Automatic Peccei–Quinn symmetry
Abstract
We present a dynamical (composite) axion model where the Peccei–Quinn (PQ) symmetry arises automatically as a consequence of chirality and gauge symmetry. The Standard Model is simply extended by a confining and chiral SU(5) gauge symmetry. The PQ symmetry coincides with a \(BL\) symmetry of the exotic sector. The theory is protected by construction from quantum gravitational corrections stemming from operators with mass dimension lower than nine.
1 Introduction
The most elegant solution to the strong CP problem is to introducte a global chiral U(1) symmetry, usually called Peccei–Quinn (PQ) symmetry [3] \(U(1)_{PQ}\), which is exact (and hidden) at the classical level but is anomalous under QCD interactions. The latter is the key to solve the problem and also the only source of the mass for the pseudoNambuGoldstone boson of the global \(U(1)_{PQ}\) symmetry: the axion.
A simple and most economical implementation would be the \(U(1)_{PQ}\) symmetry that would exist if one SM quark were to be massless. The freedom to chirally rotate that fermion would allow to fully reabsorb all contributions to \({\bar{\theta }}\), making it unphysical. This interesting possibility [4] does not seem to be realized in nature after the constraints stemming from lattice computations and we disregard it, even if the option is not completely excluded [5, 6, 7, 8, 9, 10, 11].
It is still possible to solve the strong CP problem with massless quarks, though, if extra exotic massless fermions charged under QCD exist in Nature. As the latter are not observed, the idea [12] is to charge them in addition under a new confining force [13, 14, 15], often called “axicolor” [12], whose scale is much larger than that of QCD, \(\Lambda _{QCD}\). A new spectrum of confined states results composed of those massless fermions, including mesons which play the role of axions. They are often referred to as “dynamical” or composite axions.
In the original composite axion proposal [12] the confining sector of the SM was enlarged to \(SU(3)_c \times SU({\tilde{N}})\), where \(SU({\tilde{N}})\) is the axicolor group. Two composite axions result, one of which must be invisible and obey Eq. (1.3), as there are only two sources of instantons for three pseudoscalars with anomalous couplings (taking into account the SM \(\eta '\)). The axicolor construction can be seen as a beautiful ultraviolet dynamical completion of the invisible axion paradigm. It has the advantage of being free from the scalar potential finetunings that hinder models of invisible elementary axions à la KSVZ [34, 35] or DFSZ [36, 37].

It confines and renders the theory free from gauge anomalies.

The exotic fermion representations are chiral, so that fermionic mass terms are automatically forbidden.

Minimality in the specific matter content will be a guideline. Two (or more) different axicolored fermions are present, with at least one of them being QCD colored as well.
We focus here on the case of chiral gauge SU(5), implemented via its lowest dimensional fermion representations, \({\bar{\mathbf {5}}}\) and \({{\mathbf {10}}}\), which together fulfil the conditions above. The SU(5) confinement scale will be assumed to be much larger than that of QCD, \(\Lambda _5 \gg \Lambda _{\mathrm{QCD}}\). It will be shown that a satisfactory \(U(1)_{PQ}\) symmetry is an automatic consequence of the chiral realization of the gauge group. Note that some models have been previously built for which PQ invariance is accidental, that is, not imposed by hand [50, 51, 52, 53, 54, 55, 56, 57]. Nevertheless, they all required extra symmetries in addition to axicolor, either gauge or discrete ones. In contrast, axicolor SU(5) will be shown to suffice because of its chiral character, rendering a particularly simple framework.
Relevant aspects to be developed include on one side the identification of the exotic fermion condensates, which in dynamical axion models are the only source of spontaneous symmetry breaking, e.g. for exotic flavour and for the PQ symmetries. Another important question is the impact of SU(5) gauge invariance on the possible nonperturbative gravitational couplings of the theory.
The idea will be implemented in two alternative realizations, selected so as to achieve minimal matter content. They will only differ in the QCD charges of the exotic \({\bar{\mathbf {5}}}\) and \({\mathbf {10}}\) fermions present: octets of QCD color in one model, while triplets in a second version.
The structure of the paper can be easily inferred from the table of contents.
2 The SU(5) chiral confining theory
Charges of exotic fermions under the confining gauge group \(SU(5)\times SU(3)_c\). The lefthanded Weyl fermions \(\psi _{{\bar{5}}}\) and \(\psi _{10}\) are massless and singlets of the SM electroweak gauge group. \({\mathbf {R}}\) denotes a pseudoreal representation
SU(5)  \(SU(3)_c\)  

\(\psi _{{\bar{5}}}\)  \({\bar{\mathbf{5}}}\)  \({\mathbf {R}}\) 
\(\psi _{10}\)  \({\mathbf {10}}\)  \({\mathbf {R}}\) 
If the exotic fermions carry also QCD color, this theory solves the strong CP problem. Indeed, the presence of (at least) two massless fermions ensures the existence of two distinct U(1) chiral global symmetries, exact at the classical level but explicitly broken by quantum nonperturbative effects. The \(\theta \)parameters corresponding to the two confining gauge groups become thus unphysical via chiral rotations of those fermions. Furthermore, the chiral character of the representations forbids fermionic mass terms and thus guarantees that those symmetries are automatic, instead of imposed on a given Lagrangian as customary. Finally, the requirement of a large confining scale \(\Lambda _5\gg \Lambda _\mathrm{QCD}\) leads to a realistic model, given the nonobservation of a spectrum of bound states composed of those massless exotic fermions.
For simplicity, we will consider that the set \(\{\psi _{{\bar{5}}}, \psi _{10}\}\) belongs to a (pseudo)real representation \({\mathbf {R}}\) of color QCD, so as automatically cancel \(\left[ SU(3)_c\right] ^3\) anomalies, see Table 1. Later on we will develop in detail two specific choices for \({\mathbf {R}}\): the case of the fundamental of QCD with reducible representation \({\mathbf {R}}= {\mathbf {3}}+ {\bar{\mathbf {3}}}\) in one case, and the adjoint \({\mathbf {R}}={\mathbf {8}}\) in the second case. In all cases, all mixed gauge anomalies in the confining sector vanish by construction as well, because only nonabelian SU(N) groups are present and the exotic fermions are electroweak singlets.
2.1 Global symmetries
Global chiral properties at the classical level, in the limit of vanishing \(\alpha _s\)
\(SU(n)_{{\bar{5}}}\)  \(SU(n)_{10}\)  \(U(1)_{BL}\equiv U(1)_{PQ}\)  

\(\psi _{{\bar{5}}}\)  \(\Box \)  \({\mathbf {1}}\)  \(\) 3 
\(\psi _{10}\)  \({\mathbf {1}}\)  \(\Box \)  1 
2.1.1 Confinement versus chiral symmetry breaking

The global symmetries can be spontaneously broken via fermion condensates. As a result, (almost) massless (pseudo)Goldstone bosons (pGBs) will be present in the low energy theory.

Conversely, they could remain unbroken and the spectrum of bound states would explicitly reflect those global symmetries via multiplets of degenerate states. In particular, massless baryons are then needed in order to fulfil the ‘t Hooft anomaly consistency conditions [59] to match the anomalies of the high and low energy theories.
To sum up, parts of the global symmetries in Eq. (2.7) with the field content in Table 2 need to be spontaneously broken by fermion condensates upon SU(5) confinement.
2.2 Fermion condensates: chiralbreaking versus PQbreaking
It will be assumed that \(\Lambda _5\) settles the overall scale for all dynamical breaking mechanisms in the SU(5) sector, which will take place through fermion condensates.
2.2.1 Chiral condensate
The chiral condensate in Eq. (2.16) is \(U(1)_{PQ}\) invariant, though, since its PQ charge is vanishing. The spontaneous breaking of the PQ symmetry (which is phenomenologically the only viable option as earlier explained) can only be achieved via higher dimensional fermionic condensates.
2.2.2 PQ condensate
2.3 The axion Lagrangian
2.3.1 Relation between \(f_{\text {PQ}}\) and \(\Lambda _5\) in naïve dimensional analysis
2.3.2 Coupling to gluons and domain walls
2.4 Planck suppressed operators
It has been argued that quantum gravity may violate all global symmetries. In particular, Planck suppressed operators which are not PQ invariant could be dangerous for axion solutions to the strong CP problem, since they can unacceptably displace the minimum of the axion potential from the CP conserving point.
Within our model, because of gauge invariance and chirality, the lowest dimensional operator of this type has mass dimension nine, as previously argued: it is the operator in Eq. (2.20), whose VEV breaks PQ spontaneously. This significantly strong Planck suppression suggest that our model can be protected from those gravitational issues. This is to be contrasted with the usual expectation in axion models which allow lower dimension effective operators of gravitational origin, e.g. dimension five couplings as in Eq. (1.5).
We will apply next the analysis above to two examples of the confining chiral SU(5) theory, which differ in the QCD charges of the exotic fermions \(\{\psi _{{\bar{5}}}, \psi _{10}\}\), corresponding respectively to a reducible and irreducible QCD representation \({\mathbf {R}}\). In the first model \(R=3+{\bar{3}}\), while \(R=8\) will be assumed in the second model. While the former requires four exotic fermions (instead of just two for the second option), its matter content is smaller in terms of number of degrees of freedom.
2.5 Planck suppressed operators and neutrino masses
3 Model I: colortriplet fermions
Model I: charges of exotic fermions under the confining gauge group \(SU(5)\times SU(3)_c\), the PQ symmetry and the spontaneously broken global U(1) symmetries. The lefthanded Weyl fermions \(\psi _{{\bar{5}}}\) and \(\psi _{10}\) are massless and singlets of the SM electroweak gauge group; their QCD representation has been indicated as an additional subscript
SU(5)  \(SU(3)_c\)  \(U(1)_{PQ}\)  \(U(1)_{V,\,{\bar{5}}}\)  \(U(1)_{V,\,10}\)  

\(\psi _{({{\bar{5}}},3)}\)  \({\bar{\mathbf {5}}}\)  \({\mathbf {3}}\)  \(\) 3  1  0 
\(\psi _{({\bar{5}},{\bar{3}})}\)  \({\bar{\mathbf {5}}}\)  \({\bar{{\mathbf {3}}}}\)  \(\) 3  \(\) 1  0 
\(\psi _{(10,3)}\)  \({\mathbf {10}}\)  \({\mathbf {3}}\)  \(+\) 1  0  1 
\(\psi _{(10,{\bar{3}})}\)  \({\mathbf {10}}\)  \({\bar{{\mathbf {3}}}}\)  \(+\) 1  0  \(\) 1 
The question of whether the QCD group \(SU(3)_c\) is indeed the surviving unbroken group after chiral symmetry breaking, as indicated in Eqs. (2.16), (2.17) and (2.21), deserves a specific discussion. To see this, let us note that an SO(6) subgroup of the global symmetry \(SU(6)_{{{\bar{5}}}}\times SU(6)_{10}\) satisfies the ’t Hooft anomaly consistency conditions. Besides, the condensates \({\langle {\mathbf {10}}\,{\mathbf {10}}\,{\mathbf {10}}\,{\bar{\mathbf {5}}}\rangle }\) and \(\langle {\bar{\mathbf {5}}}\,{\bar{\mathbf {5}}}\,{\mathbf {10}}\,{\bar{\mathbf {5}}}\, {\bar{\mathbf {5}}}\,{\mathbf {10}}\rangle \) can be SO(6) singlets. This means that the unbroken subgroup G of the global symmetry \(SU(6)_{{{\bar{5}}}}\times SU(6)_{10}\) contains SO(6), i.e. \(G \supset SO(6)\).^{11} The SU(3) subgroup of SO(6) is then obtained by identifying the vector representation of SO(6) to be \({\mathbf {3}} + {\bar{\mathbf {3}}}\). Therefore, it is clear that an SU(3) global symmetry remains unbroken below the confinement scale.
It should be noted that an SO(6) subgroup of \(SU(6)_{\bar{5}}\times SU(6)_{10}\) is not uniquely determined, and hence, the unbroken SO(6) is not in general aligned to the one which contains \(SU(3)_c\) for \(\alpha _s = 0\). However, it has been argued that, among the possible condensate channels, the minimum of the potential corresponds to the one preserving QCD for \(\alpha _s\ne 0\) [51]. Thus, we find that it is most likely that the SU(5) dynamics with the nonvanishing chiral and PQ condensates in Eqs. (2.16) and (2.20) preserves \(SU(3)_c\).
The \(U(1)_{V,{{\bar{5}}}}\) and \(U(1)_{V,10}\) symmetries are generically broken by those condensates. In fact, the chiral condensate in Eq. 2.16 breaks spontaneously \(U(1)_{{{\bar{5}}}}\times U(1)_{10}\) down to a U(1), where the number of positive and negative charges with respect to this U(1) is balanced at the QCD preserving vacuum. The PQ condensate could also break this remaining U(1) if the quarks in the condensates are all either in the \({{\mathbf {3}}}\) or in the \({\bar{\mathbf {3}}}\) representation of QCD. Accordingly, the model predicts one or two additional pGBs which obtain tiny masses from the higher dimensional gravitational operators in Eq. (2.36). As those pGBs decouple from the thermal bath at a temperature much higher than the weak scale, the contribution of each pGB to the effective number of relativistic species is suppressed, i.e. \(\varDelta N_{\mathrm{eff}} \simeq 0.03\), and hence the model is consistent with the current constraint \(N_{\mathrm{eff}} =2.99^{+0.34}_{0.33}\) [67].
3.1 Planck suppressed operators
3.2 Axion dark matter
The lower \(f_a\) value in Eq. (3.10) is about a factor of two too large to be compatible with that required in Eq. (3.7) by the neutron EDM bounds. This option requires a finetuning of the coefficient c of the Planck suppressed operator of \({\mathcal {O}}(10^{7})\), to be compared with the typical adjustment by 54 orders of magnitude in axion models with dimension five Plancksuppressed operators. Furthermore, for a misalignment angle close to \(\pi \) and low inflation scales, lower values of \(f_a\) are possible and the finetuning of c could be avoided altogether, even in this most conservative case of the NDA estimate of the effect. Conversely, would the NDA prefactors be disregarded, \({\mathcal {O}}\)(1) coefficients for the Planck suppressed operator are seen to be allowed in a large fraction of the parameter space.
4 Model II: coloroctet fermions
Model II: charges of exotic fermions under the confining gauge group \(SU(5)\times SU(3)_c\). Their PQ charges are shown as well. The lefthanded Weyl fermions \(\psi _{{\bar{5}}}\) and \(\psi _{10}\) are massless and singlets of the SM electroweak gauge group
SU(5)  \(SU(3)_c\)  \(U(1)_{PQ}\)  

\(\psi _{{\bar{5}}}\)  \({\bar{\mathbf {5}}}\)  \({\mathbf {8}}\)  \(\) 3 
\(\psi _{10}\)  \({\mathbf {10}}\)  \({\mathbf {8}}\)  \(+\) 1 
To see whether the QCD gauge group remains ultimately unbroken, note that an SO(8) subgroup of the global symmetry \(SU(8)_{\bar{5}}\times SU(8)_{10}\) satisfies the ’t Hooft anomaly consistency conditions, while the condensates \({\langle {{\mathbf {10}}}\,{\mathbf {10}}\,{\mathbf {10}}\, {\bar{\mathbf {5}}}\rangle }\) and \(\langle {\bar{\mathbf {5}}}\,{\bar{\mathbf {5}}}\,{\mathbf {10}}\, {\bar{\mathbf {5}}}\, {\bar{\mathbf {5}}}\,{\mathbf {10}}\rangle \) can be SO(8) singlets. In this case, we find that the unbroken subgroup G contains SO(8), i.e. \(G \supset SO(8)\). The SU(3) subgroup of SO(8) is realized as the special maximal embedding where the vector representation of SO(8) is identified with the octet of SU(3) (see e.g. [69]). Thus, it is again clear that an SU(3) global symmetry remains unbroken below the confinement scale, with nonvanishing \({\langle \mathbf {10}\,\mathbf {10}\,\mathbf {10}\,{\bar{\mathbf {5}}}\rangle }\) and \(\langle {\bar{\mathbf {5}}}\,{\bar{\mathbf {5}}}\, {\mathbf {10}}\,{\bar{\mathbf {5}}}\, {\bar{\mathbf {5}}}\,{\mathbf {10}}\rangle \) condensates. Finally, the SO(8) symmetry is aligned with that containing \(SU(3)_c\) once \(\alpha _s \ne 0\) is taken into account. This shows that, also in this model, it is most likely for the SU(5) dynamics to preserve \(SU(3)_c\).
4.1 Planck suppressed operators
Figure 1 (right panel) shows the displacement induced by the operator in Eq. (2.37) on the QCD vacuum parameter, for the value of N expected from NDA, see Eq. (4.3), which implies
4.2 Axion dark matter
The comparison between Eq. (4.4) and the \(f_a\) ranges in Eqs. (3.9) and (3.10) shows that this model with exotic fermions in the adjoint of QCD is more in tension than model I, if axions are to explain all the dark matter of the universe without recurring to fine tunings. Figure 1 (right panel) illustrates this situation. For the NDA estimation of Planck suppressed couplings, \(f_a\) as required by dark matter is a factor of five too large with respect to the neutron EDM constraint; this translates into the requirement of a \({\mathcal {O}}(10^{10})\) finetuning of the coefficient c of the Planck suppressed operator, as illustrated in Fig. 2 (right panel). Alternatively, the present model could explain a subdominant fraction of the dark matter content.
A comparison without NDA power counting estimates is also illustrated: nonfine tuned values of the coefficient c are then compatible with the axion accounting for the ensemble of dark matter, while complying with EDM limits. Overall, the uncertainty on the estimations of nonperturbative gravitational effects, and on the \(f_a\) values required to account for dark matter, is large enough to still consider this model as a candidate scenario for purely axionic dark matter.
5 Conclusions

A gauge confining symmetry which is chiral, unlike usual axicolor models which use vectorial fermions. In consequence, the PQ symmetry is automatic, without any need to invoke extra symmetries.

Exotic SU(5) fermions in (pseudo)real representations of QCD.

Inherent protection from dangerous quantum nonperturbative gravitational effects.
We showed that the ‘t Hooft anomaly conditions for the global symmetries of the exotic fermionic sector imply that the nonabelian global symmetries must be spontaneously broken. The global abelian symmetries, e.g. the PQ symmetry, must also be spontaneously broken for the theory to be phenomenologically viable, resulting in a dynamical invisible axion. Furthermore, the PQ invariance is the analogous of the \(BL\) symmetry in SU(5) Grand Unified Theory (GUT).
We have determined the fermionic operators with lowest dimension which may condense and induce spontaneous breaking. Because of SU(5) gauge invariance, six is the minimal dimension for the operator whose VEV may break the exotic flavour symmetries. An even higher dimensional condensate is needed in order to break PQ invariance: the VEV of a dimension nine operator. The latter is also the lowest dimensional effective operator which could result from gravitational quantum contributions, breaking explicitly the PQ symmetry, as these effects must respect gauge invariance. Its high dimensionality is at the heart of the inherent protection of this theory with respect to the gravitational issue.
We have developed two complete ultraviolet completions of the chiral confining SU(5) theory, which only differ in the (pseudo)real QCD representations chosen for the exotic fermions: a reducible \({\mathbf {3}}+{\bar{{\mathbf {3}}}}\) representation for Model I, and the irreducible adjoint in model II. The former is more economical in terms of the total number of degrees of freedom. Both models are phenomenological viable and largely protected from quantum gravitational concerns. Remarkably, in the case of exotic fermions in the fundamental of QCD, the \(f_a\) range allowed if axions are to explain the full dark matter content of the universe can be compatible with that required to avoid a finetuned coefficient for the Planck suppressed operator. For octetcolour fermions the compatibility is marginal but still possible.
The basic novel idea of the construction is to use a chiral confining group, which provides an automatic implementation of PQ invariance. The most economic avenue is to implement it via just two exotic fermions in (pseudo)real representations of QCD. In this perspective, we have briefly explored other confining groups as well. For instance, a chiral and confining gauge SU(4) symmetry would be a viable alternative, although it does not enjoy a sufficient protection from gravitational issues, at least in the case of only two exotic fermions. Even the smaller chiral confining SU(3) symmetry is possible, although the versions with only two exotic fermions require very highdimensional representations of the confining group and, again, they are less protected from gravitational issues than the SU(5) case (see Appendix A). Nevertheless, as the estimation of gravitational effects is somehow uncertain, it may be pertinent to dedicate specific studies to these alternative directions.
Footnotes
 1.
 2.
Here, the Planck mass does not denote the reduced Planck scale but the one given by \(M_{\mathrm{Pl}} = G^{1/2}\) with G being the Newton constant.
 3.
They can be avoided, though, in some invisible axion constructions with a variety of extra assumptions or frameworks [50, 51, 52, 53, 54, 55, 56, 57], or be arguably negligible in certain conditions [58]. It is also possible to avoid the dangerous terms in “heavy axion” models [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], as their \(f_a\) scale can be very low, e.g. not far from the TeV range.
 4.
This can be for instance, the orthogonal combination corresponding to \(\{Q_{{\bar{5}}}=1, Q_{10}=3\}\), although any combination different from that free from anomalous SU(5) couplings can play this role.
 5.
In the thermal bath, for example, the QCD breaking vacua have higher energy density than the QCD preserving one due to the thermal potential proportional proportional to \(m_{\mathrm{gluon}}^2T^2\), where \(m_{\mathrm{gluon}}\) denotes the gluon mass on the QCD breaking vacua.
 6.
Its VEV also breaks the nonabelian chiral symmetry, but this effect should be subdominant with respect to that of the lower dimension operator in Eq. (2.16).
 7.
Consistently, this would correspond to a combinatorial factor of 1 in the corresponding Feynman rules.
 8.
The QCD axion potential is approximated here by a cosine dependence, since we are only interested in the displacement of the minimum where that approximation is perfectly valid. For the correct dependence using chiral Lagrangians at NLO see Ref. [66].
 9.
We thank the referee for pointing out this coupling.
 10.
Because any global symmetry is expected to be broken by gravitational effects, BL may not be an exact low energy symmetry and a Planck suppressed Majorana contribution to neutrino masses may be present, although numerically negligible.
 11.
Our arguments do not depend on whether \(G = SO(6)\) or \(G \supsetneq SO(6)\), although we expect that \(G = SO(6)\).
 12.
The explicit breaking can be further suppressed if, for example, we assume supersymmetry with Rsymmetry. In such cases, \(f_a\) in the preferred value for the DM relic density is also allowed, though we do not pursue such possibilities further in this paper.
 13.
Assuming the fiducial density profile for the protoneutron star in Ref. [31], the bound reads \(f_a>10^8\) GeV.
 14.
\(\left[ SU(4)\right] ^3\) anomaly: \(8 A({\bar{\mathbf {4}}})+A({\mathbf {10}})=0\), since \( A({\bar{\mathbf {4}}})=1,\,\text {and }A({\mathbf {10}})=8\).
Notes
Acknowledgements
We acknowledge very interesting conversations and comments from Valery Rubakov, Mary K. Gaillard, Rachel Houtz, A. Manohar, Rocio del Rey and G. Villadoro. M.B.G and P. Q. acknowledge IPMU at Tokyo University, where this work was done. T. T. Y. is grateful to Sergei Kuzenko for the hospitality during his stay at The University of Western Australia. This project has received support 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 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, as well as from the “Spanish Agencia Estatal de Investigación” (AEI) and the EU “Fondo Europeo de Desarrollo Regional” (FEDER) through the project FPA201678645P. The work of P.Q. was supported through a “La CaixaSevero Ochoa” predoctoral grant of Fundación La Caixa. This research was also supported in part by WPI Research Center Initiative, MEXT, Japan (MI, MY and TTY), and in part by JSPS GrantinAid for Scientific Research no. 15h05889, no. 16h03991, no. 18H05542 (MI), no. 26104001, no. 26104009, no. 16H02176 (TTY), and no. 17H02878 (MI and TTY). TTY is a Hamamatsu Professor at Kavli IPMU.
References
 1.C.A. Baker et al., An improved experimental limit on the electric dipole moment of the neutron. Phys. Rev. Lett. 97, 131801 (2006). arXiv:hepex/0602020 [hepex]ADSCrossRefGoogle Scholar
 2.J. Engel, M.J. RamseyMusolf, U. van Kolck, Electric dipole moments of nucleons, nuclei, and atoms: the Standard Model and beyond. Prog. Part. Nucl. Phys. 71, 21–74 (2013). arXiv:1303.2371 [nuclth]ADSCrossRefGoogle Scholar
 3.R.D. Peccei, H.R. Quinn, CP conservation in the presence of instantons. Phys. Rev. Lett. 38, 1440–1443 (1977). [328(1977)]ADSCrossRefGoogle Scholar
 4.H. Georgi, I.N. McArthur, INSTANTONS AND THE mu QUARK MASS. Preprint. http://inspirehep.net/record/164546
 5.M. Dine, P. Draper, G. Festuccia, Instanton effects in three flavor QCD. Phys. Rev. D 92(5), 054004 (2015). arXiv:1410.8505 [hepph]ADSCrossRefGoogle Scholar
 6.K. Choi, C.W. Kim, W.K. Sze, Mass renormalization by instantons and the strong CP problem. Phys. Rev. Lett. 61, 794 (1988)ADSCrossRefGoogle Scholar
 7.D.B. Kaplan, A.V. Manohar, Current mass ratios of the light quarks. Phys. Rev. Lett. 56, 2004 (1986)ADSCrossRefGoogle Scholar
 8.J. Frison, R. Kitano, N. Yamada, \(N_f=1+2\) mass dependence of the topological susceptibility. PoS LATTICE2016, 323 (2016). arXiv:1611.07150 [heplat]
 9.J. Frison, R. Kitano, N. Yamada, Topological susceptibility with a single light quark flavour. EPJ Web Conf. 175, 14017 (2018). arXiv:1710.06643 [heplat]CrossRefGoogle Scholar
 10.S. Aoki et al., Review of lattice results concerning lowenergy particle physics. Eur. Phys. J. C 77(2), 112 (2017). arXiv:1607.00299 [heplat]ADSCrossRefGoogle Scholar
 11.W.A. Bardeen, Instanton Triggered Chiral Symmetry Breaking, the U(1) Problem and a Possible Solution to the Strong CP Problem. arXiv:1812.06041 [hepph]
 12.K. Choi, J.E. Kim, Dynamical axion. Phys. Rev. D 32, 1828 (1985)ADSCrossRefGoogle Scholar
 13.S. Weinberg, Implications of dynamical symmetry breaking. Phys. Rev. D 13, 974–996 (1976). [Addendum: Phys. Rev.D19,1277(1979)]ADSCrossRefGoogle Scholar
 14.L. Susskind, Dynamics of spontaneous symmetry breaking in the Weinberg–Salam theory. Phys. Rev. D 20, 2619–2625 (1979)ADSCrossRefGoogle Scholar
 15.S. Dimopoulos, J. Preskill, Massless composites with massive constituents. Nucl. Phys. B 199, 206–222 (1982)ADSCrossRefGoogle Scholar
 16.S. Weinberg, A new light boson? Phys. Rev. Lett. 40, 223–226 (1978)ADSCrossRefGoogle Scholar
 17.H. Georgi, D.B. Kaplan, L. Randall, Manifesting the invisible axion at lowenergies. Phys. Lett. 169B, 73–78 (1986)ADSCrossRefGoogle Scholar
 18.V.A. Rubakov, Grand unification and heavy axion. JETP Lett. 65, 621–624 (1997). arXiv:hepph/9703409 [hepph]ADSCrossRefGoogle Scholar
 19.Z. Berezhiani, L. Gianfagna, M. Giannotti, Strong CP problem and mirror world: the Weinberg–Wilczek axion revisited. Phys. Lett. B 500, 286–296 (2001). arXiv:hepph/0009290 [hepph]ADSCrossRefGoogle Scholar
 20.S.D.H. Hsu, F. Sannino, New solutions to the strong CP problem. Phys. Lett. B 605, 369–375 (2005). arXiv:hepph/0408319 [hepph]ADSCrossRefGoogle Scholar
 21.A. Hook, Anomalous solutions to the strong CP problem. Phys. Rev. Lett. 114(14), 141801 (2015). arXiv:1411.3325 [hepph]ADSCrossRefGoogle Scholar
 22.H. Fukuda, K. Harigaya, M. Ibe, T.T. Yanagida, Model of visible QCD axion. Phys. Rev. D 92(1), 015021 (2015). arXiv:1504.06084 [hepph]ADSCrossRefGoogle Scholar
 23.C.W. Chiang, H. Fukuda, M. Ibe, T.T. Yanagida, 750 GeV diphoton resonance in a visible heavy QCD axion model. Phys. Rev. D 93(9), 095016 (2016). arXiv:1602.07909 [hepph]ADSCrossRefGoogle Scholar
 24.S. Dimopoulos, A. Hook, J. Huang, G. MarquesTavares, A collider observable QCD axion. JHEP 11, 052 (2016). arXiv:1606.03097 [hepph]CrossRefGoogle Scholar
 25.T. Gherghetta, N. Nagata, M. Shifman, A visible QCD axion from an enlarged color group. Phys. Rev. D 93(11), 115010 (2016). arXiv:1604.01127 [hepph]ADSCrossRefGoogle Scholar
 26.A. Kobakhidze, Heavy axion in asymptotically safe QCD. arXiv:1607.06552 [hepph]
 27.P. Agrawal, K. Howe, Factoring the Strong CP Problem. Submitted to: JHEP (2017), arXiv:1710.04213 [hepph]
 28.P. Agrawal, K. Howe, A flavorful factoring of the strong CP problem. arXiv:1712.05803 [hepph]
 29.M.K. Gaillard, M.B. Gavela, R. Houtz, P. Quilez, R. Del Rey, Color unifed dynamical axion. arXiv:1805.06465 [hepph]
 30.G.G. Raffelt, Astrophysical axion bounds. Lect. Notes Phys. 741, 51–71 (2008). arXiv:hepph/0611350 [hepph] [51 (2006)]
 31.J.H. Chang, R. Essig, S.D. McDermott, Supernova 1987A constraints on subGeV dark sectors, millicharged particles, the QCD axion, and an axionlike particle. JHEP 09, 051 (2018). arXiv:1803.00993 [hepph]ADSCrossRefGoogle Scholar
 32.I.G. Irastorza, J. Redondo, New experimental approaches in the search for axionlike particles. Prog. Part. Nucl. Phys. 102, 89–159 (2018). arXiv:1801.08127 [hepph]ADSCrossRefGoogle Scholar
 33.K. Hamaguchi, N. Nagata, K. Yanagi, J. Zheng, Limit on the axion decay constant from the cooling neutron star in Cassiopeia A. Phys. Rev. D 98(10), 103015 (2018). arXiv:1806.07151 [hepph]ADSCrossRefGoogle Scholar
 34.J.E. Kim, Weak interaction singlet and strong CP invariance. Phys. Rev. Lett. 43, 103 (1979)ADSCrossRefGoogle Scholar
 35.M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Can confinement ensure natural CP invariance of strong interactions? Nucl. Phys. B 166, 493–506 (1980)ADSMathSciNetCrossRefGoogle Scholar
 36.A.R. Zhitnitsky, On possible suppression of the axion hadron interactions. (In Russian). Sov. J. Nucl. Phys 31, 260 (1980). [Yad. Fiz.31,497(1980)]Google Scholar
 37.M. Dine, W. Fischler, M. Srednicki, A simple solution to the strong CP problem with a harmless axion. Phys. Lett. 104B, 199–202 (1981)ADSCrossRefGoogle Scholar
 38.M.S. Turner, Cosmic and local mass density of invisible axions. Phys. Rev. D 33, 889–896 (1986)ADSCrossRefGoogle Scholar
 39.L. Visinelli, P. Gondolo, Dark matter axions revisited. Phys. Rev. D 80, 035024 (2009). arXiv:0903.4377 [astroph.CO]ADSCrossRefGoogle Scholar
 40.K. Saikawa, Axion as a nonWIMP dark matter candidate. PoS EPSHEP2017, 083 (2017). arXiv:1709.07091
 41.R. Holman, S.D.H. Hsu, T.W. Kephart, E.W. Kolb, R. Watkins, L.M. Widrow, Solutions to the strong CP problem in a world with gravity. Phys. Lett. B 282, 132–136 (1992). arXiv:hepph/9203206 [hepph]ADSCrossRefGoogle Scholar
 42.M. Kamionkowski, J. MarchRussell, Planck scale physics and the Peccei–Quinn mechanism. Phys. Lett. B 282, 137–141 (1992). arXiv:hepth/9202003 [hepth]ADSCrossRefGoogle Scholar
 43.S.M. Barr, D. Seckel, Planck scale corrections to axion models. Phys. Rev. D 46, 539–549 (1992)ADSCrossRefGoogle Scholar
 44.S. Ghigna, M. Lusignoli, M. Roncadelli, Instability of the invisible axion. Phys. Lett. B 283, 278–281 (1992)ADSCrossRefGoogle Scholar
 45.H.M. Georgi, L.J. Hall, M.B. Wise, Grand unified models with an automatic Peccei–Quinn symmetry. Nucl. Phys. B 192, 409–416 (1981)ADSCrossRefGoogle Scholar
 46.S.B. Giddings, A. Strominger, Loss of incoherence and determination of coupling constants in quantum gravity. Nucl. Phys. B 307, 854–866 (1988)ADSMathSciNetCrossRefGoogle Scholar
 47.S.R. Coleman, Why there is nothing rather than something: a theory of the cosmological constant. Nucl. Phys. B 310, 643–668 (1988)ADSMathSciNetzbMATHCrossRefGoogle Scholar
 48.G. Gilbert, Wormhole induced proton decay. Nucl. Phys. B 328, 159–170 (1989)ADSCrossRefGoogle Scholar
 49.S.J. Rey, The axion dynamics in wormhole background. Phys. Rev. D 39, 3185 (1989)ADSCrossRefGoogle Scholar
 50.L. Randall, Composite axion models and Planck scale physics. Phys. Lett. B 284, 77–80 (1992)ADSCrossRefGoogle Scholar
 51.B.A. Dobrescu, The strong CP problem versus Planck scale physics. Phys. Rev. D 55, 5826–5833 (1997). arXiv:hepph/9609221 [hepph]ADSCrossRefGoogle Scholar
 52.D. Butter, M.K. Gaillard, The axion mass in modular invariant supergravity. Phys. Lett. B 612, 304–310 (2005). arXiv:hepth/0502100 [hepth]ADSCrossRefGoogle Scholar
 53.M. Redi, R. Sato, Composite accidental axions. JHEP 05, 104 (2016). arXiv:1602.05427 [hepph]ADSMathSciNetzbMATHCrossRefGoogle Scholar
 54.H. Fukuda, M. Ibe, M. Suzuki, T.T. Yanagida, A “gauged” \(U(1)\) Peccei–Quinn symmetry. Phys. Lett. B 771, 327–331 (2017). arXiv:1703.01112 [hepph]ADSzbMATHGoogle Scholar
 55.H. Fukuda, M. Ibe, M. Suzuki, T.T. Yanagida, Gauged Peccei–Quinn symmetry—a case of simultaneous breaking of SUSY and PQ symmetry. JHEP 07, 128 (2018). arXiv:1803.00759 [hepph]ADSCrossRefGoogle Scholar
 56.M. Ibe, M. Suzuki, T.T. Yanagida, \(BL\) as a gauged Peccei–Quinn symmetry. JHEP 08, 049 (2018). arXiv:1805.10029 [hepph]ADSGoogle Scholar
 57.B. Lillard, T.M.P. Tait, A high quality composite axion. arXiv:1811.03089 [hepph]
 58.R. Alonso, A. Urbano, Wormholes and masses for goldstone bosons. arXiv:1706.07415 [hepph]
 59.G. ’t Hooft, Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking. NATO Sci. Ser. B 59, 135–157 (1980)Google Scholar
 60.S. Dimopoulos, S. Raby, L. Susskind, Light composite fermions. Nucl. Phys. B 173, 208–228 (1980)ADSCrossRefGoogle Scholar
 61.N. ArkaniHamed, Y. Grossman, Light active and sterile neutrinos from compositeness. Phys. Lett. B 459, 179–182 (1999). arXiv:hepph/9806223 [hepph]ADSCrossRefGoogle Scholar
 62.A.G. Cohen, D.B. Kaplan, A.E. Nelson, Counting 4 pis in strongly coupled supersymmetry. Phys. Lett. B 412, 301–308 (1997). hepph/9706275ADSCrossRefGoogle Scholar
 63.B.M. Gavela, E.E. Jenkins, A.V. Manohar, L. Merlo, Analysis of general power counting rules in effective field theory. Eur. Phys. J. C 76(9), 485 (2016). 1601.07551ADSCrossRefGoogle Scholar
 64.A. Ernst, A. Ringwald, C. Tamarit, Axion predictions in \(SO(10)\times U(1)_{{\rm PQ}}\) models. JHEP 02, 103 (2018). arXiv:1801.04906 [hepph]ADSGoogle Scholar
 65.M. Kawasaki, K. Nakayama, Axions: theory and cosmological role. Ann. Rev. Nucl. Part. Sci. 63, 69–95 (2013). arXiv:1301.1123 [hepph]ADSCrossRefGoogle Scholar
 66.G. Grilli di Cortona, E. Hardy, J. Pardo Vega, G. Villadoro, The QCD axion, precisely. JHEP 01, 034 (2016). arXiv:1511.02867 [hepph]CrossRefGoogle Scholar
 67.Planck Collaboration, N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters. arXiv:1807.06209 [astroph.CO]
 68.M. Tanabashi et al., Review of particle physics. Phys. Rev. D 98(3), 030001 (2018)ADSCrossRefGoogle Scholar
 69.P. Ramond, Group theory: a physicist’s survey (2010). http://www.cambridge.org/de/knowledge/isbn/item2710157
 70.R. Slansky, Group theory for unified model building. Phys. Rept. 79, 1–128 (1981)ADSMathSciNetCrossRefGoogle Scholar
 71.R. Feger, T.W. Kephart, LieART—a mathematica application for Lie algebras and representation theory. Comput. Phys. Commun. 192, 166–195 (2015). arXiv:1206.6379 [mathph]ADSMathSciNetzbMATHCrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP^{3}