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REVIEW 3 major objections 4 minor 28 references

Even at a realistic top-quark width of 1.4 GeV, toponium bound states survive as T-matrix poles in the complex energy plane, with real parts matching the small-width bound-state masses.

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2026-08-02 01:43 UTC pith:T7SU7L4J

load-bearing objection The pole-survival claim is an exact artifact of the constant-width propagator: Eq. (6) forces every complex pole to Z = E_n - iΓ_t, so the real-part agreement is tautological, not a dynamical result. the 3 major comments →

arxiv 2607.14554 v1 pith:T7SU7L4J submitted 2026-07-16 nucl-th

Toponium Spectrum in the Complex-Energy Plane

Zhanduo Tang , Oliver Fast , Ralf Rapp This is my paper
classification nucl-th
keywords toponiumtop quark widthT-matrixcomplex energy planeCornell potentialbound statesheavy quarkoniumthreshold enhancement
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 asks whether top and antitop quarks, whose electroweak decay width is comparable to their expected binding energy, can form true bound states. The authors solve a T-matrix Lippmann-Schwinger equation with a Cornell potential and, crucially, identify bound states by locating poles of the amplitude in the complex energy plane rather than peaks on the real energy axis. In the small-width limit they find about twenty S-wave spin-degenerate states; for a realistic top width of 1.4 GeV the real-axis spectrum melts into one broad bump, yet the complex-plane poles survive and their real parts essentially coincide with the small-width bound-state masses. The conclusion is that toponium bound states exist in principle even though the observed threshold signal is dominated by the dense, string-driven near-threshold states rather than the ground state. This offers a rigorous criterion for when unstable constituents can still sustain bound states.

Core claim

The central discovery is that the T-matrix for top-antitop scattering retains a discrete set of poles at complex energies Z = E_R + i E_I even when the top-quark width is set to its realistic value of 1.4 GeV. At E_I comparable to the width, the pole positions have real parts that essentially agree with the binding energies obtained in the small-width limit. In other words, the would-be bound-state spectrum is not washed out by the large decay width; it is shifted into the complex plane. This means bound-state formation can be diagnosed by pole structure even when the spectral function on the real axis shows only a broad, featureless maximum.

What carries the argument

The argument runs through the T-matrix equation in operator form, T = V/(1 - G_2 V), where V is the Cornell potential (color-Coulomb plus linear confining string) and G_2 is the uncorrelated two-body propagator built from single-quark propagators. The authors approximate the folded two-body propagator as G_{t tbar}(E,k) = 1/[E - 2 eps_t(k) + i Gamma_t], then continue E to complex values, so the entire width dependence is carried by a constant imaginary shift. Poles of T in the complex energy plane then serve as the criterion for genuine bound states; their real and imaginary parts give the mass and width of the state.

Load-bearing premise

The calculation assumes a constant, purely real top-quark width with no energy or momentum dependence and no complex self-energy generated by the binding potential; if the actual width varies with momentum or develops an imaginary part from the potential, the pole positions and their survival could change.

What would settle it

Recompute the complex-energy poles with a momentum-dependent top width or with a complex self-energy in the single-quark propagator; if the poles smear out, move by more than the width, or disappear, the claim that toponium bound states survive at the realistic width is refuted.

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

If this is right

  • Bound states can be present even when no individual peaks are visible in the spectral function on the real energy axis.
  • The broad maximum in the real-axis T-matrix at realistic width sits about 2 GeV above the would-be ground state and is driven by the confining-string near-threshold states.
  • The confining string force strongly shapes the excited-state spectrum, converting Coulomb-like decreasing spacings into equidistant spacings near threshold.
  • Complex-pole positions give a direct way to extract bound-state mass and width from the T-matrix, providing a rigorous bound-state criterion for unstable constituents.
  • The small-width limit yields about twenty S-wave spin-degenerate toponium states.

Where Pith is reading between the lines

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

  • If toponium poles persist at realistic widths, the broad peak seen in top-pair production near threshold is not evidence against bound-state formation; the bound states are simply hidden under overlapping near-threshold states. Future high-statistics line-shape measurements could be compared with the predicted complex pole position to test the identification.
  • The same complex-pole criterion generalizes to any short-lived two-body system: when constituents carry a width, the distinction between a genuine bound state and an enhancement is whether a pole with the expected quantum numbers survives at complex energies, not whether a spectral peak is visible.
  • A specific extension would be to include hyperfine splitting to check whether the pseudoscalar and vector toponium channels acquire different pole trajectories, since the recent collider threshold enhancement is attributed to the pseudoscalar channel.

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 studies toponium (t-tbar) bound states using a non-relativistic T-matrix approach with a Cornell potential whose parameters are fixed by previous charmonium/bottomonium work. The authors first compute the S-wave spectrum in the small-width limit (Gamma_t = 5 MeV) and find about 20 bound states. They then increase the top-quark width and show that the real-axis spectral function melts into a broad peak already for Gamma_t ~ 200 MeV. For the realistic width of 1.4 GeV, the spectral function shows a single broad maximum about 2 GeV above the would-be ground-state mass. The central new element is a complex-energy continuation of the T-matrix, where the authors claim to find bound-state poles whose real parts essentially agree with the small-width masses, concluding that toponium bound states survive deep into the complex plane at realistic top width.

Significance. If the pole-survival claim were a genuine dynamical result, it would be important for interpreting the recent LHC threshold enhancements as evidence for quasi-bound toponium. The paper also provides a useful comparison of spectral functions for the Cornell, string-only, and Coulomb-only potentials, and the real-axis results are a solid reference calculation. A positive aspect is that the formalism is clearly rooted in a T-matrix framework that has been benchmarked on charmonia and bottomonia in earlier work. However, the central complex-pole analysis, as presented, is mathematically trivial under the constant-width approximation, and the paper does not provide a test of the key assumption. The significance therefore rests mainly on the real-axis spectral functions, which are not the paper's advertised central finding.

major comments (3)
  1. [Complex-Pole Analysis, Eq. (6)] The two-body propagator in Eq. (6) is G_{tbar t}(E,k) = 1/[E - 2ε_t(k) + iΓ_t]. Continuing E→Z and inserting into the T-matrix equation (1) gives the pole condition (Z - 2ε_k + iΓ_t)ψ = Vψ, i.e., (H0 + V)ψ = (Z + iΓ_t)ψ. Since H0+V is Hermitian, the eigenvalues are the real small-width energies E_n, and the poles are exactly Z_n = E_n - iΓ_t. The real parts are therefore identically equal to the small-width masses, not merely 'essentially' in agreement. The claimed survival deep into the complex plane is a built-in feature of the constant-width ansatz, not a result of the complex-energy analysis. To make the central claim non-tautological, the authors need to implement an energy- or momentum-dependent top-quark width (e.g., from the Wb self-energy) and show that the poles survive or move. As it stands, the pole analysis adds no information beyond the small-width spectrum.
  2. [Toponium Spectrum / Comparison with previous work] The paper quotes a ground-state binding energy of ~2.9 GeV but does not provide any uncertainty. The comparison with Refs. [6,20,21,22] shows a wide spread (1.8–3.5 GeV), indicating strong sensitivity to α_s, σ, the string-breaking scale, and the bare top mass. Without a systematic parameter variation or at least an estimate of the numerical uncertainties, the claimed agreement with earlier works and the use of 'would-be' masses as the baseline for the complex-pole analysis are not quantitatively supported. The paper should report at least the sensitivity of the ground-state and low-lying excited-state masses to reasonable variations of α_s and σ.
  3. [Complex-Pole Analysis, Fig. 5] The complex-pole analysis is presented only as a two-dimensional plot; no pole positions are tabulated. Given that the constant-width approximation predicts exact equality of the real parts with the small-width masses, the statement 'essentially agree' is too vague. A table comparing the extracted pole positions Z_n for Γ_t=1.4 GeV with the small-width energies E_n would allow the reader to verify the prediction and to check numerical convergence. This is required to support any claim that the poles are the same bound states as in the small-width limit.
minor comments (4)
  1. [Abstract] Typo: 'has been been reignited' should be 'has been reignited'.
  2. [T-matrix Approach, Eq. (3)–(6)] The derivation of Eq. (6) from Eq. (3) assumes that the top and antitop widths are equal and energy-independent; this should be stated explicitly, and the integration over k0 should be shown or referenced.
  3. [Complex-Pole Analysis, footnote 1] The sign convention for E_I is confusing. The text says the pole is at E_I = Γ_t, but the resonance convention usually places the pole at E = M - iΓ/2. Please clarify the sign and the relation Γ_{tbar t} = 2E_I^pole.
  4. [Fig. 4 and text] For the Coulomb-only scenario, the threshold is different because the constant mass shift from the string's infinite-distance limit is absent. The text mentions this but Fig. 4 is plotted on a common energy axis; it would be helpful to mark the individual thresholds for each scenario so the reader can assess the peak positions relative to threshold.

Circularity Check

1 steps flagged

The complex-pole 'survival' claim is built into the constant-width propagator (Eq. 6): poles are just zero-width poles shifted by -i Gamma_t, so the real-part agreement is by construction; the benchmarked-potential spectrum is independent.

specific steps
  1. self definitional [Section 'Complex-Pole Analysis', Eq. (6), Fig. 5]
    "With an (approximately) energy independent top quark width one can carry out the integration over the relative-energy variable in Eq. (3) to obtain G_tbar(E,k,Gamma_t)=1/[E-2 epsilon_t(k)+i Gamma_t]. This form enables a straightforward continuation to complex energies by replacing E -> Z = E_R + i E_I. ... at E_I = Gamma_t, there still exists a well defined bound-state spectrum in terms of the T-matrix poles whose real parts essentially agree with the masses on the real axis in the small-width limit."

    Substituting Z = E_R + i E_I into Eq. (6) gives G(Z) = 1/[E_R - 2 epsilon_t(k) + i(E_I + Gamma_t)], which is the zero-width propagator evaluated at E_R + i(E_I + Gamma_t). Therefore the pole condition 1 - G V = 0 is solved by Z = E_n - i Gamma_t for every zero-width pole E_n. The reported 'survival' and real-part agreement are exactly the constant-width ansatz; they are a global shift built into the input propagator, not a result of the complex-energy continuation. The paper does not vary or test the energy/momentum dependence of Gamma_t, so the central claim is forced by construction.

full rationale

The potential parameters (alpha_s=0.27, sigma=0.225 GeV^2) are benchmarked to lattice QCD and to charmonium/bottomonium spectra in earlier work, not to toponium data, so the zero-width toponium spectrum is a legitimate prediction and not circular. The self-citation to Ref. [19] supplies the standard complex-pole method but is not the load-bearing justification for the toponium conclusion. The circularity is localized to the complex-pole section: with Eq. (6), G_tbar(Z) = G0(Z + i Gamma_t), so every pole is exactly Z = E_n - i Gamma_t and the real parts necessarily equal the small-width masses. Because this is the paper's headline result and the approximation is not tested against any energy- or momentum-dependent width, the central claim reduces by construction. Score 6 reflects one central prediction reducing by construction while the rest of the calculation is independent.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The paper introduces no new entities. It relies on a standard Cornell potential with parameters from lattice QCD and quarkonium spectroscopy, plus the assumption of a constant top width. The results are therefore only as reliable as the potential model for toponium.

free parameters (4)
  • alpha_s = 0.27
    Chosen to reproduce vacuum free energy from lattice QCD and charmonium/bottomonium spectra in previous works. It sets the Coulomb strength and thus the binding energies.
  • sigma = 0.225 GeV^2
    String tension fitted to lattice QCD and quarkonium spectra. It controls the confining part and the dense spectrum near threshold.
  • string-breaking scale r_s = ~1 fm
    Smooth string breaking at ~1 fm, taken from previous work [25]. It affects the long-range part of the potential.
  • top-quark bare mass m_t^0 = 173 GeV
    Choosen slightly above PDG average; augmented by V_inf/2 = 0.55 GeV to give threshold 347.1 GeV. Affects absolute energies but not the main qualitative conclusions.
axioms (4)
  • domain assumption The T-matrix equation derived from Bethe-Salpeter in static limit is valid for toponium.
    Eq. (1) assumes small energy transfer. For a top quark with 1.4 GeV width, the static limit may be questionable, but it is standard in quarkonium models.
  • domain assumption The top-quark width is constant and independent of energy/momentum.
    Eq. (4) uses constant Gamma_Q. The paper argues this is acceptable because the quark is near mass shell, but it is a simplifying assumption.
  • domain assumption The Cornell potential with string breaking is the correct interaction for toponium.
    The potential parameters are benchmarked for charmonium/bottomonium, but its extension to toponium is assumed.
  • domain assumption The analytic continuation to complex energies is unique and the poles found are physical bound-state poles.
    The continuation E -> E_R + i E_I is straightforward for the propagator, but the choice of Riemann sheet and interpretation of poles is not discussed in detail.

pith-pipeline@v1.3.0-alltime-deepseek · 183 in / 5374 out tokens · 51721 ms · 2026-08-02T01:43:19.186641+00:00 · methodology

0 comments
read the original abstract

The electroweak decay width of the top quark of ~2 GeV is comparable to binding energies expected for the toponium system, raising the long-standing question of whether top and antitop quarks can sustain meaningful bound states. This issue has been been reignited by recent discoveries of a cross section enhancement near the t-tbar threshold by the CMS and ATLAS collaborations in pp collisions at the LHC. Here, we deploy a T-matrix approach with an underlying Cornell potential to study the properties of the toponium system by varying the top-quark widths in the intermediate propagators. In particular, we carry out a complex-energy analysis of the T-matrix to assess how its poles, as a rigorous criterion for bound-state formation, develop from the limit of small widths to realistic ones. We also revisit the impact of the confining force in the potential on the toponium T-matrix.

Figures

Figures reproduced from arXiv: 2607.14554 by Oliver Fast, Ralf Rapp, Zhanduo Tang.

Figure 1
Figure 1. Figure 1: FIG. 1: Level spectrum of binding energies for [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Imaginary part of the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

discussion (0)

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Reference graph

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