Froggatt–Nielsen Flavon Sector Overview

• The FN flavon sector is a framework that employs a scalar flavon field and horizontal symmetry breaking to dynamically generate hierarchical fermion masses and mixing patterns.
• It builds Yukawa structures through power-law suppressions determined by the ratio of the flavon VEV to a high-energy cutoff, with specific charge assignments setting the exponents.
• The sector impacts Higgs–flavon mixing, effective field theory matching, and cosmological signatures, unifying flavor physics with collider and gravitational wave studies.

The Froggatt–Nielsen flavon sector forms the core of the Froggatt–Nielsen (FN) mechanism, a framework designed to explain the origin of hierarchical fermion masses and mixing in the Standard Model and its extensions. In this approach, flavor hierarchies are generated dynamically through the spontaneous breaking of an additional horizontal symmetry, usually realized as a U(1) or discrete symmetry, by the vacuum expectation value (VEV) of one or more flavon fields. This mechanism enables Yukawa structures and other phenomenologically relevant couplings to be suppressed by powers of small ratios, determined by the flavon VEV(s) over a high-scale cutoff, with the exponents set by “flavon charges” assigned to each fermion species. The flavon sector, encompassing the flavon field(s), their interactions, associated symmetries, and dynamical features, is thus central for both model building and observable implications in flavor physics, collider phenomenology, cosmology, and ultraviolet (UV) completions.

1. Structure and Dynamics of the Flavon Sector

The archetypal FN setup introduces a complex scalar flavon field (ϕ or S), transforming nontrivially under a horizontal symmetry (typically U(1)_H, U(1)_F, Z_N^F, or their non-Abelian generalizations). The Lagrangian includes higher-dimensional operators of the form

$$\mathscr{L}_{\rm FN} \supset y_{ij} \left(\frac{S}{\Lambda}\right)^{n_{ij}} \overline{\psi_i} H \psi_j$$

where Λ is the UV cutoff or mediator scale, H is the Higgs doublet, n_{ij} are integers fixed by horizontal symmetry invariance, and y_{ij} are order-one coefficients. When S acquires a VEV (⟨S⟩ = w), the effective Yukawa matrices become hierarchical, with entries suppressed by ε^{n_{ij}}, where ε = w/Λ.

The dynamics of the flavon sector also include its stabilization, self-interactions, and couplings to the Higgs and potentially other BSM fields. The scalar potential generally has the form: $$V(S) = m_S^2 |S|^2 + \lambda_S |S|^4 + (\text{higher-order/stabilization terms})$$ In supersymmetric extensions, holomorphic superpotential terms of the form W ∼ S^N or S^m H_u H_d have important roles for flavon stabilization and for generating a μ-term in the MSSM (Higaki et al., 2019). In non-supersymmetric UV completions, the vacuum structure and field content (e.g., presence of multiple flavons for non-Abelian symmetries) are designed so that the breaking pattern yields realistic mass textures and desired cosmological properties (Rathsman et al., 2023).

2. Hierarchical Coupling Generation and Charge Assignments

The FN mechanism leverages the flavon sector to engineer texture zeros, mass hierarchies, and mixing patterns in the fermion sector. The suppression exponents n_{ij} are determined by the sum of FN charges assigned to the left- and right-handed fermions and scalars: $$n_{ij} = Q_L^i + Q_R^j + Q_H \ \text{ (or similar expressions depending on conventions)}$$ The broad parameter space for viable charge assignments can be systematically scanned and statistically ranked, as performed in Bayesian studies (Ibe et al., 27 Dec 2024), yielding both conventional and unconventional (including negative or large) charge sets compatible with flavor data.

In many models, both continuous (U(1)) and discrete (Z_N^F) symmetries are considered for the flavon sector (Higaki et al., 2019). Discrete variants can evade issues associated with global symmetry violation at high scale while preserving the predictivity for mass hierarchies, operator selection rules, and vacuum stabilization.

The exponents n_{ij}, and thus the predicted mass and mixing patterns, can also be mapped onto modular weights in models with modular invariance, where the modular weights play the role of FN charges (Kuranaga et al., 2021).

3. Scalar Potential, Mixing, and Vacuum Stability

The flavon sector is generically coupled to the SM Higgs via Higgs–portal terms: $$V(H, S) = -\mu_h^2 H^\dagger H + \lambda_h |H|^4 - \mu_S^2 |S|^2 + \lambda_S |S|^4 + \lambda_{HS} |H|^2 |S|^2$$ When both fields acquire VEVs, they generically mix, so that the mass eigenstates are linear combinations of the CP-even Higgs and flavon field. The mass matrix and mixing angle θ are determined by: $$\tan 2\theta = \frac{\lambda_{HS} v w}{\lambda_h v^2 - \lambda_S w^2}$$ This mixing leads to observable consequences, e.g., in Higgs boson couplings, lepton flavor violating (LFV) decays such as h → μτ (Huitu et al., 2016), and new scalar resonances at the LHC or HL-LHC (Arroyo-Ureña et al., 25 Oct 2024, Koivunen et al., 2023). Collider, flavor, and electroweak precision observables place upper bounds on the mixing angle, typically |θ| ≲ 10–16° depending on the flavon mass (Giese et al., 2019).

The impact of the flavon sector on vacuum stability is multifaceted. Mixing with the Higgs alters the scalar mass spectrum and can either destabilize or stabilize the vacuum, depending on parameters. RG-improved analyses involving towers of effective field theories and threshold corrections show that, after imposing experimental constraints, the FN sector does not induce greater instability in the Higgs potential than in the SM alone (Giese et al., 2019).

4. Ultraviolet Completions and Anomaly Cancellation

UV-complete models require careful treatment of gauge anomalies in the presence of the new horizontal symmetry. In anomaly-free FN constructions, such as those based on U(1)^3 or gauged U(1)_F, the SM fermions are neutral under the flavor symmetry, and the flavor structure arises from mixing with chains of heavy vector-like fermions, sometimes conceptualized as “clockwork” gears (Smolkovič et al., 2019). The effective Yukawa couplings scale as powers of (M/⟨ϕ⟩), with M being the mass of the vector-like states. Anomaly cancellation may also be achieved in extended scalar sectors or two Higgs doublet frameworks with rational charge assignments and sum rules enforcing the absence of gauge anomalies (Rathsman et al., 2023).

The UV structure impacts the predictions for new states such as heavy Z′ gauge bosons (arising from gauged horizontal symmetries) and for the available parameter space, with consequences for RG running, Landau poles, and the spectrum of the model up to high (possibly Planckian) scale (Rathsman et al., 2023).

5. Phenomenological and Cosmological Implications

The phenomenology of the FN flavon sector is rich and multifaceted. At colliders, the flavon can be produced through Higgs–flavon mixing, with subsequent flavor-violating decays emerging from the off-diagonal couplings of the flavon to fermions, potentially explaining observed anomalies such as h → μτ (Huitu et al., 2016) or eμ resonant signatures (Koivunen et al., 2023). In extended Higgs sectors (e.g., FN-2HDM), the flavon sector enforces approximate Natural Flavor Conservation and an automatic Peccei–Quinn symmetry, with predictions for light CP-odd scalars and suppressed flavor-changing neutral currents (Dery et al., 2016).

Cosmologically, the flavon sector can have multiple roles. Its dynamics during the early Universe can drive strong first order phase transitions leading to potentially observable stochastic gravitational wave backgrounds, as calculable from finite-temperature effective potentials (Ringe, 2022, Blasi et al., 11 Oct 2024). In scenarios with high-scale flavor symmetry breaking, the spontaneous breaking of a gauged U(1) flavor symmetry results in cosmic string formation, further enhancing the gravitational wave signatures with the string tension set by the scale v_φ (Blasi et al., 11 Oct 2024). Such GW signals often probe otherwise inaccessible regions of parameter space, complementary to flavor constraints.

The flavon portal also enables nonthermal connections between the SM and dark sectors, serving as a mediator for freeze-in or freeze-out dark matter scenarios (Mandal et al., 2023, Cheek et al., 2022). In such models, the relic abundance, direct and indirect detection prospects, and late time cosmological signatures are directly linked to the structure of the flavon couplings and charge assignments. Parameter space is strongly constrained by flavor-changing neutral current processes, cosmological observations (e.g., CMB, Lyman-α forest), and laboratory searches for rare decays.

The neutrino sector offers further testability: in unified or seesaw-extended FN models, charges and sequential suppression can be tuned to yield both normal and inverted hierarchies and large lepton mixing angles, reproducing the observed MNS matrix structure (Hattori et al., 2012).

6. Matching onto EFT and Operator Analysis

When integrating out the heavy flavon at a scale Λ_FN above the EW scale, FN models leave distinctive imprints in the Standard Model Effective Field Theory (SMEFT), generating corrections to operators of dimension 6 in the Warsaw basis (Loisa et al., 26 Feb 2024). SMEFT operator coefficients—both bosonic and (fermion) four-point—inherit hierarchical suppressions, textures, and selection rules from the underlying FN charges and symmetry structure. Functional matching computations, including tree-level and one-loop UV diagrams, demonstrate that both flavor-conserving and violating SMEFT operators receive calculable contributions dependent on the flavon mass, VEV, and portal couplings.

FN-motivated textures, as explored systematically in minimal or Bayesian frameworks, can be mapped onto predictions for SMEFT Wilson coefficients, providing avenues for both direct and indirect experimental testing through global fits, collider processes, and low-energy flavor observables (Ibe et al., 27 Dec 2024). This approach offers unambiguous means to probe flavor models in the era of precision effective field theory phenomenology.

7. Theoretical and Experimental Outlook

The FN flavon sector continues to serve as an essential structure for connecting model-building ambitions in flavor physics to new physics searches across multiple experimental frontiers. Its detailed role encompasses:

• Generating and constraining the full range of hierarchical Yukawa textures.
• Structuring the scalar sector, scalar mixing, and Higgs-like (and nonstandard) scalar phenomenology.
• Providing insights into anomaly cancellation and consistency in both global and gauged contexts.
• Predicting observable signals in rare decays, neutrino physics, and baryogenesis mechanisms (including baryo- and magnetogenesis through flavon decay dynamics (Elahi et al., 2020)).
• Creating correlated signals in cosmology, such as gravitational waves from phase transitions or cosmic strings.
• Allowing for connections to new sectors, including portals to dark matter, with relic abundances, indirect detection, and cosmological bounds entangled with flavor charge assignments and VEV scales.

Flavon sector properties, when explicitly matched to experimental outcomes and SMEFT analyses, underpin a predictive bridge from ultraviolet symmetries to low-energy observables, thus anchoring the continuing investigation of the flavor puzzle in a testable, interdisciplinary fashion.