General continuum clockwork

The continuum clockwork is an extra-dimensional set-up to realize certain features of the clockwork mechanism generating exponentially suppressed or hierarchical couplings of light particles. We study the continuum clockwork in a general scheme in which large volume, warped geometry, and localization of zero modes in extra dimension are described by independent parameters. For this, we propose a generalized 5-dimensional linear dilaton model which can realize such set-up as a solution of the model, and examine the KK spectrum and the couplings of zero modes and massive KK modes to boundary-localized operators for the bulk graviton, Abelian gauge bosons and periodic scalar fields. We discuss how those KK spectra and couplings vary as a function of the volume, warping and localization parameters, and highlight the behavior in the parameter region corresponding to the clockwork limit. We discuss also the field range of 4-dimensional axions originating from either 5-dimensional periodic scalar field or the 5-th component of an Abelian gauge field, and comment on the limitations of continuum clockwork compared to the discrete clockwork.


Introduction
The clockwork (CW) is a mechanism to generate exponentially suppressed or hierarchical couplings of light particles with N massive states having comparable masses near the threshold scale of the mechanism, where the suppression or hierarchy factor is given by O(q −N ) for a CW parameter q > 1 [1][2][3]. Among the many possible implementations of the mechanism [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], the CW axions [2,3] and U(1) gauge bosons [4,5] are particularly interesting as the key features of the mechanism can be understood in terms of a specific pattern of symmetry breaking of the underlying N + 1 (global or local) U(1) symmetries [U (1)] N +1 = N i=0 U(1) i , which is (explicitly or spontaneously) broken down to a U(1) CW subgroup. Furthermore, the key model parameters such as the CW parameter q and the involved axion-instanton couplings and U(1) gauge charges, are required to be integervalued (in appropriate units) by the compact [U (1)] N +1 , so the model has a built-in criterion for natural size of these model parameters. The existence of light axion or U(1) gauge boson having hierarchical couplings can be explained by the unbroken U(1) CW generated by the charge operator

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which has a localized distribution along the linear quiver of the U(1) charges Q i . As it is a subgroup of the compact [U (1)] N +1 , the unbroken U(1) CW is yet compact, but has an exponentially large range enhanced by q N relative to the range of U(1) i , which is achieved by a series of discrete monodromy between the nearest neighbor U(1) symmetries. This allows an exponentially enhanced axion field range or an exponential hierarchy among the quantized charges of the unbroken U(1) CW gauge symmetry, which might have an interesting implication for the fundamental issues such as the weak gravity conjecture [4,24,25].
Recently it has been pointed out that the CW mechanism can be implemented in an extra-dimensional setup by taking a limit N → ∞, while identifying the label i as the coordinate of an extra spacial dimension [5]. Then, depending upon which aspects of the discrete clockwork (DCW) one likes to reproduce, there can be two different approaches to the continuum clockwork (CCW) [5,26,27]: 1 one approach based on a specific form of 5D geometry involving an exponentially large volume together with an exponential warp factor [5], which can be obtained as a solution of the 5D linear dilaton model [28], and another approach based on exponentially localized zero mode profile over the 5th dimension in a field basis defined in terms of 5D U(1) or discrete gauge symmetry, which is generated by appropriately tuned bulk and boundary mass parameters [26]. As was briefly discussed in [5,26,29] and will be discussed in more detail in this paper, each approach has its own limitations, and as a result can reproduce only certain partial features of the original DCW axions and U(1) gauge bosons. For instance, assuming a 5D discrete shift symmetry which would assure that the involved axion is a periodic scalar, and also taking into account the U(1) gauge symmetry associated with the involved 5D vector field, in CCW either the localized CW symmetry to protect zero mode is not respected by gravitational interactions, or an exponential hierarchy between the zero mode couplings at different boundaries can not be generated. As other limitations relative to DCW, CCW could yield neither an exponentially enhanced trans-Planckian field range of the zero mode axion, nor an exponential hierarchy among the 4D gauge charges, while keeping the quantized nature of gauge charges of an unbroken 4D U(1) gauge symmetry.
Yet the CCW is interesting by itself as it offers a setup to realize some features of the DCW mechanism, while incorporating the well known extra-dimensional solution of the hierarchy problem [28,30]. Motivated by this, in this paper we study the CCW in a general setup involving large volume, warped geometry, and localized zero mode profile, which are described by independent parameters. Our setup makes it possible to examine how the patterns of low energy couplings (to boundary operators) and Kaluza-Klein (KK) mass spectra behave when the model parameters are varying from the CW limit to other limits such as the large extra dimension limit [31] and the Randall-Sundrum limit [30], from which we can identify the distinctive features of the CCW. We also clarify the limitations of each approach to the CCW to sharpen the differences from the DCW, particularly in connection with the possibility of trans-Planckian axion field range, and also the origin of unbroken U(1) CW which would protect the zero mode axion or U(1) gauge boson.

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The outline of this paper is as follows. In the next section, we first review the DCW with a specific model for clockwork axions and U(1) gauge bosons [2][3][4][5], and discuss the possible continuum limits following [5] and [26]. We then introduce a general continuum clockwork which may realize the CCW in more general ground involving large volume, warped geometry, and localized zero mode profile simultaneously. In section 3, we discuss a generalized linear dilaton model yielding the background metric and dilaton profile for general CCW as a solution of the model. We study also specific 5D models for CCW axions and U(1) gauge bosons in section 3, and finally conclude in section 4.

Discrete clockwork axions and Abelian gauge bosons
In this section, we review the key features of the CW mechanism using an example of discrete clockwork axions [2,3] and U(1) gauge bosons [4,5], and discuss the possible continuum limits following [5] and [26] Generic CW model involves two pieces: (i) a clockwork sector living on a quiver with N + 1 sites equipped with asymmetric nearest-neighbor interactions, and (ii) an external sector which couples to the clockwork sector through a specific site in the quiver, e.g. the first and/or the last site. For clockwork axions and U(1) gauge bosons, the CW sector lagrangian is given by where U i and Σ i are U(1)-valued fields: As usual, here we assume for simplicity that the model parameters, e.g. f, Λ, q, etc., have site-independent values. As for the external sector, we consider the Yang-Mills (YM) gauge fields G I µ and the U(1)-charged fermions ψ I (I = 0, N ), which couple to the CW sector through the 0-th and N -th sites as follows: 2 where G I µν andG I µν are the YM gauge field strength and its dual, respectively, and we ignore the U(1) gauge anomalies which are not relevant for our discussion. 3 2 One can consider more general external sector on arbitrary sites with 0 < I < N , whose feature of interactions with the CW sector can be readily inferred from the subsequent discussions for the 0-th and N -th sites. 3 One can easily avoid the U(1) gauge anomalies by introducing additional fermions with opposite gauge charge, without affecting any of our subsequent discussions.

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A key component of the model is the clockwork gear composed of asymmetric nearestneighbor interactions in the CW sector, e.g. the axion potentials and Stückelberg gauge boson mass terms in (2.1), which is responsible for the following (explicit or spontaneous) symmetry breaking: where [U (1)] N +1 denotes the N +1 compact (global or local) U(1) symmetries of the model: 5) and the unbroken U(1) CW symmetries are given by Note that α i and β i are periodic variables with the periodicity 2π, while the periodicity of the unbroken U(1) CW symmetries, i.e. the range of α and β, are enlarged to 2πq N and 2πq N , respectively. The axion potential responsible for the explicit symmetry breaking: might be generated by non-perturbative dynamics as discussed in [2], or introduced simply by hand [3], while the Stückelberg mass terms for the spontaneous symmetry breaking: [U(1)] N +1 local → [U(1) CW ] local can be achieved by introducing the complex scalar fields σ j (j = 1, . . . , N ) which carry the U(1) j−1 × U(1) j gauge charge Q σ j = (−1,q), while having the vacuum expectation values σ j = ve iω j . In order for both the effective lagrangian (2.1) and the associated symmetries (2.5) well defined over the full range of the field variables and the symmetry group, the CW parameters q andq, the U(1) gauge charges Q 0 and Q N , and the axion-instanton couplings κ 0 and κ N should be all integer-valued.
The symmetry breaking (2.4) due to the CW gear assures that there exist a massless axion and U(1) gauge boson associated with the unbroken U(1) CW , in addition to the N massive gear modes associated with the broken symmetries. For the particular case that all model parameters are site-independent, one can diagonalize the full (N + 1) × (N + 1) mass matrices to find the mass eigenstates. This results in 4

7)
4 Even when the model parameters are site-dependent, it is still straightforward to find the explicit form of the unbroken U(1)CW symmetries and the associated massless components, which are given by QCW = i Oi0Qi with Oi0 ∝ 1/ i k=1 q k for site-dependent CW parameters q k (k = 1, . . . , N ). In such generic situation, one can not get an explicit form of the mass eigenvalues and the mixing matrices for massive modes. However, as long as parameters at different sites are comparable to each other, the qualitative features of the spectrum and mixings are expected to be same as the site-independent case.

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whereφ 0 ,Â 0 µ denote the canonically normalized massless modes, whileφ n ,Â n µ (n = 1, . . . , N ) stand for the massive gear modes, and the corresponding mixing matrices are given by [3,5] One finds also the mass eigenvalues Notice that the mixing matrix element O k0 in (2.8), which is the component of the massless mode in the k-th site of the clockwork gear, is suppressed by the factor q −k (= e −k ln q ), while all the other components of the mixing matrices and λ n are essentially of order unity. This exponentially localized distribution of O i0 along the quiver of the original field variables φ i and A i µ is directly responsible for generating small or hierarchical couplings of light particles in the limit when N is moderately large, e.g N = O(10). More explicitly, one can write down the couplings to the external sector in (2.3) in terms of the mass eigenstates: where we choose the gauge Σ i = 1, and f ,ĝ 0 = gÑ 0 ,ĝ n = gÑ n sin nπ N + 1 . (2.13)

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Here one can see that the massless modesφ 0 andÂ 0 µ couple to the external sector in the N -th site with an exponentially small coupling suppressed by q −N , while all other couplings are typically of O(1/f ) for axions and O(g) for gauge bosons, although the couplings of massive modes can be suppressed by the factor 1/ √ N for large N . 5 The above lagrangian written in terms of the mass eigenstates shows that the couplings of massless modes to the operators at the N -th site are exponentially suppressed by 1/q N compared to the couplings to the similar operators at the 0-th site, if κ 0 and κ N (Q 0 and Q N ) are comparable to each other, which would be a natural choice for integer-valued κ 0,N and Q 0,N . There is another exponential hierarchy of O(1/q N ) between the massless mode couplings and the massive mode couplings to the operators at the N -th site. Notice that this second hierarchy is independent of the relative size of κ 0 (Q 0 ) compared to κ N (Q N ). We will see that this distinction is important when discussing continuum limit of the DCW.
Here the suppressed coupling of the zero mode axion to the YM gauge fields at the N -th site is a consequence of the enlarged periodicity of [U(1) CW ] global in (2.6), which results in the enlarged field range of the zero mode axion which is exponentially bigger than the original axion field range 2πf in (2.2). Another notable feature of the DCW is the spectrum of massive modes. The parameter λ n in (2.9) for the mass eigenvalues (2.10) satisfies while λ n is increasing in n. Therefore, in the limit N 1 the mass eigenvalues m n in (2.10) become approximately degenerate around the clockwork threshold scale M CW , and their mass gaps behave as δm ∼ M CW /N , which might have interesting phenomenological implications [27].
Before moving to the discussion of continuum clockwork, we summarize the key features of the discrete CW axions and U(1) gauge bosons.
1. The key model parameters such as the CW parameter q, the axion-instanton couplings κ I and the U(1) gauge charges Q I have integer values as required by the underlying compact [U(1)] N +1 symmetry. Therefore the model is equipped with an unambiguous criterion for natural size of the relevant UV parameters, providing a basis for the subsequent discussion of the hierarchical structure of the low energy effective couplings of zero mode axion and zero mode U(1) gauge boson.
2. Zero mode axion and U(1) gauge boson are protected by the unbroken compact U(1) CW generated by a charge operator Q CW ∝ i Q i /q i which has a localized distribution along the linear quiver of the charge operators Q i of [U(1)] N +1 = N i=0 U(1) i . The localized feature of the unbroken symmetry leads to a multiplicative monodromy JHEP07(2018)113 structure, enhancing the range of U(1) CW by N j=1 q = q N . This results in also an exponential hierarchy between the couplings of zero modes at the 0-th and the N -th sites, as well as an exponentially enhanced field range of the zero mode axion.
3. Exponential hierarchy (∝ q −N ) between the zero mode couplings and the massive mode couplings to the external sector at the N -th site.
4. Approximately degenerate N massive modes around the threshold scale, with m n ∼ M CW and the small mass splittings δm ∼ m n+1 − m n ∼ m n /N in the limit N 1.

Continuum limit
Starting from a discrete clockwork model with N + 1 sites, its continuum version can be obtained by taking the limit N → ∞, while identifying the site index i as the coordinate of the 5-th spacial dimension with a finite length πR = N ∆r, where ∆r is the lattice spacing. In order to keep the hierarchy factors q N ,q N finite, while having a nonzero mass gap between the zero modes and the massive gear states, one should take also q,q → 1 with the CW mass scales M CW = Λ 2 /f andM CW = gv approaching to the lattice cutoff scale 1/∆r. More explicitly, the continuum limit takes To get the continuum limit of the lagrangian densities (2.1) and (2.3), one can make the following substitutions: together with the field and parameter redefinitions:

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and finally take the limit ∆r → 0. One then finds where we changed the index for the last site from N to π, e.g. κ N → κ π .
Here we see an important difference between the original DCW and its continuum limit. In the DCW, one starts with compact [U(1)] N +1 symmetries which are broken down to U(1) CW , and both the full U(1) symmetries and the unbroken U(1) CW are perfectly compatible with the 4D diffeomorphism and Lorentz symmetries. On the other hand, in the continuum clockwork (CCW) limit, the correspondence (2.18) implies that the CW parameters q,q can not be integer-valued anymore, except for the trivial case of µ =μ = 0. This raises a question if the U(1) symmetries (2.22) and (2.23) in the CCW can be identified as a sensible compact symmetry compatible with the 5D spacetime symmetries. To answer this question, one should embed the continuum lagrangian (2.21) into a theory manifestly invariant under the 5D diffeomorphism and Lorentz symmetry. As we will see, there are two different embeddings of the continuum lagrangian (2.21) to a 5D diffeomorphism and Lorentz invariant theory, and for each embedding one can retain only different partial features of the DCW. Note that some issues in clockwork models, e.g. the periodicity (or field range) of axions and the natural size (or quantized nature) of the axion-instanton couplings κ 0 , κ π and the U(1) gauge charges Q 0 , Q π , crucially depend on how the compact nature of the associated U(1) symmetries are introduced in the underlying theory.
We remark that the above subtle feature of CCW partly originates from the fact that we replace the discrete field index i in the DCW model with the spatial coordinate y of extra spacial dimension in the continuum limit. While the index i is a frozen label in the DCW, the spatial coordinate y describes a dynamical extra dimension which imposes nontrivial restrictions on the clockwork lagrangian via the 5D spacetime symmetries. In the following, we will discuss two different prescriptions of CCW originally proposed in [5,26], and subsequently introduce general continuum clockwork which can incorporate these two prescriptions within a common framework.

CCW-I: CW from localized zero mode profile
To embed the continuum lagrangain (2.21) in a 5D diffeomorphism and Lorentz invariant theory, we first notice following [26] that the lagrangian can be rewritten as follows by where the continuum clockwork parameters µ,μ appear as boundary and bulk masses. The above lagrangian can be easily made to be invariant under the 5D spacetime symmetries by introducing appropriate metric dependence: where the continuum lagrangain (2.24) can be obtained when the spacetime metric G M N is replaced by the flat background: However, once one includes the metric dependence, certain feature of the model is lost. For the bulk and boundary masses which are tuned to be equal as in (2.25), the CW symmetry (2.23) is respected when the metric G M N is simply replaced by its background value η M N , however not by the interactions of the metric fluctuation h M N = G M N − η M N . For instance, upon ignoring the off-diagonal part of G M N , the 5D action of Φ can be rewritten as where the last term shows that the CW symmetry (2.23) is not respected by the ydependent fluctuation of G µν . This means that generically the continuum CW symmetry (2.23) cannot be identified as a good symmetry compatible with 5D spacetime symmetries. Rather, it should be regarded as an approximate accidental symmetry which holds JHEP07(2018)113 for a particular metric background, but is broken by higher dimensional interactions of the metric fluctuation. As a consequence, the particular relations among the bulk and boundary masses, which are crucial for the existence of massless modes protected by the CW symmetry (2.23), are potentially unstable against radiative corrections involving the couplings of the metric fluctuation. 6 Nevertheless, if we accept it, while ignoring the fine tuning issue on the bulk and boundary masses, the 5D model (2.25) can reproduce many, although not all, features of the DCW model as we will show in detail in section 3.
As noticed before, one needs an appropriate periodicity condition for Φ to address the field range of zero mode axion and the natural size (or quantized nature) of the boundary axion-instanton couplings κ 0 , κ π . For this, one can simply impose the condition while making the following replacement in the 5D action (2.25): Then the infinitesimal CW symmetry (2.23) is modified as follows 30) and the zero mode of the fluctuation δΦ around Φ = 0 is protected to be light by this symmetry as far as we are not concerned with the symmetry breaking by the gravitational couplings of metric fluctuation. The 5D diffeomorphism invariant theory (2.25) with the replacement (2.29) reproduces the continuum lagrangian (2.24) for a small field fluctuation δΦ around Φ = 0 when the 5D metric is fixed as the flat background (2.26). It is then expected that certain perturbative features of this 5D theory, e.g. the spectrum and couplings of small field fluctuations, are similar to those of the original DCW model. For instance, the bulk and boundary mass terms of δΦ(x, y) and δA µ (x, y) enforce their zero modes to have y-dependent profile exponentially localized at one boundary, e.g. at y = 0, which results in an exponential hierarchy between the zero mode couplings at y = 0 and their counterpart couplings at y = πR, as well as a similar hierarchy between the zero mode coupling and the massive mode coupling to the same operator at y = πR. Although the 5D theory (2.25) reproduces certain perturbative features of the original DCW model, large field behavior of the theory such as the field range of the zero mode axion can be quite different from the DCW model. We will discuss these issues in section 3 in terms of the general CCW to be defined later. For a moment, we note the key weak point of the 5D diffeomorphism invariant CCW-I theory (2.25) which realizes certain features of the DCW model through localized zero mode profile: the model has a problem with the CW symmetry (2.23) which is not respected by gravitational interactions.

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In fact, our notion of localized zero mode is not basis-independent, and the localized feature of the zero mode of δΦ disappears for instance if one makes the y-dependent field redefinition: Φ(x, y) → e −µy Φ(x, y). To avoid an ambiguity arising from such field redefinition, one needs to specify the field basis used to address the localization of zero mode in the 5th dimension. Throughout this paper, we will consider the localization in a specific field basis defined in terms of the discrete shift symmetry which would ensure the periodicity of Φ and the compact 5D U(1) gauge symmetry for A M , i.e. the localization of the zero-mode wavefunctions δΦ(x, y)/δφ 0 (x) and δA µ (x, y)/δA 0µ (x), where the 5D fields Φ(x, y) and A M (x, y) transform under those gauge symmetries as where f 5 is an y-independent constant and the U(1) gauge transformation function Λ(x, y) obeys the y-independent periodicity condition as Λ ≡ Λ + 2π. Note that in this standard field basis for 5D gauge field, the two boundary gauge charges Q 0 and Q π are quantized in a common unit as Λ(y = 0) and Λ(y = πR) have the common periodicity. The same is true for the axion-instanton couplings κ 0 and κ π as Φ(y = 0) and Φ(y = πR) has the common periodicity. One can then start with Q 0 ∼ Q π = O(1) and κ 0 ∼ κ π = O(1), which are perfectly natural in view of the underlying symmetries, and examine the exponential hierarchies between the effective couplings of zero modes at different boundaries.

CCW-II: CW from geometry
As we noticed in the previous subsection, in an approach to implement continuum CW via localized zero mode profile, the CW symmetry (2.23) is not respected by the couplings of metric fluctuation. This problem arises from non-vanishing bulk and boundary mass parameters which are identified as the origin of the CCW parameters µ,μ. In fact, one can easily avoid this problem by embedding (2.21) into a 5D theory without any bulk or boundary mass term [5]. The continuum lagrangian (2.21) can be written in the following form by redefining the fields and boundary couplings: where the redefined fields and couplings are related to the original fields and couplings as follows.Φ = e µy Φ,Â µ = eμ y A µ ,Â 5 = eμ y Ω (2.33) In this prescription, the CCW parameters µ,μ appear in the wave function coefficients, while the bulk and boundary mass parameters are all vanishing. For the case with µ =μ,

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the above form of lagrangian can be embedded into a 5D diffeomorphism and Lorentz invariant linear dilaton model as discussed in [5,29]: where the continuum lagrangian (2.32) is reproduced when the metric and dilaton field are replaced by the following solution of the linear dilaton model: 7 The CW symmetry (2.23) can be also expressed in terms of the redefined fieldsΦ and A M as can be identified as a global shift symmetry for the redefined scalar fieldΦ, [U(1) CW ] local as the 4D subgroup of a 5D U(1) gauge symmetry for the redefined gauge fieldÂ M , and both CW symmetries are obviously compatible with the 5D spacetime symmetries, e.g. respected by the couplings of the spacetime metric and dilaton field. Moreover, under the assumption that the redefined boundary couplingsκ 0,π andQ 0,π are integer-valued, one can impose the following periodicity condition without any difficulty: whereΛ is the transformation function of the underlying 5D U(1) gauge symmetry, under whichÂ Thus it seems that this construction apparently solves all problems of the CCW-I. However this construction also has its own limitation. Notice that the redefined couplingsκ 0 ,κ π andQ 0 ,Q π reveal an exponential hierarchy in (2.34) if the original couplings κ 0 , κ π and Q 0 , Q π were of order unity in order to reproduce the continuum version (2.21) of the original DCW model. On the other hand, in the 5D theory (2.35) ofΦ andÂ M obeying the periodicity condition (2.38), the redefined couplingsκ 0,π andQ 0,π , not the original couplings κ 0,π and Q 0,π , are required to be integer-valued, so have natural size of order unity. Furthermore, the zero modes ofΦ andÂ M in (2.35) have flat profile over the JHEP07(2018)113 5th dimension because of the absence of bulk and boundary masses. 8 As a consequence, the CCW-II cannot reproduce the exponential hierarchy among the zero mode couplings at different boundaries, which is one of the primary features of the original DCW model. One may wonder whether the situation can be changed by introducing dilaton-dependent couplings (∝ e −γS ) to the boundary operators in order to realize the desired exponential hierarchy among the zero mode couplings at different boundaries. However, again once one imposes the periodicity condition (2.38), such dilaton-dependent couplings are forbidden by the discrete shift symmetry ofΦ and the 5D gauge symmetry ofÂ M . Nevertheless, as was stressed in [29], many of the other features of the DCW can be successfully reproduced in the CCW-II realized through the background geometry of the linear dilaton model.
The main reason for the above limitation of the CCW-II is attributed to the absence of localization of zero mode of 5D bosonic fields without bulk and boundary masses. There is still an example that localized zero mode profile occurs by warped geometry alone: the 5th component of a 5D gauge field C M with odd orbifold-parity whose zero mode can be interpreted as a 4D axion field [33,34]. In this case, there is a corresponding localized CW symmetry protecting the zero mode axion as we will discuss in section 3. However, as we will see, the 5D gauge symmetry of C M forbids certain boundary couplings of C 5 , so still prohibits the full realization of the characteristic features of the DCW such as the exponentially enlarged axion field range and/or the exponential hierarchy among the zero mode axion couplings to instantons at different boundaries.
To summarize, both CCW-I and CCW-II have their own limitations and realize only certain partial features of the original DCW model. For appropriately chosen 5D metric backgrounds, i.e. the flat background (2.26) for CCW-I and the linear dilaton background (2.36) for CCW-II, two models are related to each other by the field and parameter redefinitions (2.33) and (2.34). This suggests that two schemes share certain features, for instance the KK spectrum which are independent of the boundary couplings in our approximation. Yet, CCW-I and CCW-II are totally different models. In CCW-I, the CW symmetry which is supposed to protect the zero mode is not respected by the gravitational interactions of the metric fluctuation, which would raise a question about the naturalness of the scheme. In CCW-II, zero modes have flat profile over the 5th dimension in the field basis for which the 5D U(1) and discrete gauge symmetries take the standard form as (2.31), so there is no hierarchy between the zero mode couplings at different boundaries. Related to this point, in CCW-I the y-independent periodicities of the 5D axion field and the 5D U(1) gauge symmetry transformation (2.31) are imposed on Φ and A M , and as a consequence the corresponding boundary couplings κ 0,π and Q 0,π are quantized in the common unit, suggesting that they all have integer values of order unity. On the other hand, in CCW-II, the same applies for the redefined fieldsΦ andÂ M and boundary couplingsκ 0,π andQ 0,π , not for the original fields and couplings. This is the reason why the two models predict different hierarchical patterns for the effective couplings of zero modes at different boundaries. In the following, we will introduce general CCW in which CCW-I

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and CCW-II are incorporated into common framework which can realize the continuum CW in a most general way.

General CCW
Inspired by the previous discussions, let us define the general continuum clockwork (general CCW) as an extra-dimensional setup yielding light mode by means of a localized symmetry. In this perspective, we can start with the following most general quadratic lagrangian which might be obtained after replacing the background geometry and dilaton with their vacuum expectation values which are supposed to have an exponential y-dependence: Requiring to be invariant under the following localized infinitesimal shift symmetry where c 0 is an infinitesimal constant, we find the model parameters should obey the following relations for which the lagrangian can be written in a form which is manifestly invariant under (2.41): Similarly one can consider a general CCW theory for U(1) gauge bosons invariant under a localized infinitesimal U(1) gauge symmetry: Yet we have a freedom to redefine the fields as while keeping the lagrangian and localized symmetry to take the same form, but with the redefined fields and parameters. This makes the physical interpretation (or the physical origin) of each of the CCW parameters, e.g. µ 1 , µ 2 , µ for the CCW axion, obscure as they are not invariant under the field redefinition (2.46). On the other hand, we are interested in the possibility that the CCW lagrangians (2.43) and (2.45) originate from a 5D theory in which Φ can be identified as a periodic 5D scalar field associated with a discrete gauge shift symmetry of the form: Once Φ and A M are identified as such field variables in the underlying UV theory, we do not have anymore a freedom to make the field redefinition (2.46), and can attempt to embed (2.43) and (2.45) into a 5D diffeomorphism and Lorentz invariant theory defined in the field basis fixed by the above form of 5D discrete shift and U(1) gauge symmetries. As we will see, in such prescription, one can make an unambiguous physical distinction between µ 1 , µ 2 and µ. Note that the 5D discrete shift and compact U(1) gauge symmetries, which appear to be inevitable in any sensible UV completion of CCW, ensure the quantization of the U(1) gauge charges and the axion-instanton couplings in the underlying UV theory, and therefore provide also a criterion for natural size of relevant UV parameters, which would be essential for a proper interpretation of the exponential hierarchies generated by the CW mechanism.
As will be explained in the next section in detail, the most natural way to implement the general CCW within the framework of 5D diffeomorphism and Lorentz invariant theory is the generalized linear dilaton model. For instance, the general CCW axion lagrangian (2.43) at quadratic order can be obtained from the following 5D action where V B (θ) and V b (θ) are even functions of the periodic angle θ ≡ θ + 2π, which are manifestly invariant under the discrete gauge symmetry (2.47), while the background geometry and dilaton turn out to have the following expectation values: Then the CCW parameters µ 1 , µ 2 are determined by the geometric parameters k 1 and k 2 describing the exponential warp factor and the exponentially large volume, respectively, while the CCW parameter µ for localized symmetry originates from the bulk and boundary masses for in the vicinity of Φ = 0, which are tuned to satisfy the following relations to have the localized symmetry (2.41): 9

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Therefore, the generalized linear dilaton model implements the general CCW with three independent parameters k 1 , k 2 and µ. We can also see that CCW-I corresponds to µ = 0 and k 1 = k 2 = 0, while CCW-II corresponds to µ = 0 and k 1 = k 2 = 0. Thus the general CCW incorporates the two schemes into a single framework and can exhibit continuous deformation among all possible implementations of the CCW. As we will show in the next section, the three independent parameters are responsible for the following three characteristic features of the CW mechanism: i) the bulk and boundary mass parameter µ determines the hierarchy between the CW zero mode couplings at different boundaries, ii) a certain combination of µ and the warp factor parameter k 1 (for a given k 2 ) determines the hierarchy between the zero mode coupling and the massive KK mode couplings at the same boundary, and iii) the difference between the warp factor and the volume factor, i.e. k 1 − k 2 , controls the mass gap between the massive gear modes (KK modes).
As noticed in the previous discussion, for non-zero µ,μ the localized symmetries (2.41) and (2.44) to protect zero modes are not respected by the couplings of metric and dilaton fluctuations, while an exponential hierarchy between the zero mode couplings at different boundaries is possible only with non-zero µ,μ. On the other hand, if µ,μ vanish, the zero modes are protected by an unbroken symmetry respected by the metric and dilaton fluctuations, while they lose the localized feature for generating an exponential hierarchy among the couplings at different boundaries.
Before closing this section, let us briefly summarize the main features of the continuum CW.
1. General CCW is defined as a generic 5D theory producing light 4D modes protected by a symmetry localized in extra dimension. It can be considered as a generalization of the naive continuum limit of the DCW models, which can accommodate both the CCW from geometry and the CCW associated with localized zero mode profile over the extra dimension, which is achieved by fine-tuned bulk and boundary masses. However, there are always certain limitations of CCW, so even general CCW can not produce all of the interesting features of DCW. For instance, general CCW could yield neither an exponentially enhanced trans-Planckian field range of the zero mode axion, nor an exponential hierarchy among the 4D gauge charges, while keeping the quantized nature of gauge charges of an unbroken 4D U(1) gauge symmetry.
2. Sensible UV theory for axions and Abelian gauge bosons should be based on a discrete gauge shift symmetry for the periodicity of axions and/or the compact U(1) gauge symmetries for gauge bosons, which are compatible with the 5D diffeomorphism and Lorentz symmetry. Specifically, one may start with a 5D theory invariant under (2.53) Then, in the above field basis defined in terms of T Φ and U(1) 5D , the physical origin and meaning of the general CCW model parameters, including the couplings to JHEP07(2018)113 instantons and charged particles at the boundaries, are unambiguously identified. In this prescription, there are two model parameters µ 1 , µ 2 describing the effects of background geometry and dilaton field profile, and one parameter µ (orμ) describing the localization of zero mode axion (or gauge boson) by fine-tuned bulk and boundary masses.
3. Only when µ =μ = 0, the infinitesimal localized CW symmetries to protect zero modes can be embedded into a sensible 5D symmetry compatible with the 5D spacetime symmetries.
4. For µ = 0 (orμ = 0), the localized infinitesimal CW symmetry for zero mode axion (or gauge boson) corresponds to an accidental approximate symmetry not respected by the metric and dilaton fluctuations over the background vacuum values.

5D models for continuum clockwork
In this section, we provide a concrete framework to realize general CCW discussed in the previous section. We first introduce the generalized linear dilaton model which is obtained by generalizing the dilaton couplings of the original linear dilaton model [28]. This model will provide the background geometry and dilaton profile necessary to implement general CCW. We then analyze the mass spectrum and couplings of the KK graviton modes in general CCW. It turns out that many of the results on KK gravitons apply also to the KK modes of CW axions and U(1) gauge bosons, reproducing many of the characteristic features of the DCW model. In the subsequent subsections, we examine more detailed features of the CCW axions originating from a 5D angular field and/or a 5D gauge field with odd orbifold-parity, and also the CCW photon originating from a 5D gauge field with even orbifold-parity.

Generalized linear dilaton model
We start from a five-dimensional (5D) model with the 5th dimension compactified on orbifold S 1 /Z 2 with the fixed points at y = {0, πR}. All 5D fields in the model obey the boundary conditions where ± denotes the orbifold parity. The most general form of 5D metric invariant under the 4D Poincare symmetry is given by 10 where Ω 1 corresponds to the warp factor of the 4D metric component, while Ω 2 = G 55 defines the physical length of the 5th dimension: L 5 = πR 0 dy √ Ω 2 . Throughout this 10 The corresponding solution of the inflationary 4D spacetime for the linear dilaton model was studied in [8,35]. This might be straightforwardly extended to our generalized model.

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paper, we consider the case that underlying 5D gravity dynamics generates an exponential y-dependence in Ω 1,2 : where k 1 and k 2 are generically independent mass parameters with the same sign. As we will see, the above form of 5D metric can be obtained as a solution of linear dilaton model which is appropriately generalized from the well known model of ref. [28]. Motivated by the UV origin of the linear dilaton model as a dual description of the little string theory [28] (see also [36,37]), we consider the following 5D model with a universal dilaton coupling, which might be interpreted as the 5D action in the string frame: where M 5 is the 5D cut-off scale, Λ 5,0,π are the bulk and boundary cosmological constants, and the normalization of the dimensionless dilaton S is fixed by the coefficient of the kinetic term. Notice that we introduce a general dilaton coupling ξ, which was taken to be 1 in the original model [28]. Here we assume that the underlying theory involves a small (string) coupling g 2 st ∝ e −ξS , which would assure that the above 5D action defined at the leading order in g 2 st provides a sufficiently good approximation. As we go to the Einstein frame with and subsequently rescale the dilaton field to arrive at the canonically normalized kinetic term, the action reads where c ≡ ξ 12 ξ 2 − 9 . (3.7) We will see that the value of c different from 1/ √ 3 (ξ = 1) accounts for the background metric and dilaton profile necessary to realize general CCW.

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one can take them to be soft symmetry breaking parameters whose scale is well below the 5D cut-off scale M 5 . Let us parameterize them as with k, k 0,π < M 5 . Then the leading order action of the dilaton-graviton system is written as (3.10) Now we can solve the equations of motion with the following ansatz 11 ds 2 = e 2k 1 |y| (η µν dx µ dx ν ) + e 2k 2 |y| dy 2 , S = k 3 |y|. (3.11) We then find a solution with k i (i = 1, 2, 3) given by while the two boundary parameters k 0,π are fixed by the boundary conditions as Note that k is real (AdS 5 bulk space) for c 2 < 4/3, while it is imaginary (dS 5 bulk space) for c 2 > 4/3. From the above solution, we can obtain the two well-known limits: c = 0 : RS geometry with k 2 = 0, c = 1 √ 3 : Linear Dilaton geometry with k 1 = k 2 . (3.14) The large extra dimension (LED) limit can be obtained also by taking c 1, which yields k 2 πR 1 k 1 πR. We are basically interested in the parameter space where the background geometry provides a solution to the weak scale hierarchy problem. This is addressed with an appropriate size of the extra dimension depending on the values of k 1 , k 2 . The 4D Planck mass is evaluated as while the physical size of the extra dimension is given by (3.16) 11 By rescaling the soft parameters, k → e cS| y=0 k, we can set S|y=0 = 0. It is assumed that the additional boundary dynamics determines the radius, R, of the 5th dimension.

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From (3.12), one can see that k 1 and k 2 must have the same sign for real c, and their sign is determined by the sign of k. If we take them to be positive, the boundary at y = 0 is identified as the IR boundary, while the boundary at y = πR becomes the UV boundary. For negative k 1,2 , the boundaries are flipped without changing physics. In the following, we will take the convention of k 1,2 ≥ 0. Then, combining (3.15) and (3.16), we can express the proper length L 5 of the 5-th dimension in terms of M Pl , M 5 , k 1 and k 2 : where 0 < k 2 /k 1 < 4 for real k. Imposing the experimental upper limit on L 5 100 µm [38], we find Therefore, if one wishes to address the hierarchy problem with the lowest possible M 5 ∼ TeV, k 2 cannot be larger than about 2k 1 .

KK spectrum and couplings
Let us now discuss the spectrum and couplings of the bulk graviton KK modes in the generalized linear dilaton background. It turns out that much of the results apply also to the CW axions and U(1) gauge bosons. 12 To quadratic order of the 4D metric fluctuation h µν (η µν → η µν + 2h µν ) in the traceless transverse gauge h µ µ = ∂ µ h µν = 0, the action (3.10) gives rise to Notice that the (bulk or boundary) graviton mass term is absent due to the 4D diffeomorphism invariance even in the presence of non-trivial quadratic order graviton couplings to the dilaton background in S gravity . As we will show later, a generic quadratic action for 5D CW bosonic field Φ (= graviton, axion or U(1) gauge boson) takes the form:

21)
12 The spectrum and couplings for KK modes of the dilaton and also radion field depend on the additional boundary dynamics that stabilize both fields. Such a model dependence is studied in [39] for the linear dilaton model.

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where k Φ is some combination of k 1 and k 2 describing the effects of background metric and dilaton profile, and µ Φ originates from the bulk and boundary masses. For graviton, k Φ = k 1 + k 2 /2 and µ Φ = 0 as can be seen from (3.20). Notice that this corresponds to the defining lagrangian (2.43) of general CCW whose parameters are identified as The field basis used in (3.21) corresponds to the one for which the 5D discrete shift or continuous U(1) gauge symmetries are defined as (2.53), and therefore both the natural size of the corresponding couplings and the origin of the mass parameters k Φ , µ Φ , k 1 − k 2 are unambiguously determined. To perform the KK analysis, however, it is convenient to use the other field basis with the field redefinition Φ c = e k Φ |y| Φ, (3.23) for which the action (3.21) reads where The equation of motion for Φ c is then given by where the equation for the mode functions f n are given by where M n denotes the 4D mass eigenvalues of the corresponding 4D field φ (n) :

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The solution is then found to be 13 (3.32) where N n is the normalization factor, J |α| (x) and Y |α| (x) are the first and second kind Bessel functions, respectively, and The zero mode φ 0 is massless as expected, and the mass eigenvalues M n of the massive modes φ n (n = 1, 2, 3, . . . ) can be determined by the following boundary condition: The above result shows that the spectrum is similar to the case of RS for κ |m Φ |, while the overall mass scale is suppressed for κ −|m Φ | as in the LED case. In figure 1, we depict how the KK masses behave as a function of (k 1 − k 2 )/m Φ for fixed values of m Φ and R.
In fact, the parameter point κ ≡ k 1 − k 2 = 0 is special in regard to the KK mass pattern. In figure 2, we plot the mass ratios M n /M 1 as a function of κ/|m Φ |, again for fixed values of m Φ and R. It shows that the mass gap among the KK states is minimal in the unit of the first KK mass M 1 for κ = 0. This feature is related to the fact that the CW gear masses in the DCW are nearly degenerate at the CW threshold scale. As we have shown in the previous section, the direct continuum limit of the DCW produces either CCW-I or CCW-II depending on which field basis is used to realize the 5D discrete shift or continuous U(1) gauge symmetry as in (2.53). Notice that both 5D models correspond to the special case with κ ≡ k 1 − k 2 = 0 in view of general CCW. Therefore, the relatively degenerate KK masses for a given KK threshold scale can be considered as a special feature of the CCW obtained as a direct continuum limit of the DCW, but not a generic feature of general CCW. 13 For κ = 0, taking the limit κ → 0 in (3.32) is rather tricky, and the corresponding explicit expression is given by  Figure 1. KK masses (n = 1, 2, · · · , 7) as a function of κ/|m Φ | ≡ (k 1 − k 2 )/|m Φ | for fixed values of m Φ and R (left: linear, right: log). For graviton (|m Φ | = k 1 + k 2 /2), the RS limit corresponds to (k 1 − k 2 )/|m Φ | = 1, while the LED limit is (k 1 − k 2 )/|m Φ | = −2. Let us now turn to the couplings of the zero mode and KK modes to the boundary operators. For this, we should go back to the original field basis Φ rather than using the canonical basis Φ c , since the natural size of the relevant boundary couplings are determined in the original field basis for which the 5D discrete shift and continuous U(1) gauge symmetries are defined as (2.53). Then the mode couplings to the boundary operators can be read off by the component of the corresponding mode in Φ. Explicitly, if we consider the couplings on the boundary at y = y b , the mode coupling C n (y b ) is extracted from where C n (y b ) is determined to be for f n (y) given by (3.32). Let us first discuss the case m Φ > 0, which applies to the graviton case with m Φ = k 1 + k 2 /2 > 0 as we take the convention k 1 , k 2 > 0. The resulting couplings at y = 0 are calculated to be In figure 3, we depict the couplings of KK modes to boundary localized fields at y = 0 as a function of κ/m Φ ≡ (k 1 − k 2 )/m Φ for fixed value of m Φ and R. 14 Notice that there is an exponential hierarchy ∼ e −m Φ πR between the zero mode couplings and the massive KK mode couplings at y = 0 for κ 0, which is one of the main features of the DCW. This hierarchy is controlled by m Φ ≡ µ Φ + k Φ , i.e. the sum of the bulk and boundary mass parameter µ Φ and the warp factor parameter k Φ in the lagrangian (3.21). This means that both CCW-I (µ Φ = 0, k Φ = 0, κ = 0) and CCW-II (µ Φ = 0, k Φ = 0, κ = 0) possess this property. On the other hand, this exponential hierarchy disappears for κ −|m Φ | which JHEP07(2018)113 approaches to the LED limit. This can be easily understood by noting that deconstructing the 5D LED model gives a 4D lagrangian with the DCW parameter q = 1 [40,41], i.e. there is no clockwork.
The zero mode and KK mode couplings at y = πR for m Φ > 0 are determined by At this boundary at y = πR, the exponential hierarchy between the zero mode couplings and the KK mode couplings vanishes for κ ≡ k 1 − k 2 0 as expected from the DCW, while the KK mode couplings for κ m Φ are exponentially smaller than the zero mode couplings as in the RS case.
For the case m Φ < 0, the couplings can be easily obtained using the relation for n ≥ 1. The resultant couplings of the KK modes (n ≥ 1) are qualitatively similar to (3.40) and (3.42), but now m Φ replaced by |m Φ |, while the expression for zero mode couplings are same as (3.39) and (3.41). In this case, the hierarchical structure of couplings turns out to be similar to flipping the boundaries to each other, while interchanging the RS limit and the LED limit in the case of m Φ > 0. Interestingly, this means that the KK couplings are similar to the LED limit while the KK spectrums are similar to the RS limit for κ |m Φ |, and vice versa for κ −|m Φ |, when m Φ is negative. Thus the sign of m Φ for a given κ determines the pattern of the KK couplings, whereas the sign of κ determines the pattern of the KK spectrum. Let us finally discuss the zero mode couplings at different boundaries. From (3.38), the zero mode coupling at y = y b is determined as showing that a non-zero µ Φ , which is obtained only by non-zero bulk and boundary masses, is necessary to obtain an exponential hierarchy between the zero mode couplings at different y b . On the other hand, a non-zero m Φ ≡ µ Φ + k Φ , which does not necessarily require nonzero µ Φ , yields an exponential suppression of the zero mode coupling at y b = 0 (πR) for a positive (negative) m Φ , which is responsible for the hierarchy between the zero mode coupling and the massive KK mode coupling at y b = 0 (πR). In the following three subsections, we will examine how the lagrangian (3.21) can be obtained from a 5D diffeomorphism and Lorentz invariant theory for CCW axions and U(1) gauge bosons, and discuss further issues such as the field range of CCW axion zero mode.

CCW axion from 5D angular field
A natural candidate of 5D field that would yield a periodic 4D axion is a 5D angular field θ(x, y), with the periodicity: Including a periodic potential of θ, as well as the couplings to gauge fields at the boundaries, the 5D action can be written as where G 0,π are the gauge field strengths at the boundaries y = {0, πR},G 0,π are their duals, κ 0,π are restricted to be integers, and the dimensionful parameter f 5 is assumed to be well below the 5D Planck scale M 5 . If V (θ) = 0, there is an obvious massless 4D mode φ 0 (x) protected by the shift symmetry θ → θ + constant, and the corresponding mode function has a flat profile, i.e. f 0 (y) = constant for θ(x, y) = n=0 f n (y)φ n (x). In the presence of nonzero bulk and/or boundary potentials, generically there is no massless 4D mode. However, as we will show in the following, if the bulk and boundary potentials are appropriately related to each other, there can be a massless mode with exponentially localized profile over the 5th dimension, which reveals many features of the CW mechanism.
To achieve a 5D axion potential yielding such a zero mode, let us assume that both the dilatonic shift symmetry S → S +constant and the axionic shift symmetry θ → θ+constant are softly broken by a dimensionful mass parameter m through the combination e −cS e iθ m. (3.48) As the exponential dilaton dependence originates from the dilaton-dependent string coupling g 2 st ∝ e −ξS , at leading order in g 2 st the bulk dilaton potential in the Einstein frame should scale as e −2cS , while the boundary dilaton potential scales as e −cS . (See eqs. (3.4) and (3.10).) In addition to the soft breaking by e −cS+iθ m of both the dilatonic and axionic shift symmetries, the generalized linear dilaton model involves additional soft breaking by ke −cS of the dilatonic shift symmetry only, where k parametrize the bulk cosmological constant as in (3.9). Then the leading order bulk and boundary potentials of the 5D angular field θ is generically given by where we assume the CP invariance under θ → −θ, and {c i } are dimensionless coefficients of order unity. In order to have a localized massless mode under the metric and dilaton JHEP07(2018)113 background given by (3.11) and (3.12), one needs the following specific values of {c i }: Taking these particular parameter values, while replacing the spacetime metric and dilaton by their vacuum values, the 5D action (3.46) now takes the following form of general CCW: which is manifestly invariant under the discrete shift θ → θ + 2π.
To proceed, we can make a field redefinition ϕ(x, y) ≡ tan(θ(x, y)/2), (3.52) and then the 5D action (3.51) is given by It is clear that the above action admits a localized massless 4D mode given by where a(x) is the canonically normalized massless 4D axion, and u(a) is a function of a introduced to make the 4D axion a have the standard canonical kinetic energy over its entire field range. For flat metric and trivial dilaton backgrounds, i.e. k 1 = k 2 = 0, which we will be focusing on, the kinetic term of a is given by We then find where f = f 3 5 (1 − e −2mπR )/2m, and sn(x|z) is the Jacobi elliptic function which has two periods as sn(x|z) = (−1) n 1 sn(x + 2n 1 K(z) + 2in 2 K(1 − z)|z) for integer-valued n 1 , n 2 and K(z) being the complete elliptic integral of the first kind.
With the above model for CCW axion, we can address the following three issues: • Wavefunction profile and the KK mass spectrum, {f n (y), M n } for a given background value of θ .
• Field range of the canonically normalized 4D axion a.
• Effective potentials of the 4D axion a generated by the model parameters {c i } slightly deviated from the values of (3.50).

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It is in principle straightforward to examine these issues for generic metric and dilaton background. However, for simplicity in the following we will focus on the simplest background with flat metric and constant dilaton, i.e. k 1 = k 2 = 0. For the KK analysis, let us divide θ(x, y) into the background vacuum value θ and a small fluctuation δθ: θ(x, y) = θ + δθ, (3.57) where the vacuum solution of θ is given by Here the parameter y 0 is introduced to describe the vacuum value of the massless 4D axion a. As we will see later, y 0 describes also the position where the zero mode wavefunction is localized. Then the action (3.51) can be written as In figure 4, we depict the wavefunction profiles of the zero and few KK modes for certain values of y 0 . Note that y 0 can have any value in the range −∞ < y 0 < ∞, while the orbifold coordinate y is defined only over the fundamental domain: 0 ≤ y ≤ πR. From the above wavefunction profile, we can obtain also the zero mode coupling to boundary operators. For this, let us introduce a boundary coupling of the form where O(x) is a local operator constructed by generic boundary fields at y = y b . The resulting coupling of the zero mode fluctuation δφ 0 ≡ δa, which corresponds to the small fluctuation of the canonically normalized 4D axion field a which was defined in eqs. (3.52), (3.54), and (3.56), is given by This then yields in the limit y 0 → −∞ where the zero mode axion is exponentially localized at y = πR.
With the above form of boundary couplings of the zero mode axion fluctuation δa, one can easily generate an exponential hierarchy between the zero mode couplings at different boundaries, i.e. y b = 0 and y b = πR. Although the analytic expression of the massive KK mode couplings to the boundary operator O(x) is rather complicated, one can generate also a similar hierarchy between the zero mode coupling and the KK mode coupling to the boundary fields at a given y b with an appropriately chosen value of y 0 . Let us now discuss the field range of the canonically normalized 4D axion a, which is embedded in the 5D angular field θ as From the periodicity of the Jacobi elliptic function: sn(x|z) = (−1) n 1 sn(x + 2n 1 K(z) + 2in 2 K(1 − z)|z) (n 1 , n 2 = integers), (3.69) where K(y) is the complete elliptic integral of the first kind, we find the field range of a, which is matched to the 2π range of θ, i.e. θ ≡ θ + 2π, is given by a ≡ a + 2πf eff for 2πf eff = 4f K(1 − e −2mπR ). Note that there is no exponential enhancement of the field range of a relative to the 5D axion scale f 5 . Yet, an exponentially localized profile of the zero mode fluctuation δa could be successfully realized, yielding an exponential hierarchy among the couplings of δa at different boundaries as dictated by (3.65) or (3.66), as well as a similar exponential hierarchy between the zero mode coupling and the KK mode couplings at the same boundary. Exponential hierarchy among the couplings of δa without an exponentially enhanced range of a yields a rather unusual form of the effective potential of a, which might be generated by a deviation of model parameters from (3.50) or by an axion coupling to the confining gauge sector at the boundaries. To be specific, here we consider the case that the model parameter c 0 and c π are slightly deviated from the values of (3.50), which would result in the following effective potential where y b = {0, πR} and we ignored irrelevant constant parts. We stress that the above axion potential is valid over the full range of a given by (3.70). In figure 5, we depict this effective potential originating from y b = 0 (red) and y b = πR (bluee). The unusual shape of this axion potential can be understood from that the zero mode fluctuation δa ≡ a − a is exponentially localized at different position for different value of a . For a = 0, δa is localized at y b = πR. However for a = πf eff , δa is localized at y = 0. For 0 < a < πf eff , δa is localized at a certain position between 0 and πR. Since δa is localized at different position in the 5th dimension for different a , its overlap with θ(x, y b ) also changes, which results in the change of the curvature of the scalar potential along different value of a .

CCW axions from 5D gauge and Stückelberg fields
Another candidate 5D field that could give a periodic 4D axion is a 5D gauge field C M with odd orbifold parity [10,33,34]. To realize continuum CW axions in more general context, in this section we consider the case that the 5D gauge field C M has a Stückelberg mass involving an angular field θ which might originate from the phase degree of freedom of U(1)-charged complex scalar field. Under the U(1) gauge symmetry, C M and θ transform as where θ and Λ are periodic as q is an integer-valued charge, and the fields and transformation function obey the following orbifold boundary conditions: Note that the gauge transformation of θ involves a parity odd step function (y) = (y + 2πR) = − (−y) with (y) = 1 for 0 < y < πR, which is necessary to be compatible with the above orbifold boundary conditions. As in section 3.1, one can start with a 5D action of C M and θ at leading order in g 2 st ∝ e −ξS in the context of generalized linear dilaton model in the string or Jordan frame, for which the 5D action including the boundary terms involves the overall dilaton factor e ξS . Moving to the Einstein frame, one finds the relevant part of the 5D action is given by whereJ µ 0,π , J µ 0,π are model-dependent boundary-localized current operators. As for the natural size of model parameters, the 5D gauge coupling g C might be estimated as 1/g 2 C ∼ M 5 for the 5D Planck mass M 5 , the bulk Stückelberg mass of C M , i.e. qg C f 3/2 5 , is supposed to be well below M 5 , and the dimensionless boundary couplings γ 0,π , κ 0,π and c 0,π can be chosen to be of order unity. We note that the U(1) C invariance of the bulk kinetic term of θ enforces that the gauge transformation function Λ should be continuous at the boundaries, so Λ(x, 0) = Λ(x, πR) = 0. (3.78)

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As a consequence, δ U(1) C θ = 0 at the boundaries, and therefore the U(1) C gauge symmetry admits an arbitrary boundary potential of θ at y = 0, πR. One can forbid such arbitrary boundary potential by assuming a global U(1) symmetry at each boundary, which is explicitly broken by the YM instantons at the boundaries. We already assumed such global U(1) symmetries in the 5D action (3.77). Note that the axion-instanton couplings κ 0,π should be integer-valued, while the derivative axion couplings γ 0,π and c 0,π can have any real value.
To proceed, we choose a specific gauge in the 5D action, introducing R ξ gauge fixing term to remove the mixing between C µ and {C 5 , θ}, and take the Feynman-'t Hooft gauge (ξ = 1). This gauge choice makes the 5D action to take CCW form, from which the KK mass spectrum and wavefunction profiles can be easily extracted. Specifically, we add to the action (3.77), where χ = e (2k 1 −k 2 )|y| , and replace the 5D metric G M N and the dilaton field S with their vacuum values (3.11). Although this removes the mixing between C µ and {C 5 , θ}, the mixing between C 5 and θ is not removed yet. We then make the field redefinition: C 5 (x, y) = e −(k 1 + k 2 2 )|y| g C (sin β ϕ + (x, y) + cos β ϕ − (x, y)), (3.81) which results in the following CCW form of 5D action: Obviously the above CCW lagrangian enjoys the localized shift symmetries

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and then the resulting mass eigenvalues and wavefunction profiles can be immediately read off from eq. (3.32) and (3.36). Here, the two zero modes δφ ±0 clearly correspond to massless 4D axions. The zero mode axion δφ +0 (x) has a wavefunction profile exponentially localized at y = πR, while δφ −0 (x) does at y = 0. With the localized shift symmetry (3.84) and the associated zero mode axions δφ ±0 , one can achieve hierarchical axion couplings at different boundaries. We stress that the localized symmetry (3.84) for CCW axions is obtained as a simple consequence of the underlying symmetries of the model, i.e. U(1) C × U(1) 0 × U(1) π , without any fine tuning of model parameters.
As for the massive modes, one combination of δφ ±n (n ≥ 1) corresponds to the longitudinal mode of massive 4D vector field, while the other is a massive 4D scalar. Since we are working at a particular gauge, at this level we can not identify which combination corresponds to the longitudinal mode of massive vector field. This can be done by solving the 5D equations of motion as in section 3.5, however we will not perform it here. Yet we can easily show that each massive vector and scalar modes have the same mass. For instance, for k 1 = k 2 , the massive vector and scalar modes have a common mass M n = m 2 C + n 2 /R 2 for n ≥ 1, Although convenient for the KK analysis, the above prescription is not ideal for a discussion of the field range of the canonically normalized zero mode axions. A straightforward way to address this issue, as well as the low energy effective couplings of zero mode axions, is to integrate out massive KK modes in a gauge invariant manner and construct the effective action of zero mode axions. The equations of motion of C µ and θ have a gauge-covariant solution obeying C µ (x, y) = ∂ µ Γ(x, y), ∂ y θ(x, y) = q (y)C 5 (x, y), (3.86) where Γ(x, y) is a scalar field with the boundary condition Γ(x, 0) = Γ(x, πR) = 0, which transforms as Γ → Γ+Λ under U(1) C . We then find that a gauge invariant solution given by θ(x, y) − q (y)Γ(x, y) = e −(k 1 + k 2 2 )y csch(m C πR) sinh[m C (πR − y)] θ 1 (x) + csch(m C πR) sinh(m C y) θ 2 (x) , (3.87) which is described by the two gauge invariant 4D scalar degrees of freedom θ i (x) (i = 1, 2). Note that the matching condition θ 1 (x) = θ(x, y = 0), θ 2 (x) = e (k 1 + k 2 2 )πR θ(x, y = πR) (3.88) determines the periodicity of 4D fields θ i (x) as The 4D effective Lagrangian of θ i can be obtained by inserting the solution (3.87) into the original 5D action (3.77). The more conventional form can be obtained in terms of the canonically normalized axion fields a 1,2 (up to kinetic mixing):

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4D axions a i . However, as long as the underlying 5D mass parameters are sub-Planckian, e.g. m C , k i < M 5 , the model can not generate a super-Planckian axion field range in the effective 4D theory, i.e. f i < M Pl . Note that a super-Planckian effective axion scale could be easily achieved in the original DCW axion. In regard to this point, the key difference between DCW and CCW is that the extra spacial dimension for CCW should be a part of gravitational dynamics, while the linear quiver for DCW is completely decoupled from gravity. It is thus likely that generically CCW can not provide a super-Planckian axion field range. Our specific 5D model (3.77) does not generate also an exponential hierarchy among the axion-instanton couplings. Finally, we provide the relation between the canonically normalized zero modes δφ ±0 (x) defined in (3.85) and the axion fluctuation δa 1 ≡ f 1 δθ 1 and δa 2 ≡ f 2 e −(k 1 + k 2 2 )πR δθ 2 for θ i defined as the massless fluctuation of the gauge-invariant combination (3.87): 3.5 CCW photon from 5D gauge field with even orbifold-parity In this section, we study a 5D model for CCW U(1) gauge boson. The model involves a U(1) gauge symmetry and the associated gauge field A M = {A µ , A 5 } which transforms as To have a light 4D vector boson with localized profile over the extra dimension, we need to introduce the bulk and boundary masses of A M , which may take the Stückelberg form for simplicity. Then, again at leading order in g 2 st ∝ e −ξS in the generalized linear dilaton model, the 5D action is given by where θ is a periodic Goldstone field for gauge invariant Stückelberg mass, which transforms under U(1) A as θ → θ + Λ, (3.102) and m B and m b are the Stückelberg bulk and boundary mass terms, respectively. The field variables and the gauge transformation function obey the following orbifold boundary

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condition and the periodicity condition: A µ (x, y) = A µ (x, y + 2πR) = A µ (x, −y), θ(x, y) = θ(x, y + 2πR) = θ(x, −y) = θ(x, y) + 2π, For a gauge covariant KK analysis, we decompose the relevant 5D fields as (h n (y) + g n (y))φ n (x), and solve the equations of motion. One can similarly decompose the gauge transformation function as and then the 4D fields A nµ , π n , φ n transform under U(1) A as A nµ → A nµ + ∂ µ α n , π n → π n + α n , φ n → φ n . This shows that π n can be identified as the longitudinal component of the massive 4D vector fields A nµ , while the gauge invariant φ n denote physical massive 4D scalar fields. After integrating over y, the lagrangian of all 4D fields take the form where the mass eigenvalues M n will be determined later. Here we assume that M n (including n = 0) is nonzero in order to take into account all physical degrees of freedom correctly. We then find from the 5D equations of motion that the mode functions f n , h n and g n satisfy 2 )y h n (y) = 0, 2 )y h n (y) − M 2 n e (k 1 + k 2 2 )y g n (y) = 0, (3.108)

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together with the boundary conditions ∂ y − m V − k 1 − k 2 2 e (k 1 + k 2 2 )y f n (y) | y=0,πR = 0, Although the boundary conditions for f n (y) and h n (y) are different, it could be verified that they share the same mass spectrum except the mass M 0 of A 0µ . For generic values of {k 1 , k 2 , m B , m b }, M 0 can be even larger than M n (n ≥ 1). However there exists a wide range of model parameters yielding M 0 M n (n ≥ 1). In particular, for m 2 B = m b (m b + 2k 1 + k 2 ), we have a massless 4D vector (M 0 = 0) with a localized wave function f 0 (y) ∝ e m b y .
For simplicity, let us consider the case k 1 = k 2 for which it is straightforward to obtain the mode functions and mass eigenvalues. We then find Focusing on the exponential y-dependence in the limit mπR 1, we have f 0 (y) ∼ e − 3 2 k 1 πR e m b (y−πR) , f n (y) ∼ h n (y) ∼ e − 3 2 k 1 y . (3.112) This behavior of mode functions shows clearly the role of each parameter. A 0µ has an exponentially localized profile only when the boundary Stückelberg mass m b is non-vanishing. On the other hand, the mode function ratio f 0 (y)/f n (y) ∼ e (m b + 3 2 k 1 )(y−πR) (n ≥ 1) indicates that a possible exponential hierarchy between the zero mode (n = 0) coupling and the non-zero mode (n ≥ 1) coupling at a boundary with y = 0 depends on both the boundary mass m b and the background geometry described by k 1 . Actually, all KK modes are exponentially localized at y = 0 in our field basis due to the background geometry. We also note that the hierarchical structure of the boundary couplings of mass eigenstate modes is independent of the existence of massless or particularly light mode, i.e. it is valid even when M 0 ∼ M 1 .

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Related to the CCW, the most interesting parameter region of the model is the one yielding 15 In such case, the 5D model (3.101) successfully provides a light A 0µ with localized profile whose boundary couplings at different boundaries reveal an exponential hierarchy 16 as g 0 (y = 0) g 0 (y = πR) ∝ e −m b πR , (3.114) together with approximately degenerate massive KK modes whose boundary couplings at y = 0 reveal another exponential hierarchy g 0 (y = 0) g n (y = 0) ∝ e −(m b + 3 2 k 1 )πR (n ≥ 1), (3.115) thereby realizing the key features of the CW U(1) gauge bosons. Note that the mass eigenvalues of non-zero modes, M n for n ≥ 1, are independent of the boundary Stückelberg mass m b , and the shape of mode functions is rather insensitive to the bulk Stückelberg mass m B .

Conclusions
In this paper, we have studied the continuum limit of the discrete clockwork (DCW) mechanism, dubbed the continuum clockwork (CCW). To be specific, we focused on the possible continuum realizations of CW axions and U(1) gauge bosons. There are two different prescriptions for the corresponding CCW lagrangian as previously discussed in [5,26]: CCW-I associated with localized profile of zero modes in a field basis for which the 5D U(1) and discrete gauge symmetries take the standard form as (2.31), which is induced by appropriately tuned bulk and boundary masses, and CCW-II associated with the background geometry and dilaton profile of the linear dilaton model [28]. We discussed in detail that each prescription has its own limitation. For instance, assuming a discrete shift symmetry which assures that the involved axion is a periodic scalar, and also taking into account the U(1) gauge symmetry associated with the involved 5D vector field, in CCW either the localized CW symmetries to protect zero modes are not respected by gravitational interactions (CCW-I), or an exponential hierarchy between the zero mode couplings at different boundaries can not be generated (CCW-II). As other limitations relative to the DCW, CCW could yield neither an exponentially enhanced trans-Planckian field range of the zero 15 One may even take the limit M0 → 0, while keeping Mn (n ≥ 1) well above the characteristic energy scale. It is in fact not clear if the Goldstone zero mode π0 has a sensible behavior when M0 → 0 as its kinetic term vanishes, while still there could be non-vanishing interactions. Such potentially singular behavior might be a consequence of another massless degree of freedom which is already integrated out at the starting point. In this paper, we do not pursue this issue further as our major concern is the CCW features of A0µ. 16 Such exponential hierarchy among the couplings of A0µ suggests that there is no globally well defined compact 4D U(1) gauge symmetry associated with A0µ.

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mode axion, nor an exponential hierarchy among the 4D gauge charges, while keeping the quantized nature of gauge charges of an unbroken 4D U(1) gauge symmetry.
Motivated by those two prescriptions, we examined a more general CCW which can incorporate both CCW-I and CCW-II within a common framework. For this, we proposed a generalized 5D linear dilaton model providing a concrete realisation of the setup, in which the three independent parameters describing the general CCW are identified as geometric warp factor, large volume, and bulk/boundary mass responsible for localized zero mode profile. We then analyzed the KK spectra and the couplings of zero and massive KK modes to boundary-localized operators. It is found that many of the characteristic features of the DCW can be reproduced in a certain parameter region which might be identified as the CCW limit, and our setup allows a continuous deformation from the CCW limit to other limits such as the well known Randall-Sundrum and large extra dimension limits.
We subsequently discussed concrete 5D models for CCW axions and U(1) gauge bosons in generalized linear dilaton background. The 4D zero mode axion originating from a 5D angular field has quite different properties from the DCW axion since the corresponding CW symmetry to protect zero mode has a different form due to the periodicity of the 5D angular field. As a result, the couplings of the zero mode axion fluctuation to different boundary operators can exhibit an exponential hierarchy, while there is no exponentially enlarged field range of the corresponding zero mode axion. We also examined the CCW axions originating from a 5D gauge field having nonzero Stückelberg mass and odd orbifold parity. It turns out that the resulting zero mode axions can have an exponential hierarchy among the derivative couplings at different boundaries, while there is no such hierarchy for the axion-instanton couplings due to the restriction imposed by the underlying 5D gauge symmetry. One can obtain also a light 4D vector field and its KK excitations, which originate from a 5D gauge field with even orbifold parity and nonzero bulk and boundary Stückelberg masses, and reproduce many of the characteristic features of the DCW U(1) gauge bosons.