Axions in a highly protected gauge symmetry model

We study QCD axion or cosmological axion-like particles (ALPs) in a model inspired by the recent interest in 4-dimensional clockwork models, with the global symmetry being accidentally enforced by a gauge abelian quiver with scalar bifundamental fields. For the QCD axion, we analyze the connection between the degree of protection of the axion mass against gravitational corrections, the explanation of the hierarchy $f_a \ll M_P$ and the number of colored fermions needed to generate anomalous couplings to gluons, all linked together by the underlying gauge symmetries. Based on that model and on the comparison with earlier models in the literature, we derive certain general conclusions on QCD axion models that use accidental global symmetries. For the ALPs, assuming that their mass is solely given by gravitational corrections, we identify the parameter space where the decay constant and the mass are consistent with the DM abundance, and we show that this clockwork-inspired model is a particularly economical model for a very light ALP DM candidate.


Introduction
New pseudo-Goldstone bosons (PGB's) may play an important role in particle physics and cosmology, since they can solve the strong CP-problem [1][2][3], explain dark matter [4,5] or inflation [6,7] and make dark energy dynamical [8][9][10][11]. Some of these models require that the PGB mass be very light, so there is a need to protect it from large explicit (e.g. gravitational) breaking of global symmetries [12][13][14]. Consequently, one often considers the symmetry from which the PGB's originate as accidental, i.e. as unbroken by any gauge-invariant operator up to a given dimension. One often achieves this by interpreting these symmetries as remnants of extra-dimensional gauge symmetries (or, more precisely, by interpreting PGB's as fifth components of vector fields which appear as scalars in 4d).
Inspired by the recent interest in 4-dimensional clockwork models [15][16][17][18][19][20][21][22][23], we discuss in this paper the properties and potential applications of the 4d quiver model obtained by latticizing a 5d (abelian) gauge theory in a linear dilaton background [24], with Dirichlet boundary conditions for the 4-d components of the gauge boson [18,20]. As a result of the 5d gauge invariance, the 4d field content is such that its most general renormalizable gaugeinvariant lagrangian preserves an accidental global symmetry. Furthermore, the specific 5d background, or equivalently the specific 4d gauge charge assignment, ensures a strong protection of this accidental symmetry from explicit breaking terms, even when the discretization is crude (i.e. when the quiver has few sites).
The plan of the paper is as follows: in section 2 we recall the 4d model with a focus on the light scalar and examine its properties. In section 3, we discuss its potential identification with a QCD axion, while in section 4 we consider its applications to cosmological models that make use of a light particle (e.g. PGB dark matter and quintessence models). We present our conclusions in the last section. Some appendices cover additional material: appendix A contains the 5d deconstruction of an abelian vector field in a linear dilaton background whose low-energy limit matches that of our 4d picture, appendix B discusses the massive states of the model of section 2, appendices C.1 and C.2 describe realizations of the QCD axion discussed in section 3, appendix D displays a calculation of the axion-photons couplings of sections 3 and 4.1 and appendix E discusses the ranges of parameters of the model which allow the axion to be (a detectable kind of) dark matter.

Gauge group and matter content
The (4d) setup we consider is an abelian quiver model with bifundamental scalar fields, first presented in [18] as the deconstruction [25,26] of a 5d abelian gauge theory on an orbifolded linear dilaton background with Dirichlet boundary conditions for the 4d gauge field 1 (see appendix A), and whose low-energy theory was derived in [20] as the 4d theory obtained after chiral symmetry breaking by some confining non-abelian gauge group. The precise matter content and charge assignment is given by the following quiver (where q and N are integers): where F i is the field strength of the abelian vector field A i , with coupling constant g i , and with the covariant derivatives D µ φ k = ∂ µ − i(1 − δ k,0 )A µ,k + iq(1 − δ k,N )A µ,k+1 φ k . This lagrangian has a U (1) N +1 invariance, with a U (1) N gauged subgroup.
This model is inspired by the so-called clockwork mechanism [15][16][17][18][19][20][21][22][23] and has been introduced in [18] as a possible realization of it, so we will comment on defining features of this mechanism if we recover them while we proceed, or discuss those which are different.

Spontaneous breaking and Goldstone mode
We are interested in obtaining Goldstone bosons, so we consider the spontaneous breaking of the full U (1) N +1 mentioned previously by choosing the parameters m 2 k and λ kl of (2.1) so that all the scalar fields φ k get vev's f k . The spectrum then consists after gauge fixing of 1 See also Appendix A of [27]. 2 We use conventions of [28], in particular signature (− + ++). N massive vectors, N + 1 massive real scalars (discussed in appendix B) and one Goldstone boson.
Since the vev's f k break all the gauge symmetries, N out of the N + 1 phases of the φ k are absorbed by the gauge vectors through the Higgs effect. The absorbed phase combinations depend on the charges and vev's (we write φ k = f k +r k √ 2 e i θ k f k ): (2. 2) The last, uneaten phase a remains in the spectrum after gauge fixing as a Goldstone boson associated to the accidental U (1) a global symmetry which is the ungauged factor of the U (1) N +1 symmetry group of (2.1). The profile of this boson along the original phases is orthogonal to the qf i−1 θ i−1 −f i θ i gauge Goldstone bosons profiles. If we canonically normalize the field and the vev's are taken to be all equal, which will be assumed from now on 3 (we then note f k = f ), it reads: Eq. (2.3) displays the exponential localization discussed in clockwork models, and the charges of the original scalar fields under the global symmetry also match those which appear in those models. Indeed, U (1) a acts here as φ k → e iq k α φ k .

Goldstone boson protection
The lagrangian (2.1), has an accidental exact U (1) a global symmetry at renormalizable level, hence the axion a is massless. We expect however that global symmetries are broken by gravity effects [12][13][14], which forces us to include all higher order operators allowed by gauge invariance in the effective theory. For the quiver of Figure 1, these operators must be combinations of Hence, operators that explicitly break the global symmetry must involve the second term and be of extremely high dimension as soon as q and N are both slightly bigger than one. 3 In the generic case, the axion profile is, up to a normalization factor, We thus obtain in this setup a pseudo-Goldstone boson with a mass very well protected by the gauge symmetry. The exponential dependence on q and N of the second operator of and M c is the cutoff of the theory, which we take close to the Planck mass M P when we consider gravity-induced breaking effects (recall however that if a large number N of particles is present, the actual cutoff of the theory cannot be more than roughly M P √ N [29][30][31][32]). Even though M c may also be the scale of other breaking effects (such as the mass of heavy fermions explicitly breaking U (1) a and running in loops, see Appendices A and C.1 for discussions on this topic), we will for simplicity focus on gravitational scale breaking.

QCD axion
We dedicate this section to the study of the compatibility of the Peccei-Quinn (PQ) idea [1][2][3] (see also [33] for a review) with the setup of section 2 4 .

Accidental Peccei-Quinn symmetry in the low-energy field theory
We study in this section the low-energy effective field theory of the axion, assuming that every other massive field has been integrated out. In order to identify U (1) a with a Peccei-Quinn symmetry, we consider the following axionic coupling 5 : 4 Accidental PQ symmetries have been studied in many different setups, see for example [34][35][36][37][38][39][40][41][42][43][44]. 5 We will not pay attention to writing dimensionless quantities in the log's since it does not affect the discussion about axions which reside in the phases of the fields. Every expression in a log to appear in the rest of the paper should then be thought of as a dimensionless one (e.g. a log(scalar) means a log(scalar divided by a mass scale)).
where G µν is the gluon field strength andG µν = µνρσ G ρσ its dual. In this section, we will not discuss the origin of this coupling, which may arise from a string theory or from a UV-complete field theory (section 3.2 deals with the field theoretic case).
The operator in the log is, as we said in section 2.3, the first gauge-invariant term capable of coupling the axion of (2.3) to the gluons that we could have written (using a gaugeinvariant term is necessary in order not to generate any U (1) i × SU (3) 2 gauge anomaly). This coupling has two major generic features: it involves all the quiver sites, and it implies a decrease in the decay constant of the axion compared to the scale of breaking f . Indeed, when we plug back the axion profile (2.3) in (3.1), we obtain: where we recognize the effective axion decay constant of (2.6). This suggests that the present setup could describe intermediate scale axion decay constant obtained from high scale physics (such as string scale physics) 6 .
When non-perturbative effects of QCD turn on, (3.2) induces a potential for the axion: We also include every gauge-invariant term to the potential, according to the discussion of section 2.3, and in particular generate a classical explicit breaking mass term (2.7) for the axion. In order to have a fa − θ QCD < 10 −10 at the minimum of the potential and solve the strong-CP problem, we must ensure [42][43][44] that: For example, when q = 3, N = 2 and M c = M P , it implies f 10 12 GeV. If now q = 3, N = 3 and M c = M P , this becomes f 10 16 GeV 7 . This constraint is less and less demanding when q or N grow, and it shows that it is easy to protect the PQ solution in our setup with a few quiver sites, even when the axion scale is high (e.g. string like).
Axion couplings to photons, which are the subject of most axion searches, are also part of this low-energy discussion. They can be derived when we consider the axionic generalizations of (3.1):

Axionic couplings from heavy fermion loops
We now discuss the UV origin of (3.1) in terms of loops of heavy fermions coupling to the axion.
Let us first recall how axionic couplings are generated via quarks loops. (Global) anomalies with respect to SU (3) c are mediated by colored fermions 8 with some charge under the (global) symmetry, which run in loops between gluons and scalars, whose phase contains part of the axion mode. The schematic procedure 9 is: 8 In our case, heavy fermions must obtain their mass from Yukawa couplings since Dirac or Majorana mass terms require vector-like representations. 9 An example of the triangle loop calculation, including the numerical coefficients, is presented in appendix D.
where σ is a scalar field, Q L,R are left and right handed colored fermions, T a are the generators of SU (3) c , A is the gluon field of field strength G and y is a Yukawa coupling.
We then see how to generate (3.1) from fermions loops, starting from the following lagrangian 10 : This procedure is actually the minimal one (with dimension four Yukawa couplings) that generates an SU (3) 2 c ×U (1) a anomaly without generating gauge anomalies (or said differently, that generates (3.1)). It requires adding colored fermions at each site, in accordance with the fact that (3.1) involves all quiver links. In terms of couplings defined in (3.5), it has C = 1 and E = 0. Note that the lagrangian (3.7) (and (3.9) below) respects the global symmetry U (1) a .
Therefore, it cannot generate the scalar potential (2.5) by quantum corrections. However, since we had to nonetheless include (2.5) as a gravity correction, we should also consider all gauge-invariant non-renormalizable fermionic operators in addition to (3.7). Those operators could classically break U (1) a and generate both the mass of the axion and its couplings to the gluons. In appendix C.1, we present a model with such fermionic operators. If one wants to circumvent the conclusions of (3.7), one can also assign gauge charges to the fermions so that their lowest gauge-invariant mass terms are of higher dimension. One example of this type is where M c is the cutoff of the theory. The action (3.9) couples the axion to the gluons via a number of additional fermions independent on N , but the high dimension of the second coupling in the first line of (3.9) lowers the mass of the Q i fermions. Since these fermions are colored and unobserved at the LHC, we must impose m Q i a few TeV, which gives, if one takes as an example f = The bound is even more stringent as soon as we decrease f in order to satisfy (3.4). It imposes in particular that we cannot reduce the decay constant of the axion using (2.6) and This discussion goes beyond the particular case of the quiver of Figure 1 and concerns every theory with a protected PQ symmetry: the higher the quality of an accidental PQ symmetry, the higher the number of fermions required to make it anomalous with axionic couplings generated by fermion loops 11 .
Indeed, any axionic coupling term in such a theory free of gauge anomalies must be of the form: where O is by construction gauge-invariant and not invariant under the anomalous global symmetry. If it arises from loops of heavy fermions, it is through the scheme discussed above: (where we assumed that we removed from the sum every pair of vector-like fermions), and

Axion-like particles
In this section, we study the case of axion-like particles (ALP's), which generically refers to pseudo-Goldstone bosons not designed to solve the strong-CP problem, and whose interactions are consequently less constrained than those of the QCD axion.
Since section 3.2 showed us that making our accidental U (1) a anomalous with respect to some external gauge group is demanding, especially when N grows, we will focus on those ALP's which do not have any anomalous couplings. We will see that, in such models, gravitational contributions are sufficient to make the ALP a good dark matter candidate.
Furthermore, there exist operators which make the dark matter ALP detectable, if for example some standard model particles are charged under the quiver gauge symmetry. This only requires limited additions to the particle content of Figure 1, even when the number of quiver sites is large. However, while the U (1) a protection by the quiver is strong enough to generate quintessence-like mass scales, the need for trans-Planckian field values of usual axion quintessence models is still present in our setup, and is exacerbated by the reduction of the axion effective decay constant.

ALP's potentials and dark sector candidates
Since most ALP's are used in cosmology (see [45] for a review), let us first discuss the cosmological relevance of our setup. In the non-anomalous setup that we chose to consider in this section, we think of any ALP potential as generated by some classical explicit breaking in a UV theory. The lowest-dimensional gauge-invariant potential of this type for the particle a of (2.3) is (2.5), which very weakly breaks U (1) a and entitle us to call a a pseudo-Goldstone boson, as we discussed in section 2.3. It is a typical periodic potential, consistent with the ALP's origin as a periodic phase degree of freedom, and such potentials are very useful in cosmology: the smallness of the masses and the specific potential they provide make ALP's good dark matter or dynamical dark energy candidates via the misalignment mechanism.
The relic density can be calculated once we are given the initial value a init of the ALP field after inflation and its mass m a (given in (2.7) for our setup): where f a (which defines the periodicity of the ALP potential) is given in (2.6) and init depends on one's taste for tuning (the spontaneous breaking of U (1) a must happen before inflation, see the discussion of appendix E). In order for these formulas to be valid, i.e. for the ALP to behave like CDM before radiation-matter equality or like dark energy today, we supplement (4.1) with: In our setup, obtaining masses as low as those which appear in (4.2) without tuning is easy (for instance, (2.7) equals ∼ 10 −33 eV when f = 0.13M P , q = 3, N = 4). However, we can see from the comparison of (4.1) and (4.2) that axion quintessence demands initial values which are higher than the Planck mass. This can be achieved with some tuning on init or when the effective decay constant of the axion is increased compared to the mass scales of the model (as in clockwork models which, however, have no mass protection mechanism built in). Since our effective decay constant (2.6) is reduced, the latter is not an option while the former is not enough to reach the correct energy density (if we insist on keeping f below the Planck mass): indeed if we impose m a 10 −33 eV, we can only obtain Ω a 0.05 and would need at least 13 of such ALPs to reach the observed dark energy density.
In contrast, natural dark matter candidates do arise in our model. In Figure 2 We see in Figure 2 that we obtain suitable DM candidates, and that the dependence on q and N of the mass (2.7) allows us to reach very low ALP's masses. These small In order to conclude that such ALP's are to play a role in the cosmic evolution, we must check that their lifetime can be comparable to or bigger than the age of the universe. In generic models, there is a decay channel of an ALP into two photons, usually coming from a U (1) P Q × U (1) 2 em anomaly. Even though there is no anomaly in the models of this section, non-anomalous, CP-even and gauge invariant operators that enable this decay exist. For instance, they can arise if we couple one of the quiver sites of Figure 1 to an anomaly-free set of electrically charged fermions displayed in Table 1, while we keep the standard model particles uncharged under the quiver gauge group. Table 1: Anomaly-free set of fermions coupling the ALP to the photon field (the three first columns indicate the gauge charges of the fields whereas the last one gives the PQ charges induced by (4.3), as functions of q 1 and q 2 which are arbitrary) With such charges, one can write Yukawa couplings y 1,2 to φ i : The effective operators describing the decay a → γγ then are (see Appendix D for the computation): and F is the photon field strength. Notice that, contrary to anomalous couplings such as (3.1), non-anomalous interactions are site-localized, and exhibit clockwork-like effects due to the profile (2.3). This feature will be present in all the operators discussed in this section. We can also see that the non-anomalous nature of the ALPphotons coupling makes this interaction of derivative type and of higher dimension than usual anomalous aFF terms, so this decay does not make our ALP's unstable over the cosmic history. Indeed these couplings give a decay rate: where α is the fine structure constant. Hence, we conclude that this decay channel is harmless with respect to the cosmic evolution of our ALP's 12 . Indeed, as guessed above, the nonanomalous nature of the ALP-photons coupling forces the m a factor to appear in the decay rate (4.5) at a higher power than in the case of usual aFF -induced decays and ensures a long ALP lifetime. The clockwork-like dependence of (4.5) only tends to weaken the ALP couplings to photons when matter is coupled to the first quiver sites. 12 For ultra-light dark matter with m ∼ 10 −21 eV, if we choose f = m 1 = 2m 2 = 0.3M P and n = 1, the decay rate is ∼ q 2i 1+q 2 +...q 2N (10 −300 s −1 ).
The example of Table 1 is a realization of the more general gauge-invariant non-anomalous operators, coupling the axion to the photon field, which we can write within the effective field theory: where D is the covariant derivative, there is no summation over the index i and Λ the scale at which this operator is generated. For instance, in the example of Table 1 Table 1, but should not be at a lower scale than the effective theory one.
From (4.6) we can calculate the decay rate of an axion into two photons: This result generalizes (4.5) and of course does not spoil the conclusions made with Λ ∼ f since the dependence on m a , responsible of the low value of Γ a→γγ , has not changed and Λ f can only weaken the decay rate.

Detection via NMR
The non-anomalous couplings to photons of (4.6), too weak to destabilize the cosmic history of our ALP's, are also too weak to be probed by current ALP's searches, which rely on a dimension 5 anomalous aFF coupling. Non-anomalous dimension 5 generic couplings of a Goldstone pseudoscalar a to a detector's matter can be parametrized [46] as follows: where g's are dimensionless coupling constants, f a is again the axion decay constant, and N and e are respectively the nucleon and electron fields. In our setup, they can be generated in field theory if we charge the first family of the standard model under U (1) i,i+1 , according to Table 2 13 , in a way which gives them U (1) a charges. At lowest order, the most general Table 2: SM charges that produce an ALP-spin coupling (the first columns indicate the gauge charges of the fields whereas the last one gives the PQ charges induced by (4.9), as functions of q Q , q H and q L which are arbitrary) lagrangian is the SM lagrangian where only the first family Yukawa terms have been modified: where v is the Higgs vev and where we assumed that these higher order Yukawa couplings come from the same physics which generated (2.5), even though the fact that the precise scale M c divides these operators is of no importance for what follows. One can make the 13 All anomalies involving at least one standard model factor are canceled with these charges. One must however add additional fermions only charged under U (1) i,i+1 to cancel the U (1) i,i+1 × U (1) i,i+1 × U (1) i,i+1 and U (1) i,i+1 -gravity anomalies. See appendix C.1 for explicit examples on how this can be done. fermion masses in (4.9) real with an appropriate chiral redefinition of the fermions, and obtain from their kinetic terms the expected couplings: where no anomalous term appeared since the U (1) a symmetry is anomaly-free and where we note that, similar to what was observed previously for axion-photons couplings, the ALP-spin coupling of (4.10) is site-dependent due to the clockwork profile (2.3).
If the mass (2.7) of the ALP is such that it constitutes part of the dark matter, these couplings may soon be tested via Nuclear Magnetic Resonance 14 (NMR) by the CASPEr-Wind experiment [49]. As an illustration, in Figure 3 we assume that the coupling (4.10) is located at site i = 0 of the quiver and restrict ourselves to the (q, N ) values displayed in Like we did in (4.6), we can generalize such couplings in the gauge-invariant effective theory: Once again, the scale Λ is a priori undetermined since (4.11) does not break U (1) a , for instance Λ = M P if (4.11) is of gravitational origin. However, taking Λ = M P does not allow to detect our DM candidates, contrary to the case of (4.10) where it is equal to f . In this paper we have investigated the scenario of the QCD axion or ultralight axionlike particles obtained as pseudo-Goldstone modes of an accidental global symmetry coming from an abelian gauge theory quiver with scalar bifundamental fields. The underlying gauge symmetries protect very efficiently the global symmetry against the contribution from nonanomalous explicit breaking effects to the axion potential, and in particular to its mass. Such contributions are a function of the gauge charge assignment for the scalars and the number of the quiver sites. Already with a few quiver sites they are sufficiently small for the pseudo-Goldstone boson to play the role of the QCD axion or a cosmologically relevant axion-like particle. We have mainly focused on gravity-breaking effects, which account for the fact that quantum gravity does not respect global symmetries, but other sources of non-anomalous explicit breaking effects can be considered as well.
In our model, a genuine QCD axion obtained from a renormalizable theory requires an additional fermionic content, with the number of heavy colored fermions growing exponentially with the number N of gauge groups, but its mass protection against gravitational effects is already suffcient for N = 2. Stable pseudo-Goldstone dark matter models can also be easily built with small number N of gauge groups. A very interesting aspect of them is that gravitational contributions, allowed but controlled by the gauge symmetry protection, can be used in a constructive way, to induce a just right axion potential without referring to any new strongly-interacting sector and its chiral anomalies. Such a dark matter axion-like  We start by considering a 5d manifold which is the product of the 4d Minkowski space with an interval of length L, with a 5d theory living on it: where we split the 5d coordinates z M into 4d Minkowski coordinates x µ , and the position along the interval y.
We discretize the fifth dimension interval down to a regularly-spaced lattice of N +2 sites.
Defining ∆ 4 = L N +1 , y i = i∆ 4 (where i runs from 0 to N + 1), this amounts to replacing: We choose to denote f (x, y i ) = f i (x) in what follows. We do not wish to study the dynamics of the background and restrict to the following static metrics: The case a = ky, b = 0 describes the so-called linear dilaton background in the conformally flat frame, whereas a = b = ky is the Randall-Sundrum metric.
We will study a five-dimensional abelian theory of lagrangian where the 4d indices are contracted using the Minkowski metric. We impose 5d Dirichlet boundary conditions for A µ and Neumann conditions for A 4 : Deconstruction now yields: 4 . Dropping the primes and using the boundary conditions, we finally obtain Specializing to the linear dilaton background for which a = ky, b = 0, the lattice action now becomes thus establishing the link between the low-energy limit of the 4d model of section 2 and the deconstruction on a linear dilaton background of an 5d abelian vector mode with boundary conditions (A.5).
Finally, the Wilson line e i dy A 4 (x,y) gets mapped to the U (1) a -violating potential of (2.4): (A.10)

A.2 Charged bulk fermion
Deconstructed fermions might be useful in order to get insights on how U (1) a can be made anomalous or classically broken [51,52]. However, as we will see below, this procedure is not applicable in our setup. Indeed, let us consider the action of a bulk fermion charged under the abelian symmetry of the previous section: We now restrict the discussion to the linear dilaton background, with the vector boundary conditions of the previous section, to supplement with boundary conditions for the fermion.
If we choose Ψ 0,L = Ψ N +1,R = 0, the deconstructed lagrangian becomes (where we defined e = e 4 √ ∆ 4 , and dropped the primes): However, we cannot UV complete this lagrangian as we did in (A.9) since its k-dependence prevents from recognizing the low-energy expansion of the φ i 's. Only when the background is flat (k = 0 ⇐⇒ q = 1) one can follow such a procedure (when e = 1): after gauge symmetry breaking, with eigenvalues and eigenvectors: All vectors are massive since all gauge symmetries are broken. We recognize in (B.2) the specific (band-like) massive spectrum of clockwork models.
The masses of the Higgs-like r k scalar fields depend on the choices of parameters in (2.1) and do not necessarily lie in a band.

C Realizations of benchmark QCD axion models
We discuss the compatibility of usual benchmark invisible QCD axion models, namely KSVZ [53,54] and DFSZ [55,56] models, with our setup. In these models, the U (1) P Q anomaly with respect to QCD is respectively carried by additional heavy colored particles or by the standard model quarks, and the PQ symmetry arises from the introduction of a SM singlet scalar field (as well as an extra Higgs doublet for the DFSZ model). The phase shift symmetry of this singlet is not gauge protected in their original realization, consequently so we replace it by the accidental symmetry of our quiver model. We will also discuss, in the case of the KSVZ model, how the additional fermions can break U (1) a and generate (2.5) as a quantum correction.

C.1 KSVZ model: anomaly mediated by additional particles
The original KSVZ model was already (anonymously and briefly) introduced in (3.6), where σ is a SM gauge singlet, and some quiver versions of it were already described in (3.7) and (3.9). There, the needed couplings were ad hoc, in contrast with the fact that we talked about an accidental Peccei-Quinn symmetry. However, we can choose the fermions charges so that the procedure of (3.7) (respectively (3.9)) is automatically implied by the most general renormalizable gauge-invariant lagrangian (respectively the lowest-order gaugeinvariant lagrangian which renders all the additional fermions massive), given the gauge charges of the different fields involved. This is for instance achieved if the fermions charges are those displayed in Table 3 (respectively Table 4). Table 3: Colored fermions charged under the quiver gauge group of Figure 1, canceling SU (3) 2 c − U (1) i anomalies and leading to a QCD axion (U (1) a charges are those imposed by (C.1), functions of q R which is arbitrary) For example, the most general renormalizable lagrangian associated with Table 3 is, with such charges: and it defines the U (1) a charges of the fermion bilinears which make U (1) a accidentally conserved, which in turn determine the U (1) a × SU (3) 2 anomaly and justify the procedure (3.7).
Along with these colored fermions, one must also add fermions only charged under the quiver gauge group to cancel the U (1) i × U (1) j × U (1) k anomalies. A way of achieving this for (3.7) is presented in Table 5. Table 5: SM-singlet fermions charged under the quiver gauge group of Figure 1, canceling cubic quiver anomalies of Table 3 (q L is arbitrary) One can check at the level of these fermionic contents that the models are gauge-anomalyfree, and at the level of their most general renormalizable lagrangian that they preserve an anomalous U (1) a global symmetry.
Still, we only considered renormalizable lagrangian, so we could ask whether Plancksuppressed fermionic terms will be generated along with (2.5), whether such terms explicitly break U (1) a and whether they can induce quantum corrections to the axion mass. In the cases discussed above, we can supplement (C.1) by:  Table 6, one can write the following lagrangian: Table 6: Colored fermions giving the major contribution to the axion mass (U (1) a charges are those imposed by (C.3), functions of q 0 which is arbitrary))

C.2 DFSZ model: anomaly mediated by standard model quarks
We focus now on the DFSZ model, since, contrary to the KSVZ model, the original model has the important feature that the anomaly is only carried by the standard model quarks.
It makes uses of two Higgs doublets H 1,2 , an extra singlet scalar σ and can be summarized as follows: The first line of (C.4) is invariant under a global U (1) which acts on the scalars as σ → e iα σ, H 1,2 → e −iα H 1,2 . The symmetry is spontaneously broken and, according to the second line of (C.4), anomalous with respect to QCD.
In order to adapt this construction to the case of our quiver, it is important to disentangle two features of (C.4): the log(H 1 H 2 ) operator originates from the Yukawa terms of the SM quarks which run into loops 16 , whereas the H 1 H 2 σ 2 term (and the rest of the tree-level lagrangian) defines which symmetry is respected 17 . Thus, if we want to apply this logic to U (1) a , we must identify gauge charges of H 1 and H 2 which will preserve the accidental U (1) a , and identify a gauge-invariant operator O, charged under U (1) a , which will induce an axionic coupling log(O)GG to the gluons. We can immediately understand from section 3.2 that O must be of high dimension, so it must be generated by more colored particles than standard model quarks alone. It would thus be more precise to talk about a mixed DFSZ-KSVZ model, where the anomaly is mediated by both standard model quarks and additional fermions. In particular, we loose the pleasant economical quark content of the original DFSZ model, since one needs a growing number of additional particles as in the KSVZ case.
As an (unoptimized) example of this procedure, we choose the matter content and gauge charges of Table 7 in addition to that of Figure 1 18 . With these charges, one has the following most general renormalizable interaction terms: where we identify O = (H 1 H 2 ) 3 φ 5 0 φ * q 1 φ * q 2 2 ...φ * q N N . U (1) a charges are assigned to H 1,2 so that H 1 H 2 φ 2 0 is invariant, and log(O)GG makes U (1) a anomalous. The axion effective decay constant displays the same asymptotic dependence than (2.6): f a ∼ f q N .
It is worth noticing that a µ 2 H 1 H 2 or µH 1 H 2 σ term was not included in (C.4) in order to maintain a global symmetry, whereas we now cannot write something else than (C.5) that would respect gauge symmetries, which was the original goal when we introduced the quiver.
The first allowed U (1) a -violating operator is again φ 0 φ q 1 ...φ q N N and the discussion around eq. (3.4) applies. 16 Actually, since three quark families run in the loops, the correct operator is log (H 1 H 2 ) 3 . 17 and thus which combination of the phases of the scalars is a genuine massless Goldstone boson. If σ is assumed to get an intermediate scale vev, this boson is mostly located on the phase of σ and evades the astrophysical constraints on an electroweak scale axion. 18 All anomalies involving a standard model factor are canceled. The cubic, as well as the mixed abeliangravitational anomalies of the quiver gauge group can be canceled by adding heavy SM-singlet fermions with charges identical to those of the additional fermions in Table 7, with SM representations turned into multiplicities, in the spirit of Table 5. Table 7: Matter content for the quiver DFSZ model (U (1) a charges are those imposed by (C.5), functions of q Q , q H , q e , q EW and q L which are arbitrary)

D Couplings of the axion to gauge vectors
We compute the axion-photon-photon coupling for the model of Figure 1 and Table 1.
However, the calculation performed here is very general and can also be seen as a derivation of (3.6).
One considers first a theory with a gauge group (which we keep unspecified until the end, where we will identify it with QCD or electromagnetism) of generators T a , coupling constant g and vector A a µ (with field strength F a µν = ∂ µ A a ν − ∂ ν A a µ + ...), a complex scalar field σ and two chiral fermions ψ L,R in the fundamental representation of the gauge group, with a Yukawa coupling to the scalar: This lagrangian has a U (1) global symmetry under which σ → e iα σ and ψ L ψ R → e −iα ψ L ψ R . The transformation of the fermion bilinear makes this global symmetry anomalous.
We choose V (|σ| 2 ) so that σ gets a vev f . We then work out the axion dynamics by where we only kept the linear terms in a and merged the two chiral fermions in a Dirac fermion.
One gets a coupling between the axion a and the gauge boson A at one loop via the two following diagrams: The effective coupling is c µν,ab aA a µ A b ν , here in momentum space with M ψ = yf √ 2 : Tr Since the first non-zero trace including γ 5 is Tr(γ 5 γ µ γ ν γ ρ γ σ ) = 4i µνρσ and that traces of an odd number of gamma matrices are zero, we have: Tr γ 5 (−i/ k + M ψ )γ µ (−i (k + p) + M ψ )γ ν (−i (k + p + q) + M ψ ) = 4iM ψ µνρσ p ρ q σ .

(D.4)
We also use the Feynman trick followed by a Wick rotation to calculate:

(D.5)
We can expand this result at first order in p M ψ , q M ψ : 6) which, with the identification pA(p) → −i∂A(x), gives finally the one-loop coupling between the axion and the vector bosons: The first term of (D.7) is the usual axionic coupling to gauge fields, while the other terms match similar calculations already performed in the literature (see for example [57]).
If one now adds to the theory (D.1) another set of fermions coupled in the following way: there is no anomaly anymore, but there remains non-anomalous couplings to the gauge fields (where we defined M ψ = y f ): Specializing to electromagnetism, normalizing the photon field A µ → eA µ and choosing We also allow in Figure 6 (which, as Figure 3, compares the sensitivity of the CASPEr-Wind experiment with the predictions of our model) for more values of q and N , but also for M c < M P . The upper panel of Figure 6 couples the standard model with the first site of the quiver while the lower panel couples it to the last site of the quiver (which, as visible in the plot, increases the coupling and thus the detectability of the setup). We see from Figure 6 that CASPEr-Wind experiments are more sensitive to high scale (e.g. gravitational) values of M c .