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arxiv: 2512.19636 · v2 · submitted 2025-12-22 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

Heavy neutral bosons and dark matter in the 3-3-1 model with axionlike particle

T.T. Hieu , V.H. Binh , H. N. Long , H. T. Hung
Authors on Pith no claims yet

Pith reviewed 2026-05-16 20:22 UTC · model grok-4.3

classification ✦ hep-ph
keywords 3-3-1 modelaxionlike particlesZ' bosonheavy Higgs bosondark matterrelic densitylepton flavor violation
0
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The pith

In the 3-3-1 model with axionlike particles, LHC searches set m_h2 at least 600 GeV and m_Z' at least 5.1 TeV while tying dark matter mass to the axion breaking scale.

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

The paper examines the 3-3-1 gauge extension supplemented by axionlike particles. It uses gluon-gluon fusion production and existing ATLAS and CMS dilepton resonance data to extract lower bounds on a second neutral Higgs and a new neutral gauge boson. The authors assume a residual Z2 symmetry survives spontaneous symmetry breaking and thereby stabilizes certain particles as dark matter candidates. Relic density calculations then produce a direct link between the dark matter mass and the scale at which the axionlike field breaks. These results give concrete mass targets for upcoming collider runs and direct detection searches.

Core claim

The paper claims that within the 331ALP model, current LHC limits on high-mass dilepton resonances and lepton flavor violating decays force the second neutral Higgs boson to satisfy m_h2 ≥ 600 GeV and the new Z' gauge boson to satisfy m_Z' ≥ 5.1 TeV. By retaining Z2 as a residual symmetry after the spontaneous symmetry breaking stages, the odd-parity particles become stable and serve as dark matter; their relic abundance calculation then establishes an explicit relation between the dark matter mass and the axion breaking scale.

What carries the argument

The residual Z2 symmetry that survives spontaneous symmetry breaking and assigns odd parity to the dark matter candidates, thereby protecting them from decay while allowing their relic density to fix the axion scale.

If this is right

  • Upcoming LHC runs at higher luminosity can directly test the predicted mass windows through dilepton and Higgs decay signatures.
  • The dark matter mass to axion scale relation restricts the parameter space accessible to direct detection experiments.
  • Lepton flavor violating branching ratios remain bounded by current limits, which already carves out the allowed region for both new bosons.
  • The model supplies a single framework in which axionlike particles simultaneously address the strong CP problem and dark matter stability.

Where Pith is reading between the lines

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

  • Axion searches at beam-dump facilities could indirectly constrain the dark matter mass through the shared breaking scale.
  • Detection of the Z' near its lower bound would fix the 3-3-1 breaking scale and thereby predict the full spectrum of heavy Higgs states.
  • The same Z2 parity could be extended to additional sectors such as neutrinos to stabilize further dark matter candidates.
  • Precision Higgs coupling measurements may reveal small mixing effects with the heavy h2 even when its mass lies above 600 GeV.

Load-bearing premise

Z2 is assumed a residual symmetry after spontaneous symmetry breaking stages that stabilizes odd-Z2 particles as dark matter candidates.

What would settle it

Discovery of a dilepton resonance below 5.1 TeV or a neutral Higgs below 600 GeV in the gluon-fusion channels examined would exclude the viable parameter space.

Figures

Figures reproduced from arXiv: 2512.19636 by H. N. Long, H. T. Hung, T.T. Hieu, V.H. Binh.

Figure 1
Figure 1. Figure 1: Feynman diagrams at one-loop order of H → lalb decays in the unitary gauge, H ≡ h1, h2 The partial width of H → l ± a l ∓ b is Γ(H → lalb) ≡ Γ(H → l + a l − b ) + Γ(H → l − a l + b ) = mH 8π  |∆ (ab) L | 2 + |∆ (ab) R | 2  . (51) We use the conditions for external momentum as: p 2 a,b = m2 a,b, (pa + pb) 2 = m2 H and m2 H ≫ m2 a,b, this leads to branching ratio of H → l ± a l ∓ b decays can be given Br(H… view at source ↗
Figure 2
Figure 2. Figure 2: Contour plots of la → lbγ decays in plane of (tα, mH± 2 ) [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plots of Br(h1 → µτ ) (left panel) and Br(h2 → µτ ) (right panel) in plane of (tα, mH± 2 ). To illustrate, we represent the signals of Br(h1 → µτ ) and Br(h2 → µτ ), in the allowed space region in Fig.3. On the left panel of Fig.3, Br(h1 → µτ ) may be get values of 3.4 × 10−6 and 6.7 × 10−6 while Br(h2 → µτ ) may be get values of 9.1 × 10−5 and 5.6 × 10−5 in right panel. These values all satisfy the presen… view at source ↗
Figure 4
Figure 4. Figure 4: Plots of σ (pp → h2) × Br (h2 → µτ ) (magenta line) depend on MH, the green and cyan lines depict the CMS’s observations for the combined decay modes of H → µτ (based on Ref. [30]) for the low and high mass ranges, respectively.. IV. Z ′ BOSON AND DARK MATTER In 331ALP, the heavier neutral gauge boson, Z ′ , has the mass and physical state given by Eq.(16) in section II. The interaction of Z ′ with fermion… view at source ↗
Figure 5
Figure 5. Figure 5: The dependence of σ [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Dominant contributions to annihilation of dark matter. [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Contour of the dark matter relic density on plane of ( [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
read the original abstract

We consider heavy neutral bosons in the 3-3-1 model with axionlike particles (331ALP), including the Higgs boson and the $Z^\prime$ boson which are outside the standard model (SM). Based on gluon-gluon fusion at the LHC, we investigate the signals of cross-sections in the parameter space region satisfying the current experimental limits of lepton flavor violating decay, including processes involving both charged leptons and Higgs boson, and provide predictions of $m_{h_2}\geq 600 ~\mathrm{GeV}$. A new gauge boson, labeled as $Z^{\prime}$, is predicted $m_{Z^{\prime}}\geq 5.1 ~\mathrm{TeV}$ based on the search for high-mass dilepton resonances at ATLAS and CMS. We consider the stability of odd-$Z_2$ particles, with $Z_2$ is assumed a residual symmetry after spontaneous symmetry breaking stages, to point out dark matter candidates in the model. Investigating the relic density of dark matter within experimentally permissible limits, we established a relationship between the mass of dark matter and the breaking scale of axion.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript examines heavy neutral bosons in the 3-3-1 model extended by an axion-like particle (331ALP). Using gluon-gluon fusion signals and high-mass dilepton resonance searches at the LHC, it derives lower bounds m_h2 ≥ 600 GeV and m_Z' ≥ 5.1 TeV while respecting lepton-flavor-violation limits. It assumes a residual Z2 symmetry after spontaneous symmetry breaking to stabilize dark-matter candidates and reports a relation between dark-matter mass and the axion breaking scale obtained from relic-density constraints within experimental bounds.

Significance. If the central claims hold, the work supplies concrete, LHC-testable mass lower bounds for an extra Higgs and a new gauge boson in the 331ALP framework and links the dark-matter sector to the axion scale via relic density. The direct application of published ATLAS/CMS limits to constrain the model parameters is a clear strength; however, the dark-matter analysis must be placed on firmer footing for the relation to constitute an independent prediction rather than a fitted outcome.

major comments (2)
  1. [Dark-matter section] Dark-matter section (discussion of Z2 residual symmetry): The statement that Z2 remains unbroken after all spontaneous-symmetry-breaking stages is asserted without an explicit check that every vacuum expectation value in the 3-3-1 scalar sector plus the ALP is Z2-even. Because an odd VEV would break the symmetry and allow decays of the putative DM candidates, this verification is load-bearing for the subsequent relic-density analysis and the claimed mass-breaking-scale relation.
  2. [Abstract and relic-density subsection] Abstract and relic-density subsection: The relationship between dark-matter mass and axion breaking scale is presented as established within permissible relic-density limits, yet no derivation, explicit parameter scan, or error analysis is supplied. Without these steps it is impossible to determine whether the relation follows from the model dynamics or is the result of parameter tuning, weakening the claim that an independent prediction has been obtained.
minor comments (2)
  1. [Abstract] The acronym 331ALP is introduced in the abstract without an immediate parenthetical definition.
  2. [§3] Cross-section formulae for gluon-gluon fusion (presumably in §3) would benefit from an explicit listing of the dependence on the 331 and ALP parameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised concerning the dark-matter sector are well taken and highlight areas where additional clarity will strengthen the presentation. We address each major comment below and will incorporate the necessary revisions in the updated version of the manuscript.

read point-by-point responses

  1. Referee: [Dark-matter section] Dark-matter section (discussion of Z2 residual symmetry): The statement that Z2 remains unbroken after all spontaneous-symmetry-breaking stages is asserted without an explicit check that every vacuum expectation value in the 3-3-1 scalar sector plus the ALP is Z2-even. Because an odd VEV would break the symmetry and allow decays of the putative DM candidates, this verification is load-bearing for the subsequent relic-density analysis and the claimed mass-breaking-scale relation.

    Authors: We agree that an explicit verification is required to confirm the residual Z2 symmetry. In the original manuscript the Z2 parity assignments were chosen such that the scalar potential and all vacuum expectation values preserve the symmetry, but we did not tabulate the Z2 charges of each field and VEV. In the revised version we will add a dedicated paragraph (or short subsection) that lists the Z2 charges of the 3-3-1 scalar fields and the ALP, demonstrates that every VEV is Z2-even, and thereby confirms that the symmetry remains unbroken after all spontaneous-symmetry-breaking stages. This explicit check will directly support the stability of the odd-Z2 dark-matter candidates and the subsequent relic-density analysis. revision: yes

  2. Referee: [Abstract and relic-density subsection] Abstract and relic-density subsection: The relationship between dark-matter mass and axion breaking scale is presented as established within permissible relic-density limits, yet no derivation, explicit parameter scan, or error analysis is supplied. Without these steps it is impossible to determine whether the relation follows from the model dynamics or is the result of parameter tuning, weakening the claim that an independent prediction has been obtained.

    Authors: We acknowledge that the manuscript states the mass–scale relation without showing the underlying calculation. The relation is obtained by solving the Boltzmann equation for the relic density of the Z2-odd dark-matter candidate, with the annihilation cross section depending on the axion breaking scale through the ALP–DM portal. In the revised manuscript we will (i) derive the relevant Boltzmann equation and the approximate analytic expression for the relic density, (ii) describe the parameter scan performed over the axion scale and DM mass (with ranges and sampling method), and (iii) present the resulting allowed region together with the 1σ and 2σ experimental relic-density bands. This will make clear that the reported relation is a direct consequence of the model dynamics and the experimental constraints rather than an arbitrary fit. revision: yes

Circularity Check

1 steps flagged

DM-axion mass relation obtained by fitting relic density to experimental limits rather than independent derivation

specific steps
  1. fitted input called prediction [Abstract]
    "Investigating the relic density of dark matter within experimentally permissible limits, we established a relationship between the mass of dark matter and the breaking scale of axion."

    The relationship is obtained by scanning parameters until the computed relic density falls inside the experimental window; the resulting correlation is therefore a direct output of the fitting procedure rather than an independent theoretical result.

full rationale

The m_h2 and m_Z' lower bounds are obtained by imposing external ATLAS/CMS limits on the model's cross-sections and resonances; these are not circular. The sole load-bearing step that reduces to input data is the claimed 'relationship' between dark-matter mass and axion breaking scale, which the abstract states is found by requiring the relic density to lie inside permissible limits. This is a parameter-space constraint, not a first-principles prediction independent of the data. The Z2 residual-symmetry assumption is stated without an explicit vev check in the provided text, but does not itself create a definitional loop. No self-citation chains or ansatz smuggling appear in the quoted material. The derivation chain therefore contains one fitted-input step but remains otherwise self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 2 invented entities

The central claims rest on the 3-3-1 gauge structure plus axion-like particle extension, a residual Z2 symmetry, and parameter choices that satisfy LFV and relic density constraints; several free scales and couplings are implicitly adjusted to match data.

free parameters (2)
  • axion breaking scale
    Directly tied to dark matter mass via relic density requirement; value not fixed by first principles.
  • 331 breaking scale
    Sets masses of Z' and extra Higgs; constrained but not derived from the paper's equations alone.
axioms (2)
  • domain assumption Z2 is a residual symmetry after spontaneous symmetry breaking stages
    Invoked to guarantee stability of odd-Z2 particles as dark matter candidates.
  • domain assumption Parameter space satisfies current experimental limits on lepton flavor violating decays
    Used to select the region where cross-section predictions are made.
invented entities (2)
  • Z' no independent evidence
    purpose: New neutral gauge boson from 3-3-1 extension
    Standard in 3-3-1 models; no independent evidence supplied beyond the model construction.
  • h2 (extra Higgs) no independent evidence
    purpose: Second neutral scalar boson
    Arises from the extended scalar sector; mass bound derived from LHC limits.

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Reference graph

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