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arxiv: 2606.20000 · v1 · pith:CRGCE4H2new · submitted 2026-06-18 · ✦ hep-ph · physics.comp-ph

Two Flavon Froggatt-Nielsen Models with Genetic Algorithms

Miguel Crispim Rom\~ao , Stephen F. King This is my paper

Pith reviewed 2026-06-26 17:05 UTC · model grok-4.3

classification ✦ hep-ph physics.comp-ph
keywords Froggatt-Nielsenflavongenetic algorithmneutrino mass orderingCKM matrixPMNS matrixCP violationneutrinoless double beta decay
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The pith

Two-flavon Froggatt-Nielsen models accommodate all fermion data in over 100000 distinct realizations found via genetic algorithms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper casts the search for viable two-flavon FN models as a multi-objective optimization task and applies the NSGA-III genetic algorithm to fit 18 FN charges, 45 Wilson coefficients, and flavon parameters simultaneously to quark and lepton data. It reports that the relative phase between the two flavon vevs supplies a generic source of CP violation. The scan yields more than 100000 unique models that satisfy all current constraints on masses, CKM and PMNS angles and phases, and neutrino mass-squared differences. Both normal and inverted neutrino orderings appear, with the relative size of the two vevs producing qualitatively different ranges for the neutrinoless double beta decay mass m_ee. Minimal charge assignments with maximum exponent equal to three are also identified.

Core claim

By formulating each experimental constraint as a separate objective and running the Non-dominated Sorting Genetic Algorithm III, the authors identify over 100000 unique phenomenologically viable two-flavon FN models. The relative phase between the two flavon vacuum expectation values generates CP violation in both sectors, and the relative hierarchy of those vevs leads to distinct predictions for m_ee in the normal and inverted orderings. Minimal realizations exist with maximal flavon exponent as small as three, and some models reproduce the charged-fermion masses to within 6 percent without dedicated continuous-parameter tuning.

What carries the argument

NSGA-III multi-objective genetic algorithm that simultaneously optimizes the discrete FN charges and the continuous Wilson coefficients and flavon vev parameters against separate experimental objectives.

If this is right

  • Both normal and inverted neutrino mass orderings are realized among the viable models.
  • The relative hierarchy between the two flavon vevs produces qualitatively distinct predictions for the effective neutrinoless double beta decay mass m_ee.
  • Minimal FN realizations exist with maximal flavon exponent as small as three.
  • Some models reproduce charged fermion masses to within 6 percent without dedicated continuous parameter optimization.
  • The low duplication rate indicates that the space of valid two-flavon FN realizations has not been exhausted.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Precision measurements of m_ee could distinguish the relative flavon vev hierarchies realized in the viable models.
  • The same multi-objective genetic algorithm approach could be applied to other high-dimensional discrete-continuous model spaces in flavor physics.
  • The large number of solutions suggests that additional theoretical constraints, such as those from UV completions, would be needed to increase predictivity.

Load-bearing premise

The genetic algorithm has explored the full mixed discrete-continuous space of charges and parameters without bias, convergence failure, or incorrect encoding of the experimental constraints.

What would settle it

A precision measurement of m_ee that lies outside the ranges predicted by all viable models in both normal and inverted orderings, or updated mixing-angle data that no model in the scan can satisfy.

Figures

Figures reproduced from arXiv: 2606.20000 by Miguel Crispim Rom\~ao, Stephen F. King.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of a Pareto front in a two-dimensional objective space spanned by the [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Illustration depicting the scheme for constructing the reference directions ( [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Discovered models with unique FN assignments in the [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Discovered models with unique FN assignments in the [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Discovered models with unique FN assignments in the [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Discovered models with unique FN assignments in the [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Predictions for the PMNS parameters, including the Majorana phases, for the unique FN [PITH_FULL_IMAGE:figures/full_fig_p034_7.png] view at source ↗
read the original abstract

We present the first systematic and comprehensive scan of two-flavon Froggatt-Nielsen (FN) models, employing artificial intelligence techniques to explore the high-dimensional, mixed discrete-continuous parameter space. Extending the standard single-flavon FN framework to a two-flavon setup in which separate flavon fields couple independently to the up- and down-type sectors, we demonstrate that the relative phase between their vacuum expectation values (vevs) provides a natural and generic source of CP violation absent in single-flavon models. To explore this enlarged model space, we cast the search for phenomenologically viable models as a multi-objective optimisation problem, formulating each experimental constraint as a separate objective, and employ the Non-dominated Sorting Genetic Algorithm III to simultaneously fit all 18 FN charges, 45 Wilson coefficients, and flavon parameters to both the quark and lepton sectors. Our approach requires no separate training phase and identifies phenomenologically viable models orders of magnitude faster than prior reinforcement learning methods. Imposing experimental constraints on CKM and PMNS mixing angles and CP phases, charged fermion masses, and neutrino squared-mass differences, we discover over $100\,000$ unique viable models with a remarkably low duplication rate, indicating that the space of valid two-flavon FN realisations has not been exhausted. Both Normal and Inverted neutrino mass squared orderings are realised, with the relative hierarchy between the flavon vevs producing qualitatively distinct predictions for the effective neutrinoless double beta decay mass $m_{ee}$. We further demonstrate the existence of minimal FN realisations with maximal flavon exponent as small as three, and of models reproducing charged fermion masses to within $6\%$ without any dedicated continuous parameter optimisation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper performs the first systematic scan of two-flavon Froggatt-Nielsen models by casting the search for viable realizations as a multi-objective optimization problem solved with the NSGA-III genetic algorithm. It simultaneously optimizes 18 FN charges, 45 Wilson coefficients, and flavon vev parameters to fit CKM/PMNS angles, CP phases, charged-fermion masses, and neutrino mass-squared differences, reporting the discovery of over 100,000 unique viable models (both normal and inverted orderings) with low duplication, distinct m_ee predictions depending on vev hierarchy, and minimal realizations with maximal exponent as small as three.

Significance. If the numerical results hold, the work would be significant for demonstrating that the two-flavon FN parameter space is substantially larger and less exhausted than previously appreciated, while providing a concrete illustration of how genetic algorithms can efficiently explore high-dimensional mixed discrete-continuous spaces in flavor model building. The reported distinction in neutrinoless double-beta-decay predictions between vev hierarchies and the existence of models fitting charged-fermion masses to 6% without dedicated continuous-parameter tuning are concrete, falsifiable outputs that could guide future model-building efforts.

major comments (3)
  1. [Abstract and §3 (algorithm description)] The central claim of >100,000 unique viable models (abstract) rests on the correctness of the NSGA-III implementation, yet no explicit formulation of the individual objective functions (e.g., how CKM angles, CP phases, or Δm² are encoded as separate objectives) or handling of integer FN charges is provided; without this, it is impossible to confirm that constraints are imposed without circularity or implementation error.
  2. [§4 (results) and §5 (discussion)] No convergence diagnostics, population size, generation count, or post-run validation (independent χ² recomputation on the output models) are reported; this leaves open the possibility that the reported count and low duplication rate are affected by incomplete exploration or sampling bias in the 18+45+flavon parameter space.
  3. [§4.2 (neutrino sector results)] The assertion that both normal and inverted orderings are realized with qualitatively distinct m_ee predictions depends on the relative flavon vev hierarchy being correctly sampled; the paper does not show that the multi-objective formulation treats the vev ratio as an independent continuous parameter without artificial constraints.
minor comments (2)
  1. [§2] Notation for the two flavon fields and their charges should be standardized across equations and tables to avoid ambiguity between up- and down-type sectors.
  2. [Abstract] The abstract states that the approach is 'orders of magnitude faster than prior reinforcement learning methods,' but no direct timing or iteration-count comparison is given in the text.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to improve clarity and reproducibility.

read point-by-point responses

  1. Referee: [Abstract and §3 (algorithm description)] The central claim of >100,000 unique viable models (abstract) rests on the correctness of the NSGA-III implementation, yet no explicit formulation of the individual objective functions (e.g., how CKM angles, CP phases, or Δm² are encoded as separate objectives) or handling of integer FN charges is provided; without this, it is impossible to confirm that constraints are imposed without circularity or implementation error.

    Authors: We agree that explicit formulations are necessary for full verification. In the revised manuscript we will add a dedicated subsection in §3 providing the precise mathematical definitions of all objective functions (including the encoding of CKM/PMNS angles, CP phases, and mass-squared differences as separate objectives) together with the exact treatment of integer FN charges and the constraints applied to them. revision: yes

  2. Referee: [§4 (results) and §5 (discussion)] No convergence diagnostics, population size, generation count, or post-run validation (independent χ² recomputation on the output models) are reported; this leaves open the possibility that the reported count and low duplication rate are affected by incomplete exploration or sampling bias in the 18+45+flavon parameter space.

    Authors: We acknowledge that these implementation details were omitted. The revised version will report the NSGA-III population size, generation count, convergence diagnostics, and results of independent χ² recomputation performed on a representative sample of the output models to substantiate the reported count and low duplication rate. revision: yes

  3. Referee: [§4.2 (neutrino sector results)] The assertion that both normal and inverted orderings are realized with qualitatively distinct m_ee predictions depends on the relative flavon vev hierarchy being correctly sampled; the paper does not show that the multi-objective formulation treats the vev ratio as an independent continuous parameter without artificial constraints.

    Authors: The optimization treats the relative flavon vev hierarchy (magnitude ratio and phase) as fully independent continuous parameters subject only to physical bounds. We will revise §4.2 to state this parameterization explicitly and to show that the multi-objective setup imposes no additional artificial constraints, thereby confirming the sampling of both mass orderings and the associated distinct m_ee predictions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; search targets external data

full rationale

The paper casts the search for viable two-flavon FN models as a multi-objective optimization problem solved by NSGA-III, with each experimental constraint (CKM/PMNS angles, CP phases, masses, neutrino mass-squared differences) formulated as a separate objective. The reported count of over 100,000 unique viable models and the qualitative distinctions in m_ee predictions are direct outputs of this search over the 18 charges + 45 Wilson coefficients + flavon parameters; they do not reduce by construction to any fitted input or self-citation. No load-bearing self-citation, uniqueness theorem, or ansatz smuggling appears in the derivation chain. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 1 invented entities

The viability claims rest on the numerical search succeeding and on the two-flavon extension being a valid phenomenological setup; many parameters are optimized rather than derived from first principles.

free parameters (3)
  • 18 FN charges
    Discrete parameters optimized by GA to satisfy constraints
  • 45 Wilson coefficients
    Continuous parameters fitted to match observables
  • flavon vev scales and relative phase
    Parameters controlling hierarchy and CP violation
axioms (2)
  • standard math Standard Model gauge group, particle content, and flavor symmetry framework
    Background for FN mechanism
  • domain assumption Experimental constraints on masses, mixings, and mass-squared differences are accurately encoded as objectives
    GA fitness depends on this encoding
invented entities (1)
  • Two independent flavon fields no independent evidence
    purpose: Separate coupling to up- and down-type sectors allowing relative phase for CP violation
    Extension of single-flavon FN introduced to generate CP violation

pith-pipeline@v0.9.1-grok · 5835 in / 1479 out tokens · 53691 ms · 2026-06-26T17:05:45.562493+00:00 · methodology

discussion (0)

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