A Common Origin for the QCD Axion and Sterile Neutrinos from SU (5) Strong Dynamics

We identify the QCD axion and right-handed (sterile) neutrinos as bound states of an SU (5) chiral gauge theory with Peccei-Quinn (PQ) symmetry arising as a global symmetry of the strong dynamics. The strong dynamics is assumed to spontaneously break the PQ symmetry, producing a high-quality axion and naturally generating Majorana masses for the right-handed neutrinos at the PQ scale. The composite sterile neutrinos can directly couple to the left-handed (active) neutrinos, realizing a standard see-saw mechanism. Alternatively, the sterile neutrinos can couple to the active neutrinos via a naturally small mass mixing with additional elementary states, leading to light sterile neutrino eigenstates. The SU (5) strong dynamics therefore provides a common origin for a high-quality QCD axion and sterile neutrinos.


I. INTRODUCTION
The QCD axion and right-handed neutrinos are wellmotivated new particle states beyond the Standard Model (SM).The QCD axion elegantly solves the strong CP problem via the Peccei-Quinn mechanism [1], where an (anomalous) U (1) PQ symmetry is spontaneously broken, giving rise to a pseudo-Nambu-Goldstone boson [2,3] that dynamically cancels the strong CP phase.Experimental bounds limit the U (1) PQ symmetry breaking scale to be f PQ ≳ 10 8 GeV [4].The QCD axion can also provide the missing dark matter component of the Universe [5][6][7], thereby solving two problems in the Standard Model.However, explicit violation of the U (1) PQ global symmetry must be entirely dominated by QCD dynamics, with any other violation, in particular from gravity, highly suppressed [8][9][10][11][12].An attractive solution to this axion quality problem is to realize the PQ symmetry as an accidental symmetry in the low-energy theory, similar to baryon and lepton number in the Standard Model.In particular, if new strong dynamics is introduced around the PQ breaking scale, then gauge and Lorentz symmetry can be used to accidentally preserve U (1) PQ up to very high dimension terms in the Lagrangian [13].
The seesaw mechanism [14][15][16][17][18] provides a similarly elegant explanation of the hierarchy between the masses of neutrinos and charged leptons.Assuming order one Yukawa couplings, this is simply achieved by introducing right-handed neutrinos with masses ≳ 10 10 GeV.As is well-known, this mass scale is similar to the PQ scale and the two can be related [19,20].Given this coincidence of scales, and the possibility of realizing the PQ symmetry as an accidental symmetry, we seek a solution that relates these two scales via strong dynamics.The strong dynamics also has the advantage of naturally generating a peter.cox@unimelb.edu.aub tgher@umn.educ paul1228@umn.edu the PQ scale via dimensional transmutation, obviating the need to introduce explicit mass scales in the scalar potentials that are normally used to UV complete the axion Lagrangian, as in the KSVZ or DFSZ scenarios.
A particularly interesting strong dynamics based on an SU (5) gauge theory with massless, chiral fermions was recently considered in Ref. [21] to realize a highquality QCD axion.The PQ symmetry was identified as a global symmetry of the strong dynamics and assumed to be spontaneously broken at a scale f PQ .This realises a low-energy QCD axion as a composite pseudo Nambu-Goldstone (NG) boson, thereby solving the strong CP problem in a similar manner to the original dynamical axion [22,23].Furthermore, the local gauge and Lorentz symmetry accidentally preserves the U (1) PQ global symmetry up to dimension nine terms in the low-energy effective Lagrangian, thereby ameliorating the axion quality problem.
In this paper, we build upon the SU (5) model in Ref. [21] to realize both a high-quality QCD axion and right-handed (sterile) neutrinos as bound states of the same UV dynamics.In particular, we show that the spontaneous breakdown of the PQ symmetry leads to QCD singlet states with Majorana masses of order the PQ scale, f PQ , which can be identified as composite sterile neutrinos.To generate the left-handed (active) neutrino masses, the composite sterile neutrinos are then coupled either directly or indirectly to the active neutrinos, realizing heavy or light sterile neutrino mass eigenstates, respectively.
In the case where the composite sterile neutrinos couple directly to the active neutrinos (with a dimensionseven Higgs-fermion coupling), light Majorana active neutrinos are obtained via a see-saw mechanism.The interaction is generated by integrating out PQ-charged scalar fields in a UV completion.Importantly, the quality of the PQ symmetry is not affected.Alternatively, an indirect coupling to the active neutrinos can occur via a naturally generated small mass mixing between an elementary, right-handed neutrino and the composite sterile neutrinos.This leads to sterile states with naturally suppressed (sub-TeV) Majorana masses and, depending on the scale of the UV completion, can realize the neutrinos as pseudo-Dirac states [24,25].The strong dynamics, therefore, plays a pivotal role not only in addressing the axion quality problem, but also in relating the axion and neutrino masses.
The composite sterile neutrinos share features similar to those previously studied in Refs.[24][25][26][27][28], however the connection with a composite QCD axion was not previously considered.The chiral UV gauge theory also provides an explicit 4D realization of the holographic 5D setups considered in Refs.[29][30][31], which solved the axion quality problem with a composite axion and partial compositeness in the SM fermion sector.Finally, previous work in Refs.[32][33][34][35] also addressed the axion quality problem with an accidental PQ symmetry of strong dynamics, although without any connection to neutrino masses.
The outline of our paper is as follows.In Section II we review the matter content of the SU (5) gauge theory, together with the global symmetry structure and IR dynamics.We then discuss the resulting bound state spectrum, which includes a composite, high-quality axion as well as QCD singlet bound states that are identified as composite right-handed neutrinos.The generation of neutrino masses is discussed in Section III, where we present models with both heavy and light sterile neutrino mass eigenstates.A holographic connection to the light sterile neutrino case is also discussed.Our concluding remarks are presented in Section IV.The Appendices contain supplementary material related to the QCD anomaly factor (App. A), representations of the NG bosons (App.B), solution of the axion quality problem (App.C), implications for axion dark matter (App.D), and mass-mixing in the light sterile neutrino scenario (App.E).
In the limit that the QCD coupling α s → 0, the SU (5) gauge theory has an SU (n f )5 × SU (n f ) 10 global symmetry, where n f = dim R ψ .In addition, there is a single SU (5)-anomaly-free global U (1) (analogous to B − L symmetry in SU (5) grand unified theories) for which the charges of ψ5 and ψ 10 satisfy Q5 = −3Q 10 .This is identified as the PQ symmetry.The representations of the fermions under the full flavor symmetry are shown in Table II.The QCD gauge group, SU (3) c , is a subgroup of the non-abelian flavor symmetry, and the latter is explicitly broken for α s ̸ = 0. Importantly, the PQ symmetry is anomalous with respect to QCD (see App. A), which will eventually lead to the composite axion obtaining a mass from non-perturbative QCD effects in the usual way.On the other hand, the fact that U (1) PQ has no SU (5) anomaly is important to ensure that the axion remains light and provides a solution to the strong CP problem.

B. IR Dynamics and Symmetry Breaking
Given the fermion content in Table I, the SU (5) gauge theory is asymptotically free in the UV and becomes strongly coupled in the IR.The dynamics of strongly coupled, non-supersymmetric chiral gauge theories are not well understood.Techniques such as 't Hooft anomaly matching, large-N , and the a-theorem can be used to place restrictions on the dynamics but do not, in general, single out a unique IR phase.(For a recent discussion of the IR dynamics of SU (N ) theories with a single flavor of antisymmetric + anti-fundamental chiral fermions see [36].) It was pointed out in Ref. [21] that for the current SU (5) model with n f flavors it is impossible to match the [SU (n f )5] 3 and [SU (n f ) 10 ] 3 anomalies if the SU (5) confines in the IR.The global symmetry is therefore spontaneously broken by the SU (5) gauge dynamics; how-ever, there remain several possible IR phases with different unbroken global symmetry groups.Furthermore, there is the possibility of forming bilinear condensates (i.e.⟨ψ5ψ 10 ⟩) that dynamically break the SU (5) gauge theory (see e.g.[37]).Following [21], we assume that (i) the gauge theory confines and no bilinear condensates form, and (ii) the flavor breaking condensate preserves at least an SU (3) subgroup, which is the weakly gauged SU (3) c .

C. Composite axion
The spontaneous PQ breaking by the condensate (3) gives rise to a composite pseudo-NG boson, which is the axion.We parameterise the Goldstone field containing the axion, a, as where a/f PQ ∈ [0, 2π).Under the PQ transformation ψ → e iQ ψ α ψ, with α an arbitrary phase parameter, the axion transforms as a → a + αf P Q , where f PQ is the axion decay constant.This constant obeys the relation f PQ = Λ 5 /g * , where g * is a typical coupling between the composite bound states (with mass scale Λ 5 ), satisfying As usual, QCD instantons generate a potential for the axion, providing a dynamical solution to the strong CP problem.The axion mass is then given by the standard expression [38,39], where The solution to the strong CP problem is spoiled if there are additional sources of explicit PQ violation, which will modify the axion potential.As is well known [9][10][11][12], gravity is not expected to preserve the global PQ symmetry and may induce higher-dimensional, Planck scale suppressed operators that contain the PQcharged fermions.Below the scale where SU (5) confines, these operators give additional contributions to the axion potential.Importantly, the combination of Lorentz symmetry and the SU (5) gauge symmetry restricts these operators to have dimension-9 or greater, such that U (1) PQ remains an approximate accidental symmetry at low energies.Planck scale induced contributions to the axion potential are then sufficiently suppressed provided that f a ≲ 10 9 GeV [21] (see App. C).This bound also has consequences for axion dark matter, which is further discussed in App.D.

D. Composite Fermion Bound States
In addition to the composite axion, there are massive fermionic bound states.As we now show, these include SM singlets that, as will be discussed further in Section III, can act as massive sterile neutrinos.We restrict our discussion to the 3-fermion, spin-1 2 bound states, for which there are two distinct SU (5) singlets, where we have written the bound states as right-handed Weyl fermions for convenience.These bound states all have PQ charge +1, and due to the spontaneous breaking of U (1) PQ by the condensate (3) obtain Majorana masses of order the resonance scale Λ 5 .
The bound states decompose into irreducible representations of SU (3) c that depend on the representation R ψ of the constituent fermions.For example, with R ψ = 3 ⊕ 3, we have: Note that in both cases Ψ 1 and Ψ 2 each contain two QCD singlet bound states3 , which can be identified as righthanded neutrino candidates.We denote these bound states by N 1,j and N 2,j , where the index j = 1, 2 represents the two different SU (3) c singlets: It was shown in [21] that the 't Hooft anomaly matching condition for U (1) PQ can be satisfied if the bound state in Ψ 1 that transforms in the R ψ representation is massless.Thus, anomaly matching provides no guidance as to whether or not U (1) PQ is spontaneously broken when the theory confines.As discussed in Section II B, we assume that it is spontaneously broken, which provides the necessary mechanism to realize a QCD axion in the low-energy theory.

III. NEUTRINO MASS
A particularly interesting feature of the model is that the same dynamics that spontaneously breaks the PQ symmetry and generates a composite, high-quality axion also produces composite, QCD singlet fermions.These can be identified as composite sterile neutrinos if there exist additional couplings that connect the stronglyinteracting SU (5) sector with the electroweak sector of the SM.The active neutrinos then mix with the spectrum of composite sterile neutrino states which, together with the effect of the PQ breaking condensate, leads to the generation of Majorana masses for the active neutrinos.The PQ symmetry therefore serves as a generalised lepton number.In the following sections we present two explicit realisations of this idea.First, we consider a model that, after integrating out the SU (5) sector (containing the heavy sterile neutrinos), reduces to the Weinberg operator at low energies.Second, we consider an alternative model that contains elementary sterile neutrinos which mix with the composite states, resulting in light sterile mass eigenstates.

A. Heavy Sterile Neutrino Model
Before presenting a renormalisable UV model, we first consider how the SU (5) and SM lepton sectors can be connected within an effective field theory (EFT) framework.The lowest dimension operators that can achieve this are dimension-7: where L i = (ν L,i , e L,i ) T are the SM SU (2) lepton doublets, which carry PQ charge +1, and H is the Higgs doublet 4 which obtains the VEV ⟨H⟩ ≡ 1 √ 2 (0, v) T , with v ≈ 246 GeV.The index j enumerates the SU (3) c singlets in (ψψψ), with j ∈ {1, 2} for either R ψ = 3 ⊕ 3 or R ψ = 8, as discussed in section II D. We assume that the scale Λ L satisfies Λ 5 < Λ L < M Pl , with the dimensionless couplings ξij , ξ′ ij allowing for flavor-dependent masses for the different neutrino flavors denoted by the index i.
The relevant low-energy effective theory below the SU (5) resonance scale, Λ 5 , contains only the SM degrees of freedom and the pseudo-NG axion, with the PQ symmetry non-linearly realised.(The heavy composite resonances, including the singlet fermion bound states, have been integrated out.)The leading term, consistent with the symmetries, that is induced by the operators in Eq. ( 12) is where ξ ij ≃ O(1) × ξij and, for simplicity, we have taken ξ′ ij = 0.The factors of f PQ and Λ 5 have been determined using dimensional analysis, assuming the strong dynamics can be described by a single mass scale and coupling (see e.g.[40]).After electroweak symmetry breaking, the above term generates Majorana masses for the neutrinos, where λ ν,i are the eigenvalues of ξ ik ξ jk in (13).Notice that the neutrino masses feature the usual see-saw factor v 2 /Λ 5 (with Λ 5 identified as the scale of the heavy SN1987A 10  sterile neutrinos), but also an additional suppression by the ratio of the resonance scale to the EFT scale Λ L .Consequently, reproducing the measured neutrino masses requires Λ 5 to be lower than the usual Type-I see-saw scale.A lower Λ 5 is also desirable to address the axion quality problem.Notice also that Eq. ( 13) leads to axion-neutrino couplings.
Comparing Eqs. ( 5) and ( 14), the ratios between the active neutrino masses and the axion mass are estimated to be This shows that if the EFT scale is close to the PQ scale, specifically when Λ L ≃ (13/g 5/6 * ) Λ 5 , the axion and neutrino masses are in approximately the same range.
The expression ( 14) is fitted to the observed neutrino mass spectrum to determine the viable parameter space.Assuming the neutrinos are normal ordered, and fixing λ ν,3 = 1, we use the mass of the heaviest neutrino to constrain Λ L in terms of Λ 5 .The lighter neutrino masses are then simply obtained by choosing appropriate λ ν,1 , λ ν,2 .The combination of neutrino oscillation measurements [41,42] and the upper bound on the sum of neutrino masses from cosmology [43][44][45] ( m ν,i < 0. The EFT description in Eq. ( 12) should remain valid up to the energy scale of the composite resonances, otherwise the new degrees of freedom in the UV completion will, in general, modify the flavor and PQ symmetry breaking dynamics discussed in Section II B. (While such a scenario could also be viable, we do not consider it here.)This corresponds to the requirement Λ L > Λ 5 .This condition is violated in the dark (light) grey regions in Fig. 1 for g * = 1 (4π).Taking into account the lower bound on f a from SN1987A, we then find that in most of the parameter space small values of the strong sector coupling, g * ≃ 1, are needed to generate the active neutrino masses in this scenario.

UV completion
A UV completion of the operators in Eq. ( 12) can be obtained by introducing two massive complex scalar fields ϕ, ϕ 2 .
We take these fields to have masses m ϕ , m ϕ2 > Λ 5 , so that they do not affect the confinement and symmetry breaking of the SU (5) strong dynamics discussed in Section II.In addition, the scalars TABLE III: Representations of the fields in the UV completion of the heavy sterile neutrino scenario.
do not obtain VEVs and therefore do not reintroduce the axion quality problem (or affect electroweak symmetry breaking).The relevant interaction Lagrangian is5 where y5, y 10 , y 2 are dimensionless couplings and m 12 ≲ m ϕ , m ϕ2 is a mass parameter.Note that we have suppressed the indices on the Yukawa couplings y5, y 10 that enumerate the different SU (3) c contractions, as well as the generation indices of the lepton doublet and coupling y 2 .The charges of the fields are listed in Table III.
Integrating out ϕ and ϕ 2 , as shown diagrammatically in Fig. 2, yields the effective Lagrangian The two terms correspond to the dimension-7 operators in Eq. (12).Assuming all dimensionless couplings are O(1), the energy scale of the effective operators is approximately given by It was shown in Fig. 1 that to generate the observed active neutrino masses Λ L cannot be significantly larger than Λ 5 .In the UV completion this corresponds to the requirement that m ϕ ∼ m ϕ2 ∼ m 12 ≳ Λ 5 .An alternative possibility is that the UV theory contains elementary, massless, right-handed neutrinos ν R with PQ charge −1.The PQ symmetry forbids explicit Majorana mass terms for the ν R , but they form the usual Dirac masses with the ν L (which here also have PQ charge −1).Tiny Yukawa couplings, y ν , would then normally be required to explain the active neutrino masses.However, the spontaneous PQ breaking in the SU (5) sector can generate Majorana masses, m R , for ν R , if there is mixing between the elementary ν R and composite operators.As we shall show, m R can be hierarchically smaller than Λ 5 , providing a means to naturally generate light sterile states and pseudo-Dirac neutrinos.
The elementary ν R can mix with the 3-fermion, dimension- 9  2 composite operators: where Λ R is the EFT scale satisfying Λ 5 < Λ R < M Pl , and ζij , ζ′ ij are dimensionless couplings, with i the neutrino flavor index and j ∈ {1, 2} enumerating the SU (3) c singlets in (ψψψ).After the SU (5) theory confines, these operators give rise to a (PQ-invariant) mass mixing between the ν R,i and the composite fermions N j .This effect is discussed further in App.E in the context of a toy model with a single composite resonance.In the lowenergy effective theory below the resonance scale Λ 5 , the effect of the above operators is to generate a Majorana mass term for ν R .Including also the Dirac mass term for the neutrinos, we obtain with where ζ ij ≃ O(1) × ζij and we have set ζ′ ij = 0 for simplicity.Notice that m R (which has again been estimated using dimensional analysis) is suppressed relative to the resonance scale, and if Λ R ≫ f PQ there will be light sterile neutrino states.Hence, both the see-saw and pseudo-Dirac limits of the neutrino mass matrix in (20) can be naturally obtained, depending on the ratio f PQ /Λ R .In the following, we assume diagonal couplings for simplicity: y ij ν = y ν,i δ ij and ζ ik ζ jk = δ ij .The see-saw limit of the neutrino masses is then given by and m sterile ν is given by (21). Figure 3 shows (green) contours of the sterile neutrino mass 6 in the f a -Λ R plane.We have fixed m active ν,3 = 0.05 eV, such that the contours correspond to different values of y ν,3 .The left and right panels are for R ψ = 3⊕ 3 and R ψ = 8, respectively.The red region is excluded by SN1987A and the region to the left of the blue lines is favoured to obtain a high-quality axion.The requirement that the EFT scale is above the resonance scale, Λ R ≳ Λ 5 , imposes an upper bound on m sterile ν (or equivalently y ν,3 ), as shown by the grey region.
The sterile neutrino masses can naturally be hierarchically smaller than the underlying scales f a and Λ R .How-ever, similar to the standard type-I seesaw, small Yukawa couplings, y ν , are then needed to obtain the active neutrino masses.With sterile neutrino masses of order the eV scale a coupling y ν ∼ 10 −12 is needed, while TeV scale sterile neutrino states correspond to y ν ∼ 10 −6 .In the pure Dirac mass limit (m R → 0), an active neutrino mass of m active ν = 0.05 eV corresponds to y ν ≃ 10 −13 .

UV completion
A tree-level UV completion for the operators in Eq. ( 19) is obtained by introducing a massive, complex scalar field ϕ with mass m ϕ > Λ 5 .Again, the scalar does not obtain a VEV and therefore does not affect the axion quality.The charge assignments of the fields are listed in Table IV, leading to the interaction terms where y5, y 10 , y R are dimensionless couplings and we have suppressed the indices that enumerate the SU (3) c contractions.For simplicity, we consider just one active neutrino flavor and one ν R flavor.The analysis can be straightforwardly generalized to three active neutrino flavors.Integrating out the massive scalar ϕ in ( 23), as FIG. 4: UV-completion of the dimension-6 operators in Eq. ( 24) arising from tree-level exchange of the heavy scalar ϕ.(The arrows represent fermion number.) shown diagrammatically in Fig. 4, yields )(ψ 10 ψ 10 ) + h.c., (24) which are the operators in Eq. ( 19), with Λ R ∼ m ϕ for O(1) couplings.

Holographic connection
Our light sterile neutrino scenario provides a possible holographic realization of the 5D model considered in Ref. [31].In the 5D model, both the axion and right-handed neutrinos are bulk fields charged under a bulk U (1) PQ gauge symmetry, which allows for hierarchically small sterile neutrino masses.According to the AdS/CFT dictionary (see e.g.Ref. [46]), the bulk U (1) PQ gauge symmetry is dual to a global symmetry in the 4D gauge theory (CFT), while the Kaluza-Klein mass eigenstates can be understood as due to a mixing between an elementary and composite sector.
For the right-handed neutrino, this would imply a mixing term ν R O, where ν R is an elementary fermion and O is an operator in the dual (CFT) gauge theory.A naturally small mixing can be generated when dim O > 4.
This is similar to what occurs in the light sterile neutrino case.This can be seen from the UV Lagrangian in (24), where the operator is O = ψ † 5ψ † 5ψ † 10 (assuming y 10 = 0); the dual 4D theory is then identified as the SU (5) gauge theory.Since dim O = 9  2 , the mixing is small (as seen in Section III B) and therefore the sterile neutrino partner of the active neutrino can be naturally light.Thus, the UV completion considered in the light sterile neutrino case provides a specific holographic realization of the 5D model.This holographic realization is not in perfect agreement with the 5D model of Ref. [31] because the left-handed (active) neutrinos were also bulk fields in the 5D model.This means that the dual theory should also feature mixing between the elementary ν L and composite operators.It would be interesting to generalize our UV completion to also incorporate this feature.Finally, the axion in the 5D model has exponentially suppressed couplings on the UV brane.This corresponds to essentially a purely composite (and high-quality) axion in the dual theory, as also occurs in the SU (5) gauge theory.

IV. CONCLUSION
The axion and right-handed neutrinos are motivated by two seemingly unrelated puzzles of the Standard Model.In this work, we have provided a common origin for the QCD axion and right-handed neutrinos as bound states arising from strong dynamics.This builds upon the chiral SU (5) gauge theory in Ref. [21], which contains a high-quality composite axion, to also include composite neutrino states.This solution also provides a possible UV description of the holographic models considered in Refs.[29,31].
Interestingly, the strong dynamics gives rise to composite sterile neutrino masses of order the PQ breaking scale.Depending on the origin of the coupling between the composite sterile neutrinos and the left-handed neutrinos, either pseudo-Dirac or Majorana neutrinos are possible.When the composite sterile neutrinos directly couple to left-handed neutrinos in a dimension-seven interaction with the Higgs, Majorana active neutrinos are obtained via a seesaw mechanism.The dimension-seven interaction can be generated by integrating out two PQcharged, massive complex scalar fields in a UV completion that preserves the quality of the PQ symmetry.Alternatively, the composite sterile neutrinos can mix with elementary right-handed neutrinos, via dimension- 9  2 operators, to induce naturally small couplings to the active neutrinos.This leads to sterile states that are hierarchically lighter than the PQ scale and can realise pseudo-Dirac neutrinos.The PQ symmetry plays the role of a generalised lepton number in realizing either the heavy or light sterile neutrino scenario.
There are number of phenomenological features of our model that could be tested in future experiments.In the pseudo-Dirac limit there is a contribution to the number of effective neutrino species; for reheating temperatures above the SU (5) confining phase transition, ∆N eff ∼ 0.1, which can be tested in upcoming CMB experiments [28,47,48].Alternatively, if the light sterile neutrinos have sub-TeV masses they could be detected at collider experiments.In the post-inflationary scenario (assuming the residual discrete PQ symmetry is broken to avoid stable domain walls), a first order SU (5) phase transition could give rise to a gravitational wave signal associated with the PQ scale (see e.g.[49]), which is worthy of further study.Our model also predicts axionneutrino couplings that could lead to effects in neutrino oscillations within the local DM axion halo [50].Finally, baryogenesis mechanisms can be straightforwardly incorporated into our model, such as the usual leptogenesis mechanism, or a cogenesis mechanism, as considered in Ref. [28].
The coincidence between the axion decay constant and seesaw mass scales can therefore be explained by strong dynamics, which naturally connects the axion and neutrinos in a way that can also address the axion quality problem.
• R ψ = 3 ⊕ 3: One of the adjoint representations (i.e.8) of SU (3) c in the RHS of both (B5) and (B6) corresponds to the generators of the unbroken SU (3) c subgroup.The remainder gives the SU (3) c representations of the NG bosons of the spontaneous symmetry breaking SU (n f )5 × SU (n f ) 10 → SU (3) c (i.e. when the group G in Eq. ( 2) is trivial).Importantly, as discussed in Section II B, all of the NG bosons are colored in the R ψ = 8 case and hence obtain masses of order √ α s Λ 5 .For R ψ = 3 ⊕ 3, there are two massless, QCD singlet NG bosons.This corresponds to the fact that, in this case, the gauging of SU (3) c preserves two residual U (1) flavor symmetries.Due to the cosmological bounds on additional relativistic degrees of freedom, this scenario is only viable if the composite sector remains out of equilibrium with, and colder than, the SM bath.On the other hand, if these U (1) symmetries are instead contained in G (i.e.not spontaneously broken by the strong dynamics) then there are no QCD singlet NG bosons, which would the cosmological requirements.

Appendix C: Explicit PQ Breaking and Axion Quality
A feature of the SU (5) chiral gauge theory is that the PQ symmetry is accidentally preserved up to very high order [21].As discussed in Section II B, the lowest dimension SU (5) and SU (3) c gauge invariant, Lorentz scalar operators that have non-zero PQ charge contain six fermion fields.This implies that the PQ symmetry is accidentally preserved up to (gravitationally-induced) dimension nine terms in the Lagrangian.The relevant operators are listed in Eq. (3).
While the leading PQ-breaking operators contain sixfermion fields in the present SU (5) model, it is interesting to consider whether there are generalisations of this model in which PQ-breaking terms arise from eightfermion (or higher) operators, since this would provide a more robust solution to the axion quality problem.A simple extension is to consider SU (N ) gauge groups with N ≥ 6 and fermions in the antifundamental and antisymmetric irreps.Using the Mathematica package GroupMath [51], we find that there always exist PQbreaking operators with either four or six fermions for all 6 ≤ N ≤ 16.

Displacement of the axion potential minimum
Planck suppressed, PQ-charged operators cause a displacement of the axion potential minimum from its CP conserving value.To determine this displacement, we consider the following Lagrangian containing the (leading) dimension nine operators, where the c i are dimensionless constants and the overall prefactor has been estimated using naive dimensional analysis (NDA) [52,53], assuming the gravitational EFT scale Λ Pl = 4πM Pl and M Pl ≈ 10 19 GeV.All of the above operators have PQ charge −2 and each term represents multiple gauge singlet combinations.For example, the decomposition of each operator into irreducible representations of SU (5) is showing that Φ PQ,1 includes six SU (5) invariant combinations, and similarly for Φ PQ,2 , Φ PQ,3 and Φ PQ,4 .There is an analogous decomposition for SU (3) c , further increasing the number of gauge singlet combinations.The Lagrangian (C1) gives rise to the following term in the low-energy effective theory below the SU (5) resonance scale, where c PQ is a constant that depends on the c i in (C1) and is assumed to be O(1).The resulting axion potential is then approximately given by where N is the QCD anomaly factor (see App. A), and f a ≡ f PQ /N .The first term is the usual QCD contribution and the second term arises from (C6), with c PQ = |c PQ | e iδ and δ representing an arbitrary phase from gravity that is not necessarily aligned with the phases in the SM.
The displacement of the axion potential minimum with respect to the CP conserving minimum is then found to be  .These bounds are similar to those given in Ref. [21]; however, we have included all the dimension nine PQ-breaking operators in our analysis.

Appendix D: Axion Dark Matter
As is well-known, the axion provides one of the bestmotivated dark matter candidates.The production of axion dark matter in the early universe depends on whether the PQ-breaking occurs before or after the inflationary era.
If PQ-breaking occurs before (or during) inflation, then axion dark matter is produced via the misalignment mechanism.The relic axion abundance Ω a is then given by [55,56] where θ i = a i /f a is the initial misalignment angle, and h ≃ 0.68 is the present-day Hubble parameter (in units of 100 km s −1 Mpc −1 ).If the total dark matter relic density, Ω DM h 2 ≃ 0.12 [43], is due to axions, the required range of f a for an initial misalignment angle in the range θ i ∈ (0.1, 3) is This range is in tension with realizing the PQ symmetry as a high-quality accidental symmetry of the SU (5) chiral gauge theory.A modification of the misalignment mechanism is to assume that the initial velocity of the axion is nonzero -the so-called kinetic misalignment mechanism [57].For an elementary axion, this mechanism can produce the correct relic abundance for any decay constant in the range 10 8 GeV ≲ f a ≲ 1.5 × 10 11 GeV.It would be interesting to explore whether this mechanism can be implemented in a composite axion scenario, particularly for values of the decay constant f a ∼ 10 8 GeV that ameliorate the axion quality problem.Both of the above scenarios assume that the universe is not reheated to temperatures above the SU (5) de-confinement transition, which would restore the PQ symmetry.
Alternatively, in the post-inflationary PQ-breaking scenario the universe is reheated to temperatures above the PQ-breaking scale.As the universe cools, topological defects form which, depending on the domain wall number, may include stable domain walls.To determine the domain wall number, we first note that the QCD contribution to the axion potential in Eq. (C7) preserves the discrete symmetry, The physical domain of the axion field is a/f a ∈ [0, 2π|N |), with the anomaly coefficient N = −2 and N = −6 for the R ψ = 3 ⊕ 3 and R ψ = 8 models, respectively (see App. A).Therefore, the number of degenerate minima of the QCD potential in the physical domain, which is equivalent to the number of domain walls, is Since N DW > 1, explicit violation of the discrete PQ symmetry, which would allow domain walls to decay, is required if the post-inflationary scenario is to be viable.In principle, such a "bias" potential [58] which lifts the vacuum degeneracy could simply arise from Plancksuppressed, higher-dimension operators that explicitly violate the PQ symmetry, as considered in (C2)-(C5) or of higher dimension.For instance, 6 fermion, dimension-9 terms reduce the domain wall number in the octet model to N DW = 2.However, going beyond 6-fermion terms does not reduce the domain wall number further and therefore a new source of breaking would be required to lift the remaining degeneracy.This may arise, for example, if SU (3) c is embedded in a larger gauge group, such as recently considered in Ref. [59].
Assuming such a bias potential, the decays of cosmic strings and domain walls contribute to the axion dark matter density.With the present state-of-the-art calculations (see e.g.Refs.[60,61] for the case of axion strings and Ref. [62] for N DW > 1), there remains significant uncertainty in the quantitative estimation of the axion abundance from the decay of topological defects, which can dominate over the misalignment contribution.Therefore, the lower bound on f a arising from the dark matter relic density in the pre-inflationary scenario (in Eq. (D2)) could be significantly relaxed in the post-inflationary scenario.In fact, a robust upper bound of f a ≲ 5.4 × 10 8 GeV (or m a ≳ 11 meV) [63], can be derived on obtaining the required axion relic abundance from domain wall decay.In this Appendix, we present a toy model of the mass mixing in the light-sterile neutrino scenario by including a single right-handed neutrino resonance N 1 in the effective low-energy theory.The Lagrangian for this simplified model of the effects of the strong dynamics is given by where ∆ R is a Dirac mass mixing between the elementary field ν R and the composite resonance N 1 , whose value depends on the parameters of the UV completion.Note that in terms of the constituent fields, the operator corresponding to the Majorana mass term for N 1 is Φ PQ,1 in Eq. ( 3), which can be split into two PQ-charged fermionic operators.
The neutrino mass eigenvalues of (E2) are determined numerically and shown in Fig. 6 as a function of the Yukawa coupling y ν , assuming f PQ = 10 10 GeV, g * = 1, and with the active neutrino mass set to 0.05 eV.In the pure Dirac mass limit for the active neutrinos (∆ R → 0), a mass of m active ν = 0.05 eV corresponds to y ν ≃ 10 −13 .As the mass m ϕ of the scalar field in the UV completion (or Λ R in the EFT) is lowered, the sterile partner of the active neutrino increases in mass via the mixing with N 1 .More generally, the sterile partner of the active neutrino will mix with all the resonances of the strong dynamics.

FIG. 1 :
FIG. 1: EFT scale Λ L versus f a in the heavy sterile neutrino model for R ψ = 3 ⊕ 3 (left) and R ψ = 8 (right).Within the green band an active neutrino mass m active ν,3 = 0.05 eV is obtained with 1 ≲ g * ≲ 4π and Λ L > Λ 5 , assuming λ ν,3 = 1.The red shaded region is excluded by the SN1987A bound on f a [4].The estimated upper limit on f a for a high-quality axion consistent with the neutron EDM bound (assuming |Im (c PQ )| ≳ 10 −3 , see App.C) is shown by the blue dashed (dotted) line for g * = 1 (4π).The breakdown of the EFT validity when Λ L ≲ Λ 5 (= g * f PQ ) is depicted by the dark (light) grey shaded region for g * = 1 (4π).

3 ≤
13 eV) leads to the 2σ range 0.05 eV ≤ m active ν,0.06 eV.This restricts the allowed values of f a and Λ L , as shown in Fig.1for R ψ = 3 ⊕ 3 (left panel) and R ψ = 8 (right panel).Within the green band an active neutrino mass m ν,3 = 0.05 eV can be obtained with a strong sector coupling in the range 1 ≤ g * ≤ 4π.The lower edge of the band corresponds to g * = 1 and the upper edge to g * = min(4π, Λ L /f P Q ), such that Λ L > Λ 5 is always satisfied within the band.The range of f a excluded by SN1987A[4] is shown in red.Values of f a to the right of the dashed (g * = 1) or dotted (g * = 4π) blue line are disfavoured, since Planck-suppressed contributions to the axion potential can destabilise the solution to the strong CP problem (see App. C).

18 FIG. 3 :
FIG. 3: EFT scale Λ R versus f a in the light sterile neutrino model for R ψ = 3 ⊕ 3 (left) and R ψ = 8 (right).The green bands depict contours of m sterile ν , with the lower (upper) edges of the bands corresponding to g * = 1 (4π); the associated y ν,3 values for an active neutrino mass m active ν,3 = 0.05 eV are shown in the legend.The red shaded region is excluded by the SN1987A bound on f a [4].The estimated upper limit on f a for a high-quality axion consistent with the neutron EDM bound (assuming |Im (c PQ )| ≳ 10 −3 , see App.C) is shown by the blue dashed (dotted) line for g * = 1 (4π).The breakdown of the EFT validity when Λ R ≲ Λ 5 (= g * f PQ ) is depicted by the dark (light) grey shaded region for g * = 1 (4π)

5 FIG. 5 :
FIG. 5: Displacement from the CP conserving minimum, |∆ θeff |, due to Planck suppressed operators as a function of f a .The contours show different values of |Im (c PQ )|, assuming g * = 1.The solid (dashed) lines correspond to the QCD representation R ψ = 3 ⊕ 3 (R ψ = 8).The blue and red regions are excluded by the upper bound on the neutron EDM and SN1987A, respectively.

TABLE I :
Representations of the chiral fermions charged under the SU (5) × SU (3) c gauge symmetry.

TABLE II :
Representations of the SU (5) chiral fermions under the global flavor symmetry SU

TABLE IV :
Representations of the fields in the UV completion of the light sterile neutrino scenario.(Note that L has opposite PQ charge compared to the heavy sterile neutrino scenario.)