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REVIEW 3 major objections 5 minor 73 references

Higgs and Z lepton-flavor decays scale with radiative lepton decays in a 3-3-1 leptoquark model.

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2026-08-01 14:35 UTC pith:CHDGHLOU

load-bearing objection The LFV h/Z versus radiative-decay correlation is real and analytic, but the near-limit rate claims rest on an unjustified CKM input freedom. the 3 major comments →

arxiv 2607.18708 v1 pith:CHDGHLOU submitted 2026-07-21 hep-ph

LFV decays in a 3-3-1 model with singlet leptoquarks

N.H.T. Nha , L.T. Hue , N. T. K. Ngan , P.T. Bich , T.T. Hong , N.T. Tham This is my paper
classification hep-ph
keywords 3-3-1 modelsinglet scalar leptoquarklepton flavor violationanomalous magnetic momentHiggs decayZ boson decaymuon g-2radiative lepton decay
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper extends a 3-3-1 model with a singlet scalar leptoquark to predict lepton-flavor-violating decays of the Higgs and Z bosons. It claims that the branching ratios of h→ℓ_bℓ_a and Z→ℓ_bℓ_a are nearly proportional to the corresponding radiative decays ℓ_b→ℓ_aγ, so tighter limits on μ→eγ would directly tighten the predicted Higgs and Z rates. The paper also finds a complementary pattern in the anomalous magnetic moments: parameter regions that explain the muon g-2 deviation allow h→τμ to approach its current experimental bound, while regions that yield a sizable electron g-2 allow h→τe and Z→τe to approach current sensitivities while suppressing the muon g-2. If correct, the model produces channel-specific, testable predictions for upcoming collider searches.

Core claim

In the 331LQ model, the dominant one-loop contributions to charged-lepton flavor-violating decays, Higgs and Z lepton-flavor-violating decays, and the anomalous magnetic moments all share the same dependence on the top-quark leptoquark Yukawa couplings. Specifically, both Br(e_b→e_aγ) and Br(h,Z→e_b e_a) scale with f_ba = |g^LQ_{3a} h^LQ_{3b}|² + |h^LQ_{3a} g^LQ_{3b}|², producing a near-linear correlation. The numerical scan shows that the leptoquark cannot simultaneously generate large contributions to both Δa_μ and Δa_e: either Δa_μ ≥ 10⁻¹¹ and h→τμ near its current limit with Δa_e negligible, or 10⁻¹⁴ ≤ |Δa_e| ≤ 8×10⁻¹³ and h→τe, Z→τe near current sensitivities with Δa_μ suppressed to O(1

What carries the argument

The central object is the singlet scalar leptoquark S with charge 1/3, which couples to up-type quarks and charged leptons through Yukawa couplings g^LQ and h^LQ. Because the light up-quark contributions are negligible, all observables are dominated by the top-quark couplings g^LQ_{3a} and h^LQ_{3b}. The one-loop form factors for cLFV, LFVh, and LFVZ decays are expressed in terms of Passarino-Veltman functions, and the divergent parts cancel between diagrams. The shared factor f_ba is what generates the approximate proportionality between the radiative and Higgs/Z decay rates.

Load-bearing premise

The scan assumes the unknown quark mixing matrices V_uL and V_dL can be chosen so that the effective top-family leptoquark couplings g^LQ_{3i} and h^LQ_{3i} are independently as large as sqrt(4π), with no constraint from CKM or other flavor observables limiting those combinations.

What would settle it

A future measurement of Br(μ→eγ) below about 10⁻¹⁴ together with a simultaneous observation of Br(h→τμ) near 10⁻³ would contradict the predicted linear correlation, since the tiny |g_{31}h_{32}| and |h_{31}g_{32}| required by μ→eγ would suppress h→τμ far below that level.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the muon g-2 deviation is explained, Br(h→τμ) can reach the current upper limit of 1.5×10⁻³, while Br(h→τe) and Br(h→μe) remain below about 10⁻⁹.
  • In the electron g-2 window, Br(h→τe) and Br(Z→τ±e∓) approach present experimental sensitivities, while Br(τ→μγ) and Br(h→τμ) stay suppressed.
  • The stringent bound on Br(μ→eγ) forces the first-generation leptoquark couplings to be tiny, thereby suppressing both Δa_e and the h→μe, τ→eγ rates.
  • Future improvements in the limits on μ→eγ and τ→eγ will directly translate into tighter predictions for the Higgs and Z flavor-violating decay rates.
  • The correlations mean that a single measurement of one LFV channel would immediately determine the expected rates of several related channels, allowing strong cross-checks.

Where Pith is reading between the lines

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

  • If a future experiment observes h→τμ at a rate near the current bound while MEG II pushes the μ→eγ limit below about 10⁻¹⁴, the predicted proportionality would be violated, signaling physics beyond this simple leptoquark framework.
  • The model's inability to simultaneously explain both the electron and muon g-2 anomalies suggests that if future data require both deviations, an additional source beyond this singlet leptoquark would be needed.
  • The proportionality between cLFV and LFVh/LFVZ rates could be tested even without directly producing the leptoquark, by comparing limits across channels at the HL-LHC and FCC-ee.
  • A dedicated scan that fixes the CKM-related rotations rather than leaving them free would clarify whether the near-limit predictions survive when quark-mixing constraints are imposed.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies lepton-flavor-violating decays h→ℓ_aℓ_b and Z→ℓ_b±ℓ_a∓ in the 3-3-1 model with a singlet scalar leptoquark, together with the charged LFV radiative decays and anomalous magnetic moments. It provides one-loop analytic formulas, verifies the cancellation of divergences in the h and Z amplitudes, and performs a numerical scan over the leptoquark mass, the hSS trilinear coupling, and the effective third-family Yukawa couplings. The central claims are a near-linear correlation Br(h,Z→e_b e_a) ∝ Br(e_b→e_aγ) and a complementarity between Δa_μ and Δa_e that selects different reachable LFV channels. The correlation is an analytic consequence of top-quark-dominated leptoquark couplings (Eq. (24)), not a fit; the numerical reach to current limits depends on scan assumptions and on the normalization of the h→ℓℓ' width.

Significance. If correct, the model yields channel-specific predictions testable at HL-LHC and FCC-ee, and it identifies which LFV channels are most promising. The paper's technical strengths are the complete one-loop PV-function expressions, the explicit demonstration of divergence cancellation, and the use of current experimental constraints including MEG II bounds on μ→eγ. However, the quantitative predictions rely on the scanning treatment of effective couplings and on a normalization issue in the h→ℓℓ' branching-ratio formula, so the numerical reach to the quoted 'near current limits' regions is not yet firmly established.

major comments (3)
  1. [Eq. (14)] The formula for Br(h→e_a e_b) is dimensionally inconsistent: the right-hand side m_h/(8π)(|Δ_L|^2+|Δ_R|^2) has mass dimension, while the left side is a branching ratio. The text defines Br ≡ Γ/Γ_h^total with Γ_h^total ≃ 4.1×10^-3 GeV, but no 1/Γ_h^total appears in the displayed equality. If the numerical code evaluates Eq. (14) as written, all h→LFV results in Figs. 4 and 6 are overestimated by a factor ~244 GeV. The authors must correct the formula and confirm that the numerical analysis uses the correct normalization.
  2. [Eq. (8) and Sec. IV, Eq. (25)] The scan treats g^{LQ}_{3i} and h^{LQ}_{3i} as independent free parameters with only |g|≤√(4π). However, Eq. (8) expresses these as linear combinations of the fundamental couplings \tilde g^{LQ}_{ki} (k=1,2) through the CKM rotations V_uL,V_dL. The paper neither scans over V_uL,V_dL with unitarity constraints nor imposes low-energy flavor bounds on \tilde g^{LQ}_{ki} (which couple to first/second generation quarks and are constrained by e.g. K_L→μe, D^0→μe, B→Kνν). Because the rotation is unitary, large g^{LQ}_{3i} requires large \tilde g^{LQ}_{ki} in the same combination, so these constraints directly limit the effective couplings. Without demonstrating that the interesting near-limit regions survive after imposing these constraints, the quantitative predictions for Br(h,Z→τμ) and the Δa_e window are not established.
  3. [Sec. III (one-loop contributions)] The calculation includes only the scalar leptoquark S in the loops, while the 3-3-1 gauge bosons (Z', W') and the other charged scalars also generate LFV and AMM contributions at one loop. The paper does not quantify when these are negligible; e.g., the Z' couplings to leptons are not suppressed by the same Yukawa factors. Since the numerical selection of parameter points uses constraints on Br(μ→eγ) and Δa_e,μ, omitting these contributions could allow points that are actually excluded. Please state the conditions under which the LQ contribution dominates, or include the additional contributions.
minor comments (5)
  1. [Eq. (10)] The second line has 'cS_(ba)R = c_(ba)R'; the superscript S is missing on the right-hand side.
  2. [Sec. IV / Abstract] The quoted window for |Δa_e| differs: the text after Eq. (25) says '10^-14 ≤ |Δa_e| ≤ 5×10^-13' while the abstract says '8×10^-13'. Unify the value.
  3. [Eq. (8)] The index structure in Eq. (8) and the surrounding text is confusing. Define all indices (k,i,b,a) and the row/column convention for V_uL and V_dL explicitly.
  4. [Eq. (14)] After correcting the missing 1/Γ_h, the factor 1/(8π) should be cross-checked against the standard 1/(16π) for a final state with two different final-state particles; clarify the definition of Δ_L,R.
  5. [Figs. 2 and 3] In the typeset version, the axis labels and legends are dense and difficult to read. Please ensure legibility and label all panels unambiguously.

Circularity Check

0 steps flagged

No significant circularity: the advertised correlations are analytic consequences of common coupling combinations, and the numerical claims are conditional scan results under external constraints, not fitted predictions.

full rationale

The central correlation Br(h,Z→e_b e_a)∝Br(e_b→e_aγ) is not circular. Eq. (24) defines f_ba as the coupling combination that dominates Br(e_b→e_aγ), and the text says: 'We will also check the interesting fact that Br(h, Z→e_b e_a)∝f_ba ∝ Br(e_b→e_aγ) as consequences can be seen directly from analytic formulas.' That is a derived consequence, not a fitted input: the numerical scan varies m_S, λ_hSS, and g(h)^LQ_{3i} and imposes external experimental constraints; it never tunes couplings to reproduce the h/Z LFV rates from the radiative rates. The muon/electron AMM complementarity is likewise a consequence of the imposed Br(μ→eγ) bound forcing g_{31},h_{31} tiny, as the paper explains: 'small experimental upper bound of Br(μ→eγ) requires both small values of |g_{31}^{LQ*} h_{32}^{LQ}| and |g_{32}^{LQ*} h_{31}^{LQ}|'. The loop machinery is frequently cited to [37],[38],[50], which include current authors, but the explicit formulas are reproduced in the text and the correlation argument relies only on top-quark dominance through f_ba; no load-bearing claim reduces to an unverified self-citation, uniqueness theorem, or ansatz imported by citation. The one notable limitation — that g^LQ_{3i} is scanned up to √(4π) because 'the unknown property of V_uL (V_dL) still allows large values' — is an input-freedom/viability concern, not circularity, since the scan does not use the target h/Z branching ratios as inputs.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 1 invented entities

The calculation rests on many freely scanned parameters and on prior model-building assumptions. The six top-family LQ Yukawa couplings plus m_S and λ_hSS are the main free inputs; the paper also assumes unknown CKM rotations allow large g^{LQ}_{3i}, that only the top quark matters, that the 3-3-1 gauge and scalar LFV contributions are negligible, and that certain master loop formulas remain valid for light fermions. These assumptions are load-bearing but largely unverified within the paper.

free parameters (5)
  • m_S (leptoquark mass) = scanned 0.5–5 TeV
    Mass of the new scalar leptoquark S; controls loop suppression and is not predicted by the model.
  • λ_hSS (trilinear h–S–S coupling) = scanned [-5,5]×v
    Sets the hSS vertex strength and enters all LFV h decay amplitudes; scanned freely over a wide range.
  • g^{LQ}_{31}, g^{LQ}_{32}, g^{LQ}_{33} = |g| ≤ sqrt(4π)
    Effective left-handed leptoquark Yukawa couplings after CKM rotation; the dominant LFV sources in Eq. (24).
  • h^{LQ}_{31}, h^{LQ}_{32}, h^{LQ}_{33} = |h| ≤ sqrt(4π)
    Effective right-handed leptoquark Yukawa couplings; scanned up to the perturbativity bound.
  • V_uL, V_dL quark rotation matrices = chosen to permit large g^{LQ}_{3i}
    The paper assumes these unknown rotations allow the third-family LQ couplings to be large; this freedom is load-bearing for the near-limit predictions.
axioms (6)
  • domain assumption Z2 symmetry prevents SM-exotic quark mixing (Ref. [34])
    Justifies the quark mass texture in Eq. (5) and the absence of exotic-quark contributions to the LFV loop amplitudes.
  • domain assumption Only top-quark contributions matter; light up-quark contributions are neglected
    Used in Eq. (24) to obtain the proportionality between LFV h/Z decays and radiative decays; if light-quark loop functions are not negligible, the correlation shifts.
  • domain assumption Master one-loop formulas from Ref. [37] are valid for light fermions
    The paper asserts this extension beyond the stated heavy-fermion validity of Ref. [40] without giving a proof.
  • domain assumption Large v_χ limit: Z couplings to SM fermions are SM-like
    Used in Table I to set the Zff couplings; assumes the 3-3-1 gauge sector decouples at the scale considered.
  • ad hoc to paper Contributions of 3-3-1 gauge bosons and charged scalars to LFV are negligible/ignored
    Only leptoquark exchange diagrams are evaluated (Fig. 1); no estimate is given for Z', W', or charged-Higgs contributions, which could mediate LFV in the full model.
  • domain assumption The leptoquark S does not mix with other charged scalars; λ_hSS from Appendix A
    Assumed from charge conservation and the Higgs vev pattern; needed for the h–S–S vertex used in LFV h formulas.
invented entities (1)
  • Singlet scalar leptoquark S ∼ (3,1,1/3) no independent evidence
    purpose: New mediator coupling up-type quarks to charged leptons; generates the one-loop Δa_μ and LFV decay amplitudes.
    Taken from Ref. [31] rather than introduced here, but it is the central new ingredient in every prediction. No direct collider or independent evidence is provided beyond the loop observables it is invoked to explain.

pith-pipeline@v1.3.0-alltime-deepseek · 60140 in / 19509 out tokens · 180691 ms · 2026-08-01T14:35:37.113078+00:00 · methodology

0 comments
read the original abstract

Motivated by a recent study of the 3-3-1 model supplemented with a singlet scalar leptoquark, which successfully explains the muon anomalous magnetic moment and the decay $\mu\to e\gamma$ within current experimental constraints, we extend the phenomenological analysis of this framework to include the lepton-flavor-violating decays of the Standard Model-like Higgs and the $Z$ boson. An interesting feature is that the branching ratios of these decays exhibit a nearly linear correlation with the corresponding charged lepton flavor-violating radiative decays, namely $\mathrm{Br}(h,Z\to e_b^\pm e_a^\mp)\propto\mathrm{Br}(e_b\to e_a\gamma)$. Furthermore, the model exhibits a complementary interplay between the charged-lepton anomalous magnetic moments: parameter regions with $|\Delta a_{\mu}|\geq10^{-11}$ favor $\mathrm{Br}(h,Z\to\tau^\pm\mu^\mp)$ reaching their current experimental upper limits while keeping $|\Delta a_e|$ negligible, whereas regions with $10^{-14}\leq|\Delta a_e|\leq 8\times10^{-13}$ instead allow $\mathrm{Br}(h,Z\to\tau^\pm e^\mp)$ to approach the present experimental sensitivities but simultaneously suppress $|\Delta a_\mu|$ to the level of $\mathcal{O}(10^{-16})$.

Figures

Figures reproduced from arXiv: 2607.18708 by L.T. Hue, N.H.T. Nha, N. T. K. Ngan, N.T. Tham, P.T. Bich, T.T. Hong.

Figure 1
Figure 1. Figure 1: FIG. 1: One-loop Feynman diagrams with leptoquark exchanges contributing to LFV decay am [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: The particular analytic formulas are derived from general results shown in Ref. [37], [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The relationship between the AMMs of muon ∆ [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The relationship between the AMMs [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The relationship between decay rates of LFV [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The relationship between decay rates of LFV [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The relationship between [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗

discussion (0)

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Reference graph

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