Leptons masses in an $SU(3)_L\otimes U(1)_N$ Gauge Model

F. Pisano; O.F. Hernandez; R. Foot; V. Pleitez

REVIEW

Leptons masses in an $SU(3)_L\otimes U(1)_N$ Gauge Model

R. Foot, O.F. Hernandez, F. Pisano, V. Pleitez

High Energy Physics - Phenomenology · Submitted 27 Jul 1992 · 2 references

Pith's verdict: The sextet fix for 3-3-1 lepton masses is correct and is the durable contribution; the vacuum-alignment argument that is supposed to support it is sign-wrong and needs redoing, not just expanding.

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arXiv IDhep-ph/9207264 (version 3)

Pith IDREKBR5XO, Pith's permanent record of this paper and its review

CitationF. Pisano, O.F. Hernandez, R. Foot, V. Pleitez. "Leptons masses in an $SU(3)_L\otimes U(1)_N$ Gauge Model." arXiv:hep-ph/9207264 (1992). https://pith.science/paper/REKBR5XO

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Editor's note

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Short version: the diagnosis is right and the fix is the part that lasted. The Higgs-sector argument that is supposed to justify the fix does not hold up as written, and the paper's own reason for why the λ5 term doesn't spoil the vacuum has the sign backwards. The stress-test note lands; the reader's report is a bit too gentle on this point, calling it an omitted analysis when the argument actually given is incorrect on a specific direction in field space. Everything else in the reader's take I agree with, including the soundness score on the group theory and the ρ-parameter check, which is right: Eq. (29) with t² = s_W²/(1 − 4s_W²) gives M_Z²/M_W² = 1/cos²θ_W at tree level, and the a′ = 2v′² entry in the mass matrix is correctly carried through the custodial argument. What's new and good: the observation that the η Yukawa has three antisymmetric contractions (Lorentz, Grassmann, SU(3) ε) so the 3×3 lepton mass matrix is antisymmetric with eigenvalues 0, −M, M is correct and cleanly explained. The sextet S ~ (1,6,0) is the right fix: 3×3 contains a 6, the VEV in Eq. (16) gives a general symmetric charged-lepton mass matrix, and this is the version of the 3-3-1 model people actually build with. Where it's soft: the vacuum alignment. The paper says the λ5 bracket is 'positive in most of the parameter space.' It isn't. 2Tr(S†SS†S) − (Tr S†S)² ≤ 0 for any S by Cauchy–Schwarz, and it is exactly zero at the desired VEV of Eq. (16) but −v′⁴/3 at S = (v′/√3)I with the same Tr(S†S) = v′². So λ5 alone prefers the identity configuration, which is precisely the one that gives the neutrino a Majorana mass through σ01 and no charged-lepton splitting. The only term that can lift it is f2 ρᵀS†χ, linear in S32, vanishing at S ∝ I but with nonzero slope there. The desired vacuum is a minimum only above a threshold f2 ~ λ5 v′³/(uw), and the paper neither derives it nor checks that the required f2 leaves the Goldstone count at eight. The 'detailed analysis shows' sentence is doing all the work, and the heuristic in front of it points the wrong way. Two smaller things: Eq. (18) is not the most general potential invariant under the discrete symmetry — terms like η†SS†η are missing — and the discrete symmetry itself is imposed without an anomaly check. No scalar spectrum and no lepton-flavor discussion, though the latter is a general feature of the model rather than a defect of this paper. Who it's for: anyone building 3-3-1 models or working on the sextet variant. Worth a serious referee; the central mechanism is correct and the Higgs gap is fixable, but the vacuum-alignment section needs to be redone.

Referee report

Major objections (4)

[Paragraph following Eq. (19)] The claim that the desired VEV of Eq. (16) minimizes the potential is not established, and the heuristic argument given for the λ5 term fails in a specific direction. For S = (v'/√3) I_3 one has Tr(S†S) = v'² and 2Tr(S†S S†S) − (Tr[S†S])² = −v'^4/3 < 0, whereas at the desired VEV of Eq. (16) the same quantity vanishes. Thus at fixed Tr(S†S) = v'² the identity configuration has lower λ5 energy than the desired VEV, contrary to the text's statement that the λ5 term is 'positive in most of the parameter space.' The only term that can lift this is f2 ρ^T S† χ, which is linear in S, vanishes at S ∝ I, and has nonzero derivative there; the desired VEV is therefore a minimum only if f2 exceeds a threshold of order λ5 v'^3/(u w). The paper asserts that such f2 exists but neither derives the threshold nor checks that the required f2 leaves the Goldstone count at eight. This is load-bearing for the central claim that the sextet VEV is actually realized.
[Paragraph following Eq. (18)] The statement that 'a detailed analysis shows' the potential is minimized by the desired VEVs and yields exactly eight Goldstone bosons with no pseudo-Goldstones is asserted, not demonstrated. Counting real scalar degrees of freedom, the four multiplets η, ρ, χ, S contain 6 + 6 + 6 + 12 = 30 real fields; eight are eaten, leaving 22 physical scalars. Establishing the absence of flat directions in this 22-dimensional space requires checking the Hessian of Eq. (18) at the VEVs of Eq. (19), which is not done. Since the paper's consistency argument rests on this count, the analysis should be supplied or at least sketched.
[Eq. (18)] The claim that Eq. (18) is the most general SU(3) ⊗ U(1) invariant, renormalizable Higgs potential for the three triplets and the sextet respecting the discrete symmetry (20) is asserted, not proved. Quartic contractions such as η† S S† η appear to be allowed by the gauge and discrete symmetries and are not obviously reducible to the terms η†η Tr(S†S) and (Tr S†S)² that are present. The authors should either justify the absence of such terms or include them and redo the vacuum-alignment analysis.
[Eqs. (20)–(21)] The discrete symmetry is imposed ad hoc to forbid the trilinear terms η^T S† η and εijk εlmn Sil Sjm Skn, and its anomaly freedom is not checked. If the discrete symmetry is anomalous it is not a symmetry of the quantum theory, and the terms it is meant to forbid may be regenerated by nonperturbative effects. The authors should either verify the discrete anomaly cancellation or explain why the symmetry is not needed for the central claim.

Minor objections (6)

[Eq. (24)] The Yukawa coupling should be written as a matrix G_ab in generation space rather than a single G_l per lepton; the present notation obscures how the three distinct masses arise.
[Paragraph following Eq. (15)] The text should state explicitly that the sextet mass matrix is a general 3×3 matrix (not necessarily symmetric), which is sufficient for three distinct eigenvalues. The antisymmetry argument for the triplet is correct, but the corresponding argument for the sextet is not spelled out and the reader may be misled into thinking the sextet gives a symmetric matrix.
[Eq. (25)] The symbols a, b, c, a', t appearing in the neutral gauge-boson mass matrix should be defined in the text before Eq. (25), not only in the line below it.
[Paragraph following Eq. (7)] The statement that there will be no fractional charged color-singlet bound states is asserted without argument; a brief justification would help the reader.
[Eqs. (26)–(27)] The approximation c >> a, b, a' used to obtain the Z and Z' masses should be stated explicitly as the regime in which Z–Z' mixing is neglected, and the size of the neglected mixing should be estimated.
[References] Reference [1] is listed as IFT-P.017/91 (July 1991), which is unpublished; the authors should clarify its status or replace it with the published version where possible.

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The paper

The paper claims

The paper shows that adding one Higgs multiplet transforming as a sextet under the model's $SU(3)_L$ factor gives the three charged leptons distinct masses, repairing a defect in the original Pisano–Pleitez model without spoiling its tree–e

The reading

The $SU(3)_c$

What carries the argument

The carrying object is a Higgs field $S$ in the $(1,6,0)$ representation — a sextet of $SU(3)_L$ with zero $U(1)_N$ charge — whose vacuum expectation value is taken to be $v'/

Extended reading notes

Core claim

The paper shows that the original Pisano–Pleitez model fails on lepton masses for a structural reason, and that a single new scalar repairs it. The only leptonic Yukawa coupling the original scalar content allows, $G_{ab}

Load-bearing premise

The load-bearing premise is that the enlarged Higgs potential of Eq. (18) really is minimized by the stated vacuum expectation values for some choice of positive couplings, and that it yields exactly eight Goldstone bosons — the paper asserts this with 'A detailed analysis shows' but does not present the analysis.

Editorial extensions

If this is right

Three distinct charged-lepton masses follow at tree level from a single sextet VEV, replacing the degenerate pair $0, -M, M$ that the triplet-only scalar sector forced.
The tree-level relation $M_W^2 = M_Z^2\cos^2\theta_W$ survives the modification, so the fix costs nothing in electroweak precision at leading order.
The heavy neutral gauge boson mass $M^2_{Z'} = \frac{g^2}{3}(1+3t^2)c$ still depends only on the $\chi$ VEV $w$, so the 40 TeV lower bound quoted from $K^0-\bar K^0$ mixing is unaffected.
The imposed discrete symmetry $\rho \to i\rho$, $\chi \to i\chi$, $\eta \to -\eta$, $S \to -S$ forbids the trilinear terms $\eta^T S^\dagger \eta$ and $\epsilon_{ijk}\epsilon_{lmn}S_{il}S_{jm}S_{kn}$, keeping $\langle \sigma^0_1\rangle = 0$ and neutrinos massless at this level.
Anomaly cancellation still holds only when the number of generations is a multiple of three, so the model's most distinctive feature is preserved by the extension.

Reading between the lines

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

The same sextet sits one step away from generating Majorana neutrino masses: the discrete symmetry is doing real work, and relaxing it reopens a neutrino-mass sector the paper deliberately closes.
Because the sextet couples through the symmetric product of two triplets, the charged-lepton mass matrix it produces is symmetric rather than antisymmetric, so tree-level lepton mixing would come from an orthogonal rotation — a structural pattern that any later extension to neutrinos would have to face.
The claim that Eq. (18) is the most general gauge-invariant renormalizable potential for three triplets plus a sextet is checkable by enumerating invariants; if additional allowed terms exist, the vacuum-alignment argument needs revisiting.
The discrete symmetry of Eqs. (20)–(21) is imposed without an anomaly check, so whether it can be a genuine discrete gauge symmetry of the full theory is left open.

Original abstract

The $SU(3)_c\otimes SU(3)_L\otimes U(1)_N$ model of Pisano and Pleitez extends the Standard Model in a particularly nice way, so that for example the anomalies cancel only when the number of generations is divisible by three. The original version of the model has some problems accounting for the lepton masses. We resolve this problem by modifying the details of the symmetry breaking sector in the model.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,942 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.49 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

The $SU(3)_c\otimes SU(3)_L\otimes U(1)_N$ model of Pisano and Pleitez extends the Standard Model in a particularly nice way, so that for example the anomalies cancel only when the number of generations is divisible by three. The original version of the model has some problems accounting for the lepton masses. We resolve this problem by modifying the details of the symmetry breaking sector in the model.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The $SU(3)_c\otimes SU(3)_L\otimes U(1)_N$ model of Pisano and Pleitez extends the Standard Model in a particularly nice way, so that for example the anomalies cancel only when the number of generations is divisible by three.	background 1.00	0.06	0.04	asserted 0.99
s1	The original version of the model has some problems accounting for the lepton masses.	background 0.60	0.12	0.04	asserted 0.98
s2	We resolve this problem by modifying the details of the symmetry breaking sector in the model.	contribution 0.71	0.93	0.61	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.86.

What was examined, and what was not

sources read

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full text
figures and tables
references and citing sentences
relations to other papers

levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Forward citations

Cited by 15 Pith papers

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

Inverse-seesaw and inflationary scale relations in a supersymmetric 331 model from an SU(6) GUT model
hep-ph 2026-09 unverdicted novelty 6.0 of 10

A supersymmetric SU(6) model with an intermediate 331 symmetry that links the inverse seesaw mechanism, gauge-mediated SUSY breaking, and inflation via common fields and scale relations.
A flipped 331 model
hep-ph 2016-06 conditional novelty 6.0 of 10

A new anomaly-free 331 gauge model puts all quarks in identical representations and splits leptons across a sextet and two triplets, predicting lepton-flavor violation in Z-prime couplings.
Large gauge invariant non-standard neutrino interactions
hep-ph 2008-09 accept novelty 6.0 of 10

Large non-standard neutrino interactions from dimension-eight operators require at least two new mediator fields and specific coupling cancellations to avoid four-charged-lepton constraints.
Fermion spectrum and $g-2$ anomalies in a low scale 3-3-1 model
hep-ph 2020-02 conditional novelty 5.0 of 10

A new 3-3-1 model with S3 x Z2 and vector-like fermions is fitted to the fermion mass hierarchy and to the electron and muon g-2 anomalies.
A variant of 3-3-1 model for the generation of the SM fermion mass and mixing pattern
hep-ph 2017-05 conditional novelty 5.0 of 10

A renormalizable 3-3-1 model with Z2 x Z4 x U(1) symmetries reproduces the Standard Model fermion mass hierarchy through sequential loop suppression, with the top at tree level, bottom and charm at one loop, and light...

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Dark Matter and Global Symmetries
hep-ph 2015-08 conditional novelty 5.0 of 10

Assuming Planck-scale breaking of global symmetries and order-one couplings, existing indirect-detection data exclude dark matter stabilized by continuous global symmetries above roughly 100 keV (scalar), 100 MeV (fer...
LHC Constraints on 3-3-1 Models
hep-ph 2015-03 accept novelty 5.0 of 10

ATLAS 8 TeV dilepton data exclude 3-3-1 Z-prime bosons below roughly 2.3 to 2.9 TeV at 95% confidence, depending on the model.
Connection of $g-2_{\mu}$, electroweak, dark matter, and collider constraints on 331 models
hep-ph 2014-08 conditional novelty 4.5 of 10

Of six 331 models, only the two-Higgs-triplet version can accommodate the muon g-2 anomaly while surviving collider and dark matter bounds; the others are excluded as explanations.
Radiative corrections to the $\rm S, T, U$ parameters and their impact on the $W$ boson mass in the 331 model
hep-ph 2025-07 conditional novelty 4.0 of 10

For the beta=-sqrt(3) 331 model, electroweak precision data allow v3 between roughly 1.5 and 2.3 TeV and can reproduce the CDF W mass shift for v3 in 1.8-2.3 TeV, provided collider bounds on new gauge bosons are model...
A 3-3-1 model with low scale seesaw mechanisms
hep-ph 2019-05 conditional novelty 4.0 of 10

A 3-3-1 gauge model with a T' family symmetry reproduces the observed quark and lepton masses and mixings, and predicts heavy W' and Z' bosons plus rare charged-lepton decays.
The Dark Side of Flipped Trinification
hep-ph 2017-10 conditional novelty 4.0 of 10

Flipped trinification with a reduced scalar sector leaves an exact matter parity as a residual gauge symmetry, yielding stable few-TeV fermion, scalar or vector dark matter candidates.
Constraining Flavor Changing Interactions from LHC Run-2 Dilepton Bounds with Vector Mediators
hep-ph 2016-08 conditional novelty 4.0 of 10

In the 3-3-1 model with right-handed neutrinos, LHC 13 TeV dilepton data imply M_Z' > 3 TeV, which is stronger than the FCNC bounds from K, D, and B_d mixing for all but one hand-constructed quark mixing texture.
Z' models for the LHCb and g-2 muon anomalies
hep-ph 2015-11 conditional novelty 4.0 of 10

Di-muon resonance data, the B_s mass difference, and perturbativity force Z-prime explanations of the LHCb B anomalies to a Z-prime heavier than about 2.5 TeV or to suppressed couplings, while a Z-prime plus vector-li...
Radiative Seesaw in Minimal 3-3-1 Model
hep-ph 2015-04 conditional novelty 4.0 of 10

A one-loop radiative seesaw inside a minimal 3-3-1 model can fit neutrino oscillation data and lepton-flavor-violation bounds over a broad parameter region, but its charged scalars cannot explain the muon g-2 anomaly.
Fermion masses and mixings in the 3-3-1 model with right-handed neutrinos based on the $S_3$ flavor symmetry
hep-ph 2013-09 conditional novelty 4.0 of 10

A 3-3-1 model with S3 x Z3 x Z3' x Z8 x Z16 flavor symmetry and 15 singlet scalars reproduces SM fermion masses and mixings and yields m_betabeta = 4 (NH) or 48 meV (IH).

How the literature uses it

44 arXiv papers cite this in 74 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 36 sentences without a confident label

Contests it 1 paper

For instance, some models studied in Ref. [ref] lead to physical inconsistencies which rule them out.

New parameterization and analysis for E6 inspired 331 model · Introduction · contest 0.88

Compares with it 2 papers

However, the $1/√(2)$ factors cannot be absorbed, so the expression in equation ([ref]) for the sextet differs in a material way from the one used in [ref] , for example, agreeing instead with [ref] .[Without these $1/√(2)$ factors, it is easy to check that a mass term $Tr(6^†6)$ will not correspond to the sum of the norm-squared of all six components.]

Lepton number violation in 331 models · Appendix: decomposition of 331 representationstocsectionAppendix A: contractions and decomposition of 331 representation · 0.80
This is distinctive from the fermion contents in several previous 331 model studies [ref] , where the flavor non-universalities were assumed at the beginning.

A two-generational ${\rm SU}(7)$ model with extended weak sector: mass hierarchies, mixings, and the flavor non-universality · The $SU(3)_W ⊗U(1)_X_1$ gauge couplings · 0.98

Cites it as background 28 papers

For its minimal version (already possibly capable of reproducing the neutrino masses and removing the arbitrariness in the number of families) [ref] , a version with exotic heavy leptons [ref] and one with right-handed neutrinos [ref] , Dias et al found a remarkably simple relation among parameters that reduces known boson masses to the expected ones.

An exact lower bound within a 331 model closing solution · Introduction · 0.90
There are several ways to arrange fermions in a $SU(3)_L$ triplet, and these multiple possibilities give rise to different 3-3-1 models [ref] .

Flavor changing interactions confronted with meson mixing and hadron colliders · THE MODEL · 0.98
We will address this issue in a particular extension of the SM based on a $SU(3)_c ⊗SU(3)_L ⊗U(1)_N$ gauge group, typically referred to in the literature as 3-3-1 model [ref] .

On the Role of LHC and HL-LHC in Constraining Flavor Changing Neutral Currents · Introduction · 0.88

Discussion

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[1] [1]

F. Pisano and V. Pleitez, “Neutrinoless double beta decay and dou bly charged gauge bosons”, IFT-P.017/91, July 1991

View on Pith 1991

[2] [2]

F. Pisano and V. Pleitez, “ SU (3) ⊗ U(1) model for electroweak interactions”, Phys. Rev. D46, 410(1992). 8

View on Pith 1992