Leptoquark-induced radiative masses for active and sterile neutrinos within the framework of the 3-3-1 model

A. Doff; C. A. de S. Pires; Jo\~ao Paulo Pinheiro

REVIEW 3 major objections 5 minor 1 cited by

Leptoquark-induced radiative masses for active and sterile neutrinos within the framework of the 3-3-1 model

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Adding two scalar leptoquarks to the 331RHN model makes both active and sterile neutrinos light through radiative corrections.

desk verdict Plausible radiative mechanism connecting active and sterile neutrino masses in 331RHN, but the headline eV-scale sterile predictions rest on a two-loop formula that is not reproducible as printed. read the letter →

arxiv 2412.15055 v2 pith:SGYMUJA3 submitted 2024-12-19 hep-ph

A. Doff , João Paulo Pinheiro , C. A. de S. Pires This is my paper · ORCID

classification hep-ph

keywords radiativeneutrinomassleptoquark3-3-1modelright-handedneutrinossterileMajoranarareBdecaysflavor-violatingHiggsdecay

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

This paper claims that a minimal addition to the 331RHN model—one triplet and one singlet scalar leptoquark—can generate tiny Majorana masses radiatively for both active and sterile neutrinos. The central result is a loop-level hierarchy: left-handed neutrinos get mass at one loop while right-handed (sterile) neutrinos get mass at two loops, so sterile neutrinos are unavoidably light rather than heavy. With benchmark Yukawa couplings, the model reproduces the measured solar and atmospheric mass splittings and mixing angles for both normal and inverted ordering, and it satisfies current bounds on rare B-meson decays and the flavor-violating Higgs decay $h\to\tau\mu$. The paper therefore offers a radiative, leptoquark-driven alternative to seesaw mechanisms inside the 3-3-1 framework.

What carries the argument

The machinery is the minimal leptoquark content: the scalar triplet $T\sim(3,3,0)$ and the scalar singlet $S\sim(3,1,1/3)$, added to the 331RHN model. The lepton-number-violating trilinear term $M\,T^\dagger\eta S$ in the scalar potential (Eq. 6) generates a mass-mixing between the charge $+1/3$ leptoquark components; the mixing angle $\sin 2\theta = \sqrt{2}\,M\,v_\eta/(M_1^2-M_2^2)$ (Eq. 9) is the single knob that turns on both mass matrices. That mixing feeds a one-loop diagram (through $d$-quark and leptoquark exchange) that yields the active Majorana mass matrix (Eq. 12), and a two-loop diagram (through $d$, $d'$, leptoquarks and $\eta'^0$) that yields the sterile Majorana mass matrix (Eq. 18). Because both matrices share the same Yukawa couplings $Y$ and $\tilde Y$, the hierarchy of loop orders plus the heavy masses $m_{d'}$ and $M$ in the two-loop amplitude produces sterile neutrinos that are inevitably light—heavier than active neutrinos, but far lighter than typical seesaw singlets.

What would settle it

A measurement that the charge $+1/3$ leptoquark states have exactly zero mixing ($\sin 2\theta=0$), or an observation of sterile neutrinos with masses far above the eV scale while active neutrinos keep the measured splittings, would eliminate the mechanism; conversely, a precise determination of the two-loop sterile masses near the predicted eV values would support it.

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Extended reading notes

Core claim

The paper's discovery is that the same Yukawa couplings that give active neutrinos a one-loop Majorana mass matrix necessarily give the right-handed neutrinos a two-loop Majorana mass matrix, because in the 331RHN model left- and right-handed neutrinos share the same lepton triplet. The mechanism is driven by the mixing between the charge $+1/3$ components of the scalar leptoquarks $T$ and $S$, induced by the lepton-number-violating term $M\,T^\dagger\eta S$ in the scalar potential. When $\eta^0$ acquires a VEV, the mixing angle obeys $\sin 2\theta = \sqrt{2}\,M\,v_\eta/(M_1^2-M_2^2)$, and every neutrino mass term is proportional to this mixing. With leptoquarks at the TeV scale and Yukawa couplings in the range $10^{-1}$–$10^{-4}$, the active neutrino masses come out at sub-eV scale, while the sterile neutrinos come out at the eV scale (about 1–50 eV in the benchmark points), making them light but heavier than the active ones.

Load-bearing premise

The entire mass generation rests on the lepton-number-violating coupling $M$ in the scalar potential term $M\,T^\dagger\eta S$: if $M$ were zero, the $S$–$T$ mixing angle vanishes and both the one-loop active and two-loop sterile neutrino mass matrices vanish; the paper fixes $M=147$ GeV as an input without deriving it from a more fundamental scale.

Editorial extensions

If this is right

Sterile neutrinos in this model are unavoidably light (eV scale), not heavy, because their mass is generated at two loops rather than by a seesaw; this is a direct prediction of the mechanism the authors present.
The model reproduces the measured solar and atmospheric mass splittings and mixing angles for both normal and inverted ordering, with Yukawa couplings of order $10^{-1}$–$10^{-4}$ and TeV-scale leptoquarks.
The sterile neutrinos do not mix with active ones, so they escape short-baseline experimental constraints; their new interactions are through 331 gauge bosons and leptoquarks, leaving cosmology ($N_{\rm eff}$, BBN, CMB) as the natural probe.
Benchmark points satisfy the current bounds on $B_s\to\mu^+\mu^-$, the Belle II $B\to K\nu\nu$ correlation, and the ATLAS limit on $h\to\tau\mu$; the predicted $\text{Br}(h\to\tau\mu)$ is about $1.56\times 10^{-4}\%$.
Leptoquarks with masses around 2 TeV are the signature of the scenario and are being probed at the LHC.

Reading between the lines

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

If $M$ ultimately originates from a spontaneously broken symmetry rather than being put in by hand at 147 GeV, the same scale would likely control the leptoquark mass splitting and could make the sterile mass pattern more predictive; the paper leaves this connection implicit.
The correlation between the active mass ordering and the sterile spectrum (NO: $m_{R2}\approx 19$ eV, $m_{R3}\approx 24$ eV; IO: $m_{R1}\approx 1$ eV, $m_{R2}\approx 50$ eV) is a testable linkage: a future determination of the sterile masses, e.g. from cosmology, would discriminate between the two orderings.
Because the two-loop sterile mass scales with $M$ and $m_{d'}$, measuring the new quark mass or the lepton-number-violating parameter in same-sign dilepton searches would quantitatively test the predicted eV-scale sterile masses; the paper does not compute those rates.
The paper does not analyze cosmological constraints on eV-scale sterile neutrinos, yet such states contribute to $N_{\rm eff}$ and could be excluded or confirmed by CMB-S4-like observations; this is the most direct external check.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the idea is worth a serious referee, but the paper's main quantitative claim—eV sterile neutrinos—is not checkable as written. The one-loop active mass mechanism is standard and the benchmark Yukawa matrices do reproduce NuFIT splittings and angles. The novelty is real: previous 331RHN sterile-mass papers used other mechanisms, and this is the first time scalar leptoquarks generate active at one loop and sterile at two loops with shared Yukawas. That correlation is a nice structural point.

What the paper does well: it identifies the minimal leptoquark content (triplet T + singlet S), shows the M T-dagger eta S term is what seeds both masses, checks B_s to mu+mu-, B to K(*) nu nu, and h to tau mu against current bounds, and the benchmark points survive those constraints. The Yukawa fits, while underdetermined (12 real unknowns for 6 equations), are at least solved explicitly and give couplings in a plausible range.

Soft spots, in increasing order. Eq. (14) has a misprint: masses p inside the sum where a factor m_p should appear. Minor. More substantial: the active fit is not a prediction; it fits the model matrix to NuFIT with many free Yukawas, so the real test is the sterile sector. There the paper falls short. Eq. (18) has two terms weighted by U_11 and U_12; when the simplifying assumption makes the h functions identical, Eq. (19) drops those factors entirely, which is only valid if cos(theta) - sin(theta) = 1, not the case for sin2theta = 10^-2. The loop function h in Appendix A is defined but never evaluated; no numerical values or code are given, so the quoted sterile masses mR2 ~ 19 eV, mR1 ~ 1 eV cannot be reproduced. And g13, g23 appear in Eqs. (20)-(21) with no derivation—they are inputs, not outputs, of the fit. Cosmological constraints are explicitly deferred to 'elsewhere', so the eV-scale claim is supported neither internally nor externally.

None of this kills the underlying idea. The M term is a legitimate explicit LNV seed, and the one-loop structure is sound. But the central two-loop result is currently an unverifiable approximation. The authors can fix this fairly easily: carry the theta factors, evaluate h numerically, show how g13/g23 are chosen, and run the cosmological bounds for the eV states.

My recommendation: send it to peer review. It will need major revision before publication, but the mechanism is plausible, the workload is honest, and the connection between active and sterile radiative masses is worth refereeing properly.

Referee Report

3 major / 5 minor

Summary. The paper adds a scalar leptoquark triplet T and a scalar leptoquark singlet S to the 3-3-1 model with right-handed neutrinos (331RHN). A Z2 symmetry and an explicit lepton-number-violating term M T† η S generate S–T mixing. The authors then obtain one-loop Majorana masses for active neutrinos and two-loop Majorana masses for right-handed (sterile) neutrinos, with the right-handed states emerging at the eV scale for their benchmark parameters. Using NuFIT best-fit values, they present benchmark Yukawa matrices Y and \tilde Y for normal and inverted ordering, and they confront the model with B_s → μ⁺μ⁻, B → K(*)νν, and h → τμ data.

Significance. The model-building idea is interesting and structurally plausible: the one-loop active neutrino mass follows the standard scalar-leptoquark radiative pattern, and the same Yukawa sector feeds a two-loop sterile neutrino mass is a neat feature of the 331RHN framework. If the two-loop computation were made fully reproducible, the claim that sterile neutrinos are unavoidably light would be a valuable result. The paper also makes a genuine effort to connect the benchmark couplings to existing B-meson and Higgs constraints. However, the active neutrino sector is fitted rather than predicted, and the sterile sector currently depends on undetermined inputs and an unevaluated loop integral; in its present form the quantitative support for the central claim is not yet established.

major comments (3)

[III.B, Eqs. (18)-(21)] The two-loop sterile neutrino mass computation, which is the central new claim of the paper, is not reproducible as printed. In Eq. (18) the loop contributions are weighted by U11(θ) and U12(θ). After the simplifying assumption that all leptoquark masses are equal to M2, the two h-functions coincide and the θ dependence collapses to U11+U12 = cosθ − sinθ, but Eq. (19) simply omits these factors; for the benchmark sin2θ = 10⁻² this is cosθ − sinθ ≈ 0.995, so the equations are formally inconsistent even if the numerical effect is small. More importantly, the function h defined in Appendix A (Eqs. A1–A7) is never evaluated numerically; no value is quoted for h(m_d′, M_η′0, m_b, M2), so a reader cannot reproduce the quoted masses m_R² ≈ 19 eV, m_R³ ≈ 24 eV (NO) and m_R¹ ≈ 1 eV, m_R² ≈ 50 eV (IO). The two-loop result therefore lacks the quantitative support needed for the paper's main claim that sterile neutrinos are inexorably light.
[III.B, after Eq. (21)] The couplings g13 and g23 are introduced as numerical inputs immediately after Eq. (21) ("we obtain g13 = 0.000119 and g23 = 0.0014"), but their origin is not explained. They are not fixed by the active-neutrino fit of Eqs. (15)–(17), which determines only Y and \tilde Y; nor are they related to the masses m_d′ or to any symmetry constraint stated in the paper. Because the sterile mass matrix in Eq. (21) is proportional to these couplings, the quoted eV-scale masses are consequences of hand-picked parameters rather than predictions. The paper should either derive these couplings, scan over the allowed range, or explicitly state the criterion used to select them, and should show how the sterile masses vary over that range.
[III.A, Eqs. (14) and (15)] The active-neutrino fit is underdetermined as presented. Eq. (15) is a system of six equations for the twelve real entries of Y and \tilde Y, yet no solution procedure is described: the manuscript does not state an ansatz, a minimization criterion, or a parametrization that selects the matrices displayed in Eqs. (16)–(17). Since the solution space is highly degenerate, the displayed benchmark is one arbitrary point, and the subsequent sterile-mass and B-physics results inherit this arbitrariness. In addition, Eq. (14) contains a misprint: the summation index p also appears as a lepton index in the term "Y_{2p}", which should presumably be "Y_{2a}", and the expression as a whole needs careful checking.

minor comments (5)

[III.A and VI] The text states in Section III.A that the model provides "eV neutrinos" while the conclusion refers to "sub-eV neutrino mass"; these statements should be reconciled.
[III.B and footnote 4] Cosmological and astrophysical constraints on the light sterile states, which are said to interact through the new 331 gauge bosons and leptoquarks, are deferred to future work; given the eV-scale masses quoted, this leaves the phenomenological viability check incomplete.
[IV.A, Eq. (36)] The phrase "assuming the current experimental limit()" has an empty parenthesis and the corresponding citation should be supplied.
[V, Eq. (44)] The couplings y32 and y33 are used in the h → τμ amplitude without being explicitly defined in the Yukawa Lagrangian of Eq. (5); a brief definition would improve readability.
[III.B, benchmark] The lepton-number-violating parameter M is fixed to 147 GeV without derivation; since all radiatively generated masses vanish in the M → 0 limit, the dependence of the quoted sterile masses on M (or equivalently on sin2θ) should be displayed.

Circularity Check

1 steps flagged · score 3.0 of 10

Active neutrino sector is fitted, not predicted; the sterile two-loop mechanism has independent content, but its quoted eV masses are set by hand-picked g13 and g23 inputs.

fitted input called prediction [Sec. III A, Eq. (15) and surrounding text]
"To estimate the order of the Yukawa matrices ˜Y and Y, we impose that Eq. (14) recover the full neutrino mass matrix in flavor basis. Then, we are solving a system with 6 equations with 12 variables ... we solve the system (mν)L ab EXP = (mν)L ab PRED (15) where ... (mν)L ab EXP represents the experimental value for the active neutrino mass matrix in flavor basis for each mass ordering and (mν)L ab PRED is the model prediction for the active neutrino mass at one loop."

Equation (15) defines the 'model prediction' to be equal to the experimental active-neutrino mass matrix by construction. The Yukawa matrices Y and Ytilde are solved from this imposed equality, so the active masses and mixings quoted later ('our model provides eV neutrinos in a conservative way') are the NuFIT input data re-expressed in model parameters, not an output of the radiative mechanism. The paper itself notes the one-loop formula has far more free parameters than the five measured neutrino quantities, so the benchmark reproduces those quantities because Eq. (15) is imposed, not because the mechanism predicts them. This is the fitted-input-called-prediction pattern in the active sector.

full rationale

The central two-loop sterile-neutrino step is not circular in the strict sense: the right-handed neutrino masses in Eqs. (18)-(21) are derived from an independent formula for a quantity that was nowhere used as an input, and their dependence on the previously fitted Y and Ytilde is a genuine transfer of information from the active to the sterile sector. However, the active sector is presented as a prediction while Eq. (15) imposes equality with the NuFIT mass matrix by construction, so the resulting active masses and mixings are constrained inputs rather than predicted outputs. The quoted sterile masses are also not parameter-free predictions: Eq. (21) is linear in the couplings g13 and g23, and these couplings are introduced with the phrase 'we obtain' but no defining equation or external constraint is given, so the eV-scale output tracks the choice of those inputs and of M, md', and M2 rather than being forced by data. The apparent inconsistency between Eqs. (18) and (19) involving the dropped U factors, and the unevaluated loop function h in Appendix A, are internal consistency and reproducibility concerns rather than circularity. The self-citation to Ref. [74] for the bound y32 ≲ 0.5 is anchored to external (g-2)_mu data and affects only the peripheral h -> tau mu check, so it is not load-bearing. Overall, one modest fitted-input-called-prediction inflation occurs in the active sector, while the sterile-sector claim retains independent content but is underdetermined by hand-picked benchmark parameters.

Assumptions & free parameters 6 free parameters · 6 assumptions · 5 invented entities

The neutrino mass mechanism rests on a long chain of model assumptions inherited from the 331RHN framework plus new ad hoc ingredients: the Z2 symmetry, the lepton-number-violating mass M, the benchmark mass spectrum and the fitted Yukawa matrices. The paper provides no independent evidence for any of these beyond internal consistency with existing measurements.

free parameters (6)

Lepton-number-violating mass parameter M = 147 GeV (benchmark)
Sets the S-T mixing that generates neutrino masses; if M=0 the masses vanish. Chosen to give sin2theta=1e-2 with M2=2 TeV.
Leptoquark mixing angle sin2theta = 1e-2
Controls the active and sterile mass scales; not predicted, chosen by hand.
Yukawa matrices Y and Y-tilde (NO and IO) = Entries in Eqs. (16) and (17), ranging from 1e-4 to 0.9
Fitted to reproduce NuFIT neutrino parameters. The system has 12 unknowns for 6 equations, so the solution is not unique.
Quark Yukawa couplings g13 and g23 = 1.19e-4 and 1.4e-3
Quoted for v_eta=v_rho without showing the quark-sector fit; they enter the sterile neutrino mass formula.
Benchmark mass spectrum = M2=2 TeV, M1=1.3 M2, md'=1 TeV, M_eta'=3 TeV
Chosen TeV-scale values; the sterile neutrino masses scale with these inputs.
Higgs decay parameters = y33=1, y32<0.5, lambda=lambda6+lambda7
Free couplings used to estimate Br(h to tau mu); the y32 bound comes from the muon g-2 solution.

assumptions (6)

domain assumption The 331 gauge symmetry and particle content, including three scalar triplets eta, rho, chi and the family-asymmetric quark sector, are assumed from prior 331 literature.
This is the basis of the model; it is not derived in the paper.
ad hoc to paper A Z2 symmetry with (T, eta, rho, e_R, u_R, d_R) odd and all other fields even is imposed.
Imposed to select the allowed Yukawa and scalar interactions and to make eta-prime inert. The symmetry is spontaneously broken by the eta0 and rho0 VEVs.
ad hoc to paper Lepton number is violated explicitly only by the M T-dagger eta S term.
Required for Majorana neutrino masses; no origin for M is given.
domain assumption Quark mixing satisfies Vd_L = V_CKM, with up-type quarks, right-handed quarks and d-prime quarks taken in a diagonal basis.
Used to simplify the one-loop formula in Eq. (14) and the two-loop sterile mass formula.
ad hoc to paper The leptoquark and new scalar masses are fixed at the TeV scale with M2=2 TeV, M1=1.3 M2, md'=1 TeV and M_eta'=3 TeV.
Benchmark mass choices are not derived from the scalar potential parameters.
domain assumption NuFIT best-fit values for neutrino mixing angles and mass-squared differences are taken as external input.
The oscillation data set the target values used in the Yukawa fit.

invented entities (5)

Scalar leptoquark triplet T with charges +1/3, -2/3 and +1/3 independent evidence
purpose: Together with S, generates one- and two-loop neutrino masses and contributes to B-meson and Higgs decays.
A TeV-scale scalar leptoquark with these couplings can be searched for at the LHC; the paper sets its mass to 2 TeV.
Scalar leptoquark singlet S with charge +1/3 independent evidence
purpose: Together with T, generates the radiative neutrino masses and mediates B and Higgs flavor-changing processes.
Scalar leptoquarks with this charge have collider signatures; the mass is set to about 2 TeV.
Heavy down-type quarks d-prime_i, i=1,2 independent evidence
purpose: Run in the two-loop sterile neutrino mass diagram and contribute to the 331 anomaly-cancellation structure.
New quarks are required by the 331 framework and could be pair-produced at the LHC; the paper sets md'=1 TeV.
Inert scalar component eta-prime-0 of the eta triplet
purpose: Appears as an internal line in the two-loop sterile neutrino mass diagram.
A neutral scalar at 3 TeV with no specific decay signature discussed in this paper, so no independent falsifiable handle is provided.
New gauge bosons Z-prime, W-prime and U0 independent evidence
purpose: Standard 331 gauge sector; they mediate additional interactions of the light sterile neutrinos with ordinary fermions.
Z-prime and W-prime searches can constrain 3-3-1 models, although this paper gives no mass predictions for them.

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Pith. "Pith review of Leptoquark-induced radiative masses for active and sterile neutrinos within the framework of the 3-3-1 model." pith.science (2026). https://pith.science/paper/SGYMUJA3

@misc{pith2026241215055,
  author       = {Pith},
  title        = {Pith review of: Leptoquark-induced radiative masses for active and sterile neutrinos within the framework of the 3-3-1 model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SGYMUJA3}},
  note         = {Machine review of arXiv:2412.15055}
}

abstract

In this work, we introduce the minimal set of leptoquarks into the 3-3-1 model with right-handed neutrinos, capable of generating radiative masses for active neutrinos. As a main consequence, the standard neutrinos acquire small Majorana masses at the one-loop level, while right-handed (sterile) neutrinos obtain small Majorana masses at the two-loop level, naturally making them light particles as well. Additionally, we discuss the viability of this scenario and several other interesting phenomenological consequences, including its impact on $B$-meson physics and rare Higgs decays, both of which are also induced by the leptoquarks.

Figures

Figures reproduced from arXiv: 2412.15055 by the authors.

**Figure 2.** FIG. 2. Feynman diagram contribution to [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** FIG. 3. Feynman diagram contribution to [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Correlation between B( [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: FIG. 5. Feynman diagrams for 331LQ contributions to [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

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Forward citations

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

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A scalar sextet generates eV-scale Dirac neutrino masses in a 3-3-1 model, and the associated right-handed neutrino population constrains the Z' boson mass to exceed 4.4 TeV.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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primes”, O9′ and O10′, are related to the “unprimed

electric charges. The original scalar sector of the model is composed by three triplets of scalars[28, 29] η =  η0 η− η′0  , ρ=  ρ+ ρ0 ρ′+  , χ=  χ0 χ− χ′0  , (3) withη andχ transforming as (1, 3,−1/3) andρ as (1, 3, 2/3). The minimal content of leptoquar...

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has an analytical ex- pression analogous to Eq. (13). With these components at hand, we now have all the necessary elements to compute the decay width for the h→τµ process. The h→τµ partial decay width in our model stands for Γ(h→τµ) = 9mhm2 t 213π5v2 F2( ˜Y)| f1(λ, M2)|2, (46) such that F2( ˜Y)≡| ˜Y33y32|2 +| ˜Y32y33|2. In this section, we investigate th...

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