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REVIEW 3 major objections 6 minor 54 references

Natural neutrino sector in a 331-model with Froggatt-Nielsen mechanism

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Seesaw in a 331 model explains all fermion masses and families.

desk verdict Nicely built double-seesaw extension of the FN331 model, but the 'natural' in the title is asserted rather than demonstrated: the benchmarks look tuned and PMNS anarchy is an input, not a prediction. read the letter →

arxiv 1908.09384 v2 pith:RE74PDPD submitted 2019-08-25 hep-ph

Katri Huitu , Niko Koivunen , Timo J. Kärkkäinen This is my paper · ORCID
classification hep-ph
keywords neutrinomasshierarchyseesawmechanism331-modelFroggatt-NielsenflavoursymmetryPMNSmatrixsterileneutrinosthreefermionfamilies
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends the FN331 model — a version of the $SU(3)_c\times SU(3)_L\times U(1)_X$ (331) gauge theory in which the Froggatt-Nielsen mechanism generates the charged-fermion mass hierarchy — by adding three right-handed neutrino singlets. It claims that this extension realizes the seesaw mechanism at tree level, so that all three light neutrinos get sub-eV masses with roughly equal, order-one mixings matching the observed PMNS matrix. Because the Froggatt-Nielsen suppression makes the neutrino Yukawa couplings tiny, the seesaw scale is set by the $SU(3)_L\times U(1)_X$ breaking scale, about 7–50 TeV, instead of a much higher scale. The resulting fermion sector simultaneously accounts for the number of fermion families and for the mass hierarchy of all fermions, including neutrinos, without fine-tuning.

What carries the argument

The load-bearing objects are the effective flavon $\rho^\dagger\chi$ — the gauge-invariant scalar combination whose vacuum expectation value $\langle\rho^\dagger\chi\rangle=v_2u/2$ supplies the Froggatt-Nielsen counting parameter $\epsilon=(v_2u)/(2\Lambda^2)$ — and the $9\times 9$ neutrino mass matrix whose block structure realizes a double seesaw. The right-handed neutrino Majorana mass matrix is $M_{ij}\simeq\Lambda\,c^M_{ij}\,\epsilon^{q(N_{R,i})+q(N_{R,j})}$, and with $q(N_R)=0$ the heavy scale is simply the messenger scale $\Lambda$. Equal FN charges $q(L^c_{L,1})=q(L^c_{L,2})=q(L^c_{L,3})=L$ guarantee that the matrices diagonalizing the light-neutrino and charged-lepton blocks have no internal hierarchy, so the PMNS matrix is anarchical, while the same $L$ controls the sub-eV scale through the factor $\epsilon^{2L+2}$.

What would settle it

A decisive test is the absolute neutrino mass scale and ordering: the model predicts normal ordering with the lightest mass at a few meV and $\sum m_\nu\simeq 0.06$ eV; observing an inverted ordering, or a cosmological sum above $0.12$ eV, would falsify the neutrino sector. Independently, discovering a fourth family of matter particles would falsify the three-family explanation.

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Extended reading notes

Core claim

In the FN331 model, three new right-handed neutrino singlets combine with the existing lepton-triplet structure to produce a $9\times 9$ neutrino mass matrix of double-seesaw type: three light sub-eV active neutrinos, three medium-mass sterile neutrinos (eV to keV in the benchmark points), and three heavy mostly right-handed neutrinos at the $SU(3)_L\times U(1)_X$ breaking scale. The light masses scale as $m_{\rm light}\sim (v_{\rm light}^2/v_{\rm heavy})\,\epsilon^{2L+2}$, where $v_{\rm light}$ is the electroweak VEV, $v_{\rm heavy}$ is the 7–50 TeV scale, $\epsilon$ is the Froggatt-Nielsen expansion parameter, and $L$ is the common FN charge of the lepton triplets; with $L=8$ or $9$ the masses land in the few-meV range. Equal lepton-triplet FN charges make both the light-neutrino diagonalization matrix and the charged-lepton diagonalization matrix anarchical, so the PMNS matrix is automatically order one, as observed. The paper presents three benchmark points that satisfy the measured mass-squared differences, the cosmological bound on the sum of neutrino masses, and the bounds on sterile-neutrino mixing; in the 7 TeV benchmark the lightest sterile neutrino can account for the short-baseline oscillation anomaly.

Load-bearing premise

The load-bearing premise is the hand-picked flavour-charge assignment — equal charges for the three lepton doublets and zero charges for the right-handed neutrinos, chosen for simplicity — since no symmetry forces it, and a one-unit charge difference would turn the predicted neutrino-mixing pattern from roughly uniform to hierarchical, contradicting experiment.

Editorial extensions

If this is right

  • The seesaw scale is tied to the $SU(3)_L\times U(1)_X$ breaking scale: heavy right-handed neutrinos sit at tens of TeV, low enough that future colliders could in principle search for them, although the benchmark points have very small active–sterile mixing.
  • The model predicts three sterile neutrinos in two mass bands: three medium states around eV–keV and three heavy states at 10–500 TeV; the medium states escape the bound on the number of light neutrinos because their $Z$-boson couplings are suppressed by $v_{\rm light}/v_{\rm heavy}$.
  • Active neutrino masses come out sub-eV with normal ordering and $\sum m_\nu\simeq 0.06$ eV, within reach of next-generation neutrinoless double-beta decay and cosmological probes for the 7 TeV benchmark.
  • The anarchical PMNS matrix follows directly from equal lepton-triplet FN charges; if future high-precision measurements found a hierarchical pattern in the PMNS entries, the charge assignment would have to be revised.
  • The lightest sterile neutrino of the 7 TeV benchmark can account for the short-baseline oscillation anomaly, so short-baseline experiments provide a direct test of this parameter region.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The equal-charge condition is doing more work than a prediction: the model does not derive $q(L_i)=q(L_j)$ from any symmetry, so an anarchical PMNS matrix is a constraint on the flavour-charge assignment, not an output of the gauge structure alone.
  • Because the flavon is a gauge-scalar combination rather than an extra singlet, $\epsilon$ is fixed by the same VEVs that set the $Z'$ and $V^\pm$ masses; a future discovery of those gauge bosons would pin down the heavy-neutrino spectrum through a calculable relation.
  • The 7 TeV benchmark has the largest non-unitarity and sterile-mixing effects; long-baseline and short-baseline oscillation experiments can distinguish it from the higher-scale benchmarks, effectively measuring the 331-breaking scale without directly producing new particles.
  • Nothing fixes the value $L=8$ or $9$: it is chosen to put light neutrinos in the sub-eV range, so a complete theory would still need to explain why the lepton triplets carry such a large flavour charge.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper extends the Froggatt-Nielsen 331 model (FN331) by adding three right-handed neutrino singlets, N_{R,i}. This allows a tree-level seesaw mechanism with the mass scale tied to the SU(3)_L × U(1)_X breaking scale. The authors derive the hierarchy of the 9×9 neutrino mass matrix using FN charges, block-diagonalize it (Sec. 6.1), and identify three light, three medium, and three heavy mass eigenstates. They choose lepton FN charge assignments (Eqs. (40)–(41), Table 3) and provide three benchmark points BP1–BP3 with VEVs and coefficients in the interval |c| ∈ [0.5,5]. They demonstrate that the light-neutrino masses, squared mass differences, and PMNS mixings satisfy experimental constraints, and discuss bounds from neutrinoless double beta decay, kink searches, and the MiniBooNE anomaly. The paper concludes that the model simultaneously explains the number of families and the mass hierarchy of all fermions, including neutrinos, without fine-tuning.

Significance. The formal structure is sound: the seesaw block-diagonalization in Sec. 6.1 is standard, and the benchmark points in Table 4 reproduce Δm²₂₁ and |Δm²₃₂| within experimental ranges and satisfy the quoted bounds. The main strength is the demonstration that FN suppression can lower the seesaw scale to the 7–50 TeV range, making the sterile sector potentially testable. However, the claimed naturalness is not established: only three hand-picked benchmark points are shown, the PMNS anarchy is an input, and the light-neutrino mass scale is fitted by the choice of L. The paper is a useful model-building exercise, but the central claim of a 'natural' explanation is overstated without a parameter scan or a quantitative fine-tuning measure.

major comments (3)
  1. [Section 8.2 and Eq. (50)] The naturalness claim is not supported by the numerical analysis. The paper states (Section 8.2) that the order-one coefficients lie in |c| ∈ [0.5,5] 'to retain naturalness of the parameters', but only three benchmark points are given, with coefficients fixed to four decimal places. No scan over the allowed interval is reported, and no fine-tuning measure is defined. This matters quantitatively: for BP1, m1 = 0.0234 meV (Table 4), whereas the natural entry scale from Eq. (51) is (v_light²/v_heavy)ε^(2L+2) ≈ 3.8 meV, a suppression factor of about 160. Since m_light is a sum of two comparable rank-2 structures (Eq. (50)), this suppression represents a cancellation that no symmetry enforces. The claim in Section 9 that the model works 'without fine-tuning' requires a scan or a quantitative measure to be credible.
  2. [Eq. (40) and Section 7] The anarchical PMNS matrix is an input rather than a prediction. Eq. (40) sets all lepton-doublet FN charges equal, which is what makes U_e^L and U_ν anarchical (Eqs. (54) and (61)). The paper itself acknowledges in Section 7 that 'this is the extent which Froggatt-Nielsen setting can predict the structure of PMNS-matrix.' If any charge differed by one unit, PMNS elements would be suppressed by powers of ε, contradicting the measured values in Eq. (38). Thus the model accommodates anarchy by construction, rather than explaining it, which weakens the claim in Section 9 that mixings are 'explained without fine-tuning.'
  3. [Section 8.1, Eq. (65), Eq. (67)] The light-neutrino mass scale and the charged lepton hierarchy are fitted inputs. Section 8.1 states that 'By setting vlight to electroweak scale and L∼8 or 9, one obtains light-neutrino masses from the correct ballpark,' and the charged lepton hierarchy is fixed by choosing q(eR,i) to produce the texture in Eq. (67). The charges L and q(eR,i) are not determined by any symmetry or anomaly condition; they are chosen to reproduce data. Consequently, the abstract's claim that the model 'explains the mass hierarchy of all the fermions, including neutrinos' is overstated; at present the model parameterizes the hierarchy with free FN charges.
minor comments (6)
  1. [Section 2.2] "Golstone boson" should be "Goldstone boson" in the paragraph following Eq. (11).
  2. [Section 2.1] "antriplets" is a typo; it should be "antitriplets" in the sentence 'This is achieved by assigning two quark families to SU(3)_L antitriplets.'
  3. [Section 3] "incredients" should be "ingredients" and "unneccesary" should be "unnecessary" in the first paragraph of Section 3.
  4. [Section 8] In the sentence 'This is in constrast to many', "constrast" should be "contrast", and later "disapprearance effect" should be "disappearance effect".
  5. [Table 4] The row labeled "NSI strength" has no entries; the text (Section 8) gives values of O(10⁻¹³), so the table should either include those numbers or the row should be removed.
  6. [Eq. (59)] The notation in Eq. (59) is unclear: the term "sinθB1†1" appears to contain a misprint, likely B_1^{1}† or B_1^1†; please clarify the index placement.

Circularity Check

3 steps flagged · score 6.0 of 10

PMNS anarchy and the sub-eV neutrino scale are put in by hand through Eq. (40) and Eq. (65); the sterile spectrum, NSI and non-unitarity are genuine outputs, so the circularity is partial (6/10).

  1. self definitional [Sec. 6, Eq. (40); Sec. 7, Eqs. (61)-(62)]
    "The anarchical hierarchy is achieved when all the lepton families are treated equally under the FN-symmetry. We will therefore choose from now on all the lepton triplets to have equal FN-charges: q(Lc L,1) = q(Lc L,2) = q(Lc L,3)≡L. ... The texture for the PMNS is therefore anarchical as well ... We note here that this is the extent which Froggatt-Nielsen setting can predict the structure of PMNS-matrix."

    The abstract and Sec. 9 claim that the model naturally explains the observed O(1) PMNS mixings without fine-tuning. But Eq. (40) simply assumes all lepton doublets carry equal FN charges, and then Eqs. (61)-(62) follow: Ue_L and Uν have no hierarchy, so UPMNS is anarchical. If any charge differed by one unit, entries would be suppressed by powers of ε and contradict Eq. (38). The paper even concedes that this is the full extent of the FN prediction for the PMNS matrix. The observed anarchy is therefore an input chosen to match data, not a derived consequence of the model.

  2. fitted input called prediction [Sec. 8.1, Eq. (65), using Eq. (51)]
    "According to Eq. (51) all the light-neutrino masses mi will be: mi∼ v2 light/vheavy ϵ2L+2, where the only free parameter is the FN-charge of the lepton-triplet. ... By setting vlight to electroweak scale and L∼ 8 or 9, one obtains light-neutrino masses from the correct ballpark."

    The sub-eV scale of the light neutrinos is the central quantity the model claims to explain, yet the only free parameter in Eq. (65), the lepton-triplet FN charge L, is fixed by requiring the right-hand side to land in that same sub-eV ballpark. The O(1) coefficients are then chosen so that the mass squared differences in Table 4 match NuFIT data. Thus the light-neutrino mass scale and splittings enter through the choice of L and the coefficients; they are fitted inputs presented as an output of the seesaw-FN mechanism rather than independent predictions.

1 more flagged steps
  1. fitted input called prediction [Sec. 8.1, Eq. (67), using Eq. (24)]
    "The FN-charges q(eR,i) are the sole source of charged lepton mass hierarchy, as all the left-handed lepton triplet FN-charges are identical. We choose the right-handed charged lepton charges so that their mass matrix texture becomes: me∼v′ ... ε9 ε6 ε4 ..."

    The claimed explanation of the charged-lepton mass hierarchy is the FN charge assignment q(eR), but those charges are chosen precisely so that Eq. (67) reproduces the observed e-μ-τ hierarchy. The exponents 9, 6 and 4 are not derived from the 331 gauge structure or from anomaly cancellation; they are read off from the data and inserted into Eq. (24). The hierarchy is therefore encoded in the input charges, and the model's explanation reduces to a bookkeeping of the chosen FN charges rather than an independent derivation.

full rationale

The seesaw block diagonalization in Secs. 5-6 is internally consistent, and several results are genuine outputs: the sterile-neutrino mass spectrum m4-m9, the non-unitarity strengths, the NSI parameters, and the V±-W± mixing angles in Table 4 are not directly fixed by the charge choices. The self-citations to [25,26] for the flavon construction and quark-sector FCNC suppression are not load-bearing for the new neutrino predictions, and no uniqueness theorem is imported from the authors' prior work. However, three load-bearing explanatory steps do reduce by construction: the anarchical PMNS matrix is imposed by the equal-charge condition Eq. (40); the sub-eV light-neutrino scale is fixed by choosing L=8 or 9 in Eq. (65); and the charged-lepton hierarchy is set by choosing q(eR) in Eq. (67). In each case the observed quantity is used to select the input charge, after which the same quantity is reported as an explained output. The absence of a scan over the allowed O(1) coefficient interval, and the unusually small BP1 m1 relative to the naive entry scale, further weaken the paper's 'without fine-tuning' claim, though that is a naturalness risk rather than a formal circularity. Overall, the model has independent predictive content in its sterile sector, but the headline claim of explaining the known fermion spectrum and neutrino mixings is substantially fitted input, giving partial circularity.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The model's central results rest on the FN charge assignments (L and the right-handed lepton charges), the identification of the rho-dagger chi combination as the flavon, the standard seesaw block-diagonalization, and the assumption that O(1) coefficients in [0.5,5] are natural. The right-handed neutrinos are the only genuinely new entities; they carry falsifiable predictions (sterile neutrino masses and mixing).

free parameters (5)
  • Lepton-doublet FN charge L = L = 8 (BP1, BP2), L = 9 (BP3)
    Chosen so that Eq. (65), m_light ~ (v_light^2/v_heavy) eps^{2L+2}, lands in the observed sub-eV neutrino mass range.
  • Right-handed charged lepton FN charges q(eR), q(muR), q(tauR) = (2,-1,-3) for L=8; (1,-2,-4) for L=9
    Chosen to produce the charged lepton mass matrix texture in Eq. (67), i.e., to fit the electron-muon-tau hierarchy.
  • Right-handed neutrino FN charges q(NR,i) = (0,0,0)
    Set to zero for simplicity in Section 6; this choice controls the hierarchy between m_N and M and affects the medium neutrino mass scale.
  • Order-one coefficients cN_eta*, cN, c'N, cM, ce = 5 matrices listed in Section 8.2 for BP1-BP3, entries in [0.5,5]
    These coefficients are free parameters of the FN expansion; they are hand-picked in the natural range and not obtained from a fit.
  • Heavy VEVs u and v2 = (48 TeV, 55 TeV)=BP1; (21 TeV, 19 TeV)=BP2; (7 TeV, 7.5 TeV)=BP3
    Set the SU(3)L x U(1)X breaking scale; chosen around or above 7 TeV to satisfy Z' collider bounds, and varied to give benchmark spectra.
assumptions (5)
  • domain assumption The 331 gauge group with the given fermion content has anomaly cancellation only for three families.
    Used in the Introduction and Section 2.1 to explain the number of families; not derived in this paper but inherited from the 331 literature (refs [8]-[22]).
  • domain assumption The Froggatt-Nielsen effective operator expansion in powers of (rho-dagger chi / Lambda^2) is valid, with a single messenger scale Lambda and order-one coefficients.
    Eqs. (19)-(22); the messenger sector is not constructed; this is the standard FN ansatz.
  • domain assumption The VEV combination rho-dagger chi acts as the flavon with nonzero VEV and eps = v2 u / (2 Lambda^2) < 1.
    Section 3, Eq. (22); all Yukawa hierarchies follow from this assumption, which is borrowed from the authors' earlier FN331 papers [25,26].
  • standard math The 9x9 neutrino mass matrix can be block-diagonalized perturbatively; m'N is invertible and hierarchies v_light << v_heavy and eps << 1 justify the expansion.
    Section 6.1 Eqs. (45)-(50); the seesaw block-diagonalization procedure is standard but the non-singularity and convergence are assumed.
  • ad hoc to paper No significant fine-tuning among the O(1) coefficients is required beyond the chosen interval [0.5,5].
    Used as the naturalness criterion in Section 8.2; it is an assertion not quantified by a scan.
invented entities (1)
  • Three right-handed neutrino singlets N_{R,i} independent evidence
    purpose: Generate seesaw and tree-level masses for all neutrinos; their presence splits the neutrino spectrum into light, medium and heavy states.
    They are new fields not present in the earlier FN331 model; the model predicts their masses (keV and TeV scale) and mixings, so direct sterile neutrino searches and future colliders can probe them. No signal has been observed yet, but the prediction is falsifiable.

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Pith. "Pith review of Natural neutrino sector in a 331-model with Froggatt-Nielsen mechanism." pith.science (2026). https://pith.science/paper/RE74PDPD

@misc{pith2026190809384,
  author       = {Pith},
  title        = {Pith review of: Natural neutrino sector in a 331-model with Froggatt-Nielsen mechanism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RE74PDPD}},
  note         = {Machine review of arXiv:1908.09384}
}
abstract

The extensions of the Standard Model based on the $SU(3)_c\times SU(3)_L\times U(1)_X$ gauge group (331-models) have been advocated to explain the number of fermion families in nature. It has been recently shown that the Froggatt-Nielsen mechanism, a popular way to explain the mass hierarchy of the charged fermions, can be incorporated into the 331-setting in an economical fashion (FN331). In this work we extend the FN331-model to include three right-handed neutrino singlets. We show that the seesaw mechanism is realized in this model. The scale of the seesaw mechanism is near the $SU(3)_L\times U(1)_X$-breaking scale. The model we present here simultaneously explains the mass hierarchy of all the fermions, including neutrinos, and the number of families.

Figures

Figures reproduced from arXiv: 1908.09384 by the authors.

Figure 1
Figure 1. Constraints for the matrix element absolute values squared describing the strength of [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Same as Fig. 1, but for mixing of muon neutrinos. [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗

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Reviewed August 14, 2026 · model on record in the stance chip above.