Factoring the Strong CP Problem

We present a new mechanism to solve the strong CP problem using $N\geq2$ axions, each dynamically relaxing part of the $\bar\theta$ parameter. At high energies $M\gg\Lambda_{QCD}$ the $SU(3)_{c}$ group becomes the diagonal subgroup of an $SU(3)^{N}$ gauge group, and the non-perturbative effects in each individual $SU(3)$ factor generate a potential for the corresponding axion. The vacuum is naturally aligned to ensure $\bar\theta=0$ at low energies, and the masses of these axions can be much larger than for the standard QCD axion. This mechanism avoids the introduction of a discrete $Z_2$ symmetry and associated 'mirror' copies of the SM fermions, and also avoids the introduction and stabilization of new light colored states to modify the running of the QCD gauge coupling found in other heavy axion models. This strengthens the motivation for axion-like particles solving the strong CP problem at points beyond the standard QCD axion curve in the $(m_a, f_a)$ plane.


Introduction
The Standard Model (SM) describes physics remarkably well at the smallest scales probed so far. With the discovery of the Higgs boson, the SM can be consistently extrapolated to very high energies. However, there are a number of puzzles in the SM that remain. The strong CP problem is one such long standing problem. The QCD Lagrangian Given that CP is badly broken by the weak interactions, we expectθ ∼ 1. However, the QCD θangle is constrained to be very small,θ 10 −10 , from limits on the electric dipole moment of the neutron/Hg [1][2][3].
The solutions to the strong CP problem can be classified roughly into two categories. One class of solutions imposes discrete symmetries like CP as in the Nelson-Barr mechanism [4][5][6][7] (see [8] for a recent discussion) or parity (P) [6,9] that forbid the θ-angle. In Nelson-Barr models there are generally no low-energy states, while parity models often require new light colored particles [10].
The second class of solutions invokes a global U (1) Peccei-Quinn symmetry [11,12]. This U (1) P Q has a mixed anomaly with QCD, such that QCD non-perturbative dynamics break PQ explicitly and the θ-angle is dynamically relaxed to zero. The most economical version of such a solution is the massless (up-) quark solution [13], in which case the θ-angle becomes unphysical (or equivalently is relaxed dynamically by the η meson). However, lattice calculations indicate that all quarks are massive [14,15], excluding the simplest model with a massless up-quark. Another elegant model that uses the PQ symmetry is the QCD axion [11,12,16,17] (see [18] for a review). The U (1) P Q is spontaneously broken in the UV, giving rise to a light pseudo-Nambu Goldstone boson, the axion. The axion makes the θ angle dynamical, and has a potential with a minimum atθ = 0 [19].
In our paper we focus on this axion solution to the strong CP problem. Given the QCD axion decay constant f a , the mass of the axion is determined by QCD dynamics [16], The axion couples to the standard model through dimension-5 operators suppressed by f a , and the precise mass relationship equation (1.3) has motivated many existing and future experimental efforts (see Ref [20] for a review). In this work, we study an extension of the QCD axion framework where instead of a single axion relaxing the vacuum toθ = 0, two or more axions naturally cooperate to solve the strong CP problem. Each of these axions can have a mass much larger than the standard QCD axion, motivating searches for axions in the (m a , f a ) plane outside of the QCD axion window given by equation (1.3). This multi-axion solution to the strong CP problem arises in models that extend the low-energy SU (3) c gauge group to be the diagonal subgroup of a parent SU (3) × SU (3) × ... product gauge group, which is broken down to SU (3) c at some high scale M . All the SM quarks are taken to be charged under a single SU(3) factor of the parent gauge group, and we introduce an axion for each individual SU (3) factors that independently relaxes the corresponding θ-angle to 0. Each SU(3) factor of the parent gauge group is more strongly coupled than the diagonal SU (3) c subgroup, and therefore the contributions to the axion potentials from UV instantons near the scale M can be larger than the non-perturbative potential generated at low energies from QCD. Even for the smallest extension, we find that the two axions can have masses significantly larger than for the standard QCD axion. In a follow-up work, Ref. [21], we will describe a related family of models where each quark generation is charged under a different SU(3) factor, and the strong CP problem can be solved without introducing any axions degrees of freedom in a spirit similar to the massless up quark solution.
Other mechanisms have been proposed that may increase the mass of the QCD axion. The models of Refs. [22][23][24][25][26] also extend SU (3) c to an SU (3) × SU (3) gauge group, but with QCD living in a single factor instead of the diagonal subgroup. In these models, a discrete Z 2 symmetry is introduced, requiring an entire mirror SM matter sector. In contrast, our mechanism does not involve any new matter in the SU (3) factors, and the only new low-energy states are the axions directly responsible for relaxingθ. The models of Refs. [27][28][29][30][31][32] introduce extra colored matter at an intermediate scale so that QCD runs back to strong coupling at a high scale M where SU (3) c is embedded in a SU (3 + n) gauge group. However, the introduction of new colored matter generally leads to either new hierarchy problems for scalars or new CP phases for fermions, spoiling the solution to the strong CP problem. In contrast, in our mechanism no new colored states are introduced below the scale M , and extra CP phases that would feed into the UV axion potential are naturally absent.
In Sec. 2, we describe in detail the structure of the model and the calculation of the nonperturbative contributions to the axion potentials at the scale M . In Sec. 3, we discuss the sensitivity of the mechanism to higher dimensional operators and the connection to the electroweak hierarchy problem. In Sec. 4, we describe the limits on these models from axion-like particle (ALP) searches.
2 Non-perturbative Effects in Product Gauge Groups We start with a simple extension of the SM where the QCD gauge group emerges from Higgsing a product group at a scale M TeV. This a similar set-up to the renormalizable coloron models of [33,34,[34][35][36], although in our case the scales will be far out of reach of the LHC. The SM quarks are charged only under the SU (3) 1 gauge factor, and there are no fermions charged under SU (3) 2 . We take the theory to have two spontaneously broken anomalous U (1) P Q symmetries at scales f i > M , giving an axion a 1,2 in each SU (3) sector: The gauge couplings g s1,2 and theta angles θ 1,2 are independent parameters not related to each other by any symmetries. The presence of two independent axion degrees of freedom will allow both physical θ-angles to be dynamically removed. An additional cross-coupling term for the axions ∝ a1 f1G 2 G 2 can not be removed by a field redefinition. It leads to a mass-mixing between the axions, but does not destabilize the solution to the strong CP problem or significantly change the phenomenology of the light axion states, so we take it to vanish for simplicity. The UV couplings of a 1,2 to the electroweak topological terms WW and BB will only be relevant for the axion phenomenology, and will be discussed in Sec. 4.
The theory will match to the SM at a scale M where the SU (3) 1 × SU (3) 2 gauge group is Higgsed to a diagonal SU (3) c . For simplicity, we take the dynamics to be a bifundamental scalar field Σ 12 : (3,3, 1) of SU (3) 1 × SU (3) 2 × U (1) Σ with a renormalizable scalar potential inducing a vev [33,36] One combination of the SU (3) gauge bosons becomes massive with M 2 V ≡ M 2 = (g 2 s1 + g 2 s2 )f 2 Σ , and the unbroken SU (3) QCD gauge symmetry is given by The additional U (1) Σ gauge factor forbids a trilinear term in the potential, and the corresponding gauge boson absorbs the otherwise massless singlet Goldstone mode * .
We can integrate out the heavy degrees of freedom to match to the SM at this scale. The tree-level matching condition gives the couplings of the two SU(3) factors in terms of the standard model QCD coupling evaluated at the scale M , with α = g 2 /4π. Each individual gauge factor is more strongly coupled than the SM QCD gauge coupling, and this will lead to enhanced non-perturbative effects compared to the SM. The SM coupling g s (M ) can be obtained with the 1-loop running from the top pole, α s (m t ) = 0.10. The tree-level value of the SM theta term is simplȳ Because the flavor symmetries are the same as the SM, the loop-level thresholds toθ proportional to the CKM phase are negligible [37,38].
Since we are integrating out the heavy degrees of freedom, we must also consider short distance non-perturbative effects in the individual SU (3) 1 and SU (3) 2 factors, which will generate the shiftsymmetry breaking potentials for the axions. We will be interested in the regime where each factor is still at relatively weak coupling near M , and therefore the dilute instanton gas approximation (DGA) gives a good approximation for the non-perturbative effects. The effective Lagrangian generated for the axions is [39][40][41] L a = Λ 4 1 cos The axions still couple to the low energy QCDGG term, and at the scale Λ QCD ∼ 1 GeV the nonperturbative SM contributions to the axion potentials will be generated. However, unlike the standard axion case, the axion potential can be dominated by the higher energy contributions to the potential with scales Λ 1 and Λ 2 generated at the scale M . Although these scales Λ 1 and Λ 2 will be suppressed by non-perturbative and chiral suppression factors, they can still greatly exceed the scale of the standard axion potential. Crucially, and as is clear from the UV PQ symmetries of the theory, the new short-distance nonperturbative contributions to the axion potentials are exactly aligned to remove the effective θ-angle, The low-energy vacuum is in-fact guaranteed to align withθ = 0 by a generalization of the Vafa-Witten argument [19]. This result is very different from theories which have new perturbative breakings of the PQ symmetry in the UV-for example, higher-dimensional operators violating the PQ symmetry give axion potentials that are generically misaligned and would spoil the axion solution to the strong CP problem [42][43][44].
We can now estimate how the mass of the two axions is affected by these UV non-perturbative contributions to the potential. In the SU (3) 2 factor, where no colored fermions are present, the axion potential is suppressed only by the non-perturbative instanton action. The scale can be calculated as [39][40][41]  where the dimensionless instanton density depends non-perturbatively on the running gauge coupling as with α(1/ρ) the running coupling evaluated at the scale µ = ρ −1 corresponding to the size of the instanton. Higher order corrections to the instanton density have been calculated in Refs. [15]. In this work, we use the leading order result, and estimate the theoretical uncertainty by varying the scale at which the density is evaluated: . Since the instanton contribution dominantly comes from the IR, near the scale M , the instanton density in the integrand 2.9 can be approximated as where the one-loop running coupling satisfies with b 1 = 13/2 and b 2 = 21/2 (including the contributions of the bifundamental scalar Σ to the running). This gives In the SU (3) 1 factor, there is a further suppression due to the Yukawa couplings and Higgs loops, as depicted in figure 1, which can be estimated as [31,32]  1 and m a2 /m a,QCD 1 simultaneously, so that neither state has properties similar to the standard QCD axion.
Where K is a chiral suppression factor capturing the breaking of the U (1) 6 axial symmetry of the individual quarks by the Yukawa couplings † , We can compare the scales Λ 1,2 of the UV contributions to the scale Λ 0 of the usual nonperturbative axion potential generated around the QCD scale [16] given in equation (1.3).
The size of the non-perturbative UV potentials generated in each sector are shown in figure 2. The sensitivity to the large scale M can overcome the non-perturbative and chiral suppression factors. The relation equation (2.6) between α s1 and α s2 implies that Λ 1 grows as Λ 2 shrinks. In much of the parameter range, Λ 1 Λ 0 Λ 2 or Λ 2 Λ 0 Λ 1 . In these cases, one axion DOF will simply decouple, and the other will behave largely like the QCD axion.
The most interesting regime is instead when Λ 1 , Λ 2 Λ 0 -in this case there will be no light state resembling the standard QCD axion. This regime generally corresponds to α s1 > α s2 to compensate for the extra chiral suppression factor in Λ 1 . As the scale M increases, the scales Λ 1,2 grow -although the running of α 3 to weak coupling at higher energies leads to stronger instanton suppression factors, this is overcome by the M 4 dependence of the potential. For M ∼ 10 8 GeV, the UV effects can become comparable in size to the IR potential, with Λ 1 ∼ Λ 2 ∼ Λ 0 . For M ∼ 10 14 GeV, effects as large as Λ 1 ∼ Λ 2 ∼ 30Λ 0 can be realized. This realizes a model with two axions, each ∼ 1000 times heavier † Ref. [31] considers the case where Yukawa couplings to a scalar state of mass ms break the chiral symmetries of the colored fermions in the theory. Their result contains an additional suppression factor of m 2 s /M 2 , which would arise in theories where a scalar mass-insertion is necessary to violate some of the chiral symmetries, e.g. a two-higgs-doublet model. In contrast, in our case there are non-decoupling effects even in the limit ms → 0. than a standard QCD axion with the same decay constant f , with a potential aligned to dynamically setθ = 0. Note that although the non-perturbative effects have a large effect on the axion masses in this regime, the individual gauge factors are still reasonably weakly coupled at the scale M , with α s1 far away from the chiral-symmetry breaking phase which would be expected to occur at α s1 0.7 − 1 [45,46]. So far we have considered the case f a M and assumed that all states in the spontaneous PQbreaking sector are decoupled. If some of these states are lighter than M , the instanton effects can be suppressed. In a KSVZ axion model [47,48] with N f flavors of vector-like quarks of mass M ψ M , there will be a suppression in the instanton calculation by a factor of ∼ (M Ψ /Λ) N f . We will take the benchmark M Ψ = f and N f = 1 for the phenomenological studies in Sec. 4, giving Other benchmarks like a KSVZ axion with N f > 1 or a DFSZ axion [49,50] would typically have more severe suppressions for the regime f ai < M .

SU (3) N products
Although each gauge factor in the SU (3) × SU (3) model is more strongly coupled than the SM SU (3) QCD , they remain relatively weakly coupled, and the additional short distance non-perturbative effects still suffer a significant exponential suppression. Extending the gauge group to SU (3) N , with the SU (3) QCD emerging from the diagonal subgroup allows the gauge coupling in each individual factor to be substantially increased,

UV Sensitivity
There are several interesting questions about the sensitivity of this multi-axion solution to UV physics above the scale M . First, as is familiar from the standard QCD axion picture, Planck-suppressed operators may spoil some of the PQ symmetries [42][43][44], naively leading to a misalignment of the axion potential by with the dimension d ≥ 5 of the operator depending on the details of the axion model. In our case, this problem is significantly alleviated compared to the standard QCD axion because smaller values of f a and larger values of m a can be realized. Fig. 4 shows that large parts of parameter space remain open even assuming the most dangerous d = 5 operators are present. In our model there are also PQ-preserving higher dimensional operators that can be dangerous. Focusing for simplicity on the SU (3) × SU (3) case, at D = 6 the operator can introduce a shift ∆θ ∼ M 2 /Λ 2 U V that is not canceled at the axion minima. Requiringθ 10 −10 then requires a hierarchy M (10 14 GeV) Another possibility is that non-perturbative effects may not decouple sufficiently rapidly to protect the theory from CP violation at scales far above M . For example, if there are new unsuppressed sources of CP violation in the theory at the scale Λ U V , then non-perturbative effects at this scale can generate additional misaligned contributions to the axion potential. Because each gauge factor is asymptotically free, the non-perturbative effects decouple at high energies as α si run to weak coupling.
The SU (3) 1 factor contains the SM quarks, and therefore runs to weak coupling the most slowly. From equations (2.14), (2.11), the scale of the potential generated for a 1 at Λ U V is suppressed compared to the potential generated at M by a factor Assuming an O(1) misalignment of the potential generated at Λ U V , the limitθ 10 −10 requires a hierarchy M (10 15 GeV) Λ U V M pl

5/2
. Therefore without additional assumptions about the nature of CP violation at the scale Λ U V , there must be a substantial hierarchy in scales M Λ U V to protect against higher dimensional operators and sufficiently suppress non-perturbative effects. In the simple model we have presented, M is set by an elementary scalar field Σ 12 breaking the gauge group, and it seems that maintaining this hierarchy reintroduces as severe of a tuning as the tuning inθ we have set out to address. Two well-known solutions exist -this hierarchy can be protected by supersymmetry (SUSY) or by a technicolor-like mechanism.
In a SUSY model, the scale M can be protected if m soft M . When m soft M , the instantongenerated potential will be suppressed by a factor of at least (m soft /M ) 5 due to insertions of gaugino masses and the PQ-breaking bH u H d soft terms [32], so it is desirable to stay in the regime m soft ∼ M . Since m soft ∼ M 100 GeV, it is not possible to protect the weak scale with SUSY in this model, but at least the hierarchy M Λ U V can be maintained. However, the soft SUSY breaking introduces new CP phases into the theory, and the alignment of the axion potential which givesθ = 0 is no longer guaranteed. As is familiar from low-scale SUSY, we must assume that the soft terms are communicated in a way that does not introduce new CP violating phases (for a review, see Ref. [51]), since these would lead radiatively to misalignment of the short-distance axion potential generated near M (the standard QCD axion mechanism is similarly sensitive to the CP violation in the soft sector when m soft 100TeV [52]). Ref. [8] points out a similar sensitivity to SUSY-breaking CP violation in Nelson-Barr models.
In a technicolor-like solution, the Σ can be made a composite of elementary fermions under a new strong group, with the scale M generated dynamically. This resembles asymptotically free completions of the dimensional deconstruction framework [53]. However, because new fermions charged under SU (3) 1 and SU (3) 2 will be introduced, we must take care that their additional chiral symmetries are broken in a way that does not suppress the non-perturbative axion potential or re-introduce new CP phases. An example of such a model is given in Appendix A.

Phenomenology
Laboratory experiments, cosmology, astrophysics, beam dump, and collider experiments can all be sensitive probes of new light pseudoscalars, also known as axion-like particles. Their reach can be parameterized in the space of masses m a and the dimension-5 couplings of the new states, which scale as 1 fa . Although our models contain multiple axion-like states, the direct couplings in-between these states are phenomenologically unimportant and we can treat the limits on each axion-like state independently.
Since we have not specified the UV details of the spontaneous PQ-breaking, the heavy axions in our models may realize dimension-5 couplings to the electroweak gauge bosons aWW , aBB, and light fermions ∂ µ a(f † γ 5 γ µ f ) in addition to the gluon coupling aGG. When m a < m η ∼ GeV, these additional couplings to the mass eigenstates are inevitable at low energies since the axions mix with the η , η, and π 0 mesons [25], as in the standard QCD axion model [16,17,54]. When the axion is light enough that hadronic decay channels are closed, m a 3m π , the phenomenology is primarily determined by the low-energy axion-photon coupling, and this ALP case has been well studied.
For m a 1 GeV, hadronic decay channels open and typically dominate, and studies focused on an axion decaying dominantly to electroweak gauge bosons no longer apply. Limits depending on these couplings are even further weakened when UV couplings to the electroweak gauge bosons are absent, since the mixing-induced couplings become rapidly suppressed as QCD runs to weak coupling.
To study the limits on our models, we take as a benchmark an effective low-energy photon coupling of throughout the parameter space (for m a m η , the photon coupling will be entirely due to the UV physics, while for m a m η it is a result of both the UV coupling and the mixing with the neutral mesons). We use this simplified approximation as a rough guide to the phenomenology and a more realistic treatment would generically include couplings to the Z as well as to other SM particles.
We show existing limits and future projections for axion experiments as a function of m a , f a in figure 4. We also include two benchmark models in which the axions are lifted to masses much larger than the standard QCD axion.
The first benchmark model is the SU ( GeV -it would be interesting to study in further detail the limits on such a hadronically decaying state. We have chosen these two specific benchmarks to illustrate the parameter space and the correlations between the scales of the multiple axions present in each model. We note however, that the parameter space of these models motivates axions spanning the entire (m a , f a ) plane above the standard QCD axion line.
We briefly describe the constraints and projections shown in figure 4. Current constraints are shown as shaded regions. The light blue shaded region are independent of a dark matter interpretation of the axion or its cosmology. The yellow shaded regions assume that the reheating temperature was large enough to have thermally produced axions, and that there was no large entropy dump to the SM after axion decoupling. The red shaded regions assume that the axion makes up all of the dark matter.
Misalignment and Ω DM : In parts of our parameter space, axions can make up all of the dark matter. In our model the mass of the axion goes to its zero temperature value much earlier than the time it starts oscillating, making the computation of relic abundance much simpler than the standard QCD axion case. In our case the relic abundance for a single axion with mass m a and decay constant f a is given as, 0.12 f a 10 12 GeV 2 m a 0.01 eV where a(T f ) is the scale factor at the temperature T f where the axion starts oscillating, H(T f ) m a /3, θ 0 is the initial misalignment angle, and ρ crit = 3.96×10 −47 GeV 4 is the critical density of the Universe. This estimate ignores the thermal production of axions, which can become important for higher masses as well. We show the part of the parameter space in our benchmarks where heavy axions could make up the dark matter by solid lines in figure 4. It is interesting to note that in our models there is a new region of the m a -f a parameter space where axions solve the strong CP problem and constitute all of the dark matter for generic initial conditions (θ 0 ∼ 1).
Cosmology constraints The yellow region is ruled out by a combination of cosmological constraints, assuming that the reheat temperature was sufficiently high to produce an equilibrium thermal population of axions [55] which dominates over misalignment production in this region [56]. Future CMB observations can also put significant constraints on thermally produced axions from measurements of ∆N eff that constrains extra relativistic degrees of freedom present during recombination. We show the sensitivity of the CMB-S4 assuming a reheat temperature of T R = 10 10 GeV [57] with a horizontal gray line.
If the axion makes up dark matter, then its early universe values can be large enough to affect BBN [58]. This bound is superseded by the solar bound mentioned below.
Astrophysical bounds Some of the strongest constraints on the axion parameter space arise from stellar physics, where axion coupling to photons is constrained by evolution of horizontal branch stars. For lower masses of axions supernova constraints and constraints from conversion of X-ray photons to axions in cluster magnetic fields [59][60][61] impose a stronger bound. Axion helioscopes such as CAST [62] also put constraints on the couplings of axions with photons, with the future IAXO helioscope [63] improving the reach. The light blue region includes these astrophysical constraints (adapted from [64]).
Axions which lie above the QCD line in figure 4 need a tuned negative contribution to their mass to cancel the partially cancel the mass from QCD. Their potential can then flip its sign due to finite density effects in astrophysical objects, which then source the axion field. This observation was used by Ref. [65] to put bounds on this parameter space by dense astrophysical objects sourcing axion fields. In figure 4 we show the current bounds arising from the Sun and projections from neutron star mergers in LIGO (purple), which may be able to probe most of this tuned region of parameter space. We use m a = 0.1m a,QCD as a conservative projection from LIGO. Black hole superradiance [66] puts constraints on very light axions.
Collider constraints: The possibility of heavier QCD axions also opens up weak-scale decay constants, making these axions visible at colliders. We show constraints from monojet searches at the LHC [67] and from LHC dijet [68,69] and diphoton searches [70,71] for pseudoscalar states produced in gluon fusion [72]. Also show are the constraints from by monophoton and beam dump experiments (adapted from [73]). The Belle II experiment [74] will be sensitive to heavier axion-like particles at the weak scale [73]. The SHiP beam dump experiment [75] can cover intermediate mass scale axions which are not yet excluded by current experiments. Future e + e − colliders on the Z mass peak can also improve on axion couplings. We show projections [64] for the FCC-ee collider [76].
Some of the analyses above were carried out for axion-like particles which only have a coupling to photons, and hence do not directly apply to our case when decays to hadrons become important. As a crude fix we have cut off all such limits at m a = 1 GeV. It would be interesting to recast collider bounds on models of heavy QCD axions including couplings to gluons as well as photons and other SM particles.
Laboratory experiments: Laboratory experiments such as the haloscope ADMX constrains the QCD axion in the region where it can be the dark matter. We show regions of parameter space that can be covered by future experiments. The upgraded cavity experiment ADMX2 [77] will cover a larger range of QCD axion masses with higher sensitivity. The CASPEr experiments [78][79][80] propose using NMR techniques to measure time-varying EDMs induced by the axion, which will be sensitive for low-mass axions. The ABRACADABRA [81] experiment can also cover the low mass axion parameter space for axion-photon couplings. These experiments all rely on the axion being all of dark matter. The upgraded light-shining-through-wall experiment ALPS II [82] will be sensitive to larger couplings of axion-like particles with the photon.
We have included in figure 4 some dashed gray lines motivated by theoretical considerations. The line in the upper right corner signals the invalidity of the effective axion theory when m a 4πf a . The reduced sensitivity to quality of the PQ symmetry through Planck suppressed higher dimensional operators as described by equation (3.1) are also shown. In regions above these lines, generic dimension-5 and -6 operators do not spoil the solution to the strong CP problem. The vertical dashed gray line at m a = 1 GeV shows the scale where hadronic decays begin to dominate and axion mixings with the neutral mesons begin to be suppressed.
A few general comments which follow from our survey of the phenomenology: • In our plot, most of the current and future observations only probe the region above the QCD line. As noted above, this region is tuned such that there is a negative mass contribution to the axion, with a minimum which is highly aligned with the QCD minimum, and a size comparable to the QCD contribution. However, experiments which proble the axion coupling to photons can extend below the QCD line in our plot if the coupling of the axion to photons is much larger than the coupling assumed here (as in [83][84][85]). With such an enhanced couplings these experiments will also be sensitive to heavier QCD axions as considered in this paper.
• When hadronic channels are open, the branching ratio to photons is tiny. The collider signals of a heavy QCD axion therefore differ from the more-often studied case of ALPs which only have a photon coupling. Our work shows that the combination of gluon and photon couplings is very well motivated even for heavy collider-observable axions and deserve further study.
• We also note that the phenomenology of this realization of a heavy axion is substantially different from the phenomenology in a Z 2 heavy axion model, where the axion may decay to light mirrorsector particles [25], and is generically not expected to appear with multiple copies.
• A combination of heavy axions can make up the dark matter density through the misalignment mechanism in a new part of m a -f a plane, motivating searches in this region. A resonance search strategy for laboratory experiments might be less optimal if the dark matter density is not dominantly stored in axions at one mass.

Conclusions
We have described a novel mechanism for solving the strong CP problem. In the standard QCD axion mechanism, a spontaneously broken anomalous U (1) P Q symmetry results in an axion with a potential generated by non-perturbative effects near the scale Λ QCD . In the absence of any UV sources of perturbative PQ violation, the strong CP problem is solved dynamically when the axion relaxes to its CP preserving minimum. In our mechanism, we embed QCD in a SU (3) N product group at a scale M Λ QCD , and use a separate spontaneously broken PQ symmetry in each individual SU(3) factor to dynamically relax each individualθ angle. Because each individual SU(3) factor is more strongly coupled than the SM QCD at the scale M , the non-perturbative contributions to the axion potentials at the scale M can be much larger than those generated near Λ QCD for the standard QCD axion. After integrating out physics above the scale M , the theory is just the standard model with N axions, each with a PQ-violating potential that arose from non-perturbative effects near the scale M . Although the PQ symmetries are explicitly violated in this low-energy effective theory, the non-perturbative origin of the axion potentials in the full theory guarantees that the minimum relaxes the low-energyθ angle while generating a mass for each axion that is much larger than the standard QCD axion mass relationship.
Our assumptions about the existence of spontaneously broken anomalous PQ symmetries are on the same footing as the standard QCD axion, and it is encouraging that in general a large number of axions are expected in string theory models [86]. In fact, in our model, the possibility of solving the strong CP problem with a small value of f a and large value of m a can resolve the problem of maintaining a sufficient PQ quality in the presence of quantum-gravity corrections that plague the standard axion solution [42][43][44].
The main obstacle to the model we have proposed is the conflict with solutions to the electroweak hierarchy problem. Solving the electroweak hierarchy problem in this class of model suggests M ∼ TeV, while generating large axion masses requires M TeV. While we have shown that it is possible to stabilize at least the hierarchy M M pl , it requires further model building to ensure that additional sources of CP violation do not spoil the mechanism. Phenomenologically, the model motivates exploring the whole (m a , f a ) range of axion-like particles, including the range m a 3m π where hadronic decays will dominate. While the model requires the existence of multiple heavy axion states, it would be difficult to directly verify their connection to the strong CP problem, as the precision mass-coupling relationships predicted for the standard QCDaxion are no longer realized. Because there is the possibility of decoupling the heavy axions to scales m a M W , this work can also be viewed as an interesting new theoretical example of a completely decoupling solution to the strong CP problem.
We would like to thank Asimina Arvanitaki, Savas Dimopoulos, Bogdan Dobrescu, Roni Harnik, Anson Hook, Junwu (Curly) Huang, Gustavo Marques Tavares, Lisa Randall and Raman Sundrum for encouragement, helpful conversations, and comments on the manuscript. This work was initiated under a street-lamp at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293. PA is supported by the NSF grants PHY-0855591 and PHY-1216270. This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

A Composite Link Fields
are necessary to break the additional chiral symmetries of the model. The instantons in SU (3) 1 and SU (3) 2 sectors will be suppressed by the coefficients λ 1 and λ 2 , so this operator must either be generated at a nearby scale (without generating additional dangerous CP violating operators or a new hierarchy problem), or the theory must enter a conformal regime in the UV where these 4-fermion operators have large negative anomalous dimension. Field redefinitions can be used to choose λ 1,2 real, and one remaining anomalous U(1) can be used to remove one combination of the θ-angles in the SU (2) a and SU (2) b sectors. The remaining new θ angle may be removed by coupling an additional axion degree of freedom to the SU (2) a or SU (2) b sector, which will obtain a large mass ∼ λ 1 λ 2 M 2 /f .