Flavored Axion Models in Flavor Physics

• Flavored axion models are constructions where the Peccei–Quinn symmetry is flavor dependent, linking axion physics with hierarchical Yukawa patterns.
• They employ frameworks like multi-Higgs textures, Froggatt–Nielsen mechanisms, and extra-dimensional setups to produce non-universal and off-diagonal axion-fermion interactions.
• These models yield observable signatures in rare meson decays and charged-lepton flavor violation, offering practical tests for current and future experiments.

Flavored axion models (FAMs) are axion constructions in which the Peccei–Quinn (PQ) symmetry is family dependent, or else emerges from the same flavor dynamics that organize fermion masses and mixings. Their defining consequence is that the axion simultaneously participates in the solution of the strong-CP problem and in the flavor sector: the charge assignments or flavon insertions that generate hierarchical Yukawa structures also make axion couplings to fermions non-universal and, after rotation to the mass basis, generically off-diagonal. As a result, FAMs interpolate between axion physics, rare-flavor processes, texture-zero model building, Froggatt–Nielsen constructions, DFSZ- and KSVZ-type realizations, extra dimensions, modular invariance, and neutrino-mass mechanisms (Björkeroth et al., 2018, Giraldo et al., 2020, Cox et al., 2023).

1. Defining structure and effective description

At low energies, the common EFT structure of a flavored axion is a derivative coupling of the axion to fermion flavor currents together with the anomalous couplings to gluons and photons. In the notation used in the phenomenological review by Björkeroth, Chun, and King, one may write

$$L \supset \tfrac12(\partial_\mu a)^2 -\frac{\partial_\mu a}{v_{PQ}}\sum_{f,i,j}\bar f_i\gamma^\mu\left(V^f_{ij}-A^f_{ij}\gamma_5\right)f_j +\frac{\alpha_s}{8\pi}\frac{a}{f_a}G\tilde G +\frac{\alpha}{8\pi}\frac{C_\gamma}{f_a}aF\tilde F ,$$

with

$$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$

Off-diagonal entries appear whenever the flavor charges are not proportional to the identity, so flavor-changing axion couplings are not an incidental detail but a structural output of family-dependent PQ symmetry (Björkeroth et al., 2018).

This general pattern encompasses both standard PQ axions and non-standard axion-like particles in phenomenological analyses, but the model-building literature represented here is dominated by QCD axions. A recurring distinction is between flavor-universal implementations, where the axion behaves approximately as in ordinary DFSZ or KSVZ settings, and genuinely flavored implementations, where tree-level flavor-changing neutral currents (FCNCs) emerge in the axion and often also in the scalar sector. In the four-Higgs texture-zero construction of Giraldo et al., for example, the PQ charges are explicitly non-universal and therefore produce tree-level FCNCs in addition to the axion solution of strong CP (Giraldo et al., 2020).

A common misconception is that “flavored axion model” denotes a single canonical framework. The literature instead uses the term for a class of constructions sharing one central property: flavor data and axion physics are controlled by the same symmetry or by tightly coupled symmetry sectors. This includes accidental PQ symmetries from flavor groups, family-dependent DFSZ models, Froggatt–Nielsen identifications $U(1)_{FN}\equiv U(1)_{PQ}$, warped and extra-dimensional models, modular-invariant realizations, and 3-3-1 extensions with flavor-changing axion couplings (Calibbi et al., 2016, Vega et al., 2021, Bonnefoy et al., 2020, Karan et al., 21 Feb 2025).

2. Model-building architectures

The modern FAM literature is structurally diverse. Some models begin from a desired fermion texture and infer the PQ sector; others begin from a flavor symmetry and find that an axion emerges automatically or accidentally.

Framework	Defining feature	Representative papers
Multi-Higgs texture-zero FAM	Family-dependent PQ charges enforce quark texture zeros	(Giraldo et al., 2020, Giraldo et al., 9 Mar 2026)
Froggatt–Nielsen / axiflavon	$U(1)_{FN}$ identified with or related to PQ	(Calibbi et al., 2016, Vega et al., 2021, Berenstein et al., 2010, Babu et al., 27 Feb 2026)
Flavoured DFSZ classification	Three-family DFSZ models with $N_{DW}=1$	(Cox et al., 2023)
Warped / extra-dimensional	Bulk localization generates flavor hierarchies and axion profiles	(Bonnefoy et al., 2020, Ahn, 2024)
String, modular, discrete-flavor	PQ tied to $A_4$, $T'$, $SL(2,\mathbb Z)$, or anomalous $U(1)$ sectors	(Ahn, 2016, Carone et al., 2019, Carone et al., 2020, Ahn, 9 Nov 2025)
Gauge extensions	Flavor-changing axions in 3-3-1 or hadronic $N_{DW}=1$ setups	(Karan et al., 21 Feb 2025, Alonso-Álvarez et al., 2023)

In the quark-texture approach of Giraldo et al., the Standard Model is extended by four $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$0 doublets $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$1, two PQ-charged singlets $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$2, and one heavy color-triplet $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$3. Requiring a realistic Hermitian five-zero ansatz for the quark mass matrices yields a unique solution that requires four Higgs doublets, with PQ-selection rules fixing which Yukawa entries vanish and which survive (Giraldo et al., 2020).

In Froggatt–Nielsen-based realizations, the axion often coincides with the phase of the flavon that suppresses lighter-generation Yukawas. The “axiflavon” framework of Calibbi et al. uses a global horizontal $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$4 broken by a flavon $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$5 with $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$6, so that the same insertions that reproduce fermion mass hierarchies generate the axion and predict a narrow range for $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$7 and $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$8 (Calibbi et al., 2016). A UV-complete realization with heavy messenger fields identifies the global $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$9 simultaneously as $U(1)_{FN}\equiv U(1)_{PQ}$0 and $U(1)_{FN}\equiv U(1)_{PQ}$1, generating nearest-neighbour-interaction (NNI) quark textures and an $U(1)_{FN}\equiv U(1)_{PQ}$2 neutrino texture (Vega et al., 2021).

The DFSZ branch of the subject has also been systematized. Cox et al. classified three-family flavoured DFSZ models with no cosmological domain-wall problem and found exactly 17 inequivalent representative Yukawa-texture patterns, labelled $U(1)_{FN}\equiv U(1)_{PQ}$3, $U(1)_{FN}\equiv U(1)_{PQ}$4, and $U(1)_{FN}\equiv U(1)_{PQ}$5. In all of them the color anomaly sum satisfies $U(1)_{FN}\equiv U(1)_{PQ}$6, so $U(1)_{FN}\equiv U(1)_{PQ}$7, and known variants such as the top-specific model emerge as special cases (Cox et al., 2023).

Other branches move away from purely four-dimensional weakly coupled settings. The 5D warped DFSZ construction places the axion in a bulk complex scalar in a slice of $U(1)_{FN}\equiv U(1)_{PQ}$8, with all Standard-Model fermions in the bulk and the Higgs doublets either UV-localized or bulk-propagating; flavor off-diagonal axion couplings arise only in the bulk-Higgs case through overlap integrals of fermion and axion profiles (Bonnefoy et al., 2020). String- and modular-inspired realizations similarly use anomaly constraints, Green–Schwarz structure, modular weights, or non-Abelian discrete symmetries to fix both flavor textures and axion properties (Ahn, 2016, Ahn, 9 Nov 2025).

3. Flavor textures, mass hierarchies, and charge assignments

The central technical role of the PQ symmetry in FAMs is to enforce restricted Yukawa structures. In the four-Higgs model, the Hermitian quark mass matrices take the five-zero form

$U(1)_{FN}\equiv U(1)_{PQ}$9

with the vanishing entries fixed by charge-selection rules of the form $U(1)_{FN}$0 or $U(1)_{FN}$1 for allowed terms. In that construction, fitting quark masses at $U(1)_{FN}$2 while keeping a minimal choice of $U(1)_{FN}$3 Yukawas gives

$U(1)_{FN}$4

illustrating how the observed inter-family hierarchies can be shifted partly into a hierarchical Higgs-VEV pattern rather than extremely small Yukawa parameters (Giraldo et al., 2020).

The Froggatt–Nielsen branch encodes flavor hierarchies through powers of a small parameter. In the UV-complete flavored-axion model, the leading quark operators generate NNI-type mass matrices

$U(1)_{FN}$5

while the neutrino sector yields an $U(1)_{FN}$6 texture in the basis where $U(1)_{FN}$7 is diagonal (Vega et al., 2021). In the axiflavon framework, the effective Yukawas satisfy

$U(1)_{FN}$8

and determinant relations involving $U(1)_{FN}$9 and $N_{DW}=1$0 imply the sharp prediction $N_{DW}=1$1 at $N_{DW}=1$2 CL, clustered around the DFSZ value $N_{DW}=1$3 (Calibbi et al., 2016).

In modular and discrete-flavor realizations, Yukawa suppression is not just a power of a flavon VEV but is tied to modular forms or non-Abelian representations. The extra-dimensional modular-$N_{DW}=1$4 model with $N_{DW}=1$5 writes canonically normalized Yukawas as

$N_{DW}=1$6

with $N_{DW}=1$7, and uses the $N_{DW}=1$8 charge assignment to make the axion intrinsically flavored (Ahn, 2024). The modular-invariant supergravity construction instead constrains Yukawa coefficients to unit-magnitude complex numbers and derives the quark and lepton flavor structures from anomaly-free $N_{DW}=1$9 assignments (Ahn, 9 Nov 2025).

This diversity has two important implications. First, FAMs do not require a single universal flavor mechanism: texture zeros, Froggatt–Nielsen suppression, partial compositeness, modular forms, and discrete-family symmetry all appear in viable realizations. Second, the axion sector is not merely appended to an existing flavor model; in these constructions it is usually algebraically entangled with the operators responsible for the observed flavor pattern.

4. Anomalies, axion couplings, and the low-energy spectrum

All FAMs retain the standard anomalous axion coupling to QCD. In the four-Higgs texture-zero model the axion is defined below the PQ-breaking scale by the singlet phase direction carrying the nonzero QCD anomaly $A_4$0, with normalization

$A_4$1

and the QCD-induced mass

$A_4$2

After field redefinitions that remove axion-Higgs mixing from kinetic terms, the remaining axion couplings to quarks are derivative and flavor off-diagonal in the mass basis: $A_4$3 with

$A_4$4

The phenomenologically relevant entries are the $A_4$5 coupling $A_4$6 and the $A_4$7 coupling $A_4$8 (Giraldo et al., 2020).

The warped DFSZ model displays a particularly clean distinction between flavor-diagonal and flavor-off-diagonal regimes. If the Higgs doublets are UV-localized, the axion induces only flavor-diagonal axial couplings, $A_4$9 and $T'$0. If the Higgs doublets propagate in the bulk, the overlap of the axion and fermion profiles generates generic off-diagonal $T'$1. For a benchmark with $T'$2, $T'$3, $T'$4, $T'$5, and $T'$6 GeV, the effective off-diagonal scales lie in the range

$T'$7

including $T'$8 GeV and $T'$9 GeV. In that setup the anomaly ratio is the standard DFSZ value $SL(2,\mathbb Z)$0 (Bonnefoy et al., 2020).

The anomaly structure can also alter flavor predictions in the opposite direction, suppressing the dangerous couplings. In the modular-invariant flavored-QCD axion model, the axion mass and photon coupling are predicted as

$SL(2,\mathbb Z)$1

while flavor-violating couplings to $SL(2,\mathbb Z)$2 quarks and $SL(2,\mathbb Z)$3 leptons are suppressed to $SL(2,\mathbb Z)$4, with $SL(2,\mathbb Z)$5 the Cabibbo angle (Ahn, 9 Nov 2025).

The 3-3-1 realization provides a different anomaly pattern. There the axion-photon ratio satisfies $SL(2,\mathbb Z)$6, giving an enhanced photon coupling relative to canonical KSVZ and DFSZ implementations, and the family-nonuniversal embedding of quarks into $SL(2,\mathbb Z)$7 produces tree-level flavor-changing axion couplings in $SL(2,\mathbb Z)$8, $SL(2,\mathbb Z)$9, and $U(1)$0 decays (Karan et al., 21 Feb 2025).

5. Flavor observables and experimental tests

Rare meson decays are the most characteristic probes of FAMs. In the general phenomenological treatment, a two-body decay $U(1)$1 mediated by an off-diagonal vector coupling obeys

$U(1)$2

with benchmark form factors $U(1)$3 for $U(1)$4, $U(1)$5 for $U(1)$6, $U(1)$7 for $U(1)$8, and $U(1)$9 for $N_{DW}=1$0 (Björkeroth et al., 2018). The dominant constraint usually comes from kaons. The same review quotes

$N_{DW}=1$1

with future NA62 sensitivity $N_{DW}=1$2 corresponding to $N_{DW}=1$3 GeV (Björkeroth et al., 2018).

The four-Higgs texture-zero model arrives at the same hierarchy of constraints in a different notation. Using

$N_{DW}=1$4

and

$N_{DW}=1$5

the quoted limits are $N_{DW}=1$6 from E949+E787 and $N_{DW}=1$7 from Belle. These translate into bounds in the $N_{DW}=1$8-$N_{DW}=1$9 plane, with typical viable values $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$00 GeV for $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$01, and a preferred combined window $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$02 GeV with $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$03 (Giraldo et al., 2020).

Charged-lepton flavor violation supplies the lepton-sector analogue. The generic two-body decay rate

$$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$04

implies $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$05 constraints at the $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$06 GeV level from TRIUMF, while $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$07 gives $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$08 GeV from Crystal Box (Björkeroth et al., 2018). In the warped model, the quoted current lepton bound from MEG is $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$09 GeV, with Mu3e/MEG-II-fwd projected to reach $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$10 GeV (Bonnefoy et al., 2020).

Some models make sharper, fit-dependent predictions. In the “A to Z” Pati–Salam model, where the PQ symmetry arises accidentally from the discrete flavor sector, the couplings are fixed by the fermion fit. For $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$11 GeV the model predicts

$$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$12

as well as the $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$13-independent correlation

$$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$14

at the best fit (Björkeroth et al., 2018).

Dark-matter-motivated hadronic constructions reinforce the same experimental message. Requiring a consistent post-inflationary cosmology with $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$15 and no stable exotic relics singles out two KSVZ-type realizations in which flavor-violating right-handed axion couplings are generically unsuppressed. In that setting,

$$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$16

implies

$$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$17

and the resulting $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$18 rates place the axion dark-matter window squarely within the reach of NA62 and KOTO (Alonso-Álvarez et al., 2023).

6. Cosmology, domain walls, neutrino masses, and recent extensions

Cosmology enters FAMs in two distinct ways: through the usual invisible-axion constraints on $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$19, and through the stronger requirement that the flavored realization itself avoid the domain-wall problem. The cleanest result is the DFSZ classification of Cox et al., where the three-family anomaly condition

$$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$20

gives $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$21 and therefore $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$22 in each of the 17 viable classes (Cox et al., 2023). Unit domain-wall number also appears in several other flavored constructions, including the non-supersymmetric $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$23 model, the gauged-$$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$24 high-quality axion models, and the gravitational-wave flavored-axion scenario (Carone et al., 2019, Babu et al., 27 Feb 2026, Babu et al., 3 Jun 2026).

A second recurrent theme is the unification of the PQ and neutrino sectors. In the extra-dimensional $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$25 model, cancellation of the mixed gravitational anomaly forces electrically neutral mirror bulk fermions to couple to the brane neutrino field, and bulk exchange induces a brane-to-brane Weinberg operator. Requiring $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$26 eV yields

$$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$27

while the choice $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$28 gives $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$29 (Ahn, 2024). The recent four-Higgs FAM extension with right-handed neutrinos similarly ties the heavy Majorana masses to the PQ-breaking scalar through a type-I seesaw, so the neutrino and axion scales are intrinsically connected (Giraldo et al., 9 Mar 2026).

The cosmological scope of FAMs has recently widened beyond misalignment and rare decays. In the high-quality flavored-axion framework based on gauged $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$30, the axion acts as a flavon field, the right-handed neutrino mass scale is identified with the Froggatt–Nielsen scale, the axion can account for the dark-matter abundance without domain walls, and baryon asymmetry is realized through calculable leptogenesis (Babu et al., 27 Feb 2026). A related 2026 analysis argues that the evolution and decay of mixed gauged flavonic and global axionic string networks generate a distinctive plateau–valley–plateau stochastic gravitational-wave spectrum. In that work, current NA62 data are quoted as

$$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$31

with HIKE aiming at $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$32, while the gravitational-wave signal is presented as a complementary probe of the same high-quality flavored-axion dark-matter parameter space (Babu et al., 3 Jun 2026).

Taken together, the literature presents FAMs not as a narrow variant of DFSZ or KSVZ axion physics, but as a broad organizing principle: the PQ symmetry is promoted into the flavor sector, or the flavor sector generates PQ as an accidental or emergent symmetry. The resulting theories can enforce texture zeros, realize Froggatt–Nielsen suppressions, classify all domain-wall-free three-family DFSZ patterns, embed naturally in warped or modular constructions, or connect directly to neutrino masses, dark matter, leptogenesis, and even gravitational-wave phenomenology. Their characteristic experimental signature remains the same across this diversity: flavor-dependent and often flavor-changing axion couplings, with $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$33 and $$V^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}+U_{Rf}^\dagger x_{f_R}U_{Rf}),\qquad A^f=\tfrac12(U_{Lf}^\dagger x_{f_L}U_{Lf}-U_{Rf}^\dagger x_{f_R}U_{Rf}) .$$34 continuing to provide the most discriminating low-energy tests (Björkeroth et al., 2018, Alonso-Álvarez et al., 2023, Babu et al., 3 Jun 2026).