Dynamical Clockwork Axions

The clockwork mechanism is a novel method for generating a large separation between the dynamical scale and interaction scale of a theory. We demonstrate how the mechanism can arise from a sequence of strongly-coupled sectors. This framework avoids elementary scalar fields as well as ad hoc continuous global symmetries, both of which are subject to serious stability issues. The clockwork factor, $q$, is determined by the consistency of the strong dynamics. The preserved global $U(1)$ of the clockwork appears as an accidental symmetry, resulting from discrete or $U(1)$ gauge symmetries, and it is spontaneously broken by the chiral condensates. We apply such a dynamical clockwork to construct models with an effectively invisible QCD axion from TeV-scale strong dynamics. The axion couplings are determined by the localisation of the Standard Model interactions along the clockwork sequence. The TeV spectrum includes either coloured hadrons or vector-like quarks. Dark matter can be accounted for by the axion or the lightest neutral baryons, which are accidentally stable.

The principle of naturalness [1] has been one of the most important guidelines in searching for physics beyond the Standard Model (SM). According to this principle, small parameters are not expected in a theory unless the theory becomes more symmetric in the limit that those parameters vanish. By applying this principle to higher dimensional interaction terms in a low energy theory resulting from new dynamics, interaction scales are usually identified as the energy scales of the dynamics. The pion decay constant and the Fermi constant are good examples of where interaction scales are close to the energy scales of the underlying dynamics.
The clockwork mechanism in [2][3][4] provides a novel method of circumventing the principle of naturalness (see [5] for an earlier work in the context of natural inflation). It allows a large separation between the interaction scale and the dynamical scale in a theory with only O(1) parameters. Recently, this mechanism has been harnessed in models of inflation [6], the axion [7,8], composite Higgs [9], WIMPs [10], and the relaxion [11][12][13].
The basic clockwork idea involves many sites of spontaneous U (1) symmetry breaking at a scale f , each of which is associated with a Nambu-Goldstone boson (NGB). The U (1) symmetries are also explicitly broken down to a single U (1) symmetry, U (1) 0 , by couplings between the sites. Accordingly, only one linear combination of the NGBs remains massless.
Remarkably, the effective decay constant of the remaining NGB, F , can be much larger than f when the U (1) 0 charges are appropriately chosen. In particular, the clockwork mechanism is achieved by a very efficient arrangement of the symmetry, which leads to an exponentially enhanced effective decay constant, F ∼ q N f , where q > 1 is the clockwork factor and N the number of sectors. 1 We note that essentially the same idea of separating the interaction scale and dynamical scale by multiple U (1) symmetry breaking was previously proposed in Refs. [16], under the name of the phase-locking mechanism. In this mechanism, however, the charge assignment of the U (1) symmetries was not specified, and hence no exponential hierarchy was discussed.
The authors of Refs. [3,4] showed how the large number of approximate global U (1) symmetries, as well as the U (1) 0 charges in geometric progression with ratio q, can be attributed to extra-dimensional setups. In this paper, we discuss alternative routes to achieve these important features for the clockwork mechanism by utilising strong dynamics at each site. In our models, the clockwork factor is solely determined by the consistency of the strong dynamics. We also discuss models where the global U (1) 0 symmetry appears as an accidental symmetry, resulting from discrete symmetries or U (1) gauge symmetries.
As for the phenomenological applications of the dynamical clockwork, we will focus on the axion solution to the strong CP problem, which requires spontaneous breaking of a Peccei-Quinn symmetry, U (1) P Q , with a decay constant large enough to comply with experimental/astrophysical searches of the axion, f P Q 4 × 10 8 GeV [17, 18, for review]. 2 It has been known for a long time [21] that strong dynamics can realise this mechanism with heavy fermions charged under QCD as well as under a new gauge interaction that confines at scale Λ f P Q : the heavy-fermion condensate breaks U (1) P Q spontaneously and yields a composite axion. By using the clockwork, we will show how Λ can be lowered to the multi-TeV scale while keeping the axion sufficiently 'invisible'. 3 It is notable that a low PQ symmetry breaking scale guarantees better protection of the axion potential against explicit breaking effects from the Planck scale, M P , that are suppressed by powers of Λ/M P [27][28][29][30][31]. In general, for Λ in the multi-TeV scale, this suppression is close but not quite sufficient to solve the strong CP problem, therefore we will discuss additional ways to protect the axion potential, specific to our dynamical clockwork models. Different ideas to screen quantum gravity corrections can be found in Refs. [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49].
In section II, we present models where strong dynamics provides both spontaneous symmetry breaking at each clockwork site and the link between neighbouring sites. In section III, we construct alternative models where each strong sector is confined to a single site and the connection between sites is enforced by other exact symmetries. In section IV, we discuss the coupling between our dynamical clockwork models and QCD, in order to realise the axion solution to the strong CP problem. Finally, in section V, we describe the phenomenological and cosmological implications of the dynamical clockwork axion. 2 Here, we assume for definiteness the KSVZ axion model [19,20]. 3 An additional motivation for strong dynamics in the multi-TeV range is to solve the gauge hierarchy problem by the compositeness of the Higgs boson. Indeed, the simplest ultraviolet-complete composite Higgs models (see e.g. Refs. [22][23][24][25][26]) have some similarities with composite axion models: they require heavy fermions charged under the new confining interaction as well as QCD to yield composite topquark partners.

II. CLOCKWORK LINKED BY STRONG DYNAMICS
The standard clockwork mechanism [2][3][4] involves a large number of fields which obey a U (1) N +1 global symmetry that is broken spontaneously. The U (1) N +1 global symmetry is also broken explicitly to a single global symmetry, U (1) 0 , by "nearest neighbour" couplings between the fields. In the simplest case, these fields are N + 1 complex scalars, and the global U (1) N +1 symmetry is spontaneously broken at some scale, f , then explicitly broken down to the U (1) 0 at a much lower scale. The potential takes the form [3] such that λ. It is assumed here that f is the same for each φ j , and relaxing this assumption does not alter the implementation of the clockwork mechanism.
Below the scale f , we can rewrite the potential (1) in terms of the N + 1 Goldstone , which are associated with the spontaneously broken symmetries. This gives The theory contains one massless exact NGB, corresponding to the remnant global U (1) 0 symmetry. Rotating into the mass basis, π j = O jk a k , where O ∈ O(N + 1), we denote the Goldstone boson as a 0 and consequently O j0 ∝ q −j . This is the crucial feature of the clockwork mechanism: the massless axion component of each π j shrinks exponentially with increasing j, for q > 1.
By coupling the SM to one end of this chain of fields (φ N in the case q > 1, φ 0 in the case q < 1), this mechanism can create an effective coupling much smaller than one (q −N for q > 1, q N for q < 1) from O(1) interactions in the Lagrangian. For q = 1 and large N , it is possible to generate very substantial differences in scale.
The notable features of the clockwork mechanism are a) the nearest neighbour nature of the interactions, and b) the large number of approximate global U (1) symmetries.
The nearest neighbour nature of the interactions is technically achieved by a special arrangement of the U (1) 0 charges in geometric progression with a ratio q. To make the construction more convincing, however, some justification of the origin of such a wellorganised charge assignment is required. Besides, given the arguments that all global symmetries are broken by quantum gravity effects [27][28][29][30][31]50], models may seem more plausible if they do not rely on global symmetries.
In the following, we discuss how the well-organised U (1) 0 charge assignment is achieved by utilising strong dynamics. The spontaneous symmetry breaking is achieved by the confinement of asymptotically free gauge theories at each site. In the first class of models, we find that the U (1) 0 charges in geometric progression are automatically obtained from the fermion content of the chain of gauge theories. We also show that such dynamical clockwork can be realised without imposing any continuous global symmetry.

A. Dynamical phase locking
First, we consider a model with an SU (N c ) N gauge symmetry containing N + 1 vectorlike fermions, ψ j , j = 0, . . . , N , which transform as the fundamental, F, of SU (N c ) j+1 and as some representation, R, of SU (N c ) j other than the fundamental representation. We note the "boundary conditions" that ψ 0 and ψ N transform only under a single SU (N c ). This is shown in Table I. We assume that each SU (N c ) j is asymptotically free and exhibits confinement at the dynamical scale Λ j . We also assume that the fermion mass terms allowed by the gauge symmetries are all vanishing, which is natural in the 't Hooft sense (we will deal later with global symmetry protection against gravity).
Due to confinement, the axial symmetry, U (1)

(j)
A , associated with each ψ j is spontaneously broken by where we omit gauge indices and defineΛ j max{Λ j , Λ j+1 }, with Λ 0 = Λ N +1 = 0. 4 The quantum anomaly of each SU (N c ) j gauge symmetry also breaks the axial symme- 4 More precisely, when the condensate forms atΛ j , the naive-dimensional-analysis, large-N c estimate is , whereÑ c is the dimension of ψ j w.r.t. the confining group, while a, b are 'flavour' indices of the other SU (N c ) group, which confines at a smaller scale. The associated NGB decay constant is f j = Ñ cΛj /(4π). In the following we will adopt the schematic notation of Eq. (3), which is sufficient for our purposes.
A explicitly. Thus, the Goldstone bosons, π j , associated with the chiral symmetry breaking of U (1) j e iπ j , obtain non-vanishing mass terms due to the anomalous breaking. 5 There exists, however, a linear combination of the U (1) A axial symmetries which survives the anomalous breaking of SU (N c ) N . Accordingly, there is one remaining massless NGB, the axion a. Other pseudo-Goldstone modes, on the other hand, obtain masses at the dynamical scale Λ (we assume for simplicity Λ k ∼ Λ for all k), as in the case of the η meson in QCD. 6 Let us see more closely how the clockwork mechanism is realised. The anomaly-free U (1) A is given by a linear combination of the axial currents, where j µ j is defined by The q j coefficients satisfy where T (R) and d(R) are respectively the Dynkin index and the dimension of the representation R, and we used T (F) = 1/2 and d(F) = N c . The solution to these conditions is given by where up to an overall normalisation. In this way, the U (1) A charge assignment in geometric progression with a ratio q is obtained.
The fermion bilinear terms contain the axion component as where the axion field resides in the conserved current of Eq. (4), with f a being the axion decay constant. Given the current and charge normalisations in Eqs. (4) and (9), for q > 1 the axion decay constant f a is determined by the dynamical scale of the first site, therefore one expects f a f ∼ Λ/4π. 7 Thus, when the SM couples to the axion through the N -th site, the axion interactions are suppressed not by f a but by F a q N f a . Thus, our model provides a dynamical realisation of the clockwork mechanism, where the phase rotations of fermion bilinears are locked by strong dynamics.
We have not yet specified the representation R. Since we assume asymptotically free gauge theories, we require that the one-loop β functions are negative, that is, We also require that the clockwork factor is larger than 1, From these two conditions, we find that R should be the two-index anti-symmetric rep- and N c = 4 or N c = 5. In these cases, we find that the clockwork factor is which provide an exponential separation of scales Indeed, in this class of models it is difficult to achieve a large clockwork factor, but one should remark than even a mild separation of scales may have interesting phenomenological consequences for the axion. The smallness of q is simply due to the group theoretical reasons above, and replacing SU (N c ) with different gauge groups does not change the situation much. In section III, we will present a different realisation of the dynamical clockwork with a larger q, which is of order N c or even independent of the strong dynamics.
Before closing this section, let us rephrase the above arguments in terms of the effective Lagrangian of the NGB modes, π j , associated with spontaneous U (1)

(j)
A breaking. Strictly speaking, the π j 's are not well defined since the U (1)

(j)
A symmetries are broken by anomalies of the strong dynamics. Still, the effective Lagrangian provides us with a convenient description of the low energy theory below the dynamical scale, as in the case of chiral perturbation theory with the η . The effective theory of π j 's is given by Here, we normalise π j to be dimensionless variables whose shifts correspond to the axial rotations of ψ j . The decay constants are expected to be of order f j ∼Λ j /(4π). The scalar potential, F anom , represents the explicit breaking of U (1)

(j)
A symmetries by the quantum anomalies, which depends periodically on π j 's through the combinations corresponding to the anomalies with respect to SU (N c ) 1 , . . . , SU (N c ) N , respectively. As expected, the scalar potential has a flat direction, i.e. the axion direction, where q j is given in Eq. (9), is in agreement with Eq. (11). Finally, by substituting Eq. (22) into Eq. (18), we also find that the axion decay constant is for a canonically normalised axion. When the dynamical scale in each site is approximately the same, i.e. f j ∼ f , we find f a ∼ f .

B. Exact discrete symmetries
So far, we have assumed the global U (1) A symmetries. These are explicitly broken by non-zero mass terms for the fermions, which could be induced in general by quantum gravity corrections. By modifying the model slightly, we can replace the U (1)
For this purpose, we change the representation of ψ 0 from F to some other representation R 0 and allow ψ N to have N f flavours. 8 The anomaly-free conditions of U (1) A in Eqs. (6) and (8) are also changed accordingly. Now, we impose a Z N +1 m discrete symmetry, where the left-handed components of the ψ's transform as where R = A 2 . Thus, for N c = 4, the model admits an anomaly-free Z N +1

III. CLOCKWORK WITH CONNECTED STRONG-DYNAMICS MODULES
In the class of models in subsection II A, the dynamics of every gauge group was intrinsically linked by a chain of fermions charged under both SU (N c ) j and SU (N c ) j+1 .
This nice feature can perhaps be considered the gauge group equivalent to the nearest 8 For N f > 1, the chiral symmetry breaking by SU (N c ) N confinement leads to multiple NGB modes, in the adjoint of SU (N f ). We will discuss this in section IV. 9 The exact discrete symmetries forbid the fermion mass terms, but they may allow higher-dimensional operators, suppressed by powers of M P , that will break U (1)

(j)
A at some level. 10 Here ψ 0 has independent left-handed and right-handed components, both in the adjoint representation.
neighbour coupling of fields outlined in the original clockwork model. In this class of models, however, the clockwork factor is rather small due to the requirement of asymptotic freedom of the strong dynamics at each site. In this section, we discuss alternative links between the sites which also permit the clockwork mechanism. Each site is associated with a module of strong dynamics, and the connection is realised with the help of discrete 'gauge' symmetries, or U (1) gauge symmetries, such that the axion U (1) A arises as a residual, accidental symmetry.

A. Contact connection
As a module of strong dynamics, we consider a model with an SU (N c ) gauge symmetry containing two vector-like fermions, Q and ψ, which transform as the representations R Q and R ψ of SU (N c ), respectively. Several modules, numbered from 1 to N , are illustrated in Table II. As in the previous section, we assume that each SU (N c ) j gauge theory is asymptotically free and exhibits confinement at the dynamical scale, Λ j .
In each module, there is an anomaly-free axial current, where the charges q Q and q ψ satisfy the anomaly-free condition Here the axial charges are defined by q f ≡ q(f L ) = −q(f R ), for f = Q, ψ. Due to confinement, this anomaly-free axial U (1)

(j)
A symmetry is broken spontaneously, and we obtain a massless NGB in each module.
In order to connect the modules with each other, we introduce higher-dimensional contact-interaction terms, where M * denotes a dimensionful parameter which is larger than the dynamical scale.
In the presence of the contact interactions, the U (1) N A symmetries are broken down to a single U (1) A symmetry, whose charge assignment satisfies By solving Eqs. (29) and (31), we obtain the conserved U (1) A current, where the charges are in geometric progression, up to an overall normalisation, with the clockwork factor As in the models of the previous section, the fermion bilinear terms contain the axion component as where the axion field resides in the conserved current in Eq. (32), with j µ A ∼ f a ∂ µ a, f a being the axion decay constant. Thus, again, when the SM couples to the axion through the N -th site, the SM-axion interactions are suppressed by F a q N f a .
We may rephrase the above clockwork mechanism in terms of the effective theory of the Goldstone modes, Here, the π's and ξ's are the dimensionless NGBs associated with the axial rotations of Q's and ψ's, respectively. The decay constants in each module are estimated to be The κ-term represents the explicit breaking due to the SU (N c ) anomaly, while the κ -term describes the explicit breaking due to the contact interactions in Eq. (30). The coefficients κ and κ parametrise our inability to calculate the strong dynamics effects (one expects κ ∼ κ ∼ (4π) 2 , up to order one numbers). Those terms give masses to 2N − 1 Goldstone modes, while leaving the axion massless.
From the effective Lagrangian, we find that the massless axion is distributed in the 2N Goldstone modes according to where q's are given in Eq. (33). By substituting these solutions in the NGB kinetic terms, we obtain the axion decay constant, The respectively. The anomaly-free conditions for these transformations are which can be satisfied by choosing R Q , R ψ and m appropriately. Then the contact interactions are restricted to the ones in Eq. (30) by imposing invariance with respect to It should be noted that the axial U (1) A symmetry is now regarded as an accidental symmetry once the exact Z N −1 m symmetry is imposed.
As an example, let us consider the case with R Q = A 2 and R ψ = Adj: If, for instance, N c = 4, the Z N −1 2 symmetry is anomaly-free and hence it can be regarded as a gauged discrete symmetry. Incidentally, this choice leads to a rather large clockwork factor, As another example, we may consider If, for instance, N c = 11, the Z N −1 9 symmetry is anomaly-free, with again q = −4.
In both examples one can easily check that the SU (N c ) gauge theory is asymptotically free.
The contact interaction terms in Eq. (30) can be replaced straightforwardly with heavy mediator interactions. For that purpose, one can introduce N − 1 complex scalar fields, φ j , with Yukawa couplings to the fermions, Here, we assume that φ j 's are appropriately charged under the Z N −1 m symmetry in Eq. (44).
Then the contact interactions in Eq. (30) are obtained by integrating out the scalar fields with masses of O(M * ).

B. WiFi-connection
We may consider axial U (1) gauge interactions as another way to connect modules.
For this purpose, in each module there are two fermions, Q j and ψ j , which are in the fundamental representation of an SU (N c ) j , as illustrated in Table III. Thus, there are 2N axial phase rotations of the fermions, of which N are broken by SU (N c ) N anomalies. In each module, there is an anomaly-free axial current, The associated U (1)

(j)
A is also free from cubic and gravitational anomalies, as Q j and ψ j have opposite charges.
For reasons that will be clear in a moment, one needs an additional interaction to forbid bilinears Q R ψ L and ψ R Q L . A technical way to achieve this goal, without affecting the anomaly cancellation conditions, is to assign Q j and ψ j to the fundamental and antifundamental representation of an auxiliary symmetry SU (N c ) that also confines at some dynamical scale, Λ ∼ Λ. An alternative way is to gauge a vector symmetry, U (1) V , defined by the current j µ V = (Qγ µ Q − ψγ µ ψ)/2. Either way, one enforces that the vacuum direction satisfies Q R ψ L = ψ R Q L = 0. These auxiliary symmetries are also shown in Table III.
To connect the modules with each other, we introduce U (1) gauge interactions, where the charges are arranged so that the corresponding currents are Once the U (1) j+1 . Each module consists of two fermions, Q j and ψ j , transforming under a gauge interaction SU (N c ) j . We also show two possible auxiliary interactions, SU (N c ) j and U (1) are absorbed by the gauge bosons via the Higgs mechanism. At the same time, the global U (1) A symmetry is also broken spontaneously, which results in a massless axion.
To find out the axion direction, it is particularly transparent to use the effective field theory of the Goldstone modes. In the WiFi model, the effective Lagrangian is given by Here, A µ j=1···N −1 denote the gauge fields of the U (1) (j=1···N −1) g symmetries, and g j denotes the gauge coupling constant. 11 The NGBs, π j and ξ j , correspond to the axial components of Q j and ψ j , respectively. The last term shows the effect of the SU (N c ) j anomalies, where κ parametrises our inability to calculate the effects of strong dynamics. 11 We set A µ 0 = A µ N = 0 identically.
From the effective Lagrangian, we find the axion components in each Goldstone mode, which are not absorbed by the Higgs mechanism. The U (1) A charges, q Q,q , are given by We are guaranteed that U (1) A is free from all anomalies because each U (1)

(j)
A is anomalyfree by construction. For a canonically normalised axion, the decay constant is selected to be which is dominated by the highest charge contribution, assuming f j f for all sites.
As in the previous models, the axion interactions are suppressed not by f a but by F a q N f a , if the SM couples to the axion through the N -th site. It should be noted that the axion clockwork factor, q, is not given by a dynamical reason, rather by the choice of the charge ratio between the neighbouring sites.
We now confirm that the axion direction is gauge invariant. Recalling that the Goldstone modes, f j π j and f j ξ j , have canonically normalised kinetic terms, the relation in Eq. (52) can be reverted to Thus, the axion is invariant under the gauge transformations where α j denotes the gauge transformation parameter of U (1) Finally, let us discuss how the global U (1) A symmetry can be broken explicitly. Higher dimensional terms localised at each site necessarily preserve it, because U (1) A charges are proportional to the gauge U (1) (j) g charges in each module. In fact, the global U (1) A is broken only by "non-local" terms such as which carry a U (1) A charge of order q N −1 . Therefore the explicit breaking of the global U (1) A symmetry is highly suppressed, guaranteeing in particular a strong protection against quantum gravity corrections. 12

IV. CLOCKWORK EXTENSION OF INVISIBLE AXION MODELS
The dynamical clockwork models discussed above may have different applications.
Here we discuss the possibility of implementing the axion solution of the strong-CP problem [55][56][57][58]. For successful models, the axion coupling to QCD should be suppressed For our discussion, it is useful to recall the main features of the KSVZ invisible axion model [19,20]. In the simplest KSVZ model, the PQ symmetry is spontaneously broken by the VEV of a complex scalar, φ, whose phase plays the role of the QCD axion, with the PQ charge of φ being 1. The complex scalar couples to N f vector-like quarks in 12 Here, we assume q is an integer for simplicity.
The PQ symmetry is identified as the axial U (1) symmetry of the Q's. Below the scale of spontaneous breaking of the PQ symmetry, the axion couples to QCD as due to the U (1) P Q anomaly with respect to QCD. Here, g s denotes the QCD gauge coupling and G is the QCD field strength, whose Lorentz and colour indices are understood.
As long as the PQ-symmetry is not broken by other sources than the QCD anomaly, the strong CP problem is successfully solved. In particular, the vector-like quarks, Q, should have no bare mass.
The KSVZ axion model can also be realised as a composite Goldstone mode [21]. To this end, an SU (N c ) gauge theory is introduced with vector-like fermions charged under SU (N c ) × SU (3) QCD according to Table IV. This model possesses an axial U (1) symmetry with charges which is free from the SU (N c ) anomaly but broken by the QCD anomaly. 13 Therefore, this symmetry is a PQ symmetry, which is spontaneously broken due to the chiral condensation of SU (N c ) at scale F a , where the axion appears as a composite NGB. 14 13 The corresponding axial currents are normalised as Qγ 5 γ µ Q/2 and Sγ 5 γ µ S/2, respectively, so that Eq. (62) holds with N f = N c . 14 As there are 4 flavours charged under SU (N c ), S and the three colours of Q, the chiral symmetry breaking of the axial SU (4) flavour symmetry leads to 15 pseudo-Goldstone modes. The QCD-singlet one is the axion, while the 14 additional modes become massive due to the explicit breaking of the axial SU (4) by the QCD gauge interactions.
As we will see shortly, these KSVZ axion models are implemented in the dynamical clockwork models straightforwardly by identifying the unbroken U (1) A symmetry of the previous sections with the PQ symmetry. By appropriately introducing fields charged under QCD, U (1) A is broken only by the QCD anomaly and the associated pseudo-NGB plays the role of the QCD axion. The effective coupling of the axion to QCD is suppressed by F a ∼ q N × f if the QCD charged states couple to the axion through the N -th site.

A. Invisible QCD axion in the dynamical phase-locking model
In the clockwork model of section II linked by strong dynamics, the massless axion appears as the NGB associated to the U (1) A current in Eq. (4). To couple the axion to QCD, we slightly modify the N -th site, so that ψ N is charged under SU (3) QCD as the fundamental representation. This has been already indicated in Table I. Consequently, U (1) A is broken by the QCD anomaly, Here, q N denotes the U (1) A charge of the N -th site, which is given by which is slightly modified from the one in Eq. (9). As a result, the effective coupling of the axion to QCD is suppressed by an enhanced decay constant, for N large.
The model predicts a QCD octet NGB, associated with the chiral symmetry breaking of the axial SU (3) flavour symmetry of ψ N , that obtains a mass from QCD loops, where α s = g 2 s /4π and the quadratic Casimir for the adjoint representation is C 2 = 3.
The model also predicts baryonic states charged under QCD, such as whose mass is expected to be O(N c Λ N ). 15 Additional coloured hadrons including ψ 1,··· ,N −1 are also possible (note that ψ j -number is a conserved quantity, for each j).
Let us comment on the explicit breaking of the PQ symmetry. In general composite axion models, the U (1) A symmetry, and hence the PQ symmetry, can be explicitly broken by the vector-like fermion mass terms. This easily spoils the PQ mechanism. As we discussed in section II B, however, the model allows discrete gauge symmetries which forbid the mass terms, so that the PQ symmetry appears as an accidental symmetry, at least at the renormalisable level.
In the N c = 4 model of section II B, the lowest dimensional operators which are allowed by the Z N +1 2 symmetry are with M P 2.4 × 10 18 GeV. Such higher dimensional terms lead to additional terms in the axion potential, where δP Qi denotes the phase of the coefficient of the term in Eq. (69). According to section II, π i = q i a/f + . . . with q i ∼ q −i , therefore the most relevant correction to the axion potential comes from the 0-th site, and we will implicitly take i = 0 below. This correction should be added to the QCD-anomaly contribution, V P Q ∼ m 2 a F 2 a cos(a/F a ). As a result, the effective QCD theta angle at the minimum of the axion potential is shifted, where we used In this case, the effective theta angle of QCD at the minimum of the axion potential is shifted by which is highly suppressed, allowing for a dynamical scale as large as Λ ∼ O(10 12 ) GeV.
A caveat is that the SU In this case, there are 24 Goldstone modes in addition to the axion. Among them, one colour octet and four colour triplets become massive due to QCD radiative corrections.
The remaining four colour-singlet Goldstone modes can be lifted by also gauging the SU (N f − 3) = SU (2) subgroup of the flavour symmetry, with a dynamical scale much larger than QCD. Then an SU (2) triplet NGB receives a mass from SU (2) radiative corrections and the remaining singlet NGB receives a mass from the SU (2) anomaly. 16 We note in passing that the dynamical clockwork model is different in many respects to the composite accidental axion model [48] based on the moose theory [60]. In the latter 16 It is tempting to embed the minimal gauge symmetry of the grand unified theory, SU (5) Table V. This last module interacts with ψ N via a contact interaction, as in Eq. (30), that implies q Q N +1 = q ψN = q −N .
Since Q N +1 does not participate in strong dynamics other than QCD, this model corresponds to the original KSVZ axion model, with the scalar φ replaced by ψ N ψ N .
The axion coupling to QCD is therefore the effective axion decay constant is enhanced by a factor q N with respect to does not participate to the strong dynamics, the model also predicts vector-like coloured There is another simple way to couple the contact-connection clockwork axion to the QCD anomaly that mimics the original composite axion model. It consists of modifying the N -th module by charging Q N under QCD, as shown in Table VI, which indeed has the same structure as in Table IV. In this case the anomalous coupling of the axion to the QCD is given by invariant term localised in one site can break the U (1) A symmetry. This implies that the unbroken U (1) A cannot be identified with U (1) P Q if QCD is coupled to just one site.
To explicitly illustrate the impossibility of breaking the U (1) A symmetry, we introduce a (N + 1)-th module as in Table VII, connected to the clockwork chain by a gauged symmetry. Once this module is introduced, the axial rotations are broken by the QCD anomaly, The effective Lagrangian of the NGB modes contains where the first term comes from the N -th site contributions after substituting the axion components of π N and ξ N in Eq. (52). Thus, U (1) (N ) g gauge invariance implies that the axion components in π N +1 and ξ N +1 are Since q Q N = −q ψ N , the axion does not couple to the QCD anomaly.
We conclude that in the WiFi-connection model it is not possible to couple the U (1) A axion to QCD through the last site of the clockwork. Nonetheless, independently from QCD, the WiFi models provides a clockwork axion mode that is exponentially localised with decay constants decreasing from f a in the first site to F a ∼ q N × f a in the last one. Due to the gauge protection, this axion potential is extremely flat, which may have different applications.

V. PHENOMENOLOGICAL AND COSMOLOGICAL IMPLICATIONS
As we have discussed above, the dynamical clockwork mechanism can be used to achieve invisible QCD axion models, for which the effective decay constant is F a > 4 × 10 8 GeV [17,18]. Due to the possibly large separation between the dynamical scale and F a , it is obviously interesting to ask whether the models are directly testable at collider experiments by taking the dynamical scale to be O(1) TeV. We first outline the relevant collider searches, before describing the cosmological features of the models.
Here, we have used naive dimensional counting [62,63]. From the di-jet searches at the LHC Run-I, the production cross section of an octet scalar with a mass around 1 TeV is constrained to be O(1) pb [64,65]. The Run-I upper limit is much larger than the actual pair production cross section [66] as well as than the single production rate via Eq. (80). 17 The di-jet searches in Run II have improved the constraint on the production cross section down to O(0.1) pb, with which an octet in the TeV range could be testable.
A more detailed, recent analysis can be found in Ref. [67].
Besides the octet scalars, the model also predicts the stable coloured baryons of 17 The limit on the octet mass in [64] is based on the higher dimensional interaction in Eq. (80) with the coefficient g 2 s /(4πΛ) replaced by 1/m 8 .
Eq. (68), 18 Once these stable particles are produced inside colliders, they leave visible tracks, whose production cross section is constrained to be O(10 −2 ) pb for masses in the TeV range [68]. However, the masses of the baryons are O(N c Λ), which goes beyond the reach of the LHC for f ∼ Λ/4π = O(1) TeV. Several other coloured hadrons are predicted, either stable or not, but in any case with masses of O(Λ) or higher.
In the contact-connection model of section IV B, with the last module charged only other QCD as in Table V, one predicts elementary, vector-like quarks Q N +1 , whose masses are of the order of O(Λ 3 /M 2 * ). The vector-like quarks can, for example, decay into the SM particles via the interactions where q L and H denote the doublet quark and the Higgs boson in the SM. 19 Here, we assume that the U (1) Y hypercharge of Q R,N +1 is −2/3, corresponding to up-type singlet vector-like quarks. We may also assume down-type singlet vector-like quarks by taking the hypercharge to be 1/3. The coupling constants, f , can be taken to be much smaller than the Yukawa coupling constants in the SM and still allow the extra quarks to decay quickly.
For an analysis of electroweak precision constraints on vector-like quarks with generic charges, see Ref. [69]. ATLAS and CMS have performed analyses on the production of vector-like quarks [70][71][72][73]. Assuming that there is no other physics that affects their branching ratios, ATLAS set a 95% the dynamical scale Λ or larger, with the exception of the pseudo-NGBs. These are a colour octet, whose phenomenology has been already sketched above, as well as colour singlets coupled to the QCD anomaly. Indeed, in the limit where the contact interaction is small, i.e. Λ M * , the 2N modes π j , ξ j , j = 1, . . . , N , defined in section III A, split into N modes η j = (π j − qξ j )/ 1 + q 2 , that receive a mass of order Λ from the SU (N c ) j anomalies, and N massless modes π j = (qπ j + ξ j )/ 1 + q 2 . Once the contact interaction is taken into account, the mass matrix for π j can be obtained from Eq. (37) by integrating out the heavy η j states. At leading order in f /M * , one recovers exactly the same mass matrix as the minimal clockwork realisation in Eq. (2) (without the site j = 0), with the . The matrix diagonalisation [4] issues a massless axion a ≡ a 1 , with couplings to gluons given in Eq. (76), and N −1 massive pseudo-NGBs, a k , k = 2, . . . , N , whose masses and couplings to gluons are defined by where we took the limit q 1 for simplicity. These states are close in mass (splitting ∼ 1/q, see [4]) and have all couplings to gluons of the same order. If light enough, they can be produced in gluon fusion and decay back into two gluons, thus the signature is a set of N − 1 close resonances in di-jet searches. A recent analysis for one singlet pseudoscalar coupled to gluons can be found in Ref. [67]. The present LHC bound on the production cross-section is O(1) pb for m a k = 1 TeV, which roughly corresponds to f d(R Q )/5 TeV, where we assumed the N − 1 states are not resolved.

B. Axion dark matter and cosmology
The invisible axion is of particular interest as it can be a component of cold dark matter. The coherent oscillation of the axion provides a relic axion density [75], where α i ∈ [−π, π] is the initial misalignment angle. We assume that PQ symmetry breaking occurs before the primordial inflation, so that the axion takes a unique field value in the whole observable universe. By assuming α i = O(1), the observed dark matter density, Ω DM h 2 = 0.1197 ± 0.0022 [76], can be achieved for F a ∼ 10 12 GeV.
In the clockwork axion, the effective decay constant is greatly enhanced, F a ∼ q N f a , and hence the required axion dark matter can be achieved by dynamics at the TeV scale.
For example, let us set f a = 1 TeV. In the dynamical phase-locking model, F a is given in Eq. (66) and therefore the dark matter density is obtained when N ≈ 75 (54) for N c = 4 (5). These large values are unsurprising, since the value of q is close to one in this model. Once can search for axion dark matter using a microwave cavity [77,78], in which the axion is converted to a radio wave with the frequency of the axion mass, via the axion-photon coupling, Here C aγγ is a model dependent coefficient which, for instance, takes the value when one assumes that the extra quarks in the KSVZ models do not carry U (1) Y charges [17, 18, for review]. Again, z is the ratio of the masses of the up and the down quarks. So far, the ADMX experiment has put a constraint on the axion-photon coupling of for an axion with a mass of a few µeV, assuming that the axion is the dominant component of dark matter [79,80]. The next generation of the ADMX experiment is predicted to search for axion dark matter in a mass range up to 40 µeV.
Remarkably, the clockwork axion can be far more visible than in the conventional axion models. It is possible that the vector-like fermions in the i-th clockwork site carry a U (1) Y charge of O(1). When i N , the axion-photon coupling is enhanced by because the contribution to the axion direction from the i-th Goldstone mode is q N −i times larger than the one from the N -th mode, which couples to QCD. Consequently, the clockwork axion can have a much larger coupling to photons than conventional, single-site models. This of course enhances the detectability of axion dark matter.
Now we turn to the early Universe cosmology of the models. As we have discussed in sections II B and III A, discrete symmetries play an important role in achieving U (1) A charges in geometric progression. Since the discrete symmetries are assumed to be exact, the axion potential generated by the QCD axion also respects the discrete symmetries.
The discrete symmetries are spontaneously broken as the temperature of the universe goes below the QCD scale. Thus, if the PQ symmetry breaking takes place after the end of the primordial inflation, the axion field takes different field values in each Hubble volume at the QCD temperature, which causes a domain wall problem. To avoid the domain wall problem, the PQ symmetry breaking is required to take place before the primordial inflation, and never to be restored after inflation. 20 We note that the domain wall number can be exponentially large in clockwork axion models when the clockwork factor, q, is fractional. Let us consider, for example, the contact-connection model of section III A, where the phases of the quark-bilinear terms rotate by when the axion field is shifted by Then the domain of the axion is defined by where δ is determined by the requirement that all the factors q Qj,ψj × δ become a multiple of 2π for the first time. If the clockwork factor is an integer, we find that the domain of the axion is given by On the other hand, the axion potential in Eq. (75) has a period Hence, the axion potential possesses degenerate vacua, which determines the domain wall number. Note that we took N f = m, in order for the model to possess an exact Z N m symmetry. The degeneracy of the vacua reflects an underlying discrete symmetry Z m , which is left after the spontaneous breaking of Z N m . On the other hand, when the clockwork factor is fractional, the domain wall number becomes exponentially large. Indeed, when q = r/s, with r, s coprime integers, the domain of the axion is given by Thus, we find that the degeneracy of the vacua is given by which is exponentially larger than Eq. (93). This larger degeneracy is due to an accidental enhancement of the discrete symmetry of the axion potential, even if the exact Z N m symmetry is spontaneously broken to Z m only. The exponentially large N DW is a generic feature of clockwork mechanism models with a fractional clockwork factor.

C. Baryonic dark matter
The contact-connection model of section III A also provides the possibility of SU (N c ) N baryons as dark matter candidates. Since coloured stable baryons are strongly constrained cosmologically, we focus on the case where strongly-coupled fermions have no QCD charge, as in Table V. Since there are two fermions, Q j and ψ j , charged under each SU (N c ) j gauge group for j = 1, . . . , N , there is an exact fermion number symmetry U (1) 2N V . Hence, the j-module bound states divide into classes with given charges under Since this symmetry is a generalisation of baryon number, we call baryons all bound states with non-vanishing U (1) V,Q j × U (1) V,ψ j charge. The lightest baryon in each class may be stable, if the decay into two or more baryons of other classes is kinematically forbidden.
In general, one is left with O(2N ) stable baryons. 21 Let us roughly sketch how such baryons can be dark matter candidates alternative to the axion. As discussed above, to avoid domain walls we require that the PQ-symmetry is never restored after inflation, so the cosmic temperature after inflation is below Λ.
The baryon masses are m B ∼ O(N c Λ). Thus, the thermal relic baryonic dark matter is possible only when the reheating temperature after inflation is in a small window, 22 where T F is the freeze-out temperature, Here, all the N sites are in thermal equilibrium via scattering with the lighter pseudo-NGBs π 's (see the discussion at the end of subsection V A). 23 Once the reheating temperature is in the window of Eq. (96), the thermal relic density 21 For definiteness, we assume in the following Q Nc j and ψ Nc j stable baryons. In general, stable states may include both species Q j and ψ j , and N b constituent fermions with N b = N c . Given the similar annihilation cross-sections, one expects that the contribution to the relic density is dominated by the heaviest stable baryons, with mass O(N b Λ). 22 We assume instantaneous reheating at the end of inflation. 23 The heavier η modes decay into three π 's or gluons immediately, while π 's decay into QCD jets via Eq. (82) in the model in Table VI or into Q N +1 in the model in Table V . Both can also decay into axions. Thus, each sector does not leave stable particle other than the baryons. of the baryonic dark matter can be roughly estimated as where we assumed that the 2N stable baryon states contribute equally and we approximated the annihilation cross section of B into π 's by the unitarity limit [84] (see also [85]), Thus, in a model with, for example, N c ∼ 10 and N ∼ 15, it is possible to explain the observed dark matter density by the thermal relic density of the stable baryons.

D. Further applications
The clockwork mechanism has also been used in models of natural inflation [86] and for the relaxion mechanism [11]. The application to the magnetogenesis mechanism in natural inflation models can be particularly interesting. Recently, the existence of cosmological magnetic fields even in void regions has been suggested by the gamma-ray observations from blazers, which puts a lower limit on the present magnetic field, B eff 10 −15 G [87][88][89][90][91][92][93][94]. Although it is generically difficult to generate such magnetic fields with a long correlation length, there are arguments that it is possible to achieve it in natural inflation models where the inflaton is an axion-like particle with an anomalous coupling to the U (1) Y gauge field [95,96]. One needs a hierarchy between the decay constant appearing in the inflaton potential, and the one in the anomalous coupling to the U (1) Y gauge field.
Such a hierarchy can be easily achieved in the clockwork mechanism, as we have shown in section V B.

VI. CONCLUSIONS
We have remarked that a sequence of N strongly-coupled sectors may respect some accidental global symmetries that are collectively carried by the N sites. We focused on the case of an anomaly-free, axial U (1) A symmetry. When strong dynamics confines at the dynamical scale Λ, the U (1) A symmetry is spontaneously broken and the decay constant of the associated Goldstone mode, the axion, a, is exponentially enhanced at the end of the sequence, F a ∼ q N f a , where f a ∼ Λ/(4π) and the factor q > 1 is the ratio between the U (1) A charges of neighbouring sectors. Clearly, this is an implementation of the clockwork mechanism, which induces a separation between the dynamical scale and the effective interaction scale of the axion due to a geometric progression of the U (1) A charges.
In such realisations of the clockwork mechanism, the quantum stability of the dynamical scale is guaranteed by dimensional transmutation, as in QCD. As the N gauge groups SU (N c ) have the same matter content, similar couplings in the ultraviolet nicely imply similar confinement scales. As an additional bonus, the occurrence of the global symmetry U (1) A is enforced by the (discrete) gauge symmetries of the theory, which guarantee that any explicit U (1) A breaking effect must be suppressed by a large power of the cutoff.
Indeed, such protection of the axion potential is essential for the most relevant applications, such as the solution of the strong CP problem by a QCD axion, or inflation via an axion-like scalar field. We presented three different realisations of the dynamical clockwork, which correspond to specific predictions for q, to distinct sets of new physics states close to the scale Λ, as well as to different thermal histories. Let us summarise their main features in turn.
In the phase-locking model (Table I), the link between neighbouring sites is provided by the strong dynamics via fermions charged under different representations of two adjacent gauge groups. The clockwork factor q 2 is completely fixed by group theoretical coefficients, with the upper bound coming from the requirement of asymptotic freedom.
The sequence configuration is secured by an anomaly-free discrete symmetry Z N m , with m also determined by group theory. When the last site of the clockwork is charged under QCD (Table I), the U (1) A Goldstone mode immediately becomes a composite QCD axion with an exponentially large decay constant. The smallness of the θ-parameter is effectively protected from gravity corrections by the combined effects of the discrete symmetry and of the hierarchy of scales, Λ M P . When Λ is close to LHC energies, the model predicts a colour octet of NGBs as well as (heavier) stable coloured hadrons, and the observed dark matter relic density is obtained from the axion misalignment mechanism for N ∼ 50. On the other hand, since the discrete symmetry implies a large domain wall number, one should not restore U (1) A after the end of inflation, and therefore the reheating temperature should lie below Λ.
In the contact-connection model (Table II), there are two different representations of fermions for each strongly-coupled gauge group, and q is the ratio of their respective Dynkin indices, which can be much larger than one, for example O(N c ). The link between adjacent modules is provided by four-fermion interactions, suppressed by a scale M * > Λ. The configuration of these interactions is secured by charging each pair of adjacent modules under a Z m symmetry, free from gauge anomalies. By adding a last module charged only under QCD (Table V), one easily realises a composite QCD axion with clockwork-suppressed couplings. In this case there are no coloured bound states, rather one predicts elementary vector-like quarks with mass suppressed by a factor Λ 2 /M 2 * with respect to Λ. The axion relic density may match the dark matter density, for e.g. Λ ∼ 10 TeV and N ∼ 10. In addition, there are O(N ) stable neutral hadrons that also acquire a relic density in the desired range for dark matter, provided that the reheating temperature is just below Λ (not larger, in order to avoid domain walls). Alternatively, the strong-CP problem can be solved when the last module involves both QCD and the new strong dynamics (Table VI). In this case the lightest states are N − 1 pion-like mesons with masses suppressed by a factor Λ/M * with respect to f , which couple to gluons through the QCD anomaly.
In the WiFi connection model (Table III), the two fermions of each module transform in the same way under the strong dynamics, but they have opposite charge under an anomaly-free axial U (1) symmetry. The link between two adjacent modules is provided by gauging a linear combination of the two U (1) symmetries, with q being the arbitrary ratio between the U (1) charges of the two modules. In this case, each module contains an exact Goldstone mode, but N − 1 of those are eaten by the U (1) gauge bosons and one is left with a massless clockwork axion. Remarkably, the accidental axion U (1) A is extremely well protected by the sequence of gauged U (1)'s: it can be broken only by operators involving fermions from all the N sectors at the same time. This phenomenon implies that, even if QCD is coupled to one sector, U (1) A remains free from the QCD anomaly, and therefore this type of axion cannot be used to address the strong CPproblem. Still, this WiFi dynamical clockwork emerges as an outstanding mechanism to generate extremely flat axion potentials.
In summary, we proposed the dynamical clockwork as a flexible mechanism to supply very light scalars with couplings suppressed by a scale much larger than the strongcoupling scale. We explored several model-building possibilities and applied the mechanism to the QCD axion, outlining its main phenomenological features. Clearly, several interesting directions are left to investigate, from the underlying theoretical origin to the connection with models of strongly-coupled electroweak-symmetry breaking, from a more quantitative study of the QCD axion phenomenology to different cosmological applications of the dynamical, axion-like particles.