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REVIEW 3 major objections 4 minor 62 references

A Bethe-Salpeter calculation places S-wave top–antiquark states a few GeV above the top-quark mass

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 02:42 UTC pith:QENSLYZD

load-bearing objection A transparent BS calculation of single-top mesons whose mass numbers are likely washed out by the top width; worth refereeing with a request for a complex-mass treatment. the 3 major comments →

arxiv 2602.09684 v2 pith:QENSLYZD submitted 2026-02-10 hep-ph

The S-wave topped meson

Jun-Hao Zhang , Shuo Yang , Bing-Dong Wan This is my paper
classification hep-ph
keywords topped mesonBethe-Salpeter equationtop quark bound statesS-wave spectrumscreened Cornell potentialheavy-light mesonsnear-threshold enhancementtop quark hadronization
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.

The paper computes the S-wave mass spectrum of 'topped mesons' — quark–antiquark systems containing one top quark — using the instantaneous Bethe-Salpeter equation with a screened Cornell potential. For the top–bottom-antiquark system (t-bbar), it finds 1S, 2S, 3S, and 4S states at about 5.08, 5.37, 5.57, and 5.74 GeV above the top-quark mass; for the top–charm-antiquark system (t-cbar) the corresponding offsets are 1.90, 2.23, 2.45, and 2.60 GeV. The authors are careful to label these as model-dependent reference positions for possible quasi-bound configurations, not predictions of fully formed hadrons, because the top quark decays before hadronization. The numbers give experimentalists concrete windows in which to look for small enhancements in top-plus-light-jet invariant-mass distributions.

Core claim

Within the instantaneous approximation to the Bethe-Salpeter equation and a scalar-plus-vector Cornell-like potential with color screening, the paper obtains discrete S-wave eigenvalues for t-bbar, t-cbar, t-sbar, t-dbar, and t-ubar up to the fourth radial excitation. For identical quantum numbers the 0^- and 1^- (1S0 and 3S1) masses coincide numerically, confirming that spin-splitting is negligible at the top-quark mass scale. The paper's central numerical claim is Table I: for a top quark of mass 172.76 GeV, the t-bbar 1S mass is about 177.8 GeV (5.08 GeV above the top quark), with excited states at roughly 5.37, 5.57, and 5.74 GeV above; the t-cbar ground state is about 174.7 GeV (1.90 Ge

What carries the argument

The machinery is the three-dimensional Salpeter equation — the instantaneous (potential) reduction of the Bethe-Salpeter equation for a quark–antiquark bound state. The interaction kernel is a Cornell-type potential: a linear scalar confining term with an exponential color-screening factor e^{-αr} and a short-range one-gluon-exchange vector term, Fourier-transformed to momentum space. The top quark enters through a projection-operator decomposition of its propagator, and the eigenvalue equation is solved for the S-wave (0^-) and (1^-) states. The work this machinery does is to convert a QCD-motivated potential into a discrete mass spectrum for each quark–antiquark pair; the equal masses of t

Load-bearing premise

The calculation treats the top quark as a stable constituent (zero width) inside the BS equation, even though its measured width of about 1.4 GeV and its lifetime far shorter than the hadronization time mean the discrete eigenvalues may not correspond to observable resonances.

What would settle it

Measure the t-bbar invariant-mass distribution near 177.8 GeV in high-luminosity collider data: if no excess, peak, or threshold distortion appears at the predicted 1S position while the same analysis finds the known top-pair near-threshold excess, the reference positions would not be realized as physical states. A calculational falsifier would be solving the same equation with a finite top width and finding the real part of the eigenvalue shifted by more than about 1 GeV.

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

If this is right

  • If the reference positions are realized even as transient enhancements, searches in t-plus-light-jet invariant mass have concrete windows: for t-bbar near 177.8, 178.1, 178.3, and 178.5 GeV, and for t-cbar near 174.7, 175.0, 175.2, and 175.4 GeV.
  • The near degeneracy of the 0^- and 1^- states for each n means any S-wave signal should appear as a single cluster in invariant mass, without needing to separate two close spin peaks.
  • The dominant decay chain (t -> W b) implies a topped-meson candidate should appear as W + b + light jet with invariant mass near the top-quark mass, distinguishable from ordinary top-pair background by the additional spectator jet.
  • Because the t-bbar binding (about 5 GeV) exceeds the top-quark width (about 1.4 GeV), the t-bbar system is the most promising for a first experimental look.

Where Pith is reading between the lines

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

  • Given that the top width is comparable to the computed binding energies, the more realistic observable may not be a narrow resonance but a broad distortion of the invariant-mass distribution; the paper's discrete masses could best be read as positions where the slope of the distribution changes.
  • A direct testable extension is to re-run the same eigenvalue problem with a complex top-quark mass or a width term in the propagator and check whether the real parts of the eigenvalues survive; shifts larger than about 1 GeV would indicate the reference positions are not robust.
  • The equal 0^-/1^- masses suggest spin-dependent forces are negligible at this scale, so any experimentally resolved structure should show no spin-polarization asymmetry; comparing angular distributions in the candidate window would test this.
  • The same instantaneous-BS framework with a screened Cornell potential could be applied to single-bottom and single-charm systems, which have much smaller widths, providing a consistency check of the potential parameters against known heavy-light meson spectra.

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 / 4 minor

Summary. The paper applies the instantaneous Bethe-Salpeter equation to S-wave bound states of a single top quark with a light, charm, or bottom antiquark (t̄q, t̄c, t̄b), computing masses for radial excitations n = 1–4. The resulting masses lie within a few GeV of the top-quark mass: e.g., for t̄b the 1S–4S states are 5.08, 5.37, 5.57, 5.74 GeV above m_t, and for t̄c they are 1.90, 2.23, 2.45, 2.60 GeV. The authors also verify the singlet–triplet degeneracy, present BS wave functions for all states, and qualitatively discuss production and decay patterns. The paper explicitly frames the discrete eigenvalues as 'model-dependent reference positions' rather than predictions for fully formed hadrons.

Significance. If the computed spectrum is robust, the paper provides a concrete, systematically derived reference point for possible quasi-bound top–antiquark structures, which could be used in LHC searches and prospective studies. The work extends a standard and internally consistent formalism (instantaneous BS equation with a screened Cornell potential) to a system that is rarely treated in this way. It also honestly acknowledges the strong caveat posed by the short top lifetime. However, the central numerical claim is undermined by the neglect of the top-quark width, which is larger than the level spacings, and by the absence of any uncertainty estimate. The paper is therefore a useful but incomplete phenomenological reference; additional quantitative work on the width and parameter sensitivity is needed before the specific numbers can be considered reliable.

major comments (3)
  1. [§III, Table I and Eqs. (16)–(18)] The central calculation treats the top quark as stable: the propagators in Eq. (8) contain the real mass m_t only, and the eigenvalue equation (16) yields real discrete masses. The physical top width, Γ_t = 1.42 GeV, is larger than the spacings between adjacent radial states in Table I (approximately 0.29, 0.20, 0.17 GeV for t̄b; 0.33, 0.22, 0.15 GeV for t̄c) and is comparable to the binding shifts themselves. The top can decay before the would-be bound state completes an oscillation, so the static approximation is not a small perturbation. The paper should include a complex-mass treatment (e.g., m_t → m_t − iΓ_t/2 in the propagators) or a quantitative argument showing that the real parts of the poles are stable under this modification. Without such an analysis, the numbers in Table I are not secured even as reference positions.
  2. [§III, Table I] The masses are quoted to 0.1 MeV precision but no uncertainties are given. The parameters are fitted to PDG meson masses, but there is no error propagation, no sensitivity study of the potential parameters (α, λ, α_s) or the quark masses, and no comparison with the fitted meson masses. The reader cannot judge whether the differences between t̄b and t̄c (≈3.2 GeV) or the n-spacings (≈0.15–0.33 GeV) are robust or accidental. A sensitivity table, a rough parameter-variation estimate, or a statement of the expected theoretical uncertainty is necessary to give the spectrum quantitative meaning.
  3. [§IV] The production and decay discussion is qualitative and does not make contact with the computed wave functions. For example, Eq. (28) expresses the production amplitude through ψ_T(0), but no numerical values of the wave function at the origin are extracted from the BS solutions, and no estimate of the total width of the would-be states is provided. Since the constituent top quark decays with rate ≈Γ_t, the decay width of the meson should be of the same order; the paper should state this explicitly and discuss how a ~1.4 GeV width affects the observability of the proposed invariant-mass features. Without this, the experimental comments remain only heuristic.
minor comments (4)
  1. [Abstract and Sec. I] Typos and grammar: 'investigate the the mass spectrum' and 'which containing' should be corrected. The phrase 'significantly enhanced lifetimes ... compared to toponium' is potentially misleading; a width reduced by a factor of two is still ~1.4 GeV, so clarify that these are broad states.
  2. [Sec. II heading] The section title 'BETHE-SALPETER EQUA TION' contains an unwanted space. The text just before Eq. (19) says 'mixed' but it is actually correct.
  3. [Table I] The unit 'Mass (100 GeV)' is confusing. Either give masses in GeV directly or state explicitly that the entries must be multiplied by 100 GeV. Also, the 3S1 masses are said to coincide with the 1S0 but are not tabulated; consider showing them for completeness.
  4. [References [57]–[59]] These are formatted as 'Miscellaneous references' and omit full bibliographic details. They should be expanded to include authors, titles, journals, volumes, pages, and years.

Circularity Check

0 steps flagged

No significant circularity: the top-meson spectrum is computed from BS equations with parameters calibrated to ordinary meson masses, and the target states are not among the fitted data.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The input parameters are stated explicitly in Sec. III: "These parameters are determined by fitting the experimental data; that is, they are adjusted such that the calculated meson masses are consistent with the values reported by the Particle Data Group (PDG) [1]." This calibration is to ordinary meson masses, not to the t-bar q, t-bar c, or t-bar b states that Table I then predicts. The central numerical results (e.g., ~5.08 GeV above m_t for the 1S t-bar b state) emerge from solving the instantaneous BS equations (16)-(18) with the potential (19)-(20); no equation defines a predicted t-bar mass in terms of a fitted t-bar input. The singlet-triplet degeneracy imported from Ref. [35] is not load-bearing: the paper explicitly checks it numerically and reports that the 1S0 and 3S1 masses "coincide within our numerical accuracy." The paper's own caveat that the eigenvalues are "model-dependent reference positions" and not predictions for fully formed hadrons is an honest statement of physical limitation, not a sign of circularity. The neglect of the top-quark width is a physical approximation and a correctness risk, but it is not an input-output identity; the top mass and width are not fitted to reproduce the predicted spectrum. No self-citation chain or constructed equivalence between input and output is present.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 1 invented entities

The central prediction is built on four fitted parameters (quark masses, α_s, α, λ) plus three domain assumptions: the instantaneous Salpeter reduction, the Cornell-screened potential, and the neglect of the top width. No entity with independent falsifiable evidence is introduced beyond the quasi-bound states themselves.

free parameters (4)
  • quark masses m_b, m_c, m_s, m_d, m_u = 4.960, 1.620, 0.500, 0.311, 0.305 GeV
    Adjusted so that calculated ordinary meson masses match PDG values; these directly set the mass gaps in Table I.
  • strong coupling α_s(m_t) = 0.11
    Chosen value for the Coulomb part of the potential; no uncertainty or running is given.
  • screening parameter α = 0.06 GeV
    Determines the e^{-αr} factor in the potential; fitted/selected from prior potential-model studies.
  • string tension λ = 0.18 GeV^2
    Strength of the linear confining potential; fitted to PDG meson masses.
axioms (4)
  • domain assumption Instantaneous approximation: the interaction kernel is independent of the relative energy component (V(P;q,k) ≈ V(q_⊥,k_⊥)).
    Introduced in Eq. (10) of Sec. II; this reduces the full BS equation to a Salpeter equation and is standard for heavy mesons but is an approximation.
  • domain assumption The interquark potential is a Cornell-type potential with scalar linear confinement and vector Coulomb exchange, plus a screening factor e^{-αr}.
    Eq. (19) in Sec. II; this is the phenomenological input that fixes the spectrum and is extrapolated from lattice/static-quark studies to a moving top system.
  • domain assumption The top quark is treated as a stable constituent (width Γ_t set to zero) in the BS eigenvalue equation.
    No width appears in the propagators or energy equations in Sec. II-III; this is the paper's weakest premise because the top width is comparable to the predicted binding energies.
  • domain assumption Singlet-triplet degeneracy M(0^-, n1S0) = M(1^-, n3S1) for top systems.
    Eq. (21) is asserted via Ref. [35] and then numerically 'confirmed' within the same model; this is an imported result, not derived in this paper.
invented entities (1)
  • Topped mesons T ≡ (t\bar q), (t\bar c), (t\bar b) no independent evidence
    purpose: Hypothetical quasi-bound states of a top quark and a lighter antiquark, whose masses are the central prediction.
    No experimental evidence is presented; the paper itself calls them model-dependent reference positions. The only handle is qualitative final states (W + b-jets + light jet), with no predicted cross sections or widths.

pith-pipeline@v1.3.0-alltime-deepseek · 190 in / 7704 out tokens · 110551 ms · 2026-08-03T02:42:09.433166+00:00 · methodology

0 comments
read the original abstract

Motivated by the recent near-threshold enhancement in top-quark pair production reported by CMS and ATLAS, we study the S-wave spectral structure of heavy-light systems containing a single top quark, namely $t\bar{q}$, $t\bar{c}$, and $t\bar{b}$, within the instantaneous Bethe-Salpeter formalism. Because the top quark decays on a timescale much shorter than the typical hadronization time, the discrete eigenvalues we obtain should be interpreted as model-dependent reference positions of possible quasi-bound heavy-light configurations, rather than as predictions for fully formed conventional hadrons. The numerical results indicate that the masses of these configurations lie close to the top-quark mass. For the $t\bar{b}$ system, the masses of the first four S-wave $0^{-}$ radial states are about $5.1$, $5.4$, $5.6$, and $5.7$~GeV above the top-quark mass, respectively. For the $t\bar{c}$ system, the corresponding values are about $1.9$, $2.2$, $2.5$, and $2.6$~GeV. We also briefly discuss possible production and decay patterns at a qualitative level, which may serve as a reference for future dedicated phenomenological studies or for experimental constraints.

Figures

Figures reproduced from arXiv: 2602.09684 by Bing-Dong Wan, Jun-Hao Zhang, Shuo Yang.

Figure 1
Figure 1. Figure 1: FIG. 1: The BS wave functions for [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The BS wave functions for [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The BS wave functions for [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The BS wave functions for [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The BS wave functions for [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗

discussion (0)

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Reference graph

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