Hierarchical Axion Masses & Decay Constants

• The paper synthesizes methods like gauged U(1) anomalies, nonperturbative mass generation, and random matrix ensembles to explain the broad span in axion masses and decay constants.
• The topic outlines how hierarchical axion properties emerge from mechanisms such as multiplet mixing, Froggatt-Nielsen schemes, and string theory compactifications with clear cosmological implications.
• Phenomenologically, these hierarchies affect dark matter composition, inflationary dynamics, and rare flavor decay signatures, thereby guiding experimental searches.

Hierarchical axion masses and decay constants refer to the emergence of large variations—sometimes spanning many orders of magnitude—in both the masses $m_a$ and the decay constants $f_a$ of axion and axion-like particles (ALPs). These hierarchies are ubiquitous in string-inspired extensions of the Standard Model, multi-axion dark matter frameworks, and models aiming to unify flavor physics with the axion sector. Phenomenologically, hierarchical structures inform both experimental feasibility and cosmological roles for axions, from dark matter to inflationary mechanisms. This article synthesizes the construction, mechanisms, phenomenological implications, and model dependencies defining hierarchical axion masses and decay constants.

1. Theoretical Construction of Hierarchical Axion Parameters

Fundamental models often realize axion fields as pseudo-Goldstone bosons associated with spontaneously broken $U(1)$ symmetries. In string theory and related high-scale frameworks, several mechanisms drive hierarchies:

• Gauged Anomalous $U(1)$ Symmetries: As in (Berenstein et al., 2010), string-theoretic D-brane setups introduce extra $U(1)$ symmetries, often anomalous. The Green-Schwarz mechanism cancels anomalies by introducing axion-like fields, while St\"uckelberg terms give heavy masses to the anomalous gauge bosons. After integrating out these fields, approximate global symmetries survive, spontaneously broken by the vev of a complex scalar (typically a Froggatt-Nielsen field $\phi$), whose phase becomes the physical axion.
• Multiplicities in String/Axiverse Models: Compactifications generate $\mathcal{O}(10-100)$ axion fields, each associated with geometric moduli and nonperturbative potentials, yielding a spectrum of $m_a$ and $f_a$ controlled by the compactification data, moduli stabilization, and instanton actions (Stott et al., 2017, Li, 26 Oct 2025).
• Flavor Unification and Axiflavon Mechanisms: Flavor symmetries (e.g., $U(1)_{\rm FN}$ or $U(1)_H$) used to generate Standard Model fermion mass hierarchies also furnish a QCD axion candidate—the axiflavon—whose decay constant is tightly linked to the underlying flavor-breaking scale, establishing a direct relation between observed SM mass hierarchies and axion parameters (Calibbi et al., 2016, Alanne et al., 2019).

2. Mechanisms Generating Hierarchies in $m_a$ and $f_a$

Hierarchies emerge from multiple structural sources:

• Nonperturbative Mass Generation: Axion masses scale as $m_a \sim \Lambda_{\text{conf}}^2 / f_a$, with $\Lambda_{\text{conf}}$ determined by the instanton action of the coupled gauge sector, potentially spanning many orders of magnitude even if $f_a$ is held fixed (e.g., GUT scale) (Foster et al., 2022).
• Multi-Modal Mixing: In multi-axion models, mixing among axion fields with non-diagonal mass matrices redistributes the DM abundance and produces an emergent hierarchy in the physical eigenstates (Li, 26 Oct 2025). Mixing can invalidate canonical single-component axion DM scenarios, shifting dominance to ALPs with suitable $f_i$ and $m_i$.
• Random Matrix Ensembles: The spectra of mass and kinetic matrices for many axions follow statistical distributions (e.g., Marčenko-Pastur/Wishart, log-flat), yielding broad and potentially spiked hierarchies in $f_a$, $m_a$ (Stott et al., 2017). The hyperparameters of these distributions control the favored scale and spread, which is often constrained by cosmological observables.

Source/Mechanism	Controls/Produces	Hierarchy Manifestation
Instanton action	$m_a$	Exponential span of $m_a$
Moduli stabilization	$f_a$	Sub-Planckian vs. GUT-scale $f_a$
Froggatt-Nielsen sector	$f_a$ tied to flavor	Direct function of SM masses
Axion mixing	Physical $m_a$, $f_a$	Redistribution, dominance shifts
Matrix randomness/statistics	$m_a$, $f_a$	Broad/log-normal distributions

3. Model-Dependent Hierarchies and Predictions

Concrete models demonstrate how predictive constraints and choices yield specific hierarchical outcomes:

• Froggatt-Nielsen Models: In (Berenstein et al., 2010), different assignments for quark/lepton charges and operator patterns yield $f_a$ values ranging from $\sim 5 \times 10^3$ GeV (excluded) up to $10^{11}$ GeV (phenomenologically viable), demonstrating sensitivity to flavor structure and operator coefficients.
• Axiflavon-Higgs Unification: Both minimal and RH neutrino-augmented implementations (Alanne et al., 2019, Alanne et al., 2018) tightly constrain $f_a$: minimal versions with $f_a \sim 10^{11}-10^{12}$ GeV allowed by flavor and Higgs-potential matching, while heavy axion extensions can reduce $f_a$ to $\sim$ TeV, generating further mass hierarchies and decoupling axion DM persistence.
• Statistical Axiverse Models: Cosmologically consistent models with many axions are characterized by log-flat or random-matrix ensembles for $m_a$ and $f_a$, which can favor broad distributions but typically center around sub-Planckian $f_a$ ($\sim 10^{-2}-10^{-1} M_{\text{Pl}}$) (Stott et al., 2017).

4. Implications for Cosmology and Particle Phenomenology

Hierarchical axion masses/decay constants have critical consequences:

• Dark Matter Composition: In multi-component axion DM scenarios (Li, 26 Oct 2025), whether the QCD axion or one/multiple ALPs dominate is controlled by the spectrum of $f_i$. In generic string axiverse regimes with many axions and hierarchical $f_i$, the energy density after mixing is usually dominated by the lightest ALP (in the light QCD axion case), invalidating canonical single-component interpretations.
• Axion Window Constraints: Viable decay constants for the QCD axion are set by cosmological and astrophysical bounds ($10^{9} \text{ GeV}\leq f_a \leq 10^{12}$ GeV). Models outside this window (e.g., $f_a \sim 5 \times 10^3$ GeV) are excluded, establishing a phenomenologically enforced hierarchy (Berenstein et al., 2010).
• Flavor Physics and Rare Decays: Axiflavon and flavor-based models sharply predict ratios such as $g_{a\gamma\gamma}/m_a$ in a narrow band (e.g., $1-2 \times 10^{-16}~\mu{}$eV${}^{-1}$) due to the direct mapping from SM mass hierarchies to anomaly coefficients (Calibbi et al., 2016). Enhanced or suppressed flavor-violating decays—such as $K^+ \rightarrow \pi^+ a$—provide experimental signatures tightly tied to the underlying axion hierarchy.

5. Hierarchical Axion Inflation and Trans-Planckian Excursions

Field-theoretic inflationary mechanisms leverage hierarchical decay constants for phenomenological compatibility:

• Hierarchical Axion Inflation: Two-axion models (Ben-Dayan et al., 2014), without alignment tuning, use simple ratios (e.g., $f_{r_2} \ll f_{r_1}, f_{\theta_2}$) to enhance the effective inflaton decay constant:

$$f_{\mathrm{eff}} = \frac{f_{r_1} f_{\theta_2}}{f_{r_2}}$$

resulting in parametrically trans-Planckian field excursions. This mechanism is structurally distinct from alignment (KNP, clockwork) and is compatible with string-theory compactifications due to geometric discreteness in cycle selection and instanton charges.

• Warped/Throat Constructions: Axion decay constant hierarchies arise directly from geometric (e.g., throat lengths $L_i$) parameters in warped extra dimensions, with exponentials generating super-Planckian effective $f'$ for physical axion states (Fonseca et al., 2019). The calculable nature of these constructions enables controlled model-building for ultra-light ALPs, inflationary scenarios, and relaxion models.

6. Model-Independent Relations and Constraints

Despite the diversity of models and mechanisms, core scaling relations enforce universal hierarchy features:

• Axion Mass Relation:

$m_a \sim \frac{\Lambda^2}{f_a}$

with $\Lambda$ set by the nonperturbative effect generating the mass (QCD, hidden sector, dark gauge group).

• Axion-Photon Coupling:

$$g_{a\gamma\gamma} = \frac{\alpha_{\text{em}}}{2\pi f_a}\left( \frac{E}{N} - \text{hadronic term}\right)$$

where $E/N$ is the electromagnetic-to-QCD anomaly ratio, set by PQ or flavor assignments.

• Decay Constant Determination: In string, supersymmetric, or flavor models, $f_a$ is always set by the largest scale in the spontaneous breaking sector—with substantial sensitivity to moduli geometry, anomaly coefficients, and operator mixing.

7. Summary Table: Hierarchy Realizations in Major Constructions

Model/Framework	Origin of Hierarchy	$f_a$ Range	$m_a$ Range	Phenomenological Criterion
Gauged flavor (FN) (Berenstein et al., 2010)	Charge assignments, operator pattern	$5\times 10^3$–$10^{11}$ GeV	$<10^{-4}$ eV (QCD-induced)	Axion window, flavor fit
Axiflavon-Higgs (Alanne et al., 2019)	Unified symmetry, Higgs matching	$10^{11}$–$10^{12}$ GeV	$6$–$60~\mu$eV (QCD DM)	Potential matching, kaon decays
String axiverse (Stott et al., 2017)	Moduli/instanton randomness	$10^{-2}$–$10^{-1} M_{pl}$	$10^{-33}$ eV–TeV (broad)	Cosmological fit, matrix statistics
Orbifold GUT (Foster et al., 2022)	Unification, dark confining sector	$10^{15}$–$10^{17}$ GeV	keV–PeV (dark ALPs)	DM relic, glueball dilution, baryo.
Hierarchical inflation (Ben-Dayan et al., 2014)	Two-axion simple ratio	sub-Planckian constituents, $f_{\mathrm{eff}} \gg M_{pl}$	Inflaton mass as needed	Trans-Planckian excursion, string emb.
Multi-component DM (Li, 26 Oct 2025)	Mixing plus geometric moduli	Geometry/log-flat spread	Geometry/log-flat spread	Multi-ALP DM abundance redistribution

References and Significance

Papers (Berenstein et al., 2010, Calibbi et al., 2016, Stott et al., 2017, Alanne et al., 2019, Bonnefoy et al., 2018, Ben-Dayan et al., 2014, Fonseca et al., 2019, Foster et al., 2022), and (Li, 26 Oct 2025) demonstrate the variety and depth of hierarchical axion mass/decay constant constructions. Across these studies, a plausible implication is that observed hierarchies in SM fermion masses, cosmological abundances, and flavor patterns are reflected—often directly—in axion phenomenology. These models also indicate that achieving both a solution to the strong CP problem and a robust DM candidate naturally involves tight constraints and limited windows for $f_a$, while multi-axion theories generically entail broad hierarchies and multi-component DM. The calculability of these hierarchies in geometric, gauge-theoretic, and statistical terms connects UV theory to experimental signatures.