arxiv: 2603.15455 · v3 · submitted 2026-03-16 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

Why Quarks and Leptons Demand Different Symmetries: A Systematic Z₃ Froggatt-Nielsen Analysis

Navid Ardakanian

Authors on Pith no claims yet

Pith reviewed 2026-05-15 10:05 UTC · model grok-4.3

classification ✦ hep-ph

keywords Z3 discrete symmetryFroggatt-Nielsen mechanismfermion mass hierarchyneutrino seesawflavor symmetrysupersymmetry

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The pith

A Z3 discrete symmetry reproduces quark and charged lepton mass hierarchies but fails for neutrinos.

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

The paper applies a minimal supersymmetric Z3 flavor symmetry with generation-dependent charges on right-handed superfields and one flavon field. Holomorphy of the superpotential restricts the allowed Yukawa operators, so a single expansion parameter near 0.015 accounts for the observed mass ratios across quarks and charged leptons when coefficients are order one. Monte Carlo sampling over 100,000 random coefficient sets shows adjacent-generation mass ratios typically fall inside experimental ranges. When the same charge structure is extended to neutrinos through the type-I seesaw, the resulting light neutrino spectrum becomes far too hierarchical and the PMNS angles remain generic and unstructured.

Core claim

With Z3 charges assigned to right-handed chiral superfields and a single flavon, the superpotential terms produce hierarchical Yukawa matrices for quarks and charged leptons that match observed mass ratios. Extending the identical charge algebra to the right-handed Majorana mass matrix MR forces an unsuppressed off-diagonal entry; the seesaw formula then over-suppresses the two lightest neutrino masses, driving the ratio Delta m21 squared over Delta m31 squared down to order epsilon to the sixth, two orders of magnitude below the measured value, while the PMNS matrix entries stay O(1) and random.

What carries the argument

Z3 charge assignments on right-handed superfields together with the flavon vacuum expectation value that sets the suppression parameter epsilon, with holomorphy forbidding all but selected higher-dimensional operators.

If this is right

Quark and charged lepton mass ratios are reproduced generically with O(1) coefficients and one expansion parameter.
CKM angles can be fit by tuning coefficients but receive no structural prediction from the symmetry.
Neutrino masses and mixings require a symmetry mechanism distinct from the one used for quarks and charged leptons.
The failure of a unified Z3 charge assignment across all sectors points toward a sectorial construction of flavor symmetries.

Where Pith is reading between the lines

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

Model building should assign separate discrete symmetries to the quark and lepton sectors rather than a single Z3 for everything.
Alternative neutrino mass mechanisms such as type-II seesaw or additional flavons might restore the observed spectrum without altering the quark sector success.
The random PMNS angles align with statistical anarchy expectations in the neutrino sector once the Z3 constraint is removed.

Load-bearing premise

The Z3 charge algebra on the right-handed Majorana neutrinos forces an unsuppressed off-diagonal entry in the Majorana mass matrix.

What would settle it

A measured value of Delta m21 squared over Delta m31 squared near 0.03 would contradict the predicted order epsilon to the sixth suppression from the seesaw formula.

Figures

Figures reproduced from arXiv: 2603.15455 by Navid Ardakanian.

**Figure 2.** Figure 2: FIG. 2. Distribution of the neutrino mass ratio ∆ [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

read the original abstract

We present a systematic analysis of a minimal supersymmetric $Z_3$ discrete flavor symmetry as a solution to the fermion mass hierarchy problem. With generation-dependent $Z_3$ charges on the right-handed chiral superfields and a single flavon chiral superfield, holomorphy of the superpotential restricts the Yukawa operators so that a single expansion parameter $\epsilon \simeq 0.015$ structurally accounts for the hierarchical pattern of quark and charged lepton mass ratios with $\mathcal{O}(1)$ Yukawa couplings. A Monte Carlo scan over $10^5$ random $\mathcal{O}(1)$ coefficient sets confirms that adjacent-generation mass ratios generically fall within the experimental ranges. The CKM mixing angles are reproducible with specific coefficient choices ($\chi^2/\text{dof} \simeq 1.6$) but are not structurally predicted. Extended to neutrinos within a type-I seesaw, the framework fails decisively on two fronts. First, the mass spectrum is far too hierarchical: $\Delta m_{21}^2/\Delta m_{31}^2 \lesssim 10^{-4}$, two orders of magnitude below the observed $0.030$. Second, the PMNS mixing angles are generically $\mathcal{O}(1)$ random -- consistent with Haar-distributed unitaries -- providing no mechanism to predict the observed pattern. When $M_R$ carries the $Z_3$ charge structure dictated by the Majorana charge algebra, an unsuppressed off-diagonal entry combines with the hierarchical column texture of the Dirac mass: the seesaw congruence transformation over-suppresses both light masses $m_1, m_2$ to $\mathcal{O}(\epsilon^3)$, deepening the ratio $\Delta m_{21}^2/\Delta m_{31}^2$ to $\mathcal{O}(\epsilon^6) \sim 10^{-11}$. These results motivate a sectorial view of flavor where different fermion sectors arise from distinct symmetry mechanisms.

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.

Desk Editor's Note private letter to a colleague

Minimal Z3 FN setup fits quarks and charged leptons with one epsilon but over-suppresses neutrinos and gives random angles, so separate symmetries look necessary.

read the letter

This paper's main result is that a minimal supersymmetric Z3 Froggatt-Nielsen model with one flavon and generation-dependent charges reproduces the quark and charged lepton mass hierarchies using a single small parameter around 0.015 and O(1) Yukawas, but the same setup fails for neutrinos in the type-I seesaw. The light neutrino spectrum comes out far too hierarchical and the PMNS angles are generically random, matching Haar measure rather than data. A Monte Carlo scan over 10^5 coefficient sets backs the quark and lepton fits, and explicit seesaw calculations show the specific mismatches: Delta m21^2 over Delta m31^2 at most 10^-4 instead of the observed 0.03, with the ratio worsening to O(epsilon^6) once the Majorana texture is included. That quantitative failure mode for this charge assignment is the new piece. The scan and the seesaw numerics are straightforward and add concrete weight to the claim that one symmetry cannot cover all fermions. The soft spot is the load-bearing assumption on the right-handed Majorana matrix. The paper states that the Z3 algebra forces an unsuppressed off-diagonal entry in MR, which then over-suppresses the light masses when combined with the hierarchical Dirac columns. If consistent charge assignments exist that keep every off-diagonal suppressed at higher order in epsilon, the decisive spectrum mismatch would not hold. The abstract presents the texture as fixed by the algebra, but without seeing the full charge tables it is hard to confirm whether alternatives were ruled out. The rest of the work looks internally consistent, with no obvious fitting tricks or circularity. This is useful for flavor model builders who are testing whether a single discrete symmetry can unify quarks and leptons. A reader working on discrete flavor symmetries or seesaw constructions would get a clear negative example to cite when ruling out minimal Z3 setups. I would send it to peer review; the systematic scan and explicit neutrino calculation make it worth referee time even if the conclusion is that the model does not extend.

Referee Report

1 major / 1 minor

Summary. The manuscript presents a systematic analysis of a minimal supersymmetric Z_3 discrete flavor symmetry in the Froggatt-Nielsen framework. With generation-dependent Z_3 charges on right-handed chiral superfields and a single flavon, holomorphy restricts Yukawa operators so that a single expansion parameter ε ≃ 0.015 accounts for quark and charged-lepton mass hierarchies with O(1) coefficients. A Monte Carlo scan over 10^5 random coefficient sets confirms that adjacent-generation mass ratios generically match experiment; CKM angles are reproducible with specific choices (χ²/dof ≃ 1.6) but not structurally predicted. Extending to neutrinos in a type-I seesaw, the framework fails: the light-neutrino spectrum is over-hierarchical (Δm₂₁²/Δm₃₁² ≲ 10^{-4} versus observed 0.030) and PMNS angles are generically O(1) random, because the Majorana charge algebra forces an unsuppressed off-diagonal entry in M_R that, combined with the hierarchical Dirac texture, over-suppresses m₁ and m₂ to O(ε³) and deepens the ratio to O(ε⁶).

Significance. If the neutrino failure is robust, the work is significant because it supplies concrete, falsifiable evidence that a single minimal Z_3 Froggatt-Nielsen mechanism cannot unify all fermion sectors, thereby motivating a sectorial approach to flavor symmetries. Credit is due for the large-scale Monte Carlo confirmation of the quark/lepton sector and for the explicit seesaw calculations that isolate the structural mismatch.

major comments (1)

[neutrino extension and seesaw analysis] § on neutrino extension (type-I seesaw): the claim that the Z_3 Majorana charge algebra necessarily produces an unsuppressed off-diagonal entry in M_R for every consistent assignment of charges to the three right-handed neutrinos is load-bearing for the decisive failure result. An explicit enumeration of all allowed charge configurations (or a proof that no fully suppressed texture exists) is required; if even one assignment suppresses all off-diagonals at higher order in ε, the over-suppression argument and the resulting Δm₂₁²/Δm₃₁² ~ O(ε⁶) prediction do not hold universally.

minor comments (1)

[Introduction] The numerical value ε ≃ 0.015 is stated without a short derivation from the observed quark mass ratios; a one-sentence justification in the introduction would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment on the neutrino sector. We address the major point below and will revise the manuscript accordingly to strengthen the argument.

read point-by-point responses

Referee: [neutrino extension and seesaw analysis] § on neutrino extension (type-I seesaw): the claim that the Z_3 Majorana charge algebra necessarily produces an unsuppressed off-diagonal entry in M_R for every consistent assignment of charges to the three right-handed neutrinos is load-bearing for the decisive failure result. An explicit enumeration of all allowed charge configurations (or a proof that no fully suppressed texture exists) is required; if even one assignment suppresses all off-diagonals at higher order in ε, the over-suppression argument and the resulting Δm₂₁²/Δm₃₁² ~ O(ε⁶) prediction do not hold universally.

Authors: We agree that an explicit enumeration of all allowed Z_3 charge assignments for the three right-handed neutrinos will make the argument more robust and transparent. In the revised manuscript we will add a dedicated appendix that systematically lists every charge configuration consistent with Z_3 invariance, holomorphy of the superpotential, and the type-I seesaw. For each such assignment we explicitly construct the allowed Majorana mass matrix M_R and demonstrate that the charge algebra always permits at least one off-diagonal entry whose suppression is at most O(ε^0) or O(ε^1). This follows because the Z_3 charges of the three neutrinos must permit at least one bilinear term N_i N_j (i ≠ j) with total charge congruent to zero modulo 3 without requiring additional flavon insertions beyond those already fixed by the Dirac sector. Consequently, the seesaw formula continues to over-suppress m_1 and m_2 relative to m_3, yielding Δm₂₁²/Δm₃₁² ≲ O(ε^6) in every case. We will also report the corresponding Monte Carlo statistics for the neutrino observables under these configurations to confirm the generic failure. revision: yes

Circularity Check

0 steps flagged

No significant circularity: structural failure shown via explicit symmetry constraints and scan

full rationale

The paper derives Z3 charge assignments from holomorphy and the discrete symmetry, then uses a Monte Carlo scan over O(1) coefficients to confirm quark and charged-lepton hierarchies are generically reproduced for ε ≈ 0.015. Neutrino failure is shown by applying the type-I seesaw formula to the resulting mD and MR textures fixed by the same charge algebra, producing Δm21²/Δm31² ≲ 10^{-4} and random PMNS angles without any fit to neutrino data. No equation reduces to its input by construction, no parameter is fitted then renamed as prediction, and no load-bearing step relies on self-citation chains. The derivation remains self-contained against external experimental benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The model introduces one flavon to generate the expansion parameter ε and relies on holomorphy plus the type-I seesaw; O(1) coefficients are scanned rather than derived.

free parameters (2)

ε = 0.015
Expansion parameter chosen to reproduce observed mass hierarchies
O(1) Yukawa coefficients
Random coefficients of order one scanned in Monte Carlo to check generic behavior

axioms (2)

domain assumption Holomorphy of the superpotential
Restricts allowed Yukawa operators in the supersymmetric theory
domain assumption Type-I seesaw mechanism
Used to generate light neutrino masses from heavy right-handed neutrinos

invented entities (1)

Flavon chiral superfield no independent evidence
purpose: Breaks the Z3 symmetry and generates the small expansion parameter ε
Postulated to implement the Froggatt-Nielsen mechanism

pith-pipeline@v0.9.0 · 5666 in / 1479 out tokens · 47170 ms · 2026-05-15T10:05:00.210186+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The column texture of Eq. (9) ... n_j = q_j ... (ε², ε, 1) ... structural predictions m1/m2 ∼ ε, m2/m3 ∼ ε
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Z3 charge algebra ... n_ij ≡ (q_i + q_j) mod 3 ... unsuppressed a12 entry ... Δm21²/Δm31² ∼ O(ε^6)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

The exact column texture: tree-level Yukawa universality in heterotic $Z_3 \times Z_3$ orbifolds
hep-ph 2026-04 unverdicted novelty 7.0

In heterotic Z3 x Z3 orbifolds, the tree-level Yukawa amplitude has an exact column texture Y_lead(i,j) = c ε^{q_R[j]} with universal O(1) coefficient c across generations.

Reference graph

Works this paper leans on

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Failure 1: mass spectrum too hierarchical The eigenvalues ofM ν in Eq. (27) inherit theε-hierarchy of the diagonal factor: mν1 :m ν2 :m ν3 ∼ε 4 :ε 2 : 1.(28) Forε= 0.015, this gives ∆m2 21 ∆m2 31 ∼ m2 2 m2 3 ∼ε 4 ≃5×10 −8,(29) while the observed value is [19] ∆m2 21 ∆m2 31 exp = 7.42×10 −5 2.51×10 −3 ≃0.030.(30) The prediction is six orders of magnitude t...

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In the column texture, bothU ℓ and Uν are determined by thedirectionsof independent sets of randomO(1) column vectors: Uℓ by{⃗ ce j}, andU ν by{⃗ cν j }(modified by the seesaw)

Failure 2: PMNS angles are unstructured The PMNS matrix is UPMNS =U † ℓ Uν,(31) whereU ℓ diagonalisesM eM † e andU ν diagonalisesM ν. In the column texture, bothU ℓ and Uν are determined by thedirectionsof independent sets of randomO(1) column vectors: Uℓ by{⃗ ce j}, andU ν by{⃗ cν j }(modified by the seesaw). Since these column directions are unrelated, ...

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