A high quality composite axion

The strong CP problem is a compelling motivation for physics beyond the Standard Model. The most popular solutions invoke a global U(1)PQ symmetry, but are challenged by quantum gravitational corrections which are thought to be incompatible with global symmetries, arguing that realistic theories contain additional structure. We explore a construction in which the U(1)PQ symmetry is protected to arbitrary order by virtue of a supersymmetric, confining SU(N)L × SU(N) × SU(N)R × U(1)X product gauge group, achieving θ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left|\overline{\uptheta}\right| $$\end{document} < 10−11 for an SU(5) model with fa ≲ 3 × 1011 GeV. This construction leads to low energy predictions such as a U(1)X gauge symmetry, and for X = B − L engineers a naturally O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O} $$\end{document}(TeV) value for the μ parameter of the MSSM.


Introduction
Despite the conceptual simplicity of the axion solution to the strong CP problem, relatively few axion models have been developed which naturally predict θ 10 −11 when confronted with gravitationally induced U(1) PQ violating operators. Models which do sufficiently protect the axion scalar potential from gravitational perturbations typically require large groups or complicated structures, leading to an ongoing search for more satisfying solutions.
In this work we present a relatively simple composite axion model in a confining supersymmetric theory, which is consistent with gauge coupling unification and compatible with current experimental results. Certain mesons in the theory are identified as composite Higgs fields, ameliorating the B/µ problem of the MSSM, and in one variant of our model the B − L global symmetry of the Standard Model is gauged.

The strong CP problem
The Standard Model (SM) contains several puzzles, one of the most pressing of which is the value of the θ parameter in the QCD Lagrangian: L = g 2 θ 64π 2 µνρσ G a µν G a ρσ ≡ g 2 32π 2 θG µνG µν . (1.1)

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Searches for an electric dipole moment of the neutron have so far resulted only in upper limits on its magnitude, implying that θ < 6 × 10 −11 [1,2], whereθ is the physically relevant combination of CP violating phases, where M Q is the quark mass matrix. As the θ term violates both P and CP , the unnaturally small value ofθ is referred to as the strong CP problem. For more complete reviews, see for example [3][4][5].
In many popular solutions of the strong CP problem,θ is rendered unphysical by ensuring that the classical Lagrangian respects a global U(1) symmetry, which is explicitly broken by the QCD anomaly. A simple example can be seen from eq. (1.2), if one sets m u = 0: an axial U(1) A symmetry emerges in this limit, so that arg det M Q (and thereforē θ) becomes unphysical. If it were not for compelling evidence that m u,d = 0, this "massless up quark solution" would naturally explain the absence of CP violation in the strong sector.
Axion models address the strong CP problem by associatingθ with the pseudo-Nambu-Goldstone boson of an approximate U(1) PQ global symmetry. This is achieved by introducing a (SM singlet) complex scalar φ together with left-handed color (anti)-triplet fermions Q and Q, along with the interaction where V (φ) is designed such that φ acquires an expectation value φ 10 9 GeV. The bare mass term mQQ is forbidden, so that L respects a U(1) PQ symmetry under which φ is charged. The SU(3) 2 c -U(1) PQ anomaly coefficient is nonzero, as can be seen from the fact that (QQ) carries a net U(1) PQ charge.
Expanding about the φ = 0 vacuum, the axion a is identified as the phase of φ: where f a ≡ √ 2 φ . The SU(3) 2 c -U(1) PQ anomaly induces an aGG coupling, and nonperturbative QCD dynamics generate a periodic potential for a which can be heuristically (up to chiral symmetry-violating corrections [6], which are unimportant for our discussion) described by The axion potential is minimized by a = f aθ , so that CP is conserved in the QCD vacuum.
In "invisible axion" models of this type [4,[7][8][9] the axion is light and weakly coupled, with a mass given by: (1.7) A lower bound f a 10 9 GeV is set primarily by astrophysical observations of stellar cooling and supernovae. In much of the parameter space, the axion provides a natural dark matter candidate: its interactions are suppressed by the decay constant f a , and it can be produced in the early universe by the misalignment mechanism [10][11][12]. For O(1) initial misalignment angles, the correct relic abundance is obtained for f a 10 12 GeV, though f a could be larger if the misalignment was smaller. The fact that the QCD axion could also play the role of dark matter is one of the reasons for its continued popularity as a solution to the strong CP problem.

Axion quality problem
A closer inspection of the simple axion model presented above reveals a new set of theoretical difficulties, namely a hierarchy problem and a fine-tuning problem. The axion model prefers a hierarchy between the scale of symmetry breaking f a and the Planck mass, M P . A number of standard solutions, such as supersymmetry or compositeness, have been proposed which would render an axion scale f a M P technically natural. However, many axion models still suffer from a more severe fine-tuning, known as the axion quality problem.
Arguments from general relativity [13][14][15][16][17][18] suggest that non-perturbative quantum gravitational effects do not respect global symmetries such as baryon number or U(1) PQ . This is highly problematic for most axion models, which rely on U(1) PQ being an exact symmetry in the α s → 0 limit, explicitly broken only by the QCD anomaly. If additional PQ-violating operators representing the short distance influence of quantum gravity such as are present, the corresponding perturbation in V (a) can shift a far away from the CPconserving value of eq. (1.6): where the phase ϕ is determined by λ k , and ∆ PQ is the U(1) PQ charge of the operator φ.
It is convenient to describe such perturbations by defining a "quality factor" Q: (1.11) Satisfying this bound requires that the theory of quantum gravity somehow produce a severe fine-tuning in the λ k , such that even the dimension-12 operators in eq. (1.9) must have λ k 1. In a truly compelling axion model, the U(1) PQ symmetry should emerge as a consequence of some other underlying structure which forbids the problematic operators. For example, a gauged discrete Z n symmetry [19] for some n 13 can forbid all PQ-violating operators smaller than (φ n + c.c.). Significantly smaller groups can be employed to the same effect in supersymmetric theories [20,21], if the discrete group is an R symmetry. Composite axion models such as [22][23][24] also protect U(1) PQ to arbitrarily high order, with the added benefit that the axion scale f a can be generated dynamically. Other constructions [25][26][27] associate U(1) PQ with a different, gauged U(1), so that many of the PQ-violating operators are forbidden. Many of these constructions are intricate and also rather delicate in the sense that the axion quality is easily ruined in extensions of the model.
In this work we present an alternative composite axion model based on an SU(N ) × SU(N ) confining supersymmetric gauge theory with simple matter content. The Standard Model matter fields and interactions are easily embedded, and we show that the axion quality is preserved even with the addition of new fields. Upon identifying the H u and H d doublets as mesons from SU(N ) confinement, we find that the µ parameter of the MSSM naturally assumes an O(TeV) value. Finally, we explore the ability of this model to mediate supersymmetry breaking via composite messengers.

Composite axion model
Conjectured dualities [28,29] allow one to analyze the low energy behavior of supersymmetric gauge theories. In particular, an SU(N c ) gauge theory with N f = N c flavors of quarks (Q + Q) in the (anti-)fundamental representation is expected to confine at a characteristic scale Λ, such that the low energy degrees of freedom are described by the gauge-singlet operators  [30]. We show that a composite axion emerges in a subset of these theories, with sufficiently high axion quality. We invoke the gauge group SU(N ) L × SU(N ) SM × SU(N ) R × U(1) X , where SU(N ) SM contains the Standard Model SU(3) c ×SU(2) L ×U(1) Y either as a gauged subgroup or as an SU(5) grand unified theory. The strongly coupled SU(N ) L,R confine at the characteristic scales Λ L,R TeV, but the Abelian U(1) X is weakly coupled. 1 The bifundamental fields Q 1,2 and Q 1,2 have U(1) X charges ±1, as depicted in the moose diagram of figure 1, with U(1) PQ charges shown in table 1.   Below the scales Λ L and Λ R , the low energy degrees of freedom are described by the composite operators satisfying equations of motion:

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In the absence of a superpotential, this model respects the global SU(N ) 1 × SU(N ) 2 symmetries shown in figure 1, as well the gauged U(1) X . There is also a conserved U(1) R , under which the gauginos have charge +1 and all of the Q 1,2 and Q 1,2 are neutral, which remains unbroken everywhere on the moduli space.
In the regime where G 0 is weakly coupled, there is another nearly exact global symmetry, U(1) PQ , which is broken only by the G 2 0 -U(1) PQ anomaly. Due to the locally conserved U(1) X , there is no unique assignment of Peccei-Quinn charges: rotations under U(1) PQ can always be combined with a global U(1) X transformation to define a new, equally valid Peccei-Quinn symmetry. This degeneracy is parameterized by the parameter α in table 1.
On the quantum-deformed moduli space described by eq. (2.3), the global SU(N ) 1 × SU(N ) 2 × U(1) X × U(1) PQ symmetry must be broken to a subgroup. Furthermore, if the JHEP11(2018)199 low energy limit of this theory is to approach the Standard Model, then it must be true that det M = det M = 0; otherwise, SU(3) c would be broken in the vacuum. The vacuum therefore must be engineered to lie on the B 1 B 2 = 0, B 1 B 2 = 0 branch of the moduli space, where U(1) X and U(1) PQ are both spontaneously broken, and the U(1) X vector supermultiplet acquires a mass by "eating" a combination of the chiral superfields. This is accomplished by including a term in the superpotential of the form: which after confinement generates a mass term for the mesons, W ∼ µM M , lifting the mesonic flat directions. If not otherwise present, this term is expected to be induced by quantum gravitational effects. A unique definition of the Peccei-Quinn charges emerges once U(1) X is broken: by canonically normalizing the kinetic terms of the (would-be) Nambu-Goldstone bosons of U(1) PQ and U(1) X , the parameter α of table 1 is related to the vacuum expectation values (VEVs) of the baryons as and where the axion decay constant f a is With this normalization, a U(1) PQ rotation by a phase θ is achieved by the linear shift a → a + θf a . (2.8) Although the products v 1 v 2 andv 1v2 are set by the quantum modified constraints, the values of the decay constants f a and f X vary along the flat directions within the allowed ranges

Axion quality
To examine the axion quality, we introduce operators characterized by M P which represent an effective field theory description of the low energy residual effects of quantum gravity.

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It is convenient to introduce a set of rescaled composite operators with mass dimension +1: (2.11) The effective gravitational superpotential violating all of the global symmetries takes the form: with parameters λ i and ρ i encoding the UV physics. Of the operators listed above, only the two associated with λ 1 and λ 2 violate U(1) PQ . All of the lower-dimensional operators such as (Q 2 Q 1 )(Q 1 Q 2 ) are neutral under U(1) PQ , and thus not harmful to the axion quality.
In a supersymmetric vacuum, the leading U(1) PQ violation appears with M 4N −6 P suppression in the Lagrangian: for example, within terms such as implying a perturbation to the axion potential on the order of More serious perturbations to the axion potential emerge when supersymmetry breaking is taken into account. Supersymmetry breaking induces an "A-term" potential, where the mass scales A i are in principle calculable once a particular mechanism of supersymmetry breaking is specified. To remain agnostic concerning the details of supersymmetry-breaking, we assume that the A i should be of roughly the same magnitude as the SU(3) c × SU(2) L × U(1) Y gaugino masses. Both the A 1 and A 2 terms in eq. (2.17) perturb the axion potential: (2.18) -7 -

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Again taking Λ L,R ≈ f a ≈ 10 11 GeV, the constraint on the quality factor eq. (1.11) can be written as for i = 1, 2, indicating that models with N ≥ 5 are free from fine-tuning as long as the characteristic scales Λ L,R and f a are not much larger than 10 11 GeV.
In figure 2 we plot the maximum values of λ i consistent with eq. (2.19), for given values of f a , N , and the other parameters, with the simplifying assumptions A 1 ≈ A 2 and λ 1 ≈ λ 2 . It is convenient to label the vacua with the following parameterization: All of the dimensionful parameters except for A i and M P are now expressed in terms of f a : so that the axion quality condition is expressed: Because β L,R label degenerate vacua on the moduli space defined by eq. (2.3), particularly large or small values of tan β L,R are typically unnatural. On the other hand, γ is primarily determined by the ratio Λ L /Λ R : so large or small values of tan γ are more easily tolerated from a naturalness perspective.
As we see from eq. (2.22), the best axion quality is achieved for tan γ ≈ 1, when f a ≈ f X and Λ L ≈ Λ R . We show the maximum tolerable λ 1 ≈ λ 2 as a function of f a for a few choices of N , tan β L = tan β R , and sin 2γ in figure 2. While effective field theory would suggest that generic theories of quantum gravity should produce λ 1,2 ∼ O(1), in [16][17][18] it is argued that wormhole-induced U(1) PQ violation yields suppressed values of λ i ∼ exp(−S w ), where the wormhole action S w depends logarithmically on the axion decay constant, S w ∼ a−b ln fa M P . For typical cases the resulting suppression in λ i is modest: values as small as λ ∼ 10 −7 are achieved in [16] for f a ∼ 10 12 GeV. For N = 5 such that G 0 is large enough to contain the SM, O(1) λ's are consistent with f a 10 11 GeV.
Generally, the high axion quality observed in eq. (2.19) is preserved even when new fields are coupled to the model provided that they are neutral under U(1) X . Problems arise if there are fields S with U(1) X charges: for which case W g includes gauge-invariant terms S p B 1,2 or S p B 1,2 for some power p < N .
(2.26) Table 2 where the subscripts indicate the B − L charge of each operator. Since none of these carry PQ charge, the superpotential operator constructed by multiplying any of them by a baryon superfield would violate U(1) PQ unacceptably. To avoid this issue, we restrict ourselves to the cases where q = ± n 5 , for n = 0, 1, 2, 3, 4, and also q = ± 1 3 .

Composite Higgs doublets
The identification of X = B −L has positive implications for the superpotential, notably by forbidding many of the operators that would mediate highly constrained B and/or L violation such as proton decay [31]. The allowed low energy effective superpotential has the form: symmetry. If generated by quantum gravitational residuals, the natural mass scale for µ and µ would thus be: This is µ ∼ O(TeV) for our benchmark choice of Λ L ≈ Λ R ≈ 10 11 GeV. The Yukawa interactions of eq. (2.28) similarly correspond to dimension five operators in the UV. Realizing the large couplings necessary for the heavy quarks requires that they be generated at a lower scale M F M P : where y t ∼ 1 requires M F ∼ Λ R (and y b requires Λ L is not much larger). Unlike the dynamics generating the µ terms, the Yukawa interactions are compatible with the U(1) R symmetry, which allows for the disparate scales to remain technically natural. The presence of the four additional M (2) and M (2) in eq. (2.30) poses a potential phenomenological problem. In the absence of any additional structure, the y u,d,e couplings of the matter fields with the heavier SU(2) L doublets will generally introduce flavor-changing neutral currents (FCNC). A number of potential solutions exist in the literature. For example, by imposing minimal flavor violation [32] on eq. (2.30), the M (2) and M (2) can have masses as small as a few TeV. Or, as we discuss in section 3, a discrete symmetry can be imposed (even if broken at M P ) to forbid the y u,d,e couplings for all of the mesons except for H u and H d .

Color-triplet mesons
As illustrated in eq. (2.29), we expect that gravitational effects induce electroweak scale O( Λ L Λ R M P ) supersymmetric masses for each of the five pairs of M Generically, color triplets with weak scale masses are very tightly constrained, especially because the interactions Higgs color triplets which typically appear in SU(5) grand unified theories. In section 3 we explore the possibility that they could (along with the extra SU(2) L doublets) serve as messengers for gauge-mediated supersymmetry breaking.

Alternatives to B − L
In addition to B − L, there are a number of other acceptable anomaly-free U(1) X charge assignments for the Standard Model matter. While none are as attractive as B − L, in this section we sketch three alternatives: a "5/-3/1" pattern of U(1) X charges within each generation; every matter superfield neutral under U(1) X ; and an L i − L j model. Table 3. Charges of the matter fundamental superfields and Higgs doublets and composite baryons and mesons in the "5/-3/1" U(1) X model.

5/-3/1 model
An alternative charge assignment is shown in table 3: Q L ,ū R andē R fields have U(1) X charge q; L andd R have charge −3q; and theν R has charge 5q to cancel the U(1) 3 X . anomaly. Forbidding all U(1) PQ -violating operators of dimension less than 10 requires: but otherwise q is a free parameter describing a family of models. With this charge assignment the undesirable baryon and lepton number violating operators LH u , LLē R , QLd R andū RdRdR are all forbidden, and proton decay occurs via the dimension 5 operator W ∼ū RūRdRēR /M P . Unlike in the B − L model, U(1) X forbids the mesons M (2) and M (2) from having Yukawa interactions with MSSM matter unless q = 0. Thus, additional fundamental Higgs doublets H u + H d with U(1) X charges ±2q must be added to generate quark and lepton masses, would allow the mesons to decay entirely into the Standard Model depend sensitively on q, with the operators permitting prompt decay also typically violating U(1) PQ and forbidden by eq. (2.32). As consequence, the lightest mesons tend to have long lifetimes, and for some values of q can be absolutely stable and bounded by the strong constraints on colored or charged cosmological relic particles.

q = 0: neutral MSSM
In the limit q → 0, the MSSM decouples from U(1) X . This assignment allows for Yukawa interactions between the mesons and MSSM matter, permitting M (2) and M (2) to play the role of the MSSM Higgs doublets, with O(Λ L Λ R /M P ) supersymmetric masses as in eq. (2.29). However, U(1) X no longer forbids the problematic operators of eq. (2.31) or Among the potentially disastrous consequences of W bad is a short proton lifetime. This problem is averted in the MSSM by imposing a Z 2 R parity, which ensures that the superpotential respects the B − L global symmetry. Upon imposing R parity or some other discrete symmetry on the q = 0 model, the superpotential comes to resemble that of the B − L axion model in all respects except one: if q = 0 the right-handed neutrino is a singlet under the gauge symmetries, at which point it can be safely omitted.

L i − L j models
The Standard Model also admits anomaly-free U(1) symmetries for which charges are not uniform across all three generations. The combinations of L µ − L τ and L e − L τ are among the phenomenologically interesting alternatives. Models of this type are typically consistent with a composite H u and H d , but as in the MSSM, an R parity must be imposed on such models to ensure that all of the B and L violating operators of eq. (2.34) are forbidden.

Gauge-mediated supersymmetry breaking
Beyond the usual MSSM superfields, there are relatively few additional light degrees of freedom: • The four baryons B 1,2 and B 1,2 contain at most two light fields in the B i = 0, B i = 0 vacuum. There is a chiral multiplet containing the composite axion.
• For U(1) X gauge coupling g X 1, there is a U(1) X vector supermultiplet with a mass m X ∼ g X f X , where f X is typically ∼ f a . In this section we explore how these mesons may be utilized as messengers of supersymmetry breaking.

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We parameterize the supersymmetry-breaking in a secluded sector as a set of one or more chiral superfields X i acquiring F -term expectation values, with F X = 0. Introducing superpotential terms of the form W ∼ XM (3,2) M (3,2) communicates supersymmetry breaking to the MSSM [33,34]. In the UV theory this superpotential originates from dimension-5 operators (Q 2 Q 1 )X(Q 1 Q 2 )/M 2 S , reducing in the IR to where the indices i, j = 1 . . . 5, for some scale M S √ Λ L Λ R which we take to be small compared to M P . It is convenient to absorb the factors of Λ L Λ R /M 2 S into the definitions of λ 2,3 : As with the Yukawa couplings of eq. (2.30), the superpotential W s respects a global U(1) R symmetry under which the mesons M and M are neutral, and X has charge +2. As discussed in section 2.2, Yukawa-like couplings between the matter fields and the four heavy M (2) + M (2) may introduce unacceptable flavor-changing neutral currents.
A standard solution is to impose a "messenger parity" on the model, under which the Higgs H u,d are even, and the messengers M (2,3) and M (2,3) are odd. Thus, the direct couplings between messenger SU(2) L doublets and the matter fields are forbidden, and the problematic flavor-changing neutral currents are avoided. 2 Imposing this Z 2 symmetry reduces eq. (3.2) to: 4) where, if the messenger parity is derived from the global symmetries of the quarks Q 2 and Q 2 , we take the SU(3) c triplets M multiplets. Following [35], the gauge coupling strength α GUT at the unification scale M GUT is modified by where N f = N c = 5. Requiring that SU(5) SM remains perturbative up to the unification scale imposes a lower bound on X :
(3.7) Performing SU(4) 1,2 ×U(1) 1,2 rotations on the fields M (2) and M (2) , the matrices (λ 2 X +µ 2 ) and (λ 2 F X ) can be simultaneously diagonalized and made real: with eigenvalues M 2 i ± F i . This basis also diagonalizes the scalar mass matrix of M (3) and M (3) in the special case λ 2 = λ 3 and µ 2 = µ 3 (but not in general). Positivity of the (squared) messenger masses imposes a constraint on the F -term VEV of the superfield X: for each pair of λ ii 2 and µ ii 2 in the diagonal basis. Note that due to the compositeness of the messengers, the couplings λ 2,3 are suppressed by a factor Λ L Λ R /M 2 S which may be much smaller than unity.
To produce the correct electroweak scale, the M 2 and F terms for H u and H d must coincide. Taking λ 1,1 2 ∼ Λ L Λ R M 2 S and µ 1,1 2 ∼ Λ L Λ R M P , this condition implies a relationship between the scales M S , X and F X : (3.10) Taking the simplifying case √ Λ L Λ R ∼ f a ∼ 10 11 GeV and M S f a in the limit X < 10 5 GeV, eq. (3.10) reduces to the condition √ F X ∼ faM S M P . An investigation of the extensions to the composite axion model satisfying this constraint would be an interesting opportunity for future work.

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The general SU(N ) L × SU(N ) R × U(1) X axion model allows the Standard Model matter fields to carry nearly any anomaly-free U(1) X charge assignment without negatively affecting the axion quality. In particular, attractive features emerge when U(1) X is associated with gauging the Standard Model B − L global symmetry. The leading terms in the superpotential are those of the MSSM, with none of the problematic B or L violating operators that would otherwise need to be forbidden by invoking a discrete "matter parity". Additionally, if the Higgs H u and H d are taken to be the lightest of the SU(2) L charged mesons from SU(5) L and SU(5) R confinement, the dimension-4 gravitationally-induced operator naturally generates an electroweak scale µ term for f a ∼ 10 11 GeV. Other choices of U(1) X charge assignments share this feature, that the SU(2) L charged mesons have the same quantum numbers as H u and H d , and could therefore produce a composite Higgs with a TeV scale µ term.
The low energy phenomenology largely resembles the MSSM plus a chiral superfield containing the standard QCD axion, axino, and a saxion. More unique are the presence of meson fields in vectorlike color triplet and electroweak doublet representations. In theories in which the lightest weak doublet pair are identified as the MSSM Higgs superfields, they will have ∼ TeV masses. Their detailed phenomenology depends on the U(1) X charge assignments and some choices of (perhaps slightly broken) global symmetries, and their presence indicates that the Large Hadron Collider could potentially uncover clues to higher scale physics. Alternatively, some of these fields could play the role of messengers, leading to a picture in which supersymmetry-breaking is mediated by gauge interactions.
Among the many opportunities for future work, some promising directions include developing the supersymmetry breaking sector, explaining the pattern of Yukawa couplings in the MSSM, or exploring the cosmological implications of the composite model in the early universe.