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arxiv: 2606.09431 · v1 · pith:2XHOFWN5new · submitted 2026-06-08 · ✦ hep-ph · hep-th

Model of Flavors

Jiri Hosek This is my paper

Pith reviewed 2026-06-27 15:59 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords quantum flavor dynamicsdynamical electroweak symmetry breakingcomposite Higgs bosonssterile neutrinosMajorana massesPagels-Stokarflavor symmetryseesaw mechanism
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The pith

Quantum flavor dynamics generates small Dirac masses whose squares set the W and Z boson masses at the Fermi scale.

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

The paper revives dynamical electroweak symmetry breaking by replacing the Higgs sector with a strongly coupled chiral gauge theory called quantum flavor dynamics. This SU(3)_f interaction at a high scale produces large Majorana masses for one right-handed neutrino per flavor and small Dirac masses for the standard-model fermions, with the latter degenerate within each flavor. Spontaneous breaking by these Dirac masses then fixes the W and Z masses in terms of the square root of the sum of all m_f squared and produces three composite scalar states at that scale. The approach also claims that an existing UV-finite dynamical perturbation theory can split the degenerate masses according to electric charge and mass ratios, making six Majorana neutrino masses calculable via the seesaw.

Core claim

The BCS-motivated idea of Weinberg and Salam on dynamical EW symmetry breaking is revived: The Higgs sector of the EW gauge model of three fermion flavors (families) is replaced with the chiral gauge SU(3)_f quantum flavor dynamics (QFD) strongly coupled at a huge scale Λ. With all chiral fermions in flavor triplets the anomaly freedom demands the welcome BSM extension of the SM fermion sector by one EW-sterile neutrino ν_R per flavor. The QFD distinguishes flavors by generating at strong coupling the chirality-prohibited fermion masses: Three different Majorana masses M_f ∼ Λ of ν_R, and three different, arguably small Dirac masses m_f of SM fermions degenerate for all species in each flavo

What carries the argument

The chiral gauge SU(3)_f quantum flavor dynamics (QFD) strongly coupled at scale Λ, which generates the Majorana and Dirac fermion masses and drives the spontaneous symmetry breakdowns.

If this is right

  • W and Z boson masses are fixed by the square root of the sum of all standard-model fermion Dirac masses squared.
  • Three composite scalar Higgs bosons appear at the electroweak scale.
  • Six Majorana neutrino masses become calculable through the seesaw mechanism after Pagels-Stokar splitting.
  • All flavor gluons acquire masses of order the large Majorana masses M_f.
  • A light pseudo-Nambu-Goldstone Majoron boson appears from the breaking of the flavor symmetry.

Where Pith is reading between the lines

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

  • Applying the Pagels-Stokar splitting in detail could yield explicit numerical predictions for the individual m_f values that can be compared with measured fermion masses.
  • The composite Higgs states would generally have different production cross sections and decay branching ratios than a single fundamental Higgs, offering a potential discriminator at colliders.
  • The presence of the Majoron and the heavy flavor gluons could leave imprints in early-universe cosmology or rare decays.

Load-bearing premise

The Dirac masses of the standard-model fermions are generated by the quantum flavor dynamics, remain small, and stay nearly degenerate within each flavor so that the Pagels-Stokar dynamical perturbation theory can be applied.

What would settle it

A direct search at the LHC or a future collider that finds only one fundamental scalar Higgs boson rather than three composite 0⁺ states near 246 GeV.

Figures

Figures reproduced from arXiv: 2606.09431 by Jiri Hosek.

Figure 1
Figure 1. Figure 1: FIG. 1. The loop-built bridge between the right-handed and [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

The BCS-motivated idea of Weinberg and Salam on dynamical EW symmetry breaking is revived: The Higgs sector of the EW gauge model of three fermion flavors (families) is replaced with the chiral gauge $SU(3)_f$ quantum flavor dynamics (QFD) strongly coupled at a huge scale $\Lambda$. I. With all chiral fermions in flavor triplets the anomaly freedom demands the welcome BSM extension of the SM fermion sector by one EW-sterile neutrino $\nu_R$ per flavor. II. The QFD distinguishes flavors by generating at strong coupling the chirality-prohibited (i.e. calculable) fermion masses: Three different Majorana masses $M_f \sim \Lambda$ of $\nu_R$, and three different, arguably small Dirac masses $m_f$ of SM fermions degenerate for all species in each flavor. 1. Complete spontaneous breakdown of $SU(3)_f \times U(1)$ by $M_f$ implies: (i) All flavor gluons acquire self-consistently masses $\sim M_f$. (ii) There is the $\nu_R$-composite pseudo-NG Majoron. (iii) There are three very heavy $0^{+}$ $\nu_R$-composite Higgs bosons. 2. Spontaneous breakdown of the EW $SU(2)_L \times U(1)_Y$ symmetry to $U(1)_{em}$ by $m_f$, in sharp contrast with the Higgs mechanism, implies: (i) The $W$ and $Z$ bosons acquire masses $\sim g(\sum m^2_f)^{1/2}$ and $\sim (g^2+g^{'2})^{1/2}(\sum m^2_f)^{1/2}$ respectively, defining the effective Fermi scale $v=246 \rm GeV$. (ii) There are three SM-fermion-composite $0^{+}$ Higgs bosons $h_f$ at this scale. III. The UV-finite EW dynamical perturbation theory of Pagels and Stokar splits the flavor-degenerate masses of SM-fermions by their electric charges and the ratios $m_f/m_{W,Z}$. Six Majorana neutrino masses are calculable by seasaw.

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 manuscript revives dynamical electroweak symmetry breaking via a strongly coupled SU(3)_f quantum flavor dynamics (QFD) at scale Λ. It generates three Majorana masses M_f ~ Λ for right-handed neutrinos and three degenerate Dirac masses m_f for SM fermions per flavor. Spontaneous breaking by M_f gives masses to flavor gluons and produces composite states including a pseudo-NG Majoron. Breaking by m_f yields W and Z masses ~ g (∑ m_f²)^{1/2} and ~ (g² + g'²)^{1/2} (∑ m_f²)^{1/2}, defining v = 246 GeV, plus three composite 0⁺ Higgs bosons h_f. The Pagels-Stokar formalism is invoked to split the degenerate m_f by charge and m_f/m_{W,Z} ratios, with six Majorana neutrino masses claimed calculable via seesaw.

Significance. If the missing derivations were supplied and the Pagels-Stokar application justified, the approach would constitute a parameter-light dynamical alternative to the elementary Higgs, with calculable flavor hierarchies and composite states. The current absence of explicit integrals, gap equations, or quantitative spectra prevents evaluation of whether it reproduces the observed fermion masses or resolves the hierarchy problem beyond the input scale Λ.

major comments (3)
  1. [Abstract] Abstract and EW symmetry-breaking paragraph: The W/Z mass expressions are written as proportional to (∑ m_f²)^{1/2} with the effective Fermi scale defined by setting v = (∑ m_f²)^{1/2} = 246 GeV through choice of the input m_f values; this renders the mass prediction circular rather than independently derived from the QFD dynamics.
  2. [Pagels-Stokar splitting] Section on Pagels-Stokar splitting: No explicit integral over a dynamical mass function, gap equation, or effective-action derivation is referenced to show how the UV-finite EW perturbation theory splits the flavor-degenerate m_f; the formalism originates in strong-coupling self-consistency, yet is applied here to perturbative splitting in the weak EW sector without justification of the regime change.
  3. [Neutrino masses and composite states] Claims of calculable neutrino masses and composite Higgs bosons h_f: The seesaw and composite-Higgs statements are asserted without the required matching conditions, mass matrices, or decay-constant integrals that would follow from the QFD gap equations at scale Λ.
minor comments (2)
  1. Notation for the three flavor indices and the distinction between M_f and m_f should be introduced with a single table or equation block for clarity.
  2. The manuscript would benefit from a dedicated paragraph contrasting the present construction with prior dynamical-breaking attempts that also invoke composite scalars.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment below with clarifications on the dynamical origin of the scales and structures in the model, and we indicate revisions to improve explicitness where the presentation was insufficient.

read point-by-point responses

  1. Referee: [Abstract] Abstract and EW symmetry-breaking paragraph: The W/Z mass expressions are written as proportional to (∑ m_f²)^{1/2} with the effective Fermi scale defined by setting v = (∑ m_f²)^{1/2} = 246 GeV through choice of the input m_f values; this renders the mass prediction circular rather than independently derived from the QFD dynamics.

    Authors: The m_f are not free inputs chosen to fit v; they are generated dynamically by the strong SU(3)_f dynamics at scale Λ via the gap equations, with the overall normalization fixed by requiring the resulting electroweak scale to match the observed v = 246 GeV. The individual values and ratios among the three m_f are then determined by the subsequent Pagels-Stokar splitting. We agree the abstract phrasing is ambiguous on this point and will revise it (and the corresponding paragraph) to state explicitly that v is matched to experiment while the spectrum and hierarchies follow from the QFD dynamics. revision: partial

  2. Referee: [Pagels-Stokar splitting] Section on Pagels-Stokar splitting: No explicit integral over a dynamical mass function, gap equation, or effective-action derivation is referenced to show how the UV-finite EW perturbation theory splits the flavor-degenerate m_f; the formalism originates in strong-coupling self-consistency, yet is applied here to perturbative splitting in the weak EW sector without justification of the regime change.

    Authors: The Pagels-Stokar integral is applied in the effective theory below Λ, where the electroweak couplings are perturbative. The strong dynamics sets the common m_f at scale Λ; the weak gauge interactions then induce the charge-dependent splittings via the standard Pagels-Stokar formula. We will add an appendix containing the explicit integral expression for the splitting, the relevant gap equation, and a short justification that the expansion is valid because g, g' ≪ 1 at the electroweak scale while the strong sector is integrated out at Λ. revision: yes

  3. Referee: [Neutrino masses and composite states] Claims of calculable neutrino masses and composite Higgs bosons h_f: The seesaw and composite-Higgs statements are asserted without the required matching conditions, mass matrices, or decay-constant integrals that would follow from the QFD gap equations at scale Λ.

    Authors: The composite states and seesaw structure follow directly from the symmetry-breaking pattern of the strong SU(3)_f dynamics, but the manuscript does not supply the explicit matching conditions or integrals. We will add a new section deriving the 3×3 Majorana and Dirac mass matrices from the gap equations, the decay constants f ∼ Λ for the composite Higgses h_f, and the resulting light neutrino spectrum after seesaw diagonalization. This will make the calculability quantitative. revision: yes

Circularity Check

1 steps flagged

Fermi scale v defined directly from (∑ m_f²)^{1/2} set to 246 GeV, making W/Z masses tautological

specific steps
  1. self definitional [Abstract]
    "The W and Z bosons acquire masses ∼ g(∑ m_f²)^{1/2} and ∼ (g²+g'²)^{1/2}(∑ m_f²)^{1/2} respectively, defining the effective Fermi scale v=246 GeV."

    The masses are expressed directly in terms of (∑ m_f²)^{1/2}, which is then defined to equal the experimentally known v=246 GeV. The boson masses are therefore constructed to match observation by choice of the m_f magnitudes rather than derived independently from the dynamics.

full rationale

The paper's central derivation states that spontaneous breakdown by m_f implies W/Z masses proportional to (∑ m_f²)^{1/2}, then explicitly defines that quantity as the known v=246 GeV. This reduces the mass expressions to a restatement of the input scale chosen to reproduce experiment, rather than an independent prediction. The m_f values (generated by QFD but degenerate and small) must be selected to enforce the numerical match. No other load-bearing steps (e.g., Pagels-Stokar splitting) exhibit explicit reduction to inputs by the paper's own equations in the provided text; the core EW mass claim is self-definitional.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 4 invented entities

The central claim rests on the existence of strong SU(3)_f dynamics at high scale, the requirement of sterile neutrinos for anomaly cancellation, and the applicability of dynamical mass generation mechanisms without providing independent verification of these assumptions.

free parameters (2)
  • scale Λ
    The strong coupling scale of QFD, chosen to be huge so that M_f ~ Λ are large.
  • Dirac masses m_f
    Three different small Dirac masses chosen such that ∑ m_f² reproduces the observed Fermi scale v = 246 GeV.
axioms (3)
  • domain assumption Anomaly freedom of the chiral gauge theory with all fermions in flavor triplets requires one right-handed sterile neutrino ν_R per flavor.
    Stated explicitly as demanding the BSM extension of the SM fermion sector.
  • domain assumption The QFD is strongly coupled at scale Λ and breaks SU(3)_f completely via the Majorana masses M_f.
    Core assumption enabling the spontaneous breakdown and mass generation.
  • domain assumption The UV-finite EW dynamical perturbation theory of Pagels and Stokar applies to split the flavor-degenerate masses.
    Invoked to generate the observed mass splittings by charges and ratios.
invented entities (4)
  • ν_R-composite pseudo-NG Majoron no independent evidence
    purpose: Arises from spontaneous breakdown of SU(3)_f × U(1)
    Postulated as a consequence of the symmetry breaking by M_f.
  • three very heavy 0⁺ ν_R-composite Higgs bosons no independent evidence
    purpose: Result of SU(3)_f breakdown
    Composite states from the strong dynamics.
  • three SM-fermion-composite 0⁺ Higgs bosons h_f no independent evidence
    purpose: Result of EW symmetry breakdown by m_f
    Composite states at the Fermi scale.
  • flavor gluons with masses ~ M_f no independent evidence
    purpose: Gauge bosons of SU(3)_f acquiring masses from breaking
    Self-consistent mass generation from the dynamics.

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Reference graph

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