Minimal Clockwork Axion Model

• The minimal clockwork axion model is an economical framework where a chain of axions produces a single light mode with an effective field range f_eff = jF_a, enabling enhanced couplings.
• It leverages a product of small integers in the UV alignment, generating hierarchies that underpin observable signals in QCD axions, ultralight dark matter, and collider phenomenology.
• Different realizations—including two-site alignment, confinement towers, and KSVZ-like chains—demonstrate distinct phenomenologies while satisfying unitarity and perturbativity constraints.

Searching arXiv for recent and foundational work on minimal clockwork axion models and related realizations. A minimal clockwork axion model is an economical realization of the clockwork mechanism in which a chain of aligned axionic degrees of freedom produces a single light mode with a parametrically enlarged effective field range and localized couplings. In the low-energy description emphasized in the cosmology literature, the axion is a compact field with fundamental period $F_a$, $a \equiv a + 2\pi F_a$, and the target effective theory contains

$$V(a)=\mu^4 \cos\!\Big(\frac{a}{j F_a}\Big), \qquad \mathcal{L}\supset \frac{k\alpha}{8\pi}\frac{a}{F_a}F_{\mu\nu}\widetilde F^{\mu\nu},$$

so that the potential depends on an effective field range $f_{\rm eff}\equiv jF_a$ while the anomalous gauge coupling is still normalized by the fundamental period $F_a$ (Agrawal et al., 2018). In this sense, “minimal” does not denote a unique model; it denotes the smallest field content, site number, or UV structure sufficient to generate the mismatch $f_{\rm eff}\gg F_a$, often for the QCD axion, ultralight dark matter, collider-visible axion-like particles, or cosmological defect phenomenology (Agrawal et al., 2017).

1. Core clockwork structure

The defining clockwork feature is the generation of a large integer $j$ from a product of small integers. In a UV clockwork or multi-axion alignment construction with $n$ axions $a_i$ and confining sectors, the scalar potential is

$$V(a_1,\dots,a_n)=\sum_{i=1}^{n-1}\mu_{i+1}^4 \cos\!\Big(\frac{N_i a_i}{F_i}+\frac{a_{i+1}}{F_{i+1}}\Big)+\mu_1^4 \cos\!\Big(\frac{a_1}{F_1}\Big).$$

Integrating out the heavy combinations enforces $a \equiv a + 2\pi F_a$0, and the light mode inherits

$a \equiv a + 2\pi F_a$1

with the gauge coupling still of the form $a \equiv a + 2\pi F_a$2 (Agrawal et al., 2018). The effective periodicity is therefore $a \equiv a + 2\pi F_a$3.

This same mechanism appears in confinement-tower realizations of the QCD axion. For $a \equiv a + 2\pi F_a$4 axions and $a \equiv a + 2\pi F_a$5 hidden confining groups, the heavy constraints $a \equiv a + 2\pi F_a$6 imply $a \equiv a + 2\pi F_a$7, so the light axion has

$a \equiv a + 2\pi F_a$8

while the photon anomaly can be enhanced by an integer factor $a \equiv a + 2\pi F_a$9, with $$V(a)=\mu^4 \cos\!\Big(\frac{a}{j F_a}\Big), \qquad \mathcal{L}\supset \frac{k\alpha}{8\pi}\frac{a}{F_a}F_{\mu\nu}\widetilde F^{\mu\nu},$$0 in the schematic tower and $$V(a)=\mu^4 \cos\!\Big(\frac{a}{j F_a}\Big), \qquad \mathcal{L}\supset \frac{k\alpha}{8\pi}\frac{a}{F_a}F_{\mu\nu}\widetilde F^{\mu\nu},$$1 in the explicit KSVZ assignment (Agrawal et al., 2017).

A common shorthand writes the anomalous interaction as $$V(a)=\mu^4 \cos\!\Big(\frac{a}{j F_a}\Big), \qquad \mathcal{L}\supset \frac{k\alpha}{8\pi}\frac{a}{F_a}F_{\mu\nu}\widetilde F^{\mu\nu},$$2. In the canonically normalized basis, however, the coupling strength always carries the gauge coupling, and one has $$V(a)=\mu^4 \cos\!\Big(\frac{a}{j F_a}\Big), \qquad \mathcal{L}\supset \frac{k\alpha}{8\pi}\frac{a}{F_a}F_{\mu\nu}\widetilde F^{\mu\nu},$$3. Large axion–gauge coupling therefore requires

$$V(a)=\mu^4 \cos\!\Big(\frac{a}{j F_a}\Big), \qquad \mathcal{L}\supset \frac{k\alpha}{8\pi}\frac{a}{F_a}F_{\mu\nu}\widetilde F^{\mu\nu},$$4

not merely $$V(a)=\mu^4 \cos\!\Big(\frac{a}{j F_a}\Big), \qquad \mathcal{L}\supset \frac{k\alpha}{8\pi}\frac{a}{F_a}F_{\mu\nu}\widetilde F^{\mu\nu},$$5 (Agrawal et al., 2018).

2. Minimal realizations in the literature

The phrase “minimal clockwork axion model” is used for several distinct economical constructions. They differ in whether minimality refers to the number of axions, the number of scalar sites, the number of confining sectors, or the amount of extra matter.

Realization	Minimal content	Characteristic outcome
Two-site alignment	Two axions, two confining groups	$$V(a)=\mu^4 \cos\!\Big(\frac{a}{j F_a}\Big), \qquad \mathcal{L}\supset \frac{k\alpha}{8\pi}\frac{a}{F_a}F_{\mu\nu}\widetilde F^{\mu\nu},$$6, enhanced periodicity $$V(a)=\mu^4 \cos\!\Big(\frac{a}{j F_a}\Big), \qquad \mathcal{L}\supset \frac{k\alpha}{8\pi}\frac{a}{F_a}F_{\mu\nu}\widetilde F^{\mu\nu},$$7
Confinement tower	$$V(a)=\mu^4 \cos\!\Big(\frac{a}{j F_a}\Big), \qquad \mathcal{L}\supset \frac{k\alpha}{8\pi}\frac{a}{F_a}F_{\mu\nu}\widetilde F^{\mu\nu},$$8 axions, $$V(a)=\mu^4 \cos\!\Big(\frac{a}{j F_a}\Big), \qquad \mathcal{L}\supset \frac{k\alpha}{8\pi}\frac{a}{F_a}F_{\mu\nu}\widetilde F^{\mu\nu},$$9 hidden groups	$f_{\rm eff}\equiv jF_a$0, integer photon-coupling enhancement
KSVZ-like scalar chain	$f_{\rm eff}\equiv jF_a$1 complex scalars, one vector-like quark at site $f_{\rm eff}\equiv jF_a$2	Invisible zero-mode plus $f_{\rm eff}\equiv jF_a$3 heavy ALPs
Heterotic M-theory $f_{\rm eff}\equiv jF_a$4	Two 4D zero modes $f_{\rm eff}\equiv jF_a$5	QCD axion plus ultra-light companion
Three-scalar GW model	Three complex scalars with $f_{\rm eff}\equiv jF_a$6	QCD axion plus two tensionful gear-wall sectors

In two-site KNP alignment, the potential

$f_{\rm eff}\equiv jF_a$7

leads, after integrating out the heavy combination, to

$f_{\rm eff}\equiv jF_a$8

with $f_{\rm eff}\equiv jF_a$9, $F_a$0, and $F_a$1 (Agrawal et al., 2018).

A different minimality criterion is adopted in collider-oriented QCD-axion models. There the clockwork sector consists of $F_a$2 complex scalars $F_a$3, a nearest-neighbour interaction with gear parameter $F_a$4, and a single KSVZ-like vector-like quark generation $F_a$5 localized at site $F_a$6. The exact Goldstone is the axion zero-mode, while the remaining $F_a$7 pseudo-Nambu–Goldstones are heavy ALPs (Bhattacharya et al., 2024).

In ultralight-dark-matter constructions, minimality means a short chain with one confining group per link and only an endpoint coupling to photons. The effective theory takes the form

$F_a$8

so that

$F_a$9

(Dror et al., 2020).

The heterotic M-theory realization is minimal in a different sense. For $f_{\rm eff}\gg F_a$0, the 4D theory contains two axions, $f_{\rm eff}\gg F_a$1 and $f_{\rm eff}\gg F_a$2, obtained from the continuous clockwork of the 11D three-form sector. This two-site structure is sufficient to realize a QCD axion with $f_{\rm eff}\gg F_a$3 together with an ultra-light hidden-sector axion (Im et al., 2019).

3. Spectra, localization, and anomalous couplings

In scalar-chain realizations, spontaneous symmetry breaking at scale $f_{\rm eff}\gg F_a$4 and nearest-neighbour interactions produce one localized zero-mode and a tower of heavy gears. With

$f_{\rm eff}\gg F_a$5

the pseudoscalar spectrum is

$f_{\rm eff}\gg F_a$6

The zero-mode profile is

$f_{\rm eff}\gg F_a$7

so it is exponentially localized at one end of the chain, while the heavy ALPs are delocalized (Bhattacharya et al., 2024).

The anomalous couplings are inherited from the site that talks to the heavy KSVZ fermion. Above QCD confinement, the site field $f_{\rm eff}\gg F_a$8 couples as

$f_{\rm eff}\gg F_a$9

After diagonalization, the physical modes satisfy

$j$0

Hence the zero-mode has $j$1 and is effectively invisible at colliders, whereas the heavy ALPs have $j$2 and comparatively unsuppressed couplings (Bhattacharya et al., 2024).

The same localization principle appears in continuum clockwork. In the generalized 5D linear-dilaton background, the zero-mode profile behaves as

$j$3

and the ratio of boundary couplings is exponentially hierarchical, $j$4 (Choi et al., 2017). This reproduces the discrete clockwork pattern of an exponentially localized light mode and a nearly degenerate KK gear spectrum when $j$5.

4. Consistency conditions and theoretical limitations

The central consistency condition for minimal clockwork axion models is the unitarity bound on the cosine potential. For a single axion with

$j$6

expanding around the minimum gives an $j$7 amplitude of order $j$8, implying $j$9 and $n$0. The conservative form adopted in the clockwork analysis is

$n$1

Applied link-by-link in alignment or clockwork chains, this yields the stronger statement

$n$2

for the surviving light potential (Agrawal et al., 2018).

This bound sharply constrains attempts to obtain very large axion–gauge couplings. Clockwork enlarges $n$3, but it does not change the fact that the anomalous coupling is normalized by $n$4. In weakly coupled regimes $n$5, so $n$6 typically requires very large $n$7 (Agrawal et al., 2018).

Enhancing $n$8 by large-charge matter is limited by perturbativity. If the anomaly arises from $n$9 fermions in a representation with Dynkin index $a_i$0, then

$a_i$1

Thus one cannot make $a_i$2 arbitrarily large in a weakly coupled theory (Agrawal et al., 2018).

Kinetic mixing offers a different route. For a heavy axion $a_i$3 kinetically mixed with a lighter axion $a_i$4,

$a_i$5

the heavy mode inherits an effective coupling with

$a_i$6

This can be parametrically large if $a_i$7, but it introduces a lighter axion and correspondingly light charged matter (Agrawal et al., 2018).

Continuum clockwork adds further limitations. In that framework, localized clockwork symmetries are accidental, are not respected by couplings to metric and dilaton fluctuations, and do not yield trans-Planckian 4D axion field ranges. The continuum construction also does not generate an exponential hierarchy among quantized 4D $a_i$8 charges in the way discrete clockwork does (Choi et al., 2017).

5. QCD axions, ultralight dark matter, and UV completions

For the QCD axion, confinement-tower clockwork extends KSVZ constructions by replacing a single PQ scalar with a chain of axions and hidden confining sectors. In the explicit KSVZ assignment,

$a_i$9

and the photon coupling becomes

$$V(a_1,\dots,a_n)=\sum_{i=1}^{n-1}\mu_{i+1}^4 \cos\!\Big(\frac{N_i a_i}{F_i}+\frac{a_{i+1}}{F_{i+1}}\Big)+\mu_1^4 \cos\!\Big(\frac{a_1}{F_1}\Big).$$0

This allows photophilic QCD axions with large quantized $$V(a_1,\dots,a_n)=\sum_{i=1}^{n-1}\mu_{i+1}^4 \cos\!\Big(\frac{N_i a_i}{F_i}+\frac{a_{i+1}}{F_{i+1}}\Big)+\mu_1^4 \cos\!\Big(\frac{a_1}{F_1}\Big).$$1 while keeping only one hypercharged pair in the tower (Agrawal et al., 2017).

A more UV-driven notion of minimality is realized by dynamical clockwork. In the contact-connection model, each site carries an $$V(a_1,\dots,a_n)=\sum_{i=1}^{n-1}\mu_{i+1}^4 \cos\!\Big(\frac{N_i a_i}{F_i}+\frac{a_{i+1}}{F_{i+1}}\Big)+\mu_1^4 \cos\!\Big(\frac{a_1}{F_1}\Big).$$2 gauge group with $$V(a_1,\dots,a_n)=\sum_{i=1}^{n-1}\mu_{i+1}^4 \cos\!\Big(\frac{N_i a_i}{F_i}+\frac{a_{i+1}}{F_{i+1}}\Big)+\mu_1^4 \cos\!\Big(\frac{a_1}{F_1}\Big).$$3 and $$V(a_1,\dots,a_n)=\sum_{i=1}^{n-1}\mu_{i+1}^4 \cos\!\Big(\frac{N_i a_i}{F_i}+\frac{a_{i+1}}{F_{i+1}}\Big)+\mu_1^4 \cos\!\Big(\frac{a_1}{F_1}\Big).$$4, giving

$$V(a_1,\dots,a_n)=\sum_{i=1}^{n-1}\mu_{i+1}^4 \cos\!\Big(\frac{N_i a_i}{F_i}+\frac{a_{i+1}}{F_{i+1}}\Big)+\mu_1^4 \cos\!\Big(\frac{a_1}{F_1}\Big).$$5

With $$V(a_1,\dots,a_n)=\sum_{i=1}^{n-1}\mu_{i+1}^4 \cos\!\Big(\frac{N_i a_i}{F_i}+\frac{a_{i+1}}{F_{i+1}}\Big)+\mu_1^4 \cos\!\Big(\frac{a_1}{F_1}\Big).$$6 TeV, this allows $$V(a_1,\dots,a_n)=\sum_{i=1}^{n-1}\mu_{i+1}^4 \cos\!\Big(\frac{N_i a_i}{F_i}+\frac{a_{i+1}}{F_{i+1}}\Big)+\mu_1^4 \cos\!\Big(\frac{a_1}{F_1}\Big).$$7 with modest $$V(a_1,\dots,a_n)=\sum_{i=1}^{n-1}\mu_{i+1}^4 \cos\!\Big(\frac{N_i a_i}{F_i}+\frac{a_{i+1}}{F_{i+1}}\Big)+\mu_1^4 \cos\!\Big(\frac{a_1}{F_1}\Big).$$8. The synthesis of the model emphasizes that $$V(a_1,\dots,a_n)=\sum_{i=1}^{n-1}\mu_{i+1}^4 \cos\!\Big(\frac{N_i a_i}{F_i}+\frac{a_{i+1}}{F_{i+1}}\Big)+\mu_1^4 \cos\!\Big(\frac{a_1}{F_1}\Big).$$9–17 is sufficient to reach $a \equiv a + 2\pi F_a$00, while the TeV spectrum contains either colored hadrons or vector-like quarks depending on how the clockwork chain is terminated (Coy et al., 2017).

For ultralight axion dark matter, clockwork is used to separate the matter-power-spectrum bound on the effective decay constant from the coupling scale relevant for detection. The robust late-universe requirement is

$a \equiv a + 2\pi F_a$01

independent of the production mechanism. Clockwork satisfies this by raising

$a \equiv a + 2\pi F_a$02

while keeping

$a \equiv a + 2\pi F_a$03

set by the smaller endpoint scale $a \equiv a + 2\pi F_a$04 (Dror et al., 2020). This makes detectable ultralight axion dark matter compatible with a matter-power spectrum close to $a \equiv a + 2\pi F_a$05CDM.

In heterotic M-theory, the continuous clockwork is UV-complete and geometrically determined. For $a \equiv a + 2\pi F_a$06, two 4D zero modes arise, and in the regime $a \equiv a + 2\pi F_a$07 the heavy state is the QCD axion with

$a \equiv a + 2\pi F_a$08

while the orthogonal state is an ultra-light axion essentially decoupled from QCD. The allowed scale

$a \equiv a + 2\pi F_a$09

naturally places the QCD axion in the conventional window and predicts an ultra-light hidden-sector companion (Im et al., 2019).

6. Phenomenology: colliders, inflation, and gravitational waves

Collider phenomenology is especially developed in the KSVZ-like scalar-chain realization. The dominant production channel for the heavy ALPs is gluon fusion, and the visible signature is

$a \equiv a + 2\pi F_a$10

For $a \equiv a + 2\pi F_a$11 GeV, $a \equiv a + 2\pi F_a$12 TeV, and $a \equiv a + 2\pi F_a$13, the full ALP spectrum is accessible and the diphoton invariant-mass distribution forms a wide band of resonances. When $a \equiv a + 2\pi F_a$14, the mass splittings $a \equiv a + 2\pi F_a$15 become comparable to detector resolution, and the spectrum mimics a single broad resonance, the “axion iceberg” (Bhattacharya et al., 2024). In the explicit light-ALP benchmark $a \equiv a + 2\pi F_a$16, $a \equiv a + 2\pi F_a$17, $a \equiv a + 2\pi F_a$18 GeV, $a \equiv a + 2\pi F_a$19 GeV, the cumulative significance is $a \equiv a + 2\pi F_a$20 at $a \equiv a + 2\pi F_a$21 and $a \equiv a + 2\pi F_a$22 at $a \equiv a + 2\pi F_a$23 (Bhattacharya et al., 2024).

In inflationary model building, minimal clockwork is not universally sufficient. For chromonatural inflation with sub-Planckian field range, the bound $a \equiv a + 2\pi F_a$24, together with $a \equiv a + 2\pi F_a$25, stability $a \equiv a + 2\pi F_a$26, and perturbativity $a \equiv a + 2\pi F_a$27, implies that obtaining $a \equiv a + 2\pi F_a$28 requires

$a \equiv a + 2\pi F_a$29

Clockwork enhances $a \equiv a + 2\pi F_a$30 but not $a \equiv a + 2\pi F_a$31, so a minimal clockwork axion with sub-Planckian field range cannot furnish a viable UV completion of chromonatural inflation (Agrawal et al., 2018).

A different phenomenological direction appears in the three-scalar minimal clockwork axion model for stochastic gravitational waves. With

$a \equiv a + 2\pi F_a$32

$a \equiv a + 2\pi F_a$33, and three scalar fields, the model yields one massless axion and two gear modes with

$a \equiv a + 2\pi F_a$34

while the effective QCD-axion decay constant is

$a \equiv a + 2\pi F_a$35

(Yin et al., 6 Jul 2025). The best-fit PTA point is

$a \equiv a + 2\pi F_a$36

which gives $a \equiv a + 2\pi F_a$37, a nano-hertz peak near $a \equiv a + 2\pi F_a$38, and $a \equiv a + 2\pi F_a$39. For the second wall, choosing $a \equiv a + 2\pi F_a$40 yields

$a \equiv a + 2\pi F_a$41

placing the signal in the LISA, Taiji, and TianQin band (Yin et al., 6 Jul 2025).

Across these realizations, a common lesson emerges. Minimal clockwork axion models efficiently generate a large effective field range or a large effective anomaly coefficient from order-one site data, but the resulting phenomenology depends crucially on how the clockwork is UV-completed. In some settings the minimal chain yields an invisible QCD axion plus heavy observable ALPs; in others it supports ultralight dark matter, a QCD axion with a hidden ultra-light companion, or dual stochastic-gravitational-wave signals from domain-wall annihilation. At the same time, unitarity, perturbativity, anomaly quantization, and continuum limitations prevent clockwork from being an unconstrained mechanism for arbitrarily large axion–gauge couplings (Agrawal et al., 2018).