QCD}}\) and \(\Lambda _{\mathrm{diag}}\), with \(\Lambda _{\mathrm{diag}}\sim {\mathcal {O}}(\# \, \hbox {TeV}) \, \gg \Lambda _{\mathrm{QCD}}\). The scale \(\Lambda _{\mathrm{diag}}\) then gives the order of magnitude of the mass of the dynamical composite axion inherent to the colorunified mechanism. Furthermore, massless (or almost massless) sterile fermions are a lowenergy 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 which is also of order \(\Lambda _{\mathrm{diag}}\) and thus low. In Model II, this second PQ scale coincides instead with the much larger colorunification scale, and the associated axion is elementary. We computed the twoloop 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, \(\Lambda _{\mathrm{diag}}\) and \(\Lambda _{\mathrm{QCD}}\), 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 \(SU(3)_{\mathrm{diag}}\), and finally the smallsize 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 \(\eta '\), the dynamical axion inherent to color unification, and the second axion (either dynamical or elementary) associated to the solution of the \(\theta '\) problem. These three bosons then acquire masses of order \(\Lambda _{\mathrm{QCD}}\), \(\Lambda _{\mathrm{diag}}\) and \(\Lambda _{\mathrm{SSI}}\), respectively, and no standard invisible axion (coupling anomalously only to QCD instantons) is left in the lowenergy 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 a TeV. The mechanism allows a wide extension beyond the invisible axion range of axion parameter space which solves the strong CPproblem. Typical axion masses are in the TeV regime and above, although strictly speaking one of the axions can become as light as the usual invisible axion.
With axion 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 in some cases also triplets in Model I, and only octets in Model II). We have recast the results from present experimental searches for heavy colored mesons to infer a \(2.9 \, \hbox {TeV}\) bound on the confinement scale of the second confining group, \(SU(3)_{\mathrm{diag}}\), which is directly related in Model I to the axion scale.
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 finetuning 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 colorunification scalar potential, and furthermore this model is a hybrid dynamicalelementary 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 nonperturbative effects of quantum gravity, as its PQ scales are \(\Lambda _{\mathrm{diag}}\sim \, \hbox {TeV}\). 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 \(\Lambda _{\mathrm{diag}}\). This last subject deserves future detailed attention in particular in view of the dark matter puzzle.
Footnotes
 1.
The SM \(\eta '\) is excluded from this definition since the \(U(1)_A\) symmetry associated to it is broken by the nonzero quark masses.
 2.
By “sterile fermion” is meant any fermion which is not charged under the SM gauge group.
 3.
This breaking will become manifest in the next section.
 4.
Through tuned Yukawa couplings of the tildequark sector to an extended scalar sector.
 5.
If SU(6) sufficed to obtain a realistic spectrum, the \(SU({{\tilde{3}}})\) group and \({{\tilde{\Lambda }}}\) scale of this section would correspond to those of the axicolor group [11] as described in the introduction. The extension of the CUT group will break this direct correspondence, although two confining groups will still be at play.
 6.
 7.
In this notation taken from unified models the contraction of the spinor indices is implicit, more precisely the first term would read \(Q_L^T C \Phi U^c_L\), where \(C=i\gamma _2\gamma _0\) 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. [23], 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.
 10.
Figure 1 assumes a zero \(\chi \) mass. As will be shown in Sect. 3.1.5, \(\chi \) acquires an effective mass due to smallsize instanton effects. Threshold effects near \(m_{\chi }\) may alter the running. Even when these effects are large enough to be noticeable, Fig. 1 still captures the qualitative behavior of the RG flow.
 11.
\(U(4)_V\) remains unbroken and contains as a subgroup the \(SU(3)_c\) QCD gauge group.
 12.
They contribute to the running of the QCD coupling constant, but given their high masses their impact is unnoticeable.
 13.
 14.
In Sect. 3.1.5, we will consider \(y_i'\) couplings of \({\mathcal {O}}(1)\) which corresponds to the regime of large smallsize instanton effects, translating to a very heavy \(\chi \) fermion which decouples from the spectrum well above the confinement regime.
 15.
For higher axion masses there are also collider constraints [41]: dijet searches at the LHC provide bounds on axions with \(m_a>1 \, \hbox {TeV}\) [42]. These searches can be extended to axion masses slightly below a TeV by searching for dijet resonances accompanied by hard initial state radiation [43]. These bounds are weak, though, and only apply in a small window of axion masses.
 16.
The PQ scale (usually denoted by \(f_{PQ}\)) and \(f_a\) differ by a modeldependent factor stemming from the relative strength of the axion coupling to gluons. Here we disregard the distinction between \(f_a\) and \(f_{PQ}\).
 17.
The physical states correspond to the following combinations: \(\eta '_{\psi ,\,phys}\simeq 1/ f_{\mathrm{d}}(2\,{\eta '_\chi }+\sqrt{6}\,\eta '_\psi ) \) and \(\eta '_{\chi ,\,phys}\simeq \,1/ f_{\mathrm{d}}(\sqrt{6}\,{\eta '_\chi }\,2\eta '_\psi )\).
 18.
The Lagrangian possesses another U(1) symmetry, namely the generalized baryon number symmetry defined in Eq. (12), which is however nonanomalous under \(SU(3')\). See footnotes 8 and 9 on the harmless consequences of the nonanomalous global symmetry.
 19.
For example, orbifold compactifications of the heterotic string have discrete symmetries that prevent the presence of some higher dimension operators, and this can strongly and safely suppress the dangerous effects under discussion [68].
 20.
Although no explicit demonstration was given in that work for the case of dynamical breaking via condensates, plausibly the result would also apply for models with dynamical axions and very high axion scales.
Notes
Acknowledgements
We acknowledge M. García Pérez, T. Yanagida, V. Sanz, K. Harigaya, S. Knapen and A. Ringwald for very interesting conversations and comments. M.B.G, R. H., R.dR and P. Q. acknowledge Berkeley LBNL, where part of this work has been developed. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreements No 690575 (RISE InvisiblesPlus) and No 674896 (ITN ELUSIVES). M.B.G, R. H., R.dR and P. Q. also acknowledge support from the the Spanish Research Agency (Agencia Estatal de Investigación) through the grant IFT Centro de Excelencia Severo Ochoa SEV20160597. M.B.G, R.dR and P. Q. acknowledge as well support from the “Spanish Agencia Estatal de Investigación” (AEI) and the EU “Fondo Europeo de Desarrollo Regional” (FEDER) through the project FPA201678645P. The work of M.K.G. was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, Division of High Energy Physics, of the U.S. Department of Energy under Contract DEAC0205CH11231, and in part by the National Science Foundation under grant PHY1316783. The work of R. H. was supported to and ESR contract of the H2020 ITN Elusives. The work of P.Q. was supported through a “La CaixaSevero Ochoa” predoctoral grant of Fundación La Caixa.
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