Flavored Axions: Properties and Implications

• Flavored axions are axion models that assign non-universal Peccei-Quinn charges to Standard Model fermions, linking the strong CP problem resolution to flavor hierarchies.
• They predict both flavor-diagonal and potentially large flavor-violating axion-fermion couplings, leading to observable signatures in rare meson and lepton decays.
• Their predictive framework connects the axion decay constant and mass to new physics scales, with constraints arising from astrophysical, cosmological, and laboratory experiments.

Flavored axions generalize the QCD axion paradigm by endowing Standard Model (SM) fermions with non-universal Peccei-Quinn (PQ) charges, thereby linking axion physics to flavor symmetries and mechanisms for fermion mass hierarchies. Such models connect the resolution of the strong CP problem with flavor dynamics, often incorporating Froggatt-Nielsen (FN), seesaw, or modular flavor structures. Flavored axion scenarios generate distinctive flavor-dependent axion-fermion couplings --- including potentially large flavor-violating terms --- which are subject to stringent constraints from both astrophysics and laboratory experiments, and exhibit unique phenomenological signatures compared to flavor-universal (KSVZ/DFSZ-like) axions. These models also frequently tie the axion decay constant to other new physics scales, such as the seesaw or FN scale, and offer predictive frameworks for axion mass, couplings, and cosmological implications.

1. Theoretical Structure and Flavored PQ Symmetries

Flavored axion models extend the SM gauge and flavor structure with global or local symmetries under which SM fermion generations carry distinct PQ charges. The PQ symmetry is typically embedded within a larger flavor symmetry group, such as $U(1)_F$ (Froggatt-Nielsen), non-Abelian structures (e.g., $SU(3)\times SU(2)\times U(1)_F$), or more intricate constructions involving modular invariance or string-theoretic ingredients.

In these models:

• PQ charges: Fermion fields are assigned PQ charges not aligned with the fermion mass matrices. For instance, in FN-type models, these charges simultaneously generate hierarchies in the Yukawa couplings and axion-fermion couplings.
• Breaking scale: The PQ breaking scale $F_A$ (or $f_a$) is set either through independent symmetry breaking or is dynamically connected to other scales in the model, such as the seesaw neutrino mass scale or the scale of modular symmetry breaking (Ahn, 2018, Ahn et al., 2019, Ahn, 9 Nov 2025).
• Anomaly structure: The color anomaly $N$, electromagnetic anomaly $E$ and the domain-wall number $N_{\rm DW}$ are determined by the charge content and often strongly constrained by anomaly cancellation conditions, string consistency, or mixed anomaly conditions (Darmé et al., 2022, Ahn et al., 2019, Ahn, 2016).

These structural features give rise to axions whose couplings are fixed by predictive, model-dependent flavor assignments, leading to characteristic patterns of both diagonal and off-diagonal interaction strengths.

2. Axion Couplings: Flavor Universal and Flavor Violating Sectors

The effective interaction Lagrangian for a flavored axion below the PQ breaking scale generically contains:

$$\mathcal{L} \supset \frac{a}{f_a} \frac{\alpha_s}{8\pi} G^a_{\mu\nu}\tilde G^{a\mu\nu} + \frac{E}{N} \frac{a}{f_a} \frac{\alpha_{em}}{8\pi} F_{\mu\nu}\tilde F^{\mu\nu} + \frac{\partial_\mu a}{2f_a} \sum_{i,j} \bar{\psi}_i \gamma^\mu (C^V_{ij} + C^A_{ij} \gamma_5)\psi_j$$

with key properties:

• Flavor-diagonal couplings $C^A_{ii}$: These are determined by the (model-dependent) difference between the left and right PQ charges after rotation to the mass basis. They typically remain $\mathcal{O}(1)$ in many flavored models, matching the strength of minimal DFSZ/KSVZ axions.
• Flavor-off-diagonal couplings $C^V_{ij}$, $C^A_{ij}$ ($i\neq j$): Generically unsuppressed if the PQ charge basis and the Yukawa/mass basis are misaligned (Ziegler, 2023, Ahn et al., 2019, Björkeroth et al., 2018, Ziegler, 2019). In paradigmatic Froggatt-Nielsen embeddings, these couplings scale as:

$$|C^{V,A}_{ij}| \sim \mathcal{O}(1) \; \epsilon^{|\Delta Q|}$$

where $\epsilon$ is a small flavor-breaking parameter, and $|\Delta Q|$ encodes charge differences (Ziegler, 2019).

• Anomaly-driven couplings: The axion-photon and axion-gluon couplings are controlled by $E/N$; values near $8/3$ (DFSZ-like) or significantly different, e.g., $E/N \approx 1.87$ in certain FNPQ models leading to up to $14\times$ suppression of $g_{a\gamma\gamma}$ (Vega et al., 2021).

In modular and string-based frameworks (e.g., $SL(2,\mathbb{Z}) \times U(1)_X$ models), further constraints from modular transformations and mixed anomalies can both determine flavor structure and strongly influence the size and pattern of axion couplings (Ahn, 9 Nov 2025, Ahn, 2016).

3. Phenomenological Consequences: Rare Decays and Laboratory Constraints

Flavored axions induce tree-level flavor-changing neutral currents (FCNCs) mediated by axion emission in both quark and lepton sectors, providing powerful direct probes:

Axion-emitting meson decays

• Kaon sector: $K^+\to\pi^+ a$ is particularly sensitive, with branching ratios scaling as

$$\mathrm{Br}(K^+\to\pi^+ a) \simeq \frac{|C^V_{sd}|^2}{16\pi}\frac{m_K^3}{f_a^2} (1-\frac{m_\pi^2}{m_K^2})^3$$

NA62 bounds typically require $f_a / |C^V_{sd}| > 3\times10^{11}$–$10^{12}$ GeV for $m_a \ll 100$ MeV (Ahn, 2018, Darmé et al., 2022, Ziegler, 2019, Alonso-Álvarez et al., 2023).

• B and D meson decays: $B\to K a$, $D\to\pi a$, and lepton flavor-violating decays (e.g., $\mu\to e a$) are similarly sensitive, with model-dependent branching ratios and reach (Ziegler, 2023, Ahn et al., 2019, Björkeroth et al., 2018, Karan et al., 21 Feb 2025).

Constraints

• Laboratory: Current and projected bounds (NA62, KOTO, Belle II, Mu3e, MEG II, etc.) already probe PQ scales up to $10^{12}$ GeV in the most optimistic scenarios, offering reach well beyond that of conventional photon-coupling searches (Ahn, 2018, Ziegler, 2019, Björkeroth et al., 2018, Ziegler, 2023, Vega et al., 2021).
• Astrophysical: Limits on $g_{aee}$ from white-dwarf/red-giant cooling and $g_{a\gamma\gamma}$ from horizontal-branch star evolution yield lower bounds on $f_a$ independent of flavor structure, though they often sit below the reach of FCNC searches in flavored axion models (Ahn, 2018, Ahn et al., 2019, Darmé et al., 2022).

The size of off-diagonal couplings is a direct consequence of the assumed PQ-flavor structure, and models engineered to suppress these couplings ("decoupled-state" $\nu$DFSZ or modular-invariant frameworks, for example) may evade some rare-decay bounds but often at the cost of increased model-building complexity (Rocha et al., 31 Mar 2025, Ahn, 9 Nov 2025).

4. Mass, Decay Constant, and Coupling Predictions

Once the axion decay constant is fixed (either via experimental constraints or model-building---e.g., matched to seesaw or FN scales), all other axion properties are sharply predicted.

Mass and coupling relations

• Axion mass: The standard chiral Lagrangian relation applies

$$m_a^2 F_a^2 = m_{\pi^0}^2 f_\pi^2 \frac{z}{(1+z)(1+z+w)},\;\; z\equiv m_u/m_d,\,w\equiv m_u/m_s$$

For $F_A \sim 3.6 \times 10^{10}$ GeV, $m_a \sim 1.5 \times 10^{-4}$ eV (Ahn, 2018, Ahn et al., 2019).

• Couplings
  • $g_{Aee} \sim \mathcal{O}(10^{-14})$ for $F_A \sim 3.6\times 10^{10}$ GeV (model-dependent) (Ahn, 2018, Ahn et al., 2019).
  • $g_{Ann} \sim 2 \times 10^{-12}$ for the same $F_A$ (Ahn, 2018).
  • $g_{a\gamma\gamma}$ varies from $10^{-14}$–$10^{-13}$ GeV$^{-1}$, with precise value depending strongly on $E/N$ (Ahn, 2018, Ahn et al., 2019, Vega et al., 2021, Darmé et al., 2022, Ahn, 9 Nov 2025).

Table: Representative quantitative predictions

Parameter	Value (typical)	Context/model
$F_A$	$3.56^{+0.84}_{-0.84}\times 10^{10}$ GeV	Axion + flavor/astro constraints (Ahn, 2018)
$m_a$	$1.54^{+0.48}_{-0.29} \times 10^{-4}$ eV	For $F_A$ above
$g_{Aee}$	$3.29^{+2.47}_{-0.98}\times 10^{-14}$	Flavored PQ (Ahn et al., 2019)
$g_{a\gamma\gamma}$	$2.15^{+1.61}_{-0.64}\times 10^{-14}$ GeV$^{-1}$	Flavored PQ, $E/N=8/3$
$g_{Ann}$	$2.14^{+0.66}_{-0.41}\times 10^{-12}$	For $F_A$ as above (Ahn, 2018)

5. Impact of Astrophysical, Cosmological, and Flavor Constraints

Flavored axions are constrained (and sometimes motivated) by multiple orthogonal sources:

• Astrophysical cooling: White dwarf/red giant cooling restricts $g_{Aee}$, which in turn bounds $F_A$; these limits often select a specific window for $F_A$, e.g., $$0.5 \times 10^{10} \text{ GeV} \lesssim F_A \lesssim 4.4 \times 10^{10}$$ GeV (Ahn, 2018).
• Rare processes: The non-observation of $K^+ \to \pi^+ a$ or $\mu \to e a$ directly excludes models with $f_a/C_{ij}$ below $10^{11}$--$10^{12}$ GeV (Ahn, 2018, Ziegler, 2019, Ziegler, 2023). The upcoming generation of experiments (NA62, KOTO, Mu3e) can probe flavored axions over nearly two orders of magnitude in $F_A$.
• Cosmological observables: Freeze-in production via flavor-violating couplings contributes to $\Delta N_{\rm eff}$; CMB-S4 projections show that cosmological bounds will soon be competitive with laboratory flavor probes for $f_a / C_{ij} \sim 10^8$–$10^{10}$ GeV (D'Eramo et al., 2021).

6. Model Variants, Suppression Mechanisms, and UV Realizations

The degree and structure of flavor violation in axion couplings is highly model-dependent. Some representative cases:

• PQ-flavor protection: Embedding PQ in a full SU(3) flavor symmetry or via radiative mass mechanisms delays the onset of dangerous PQ-violating higher-dimensional operators (e.g., det~Yukawa dimension 12), improving axion "quality" while fixing flavor structure (Cheung, 2010).
• Suppressed flavor-violating couplings: Modular-invariant models can engineer off-diagonal axion-fermion couplings to be $\mathcal{O}(\lambda^4)$ ($\lambda\approx0.22$ the Cabibbo angle), yielding strong suppression of $s\to d$ and $\mu\to e$ interactions (Ahn, 9 Nov 2025).
• String-theoretic frameworks: Axion and flavor charges may be anchored to string moduli and anomaly-cancellation conditions, resulting in the survival of flavored PQ axions with decay constants $10^{10}$--$10^{12}$ GeV and predictable anomaly coefficients (Ahn, 2016).

These mechanisms allow flavored axion models to remain consistent with the full suite of flavor and cosmological constraints for a range of $f_a$ and $m_a$ values, while offering discovery prospects at the next generation of flavor and astrophysics experiments.

7. Summary and Outlook

Flavored axion models directly intertwine flavor physics and the Peccei-Quinn solution to the strong CP problem. They generically predict:

• Non-universal and often sizable flavor-diagonal axion-fermion couplings.
• Tree-level flavor-violating couplings leading to observable FCNC processes, particularly in rare meson and lepton decays.
• Strong connections between $f_a$, $m_a$, and flavor structure, leading to precise predictions for couplings and mass once one scale is measured.
• Constraints on model-building from astrophysical cooling hints, cosmology ($\Delta N_{\rm eff}$), and laboratory rare decay limits.
• Opportunities for direct discovery or exclusion through next-generation rare decay searches (e.g., NA62, Mu3e), as well as astrophysical and cosmological observations.

The current and anticipated experimental reach will further test the viability of flavored axion scenarios, offering a unique approach to probing the flavor structure of new physics intertwined with axion phenomenology (Ahn, 2018, Ziegler, 2023, Ahn et al., 2019, Darmé et al., 2022, Ahn, 9 Nov 2025, Ahn, 9 Nov 2025).