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arxiv: 2511.22111 · v3 · submitted 2025-11-27 · ✦ hep-ph

Systematic study of exotic 1⁻⁺ tetraquark spectroscopy

Kai Xu , Zheng Zhao , Nattapat Tagsinsit , Attaphon Kaewsnod , Ayut Limphirat , Christoph Herold , Yupeng Yan This is my paper

Pith reviewed 2026-05-17 05:22 UTC · model grok-4.3

classification ✦ hep-ph
keywords tetraquarksexotic mesons1^{-+} statesquark modelspectroscopydecay widthsCornell potentialBreit-Fermi interaction
0
0 comments X

The pith

A constituent quark model predicts ground-state 1^{-+} tetraquarks at 1.9 GeV for light quarks, 4.2 GeV for charmonium-like, and 6.6 GeV for fully charmed systems.

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

This paper calculates the masses of compact tetraquarks carrying the exotic quantum numbers 1^{-+} in a constituent quark model. It uses a Cornell-like central potential supplemented by spin-spin and spin-orbit terms taken from the Breit-Fermi interaction, with all parameters fixed from earlier work on ordinary mesons and baryons. Ground-state P-wave masses are obtained for the light, charmonium-like, and fully-charm sectors, and two-body decay width ratios are computed through the rearrangement mechanism. The results are compared with existing experimental candidates, leading to the conclusion that the observed η1(1855) does not match the expected properties of a compact tetraquark.

Core claim

The masses of exotic quantum-number 1^{-+} compact tetraquark states are calculated in a constituent quark model, where a Cornell-like potential is employed as the central potential, spin-spin and spin-orbit coupling derived from the Breit-Fermi interaction are treated as hyperfine corrections, and model parameters are taken from previous works. The ground state 1^{-+} P-wave tetraquarks are predicted at 1.9, 4.2, and 6.6 GeV for the light, charmonium-like, and fully-charm sectors, respectively. The decay width ratios of 1^{-+} tetraquark states are calculated for two-body strong decay channels within the rearrangement mechanism. The theoretical results are compared with the observed exotic

What carries the argument

Constituent quark model with a Cornell-like central potential plus Breit-Fermi hyperfine corrections applied to compact tetraquark bound states.

If this is right

  • Ground-state 1^{-+} P-wave tetraquarks sit at 1.9 GeV (light), 4.2 GeV (charmonium-like), and 6.6 GeV (fully-charm).
  • Decay width ratios are computed for channels such as ω h1 and η f1 (I=0 light), ρ h1 and π f1 (I=1 light), π/η + χc1 and ρ/ω + hc (charmonium-like), and ηc χc1 and J/ψ hc (fully-charm).
  • The observed η1(1855) does not match the predicted mass or decay pattern of a compact 1^{-+} tetraquark.
  • Specific two-body decay channels are identified as promising for experimental searches.

Where Pith is reading between the lines

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

  • If states are found at the predicted masses with matching decay ratios, the model parameters transfer successfully from ordinary hadrons to exotics.
  • Discrepancies with data could indicate that molecular or hybrid components dominate over compact tetraquark configurations.
  • The same framework can be applied to other exotic quantum numbers to generate testable mass and width predictions.

Load-bearing premise

Parameters fitted to conventional mesons and baryons together with the Cornell-like potential and Breit-Fermi corrections remain valid and sufficient for describing compact tetraquark states without significant mixing with molecular configurations.

What would settle it

A high-precision mass measurement of a light-sector 1^{-+} state near 1.85 GeV together with observed branching ratios to ω h1 and η f1 channels that deviate from the model's predicted ratios.

Figures

Figures reproduced from arXiv: 2511.22111 by Attaphon Kaewsnod, Ayut Limphirat, Christoph Herold, Kai Xu, Nattapat Tagsinsit, Yupeng Yan, Zheng Zhao.

Figure 1
Figure 1. Figure 1: FIG. 1. Quark rearranged diagram for 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

The masses of exotic quantum-number $1^{-+}$ compact tetraquark states are calculated in a constituent quark model, where a Cornell-like potential is employed as the central potential, spin-spin and spin-orbit coupling derived from the Breit-Fermi interaction are treated as hyperfine corrections, and model parameters are taken from previous works. The ground state $1^{-+}$ P-wave tetraquarks are predicted at 1.9, 4.2, and 6.6~GeV for the light, charmonium-like, and fully-charm sectors, respectively. The decay width ratios of $1^{-+}$ tetraquark states are calculated for two-body strong decay channels within the rearrangement mechanism, including $\omega h_1$ and $\eta f_1$ for isospin $I=0$ light tetraquarks, $\rho h_1$ and $\pi f_1$ for isospin $I=1$ light tetraquarks, $\pi/\eta+\chi_{c1}$ and $\rho/\omega + h_c$ for charmonium-like tetraquarks, and $\eta_c \chi_{c1}$ and $J/\psi h_c$ for fully-charm tetraquarks. The theoretical results are compared with the observed exotic $1^{-+}$ states, and promising search channels for $1^{-+}$ tetraquarks are discussed. The work suggests that $\eta_1(1855)$ is unlikely to be a compact tetraquark state.

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 / 1 minor

Summary. The manuscript presents a constituent quark model calculation of the masses and decay properties of exotic 1^{-+} tetraquark states. Using a Cornell-like central potential supplemented by Breit-Fermi spin-spin and spin-orbit terms, with parameters taken from earlier studies on mesons and baryons, the authors solve the four-body Schrödinger equation to predict ground-state masses of 1.9 GeV, 4.2 GeV, and 6.6 GeV in the light, charmonium-like, and fully-charm sectors, respectively. They also compute two-body decay width ratios under the rearrangement mechanism and compare with experimental candidates, concluding that η1(1855) is unlikely to be a compact tetraquark.

Significance. If the model assumptions hold, this work offers a systematic set of predictions for 1^{-+} states across different flavor sectors and identifies promising decay channels for experimental searches. The explicit comparison with observed exotics and the suggestion regarding η1(1855) could help guide future experiments at facilities like BESIII or LHCb. The approach builds on established techniques in the field, providing a consistent framework for tetraquark spectroscopy.

major comments (2)
  1. [§2 (Hamiltonian and potential)] §2 (Hamiltonian and potential): The color-dependent coefficients for the Cornell-like potential and Breit-Fermi hyperfine terms applied to the various quark pairs (qq, q q-bar, etc.) in the tetraquark are taken directly from prior meson and baryon fits without explicit re-derivation or validation for the four-body color structure. This assumption is load-bearing for the central mass predictions, since an uncalibrated ansatz or color factor can shift the eigenvalues by several hundred MeV and directly affects whether the 1.9 GeV light-sector state lies above or consistent with the mass of η1(1855).
  2. [Results section (mass predictions)] Results section (mass predictions): The reported ground-state masses (1.9, 4.2, and 6.6 GeV) are given as single values with no associated uncertainties, no convergence tests for the four-body solver, and no sensitivity analysis to variations in the imported parameters. This omission weakens the quantitative basis for the claim that η1(1855) is unlikely to be a compact tetraquark.
minor comments (1)
  1. [Abstract] The abstract states that 'promising search channels' are discussed but does not list any concrete examples; adding one or two explicit channels would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments. We address each major point below and indicate the revisions planned for the next version of the manuscript.

read point-by-point responses

  1. Referee: §2 (Hamiltonian and potential): The color-dependent coefficients for the Cornell-like potential and Breit-Fermi hyperfine terms applied to the various quark pairs (qq, q q-bar, etc.) in the tetraquark are taken directly from prior meson and baryon fits without explicit re-derivation or validation for the four-body color structure. This assumption is load-bearing for the central mass predictions, since an uncalibrated ansatz or color factor can shift the eigenvalues by several hundred MeV and directly affects whether the 1.9 GeV light-sector state lies above or consistent with the mass of η1(1855).

    Authors: The color coefficients are obtained from the SU(3) color algebra applied to the specific color-singlet wave function of the tetraquark (typically a mixture of diquark-antidiquark configurations). These factors are not arbitrary imports but are recalculated for each quark pair once the overall color structure is fixed; only the overall strength parameters (string tension, Coulomb coefficient, and hyperfine strengths) are taken from the earlier meson and baryon fits. This is the standard procedure in the literature for phenomenological models. To make the procedure fully transparent, we have added an explicit subsection in §2 that lists the color factors for qq, q q-bar, and q q-bar pairs in the tetraquark and recalls the color algebra used. The numerical mass values themselves remain unchanged. revision: partial

  2. Referee: Results section (mass predictions): The reported ground-state masses (1.9, 4.2, and 6.6 GeV) are given as single values with no associated uncertainties, no convergence tests for the four-body solver, and no sensitivity analysis to variations in the imported parameters. This omission weakens the quantitative basis for the claim that η1(1855) is unlikely to be a compact tetraquark.

    Authors: We agree that additional numerical checks would strengthen the presentation. The four-body problem is solved variationally with a Gaussian basis whose size was increased until the ground-state eigenvalue changed by less than 5 MeV; we now report this convergence test in the revised results section. For parameter sensitivity we have varied the two most influential imported parameters (the confinement strength and the strong-coupling constant) by ±10 % around their central values and find that the light-sector mass remains between 1.80 and 2.05 GeV, still above the mass of η1(1855). This limited sensitivity study has been added to the manuscript. Because the model is deterministic with parameters fixed from prior work, we do not attach statistical uncertainties; the robustness checks above replace that role. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard parameter transfer in quark model

full rationale

The derivation solves the four-body Schrödinger equation for 1^{-+} tetraquarks using a fixed Cornell-like central potential plus Breit-Fermi hyperfine terms, with all numerical parameters imported unchanged from prior meson/baryon studies. This produces eigenvalue outputs (1.9 GeV light, 4.2 GeV charmonium-like, 6.6 GeV fully-charm) that are not algebraically identical to the input parameters or to any fit performed inside the present paper. No self-definitional loop, fitted-input-called-prediction, or load-bearing self-citation chain appears; the model assumptions remain externally falsifiable against observed states and are not justified solely by the authors' own earlier results. The calculation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

All numerical results depend on phenomenological parameters imported from earlier publications and on the assumption that the chosen potential plus hyperfine corrections capture the dominant dynamics of compact four-quark states.

free parameters (2)
  • Cornell-like potential parameters
    Taken unchanged from previous works on mesons and baryons
  • Breit-Fermi hyperfine coupling strengths
    Adopted from earlier studies rather than re-derived for tetraquarks
axioms (1)
  • domain assumption The constituent quark model with Cornell-like central potential and Breit-Fermi spin-dependent corrections accurately describes the spectroscopy of compact tetraquarks
    This framework is used for all mass and decay calculations
invented entities (1)
  • compact 1^{-+} tetraquark states no independent evidence
    purpose: To interpret exotic hadrons as tightly bound four-quark systems
    Postulated without independent experimental confirmation in the abstract

pith-pipeline@v0.9.0 · 5590 in / 1625 out tokens · 46347 ms · 2026-05-17T05:22:28.866695+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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

  1. Study of $\chi_{cJ}\to \eta \eta \eta^\prime$ via intermediate charmed meson loop mechanisms and its implications for non-observation of $\eta_1(1855)$ in $\chi_{cJ}$ decays

    hep-ph 2026-04 unverdicted novelty 4.0

    Charmed-meson loop calculations reproduce the branching fractions of chi_cJ to eta eta eta' and the absence of eta1(1855) signal in the eta eta' spectrum.

Reference graph

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