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Ensuring that toponium is glued, not nailed

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The LHC's near-threshold top-antitop excess is compatible with QCD toponium, not a new short-range force.

desk verdict A physically transparent qualitative diagnostic separating QCD toponium from a contact-bound state, but the quantitative 'compatible with CMS' claim rests on a normalization without an uncertainty budget. read the letter →

arxiv 2411.19180 v2 pith:JLYV6B6C submitted 2024-11-28 hep-ph

Felipe J. Llanes-Estrada This is my paper · ORCID
classification hep-ph
keywords toponiumtop-antitopboundstateresonancelineshapecontactinteractionsSommerfeldenhancementpNRQCDLHCtop-pairexcess
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the roughly 7 pbarn excess in top-antitop production just above threshold recently reported by CMS can be explained by ordinary QCD toponium, a top-antitop bound state held together by gluons, rather than requiring new physics. It contrasts this 'glued' state with a hypothetical 'nailed' state bound by a new short-range contact force. The decisive signature is the line shape: a single isolated resonance would leave a dip between the peak and the continuum if the binding energy reaches about 3 GeV, but the Coulomb-like tower of excited QCD states fills that dip and adds about a pbarn to the cross section. A short-range force, by contrast, concentrates the wavefunction near the origin and would produce a much larger cross section for the same binding energy, so matching the observed excess leaves no visible bound state. The paper concludes that the CMS excess is compatible with QCD toponium, and that a short-range interaction invoked to explain it or part of it would not display a bound state.

What carries the argument

The central object is the modified phase-space factor in Eq. (A.8), the ttbar relative-velocity factor that replaces the free beta_t in the hard gg to ttbar cross section. It encodes the near-threshold dynamics through the nonrelativistic Green's function: the Sommerfeld Coulomb enhancement plus a sum over n=1..5 Coulomb-like eta_t bound states, where eta_t is the ground-state toponium. For generic potentials the paper solves the same Green's-function equation numerically (Eq. A.11) on a grid, and the contact-interaction case is treated with a regulated delta potential that supports exactly one bound state. The argument turns on comparing the Green's function at r=0, which controls the line shape, between QCD and short-range binding.

What would settle it

Look at the ttbar invariant-mass line shape near threshold with resolution good enough to see the dip between the eta_t peak and the continuum. If the binding energy is near 2.5 to 3 GeV, the QCD tower predicts the excited states add about 1 pbarn and fill the dip; observing a single isolated resonance with a clean dip, or a cross-section several times larger than the NNLO QCD prediction at the same binding energy, would falsify the QCD-toponium interpretation and point to a short-range force.

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Extended reading notes

Core claim

The central claim is that the observed CMS excess is standard QCD toponium, and the tell-tale that distinguishes QCD 'glue' from an exotic short-range 'nail' is the excited-state tower. On the QCD side, the paper builds the near-threshold color-singlet gg to ttbar cross section from the nonrelativistic Green's function (Eq. A.8), which includes both the Sommerfeld enhancement and a sum over the first five Coulomb-like s-wave bound states, with the ground-state eta_t plus its n=2..5 excitations. With the binding energy in the 2-3 GeV range favored by pNRQCD, these excited states fill the threshold dip that a lone eta_t resonance would show, adding about 1 pbarn to the integrated cross section. The contrast with a contact interaction is sharp: a regulated delta potential that binds at a comparable energy produces a wavefunction concentrated at the origin, and because the Green's function in Eq. (1) is evaluated at r=0, the cross section is much larger than observed; tuning the contact coupling down to match the observed cross section leaves a state with BE around 1 GeV that is washed out by the 2 Gamma_t of about 3 GeV width. The paper therefore concludes that the CMS excess is compatible with QCD-glued toponium, and that a short-range interaction invoked to explain the excess, or a fraction of it, would not display a bound state. It also estimates the precision needed to constrain contact-interaction Wilson coefficients: about 1 percent on the eta_t mass, or better than about 10 percent on the excess cross section, would begin to be sensitive.

Load-bearing premise

The prediction assumes that near-threshold color-singlet ttbar production factorizes into a hard parton-level cross section times a modified phase-space factor built from the nonrelativistic Green's function with only the first five Coulomb-like bound states included; the paper itself notes this factorization is still under active investigation.

Editorial extensions

If this is right

  • If the toponium binding energy reaches about 3 GeV, the ground-state peak separates from threshold by more than the width, and observing the dip between peak and continuum filled by excited states would confirm the QCD tower over a single short-range bound state.
  • For smaller binding energies, the excited states add only a tenuous increase, at most about 1 pbarn, to the proton-proton cross section near threshold.
  • A contact interaction that binds at a QCD-like energy would produce a much larger cross section than observed, so the CMS excess does not support an isolated short-range bound state.
  • Constraining new-physics Wilson coefficients from the eta_t mass would require about 1 percent precision on the peak position, while about 10 percent precision on the excess cross section may already begin to be sensitive.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • As an editorial extension, a future e+e- scan across the top threshold could directly test the dip-filling prediction: a filled dip with Coulombic level spacing would confirm the glued tower, while an isolated peak would point to a short-range force.
  • The same wavefunction-at-origin comparison should carry over to any system where a contact interaction competes with a Coulomb-like force, so the glued-versus-nailed diagnostic is a template for exotic heavy-flavour or dark-matter bound states.
  • The dependence on only the first five Coulombic states is testable: computing the line shape with a full Richardson or Cornell potential, which the paper notes can push excited states above threshold, would show whether the dip remains filled, and if it does not, the comparison with the CMS excess would shift.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper argues that the near-threshold ttbar cross-section excess reported by CMS is compatible with standard QCD-bound toponium ('glued') rather than with a state bound by a new short-range interaction ('nailed'). The argument combines three ingredients: (i) the binding energy of the eta_t is estimated from pNRQCD potentials to lie in the range giving a peak separated from threshold at the level of the top width; (ii) the Coulomb-like QCD spectrum contains an excited-state tower that fills the threshold dip and contributes roughly one additional pbarn to the integrated cross-section; and (iii) a short-range contact interaction, having a wavefunction much more concentrated at the origin, either produces a much larger cross-section for the same binding energy or, if matched to the observed cross-section, yields a state too broad to be visible. The line shapes are computed by replacing the free phase space with a modified phase-space factor built from the nonrelativistic Green's function (Eq. A.8), folded with a simple gluon PDF (Eq. A.2).

Significance. If the quantitative compatibility claim holds, the paper makes a useful contribution: it sharpens a falsifiable line-shape diagnostic (an isolated resonance with a filled-in dip signals the QCD excited-state tower, whereas a single short-range bound state would not fill the dip), and it explains why the wavefunction-at-the-origin difference between Coulomb-like and contact potentials controls the cross-section ordering. The author is transparent about using leading-order parton cross-sections, a 2004-era PDF parametrization, and a factorized phase-space treatment, and explicitly flags the factorization assumption as an active research subject. These choices make the numerical results estimates rather than precision predictions, but the qualitative contrast is well motivated and worth publishing if the central quantitative claim is suitably qualified.

major comments (3)
  1. [Section 3, Table 1 and Appendix A (Eqs. A.1, A.2, A.8)] The central quantitative claim in conclusion (iii) of Section 6 rests on the absolute normalization of the cross-section, but the manuscript provides no uncertainty budget for any of the ingredients that set that normalization. The gluon luminosity is the 2004 CTEQ5-style parametrization of Eq. (A.2) with no PDF error bands; Eq. (2) is a leading-order alpha_s^2 hard cross-section; only the gg initial state is included; and the soft and hard scales are fixed at specific values with no variation. In addition, the alpha_s <= 0.5 saturation described in Appendix A introduces an unquantified artifact in the potential calculation. Since the QCD entries in Table 1 (8.3 and 9.1 pb) lie only about 1.2-2.0 pb above the CMS central value of 7.1(0.8) pb, a moderate downward shift from any of these sources would remove the claimed compatibility. An error estimate, or a reformulation of conclusion (iii) as an order-of-magnitude statement, is needed before the quantitative comparison can be considered established.
  2. [Appendix A, after Eq. (A.8)] The factorization ansatz underlying all numerical line shapes is explicitly acknowledged in the manuscript to 'remain the subject of active investigation' [37], yet no estimate is given for the size of the corrections to this factorization. The additional cross-section attributed to toponium is literally the difference between the modified phase space and the bare phase space (Table 1: 4.9 pb for Sommerfeld-only vs 8.3-9.1 pb with bound states), so factorization corrections are first-order, not a small perturbation. The n_max = 5 truncation is also not varied, and color-octet contributions are only mentioned qualitatively in Figure 4. Since conclusion (iii) is quantitative, the manuscript should either propagate an estimate of these uncertainties or explicitly downgrade the compatibility claim to a plausibility argument.
  3. [Section 4, Eqs. (5)-(7) and Table 1] The short-range cross-sections used to argue that a new interaction would not display a bound state for the observed excess are computed for two illustrative regulator scales (M = 1 TeV and M = 13 GeV), with the Wilson coefficient tuned to produce a chosen binding energy, but no regulator dependence or matching uncertainty is provided. The statement in conclusion (iii) that a short-range interaction invoked to explain the excess 'will not display a bound state' is therefore tied to these particular examples and to the same absolute normalization criticized above. A scan over mediator masses and couplings, or an analytic argument removing the normalization dependence, is required to make the exclusion robust.
minor comments (4)
  1. [Section 2, Eq. (3)] The sign convention for the binding energy is inconsistent: Eq. (3) defines '-BE' with a negative sign, while Table 2 and the text discuss positive BE values; please make the sign convention uniform.
  2. [Table 2] Table 2 is difficult to read: the raised/lowered scale-variation values are not separated clearly from the central values. Please reformat the table so that each eigenstate and its scale variation are presented in a reader-friendly way.
  3. [Figure 2 caption] The sentence 'Above 3.5 (100 GeV)^2 where beta_t ~ 0.2' is unclear; the axis unit and the physical threshold location should be stated explicitly.
  4. [Section 6 and reference [19]] The claim that the LO line shape is 'surprisingly close' to the high-order calculation of Ref. [19] is not quantified; a direct comparison or a table of peak positions and widths would make this valuable point more convincing.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-citations appear in the BSM discussion, but the central glued-vs-nailed comparison is computed from external QCD inputs and benchmarked to CMS data, so no load-bearing circularity.

  1. other [Section 5, Section 4, and Appendix A (Refs. [27], [33])]
    "Recently [27] we have examined the e ffect of such contact interactions ... for numerical work, and also for short- but not zero-range V, a regulated form of the contact potential is a Yukawa exchange, VBSM = αcontact/|r'−r| e^{−Λ|r'−r|} can be used [27] ... The NNLO one can be found in [33] and refs. therein."

    These are the only self-referential elements. Ref. [27] supplies the Yukawa regulator used in the numerical contact-potential comparison and the HEFT-bound context invoked against the nailed option; Ref. [33] supplies the NNLO potential expression used in the spectrum. They do not drive the central QCD-compatibility claim: the binding energies are cross-checked against external high-order calculations ([17,18,19]), and the CMS excess is used as a benchmark rather than a fitted input. The step is therefore a minor, peripheral self-citation rather than a load-bearing circularity, and the score reflects that mild self-reference without treating the main derivation as circular.

full rationale

The derivation chain runs from the parton-level gg->tt cross-section (Eq. 2), the gluon luminosity parametrization (Eqs. A.1-A.2), and the Fadin-Khoze modified phase space (Eq. A.8) into which the t-quark width and Coulomb/pNRQCD bound-state energies are inserted. The binding energies in Table 2 are computed from standard LO/NLO/NNLO potentials with explicit scale choices, and the CMS near-threshold excess is used only as a comparison value, not as an input fitted to fix the model. The contact-interaction ('nailed') comparison solves the Schrödinger Green's function (Eq. A.11) with a delta or Yukawa potential, with C tuned to a chosen BE; the resulting cross-section is a distinct predicted observable. The claim that a ~7 pb excess would not display a short-range bound state follows from the computed BE-dependence (BE ~1 GeV, below the 3 GeV width), not from fitting CMS. The factorization ansatz behind Eq. A.8 is explicitly flagged in Appendix A as 'remains the subject of active investigation'; this is an accuracy limitation, not a circular reduction. The paper is also anchored to external benchmarks: the QCD line shape is compared with high-order calculations [18,19], and the CMS data provide an external falsification target. The only self-referential inputs are Refs. [27] and [33], which are peripheral to the central glued-vs-nailed comparison. Accordingly, the circularity score is 2, reflecting minor self-citation, not a structurally circular derivation.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central comparison does not fit any parameter to the CMS excess. It relies instead on standard QCD inputs and a few modeling choices, listed above. The principal risk is the factorized phase-space approximation rather than the value of any fitted constant, and the paper does not introduce new particles, forces, or conserved quantities.

free parameters (4)
  • nmax (number of included toponium excited states) = 5
    The bound-state sum in Eq. (A.8) is truncated at nmax=5, motivated by the near-threshold Coulomb spectrum but not derived from data. It directly affects the dip-filling and the size of the cross-section excess.
  • Renormalization scales (mu_soft and mu_hard) = mu_soft ~ 25 GeV (alpha_s ~ 0.147); mu_hard ~ 172 GeV (alpha_s ~ 0.108)
    Chosen by pNRQCD/BLM scale-setting logic. The spectrum and cross-section depend on these choices; scale variation is shown in Table 2 for the spectrum but is not propagated to the final cross-section numbers.
  • alpha_s saturation ceiling = 0.5
    An ad hoc numerical guard in the spectrum diagonalization. The paper states it should not affect the lowest states, but it is not physically motivated and could shift excited-state energies.
  • Contact-interaction Wilson coefficient C(Lambda) in illustrative BSM cases = Tuned to produce BE=1 or 2 GeV at M=13 GeV (with C=3.7 in one case); C(1 TeV) values tuned to BE around 2.5 GeV but…
    In the 'nailed' scenarios, C is tuned to a chosen binding energy to enable an equal-BE comparison with QCD. This is a parameter scan for illustration, not a fit to the CMS excess.
assumptions (7)
  • domain assumption Nonrelativistic Schrodinger equation with pNRQCD static potentials (Eq. A.3-A.4) describes the ttbar bound-state spectrum near threshold.
    Used to compute the Table 2 spectrum and the excited-state energies that fill the threshold dip; assumes soft-scale pNRQCD and neglects relativistic and nonperturbative corrections beyond the stated potentials.
  • domain assumption The color-singlet ttbar production cross-section factorizes as the parton-level Born term times a modified phase-space factor built from the nonrelativistic Green's function (Eq. A.8).
    The paper itself cites [37] stating this factorization remains under active investigation; if it fails, the line-shape and cross-section conclusions change.
  • domain assumption The gluon luminosity is well described by the CTEQ5 parameterization in Eq. (A.2).
    An old 2004 PDF fit; no PDF uncertainty is propagated, yet all proton-proton cross-section numbers in Table 1 depend on it.
  • standard math Watson's final-state interaction theorem ensures the eta_t pole position is the same in e+e- and gg production.
    Invoked in Section 2 to transfer binding-energy calculations between production channels.
  • domain assumption A regulated delta-function potential with renormalized coupling (Eqs. 4-5) and a Yukawa realization represent a TeV-scale BSM contact interaction.
    Used for the 'nailed' scenario; assumes the nonrelativistic contact effective theory is applicable at the top-quark scale and that the regulated potential captures the relevant BSM short-range physics.
  • domain assumption The line shape near threshold is controlled almost entirely by m_t, Gamma_t, and m_eta_t.
    Used in Section 6 to explain why leading-order results track high-order calculations; this reduction is asserted, not derived in the paper.
  • standard math The running coupling is given by Eq. (A.6) with Lambda_QCD=0.22 GeV, and is saturated at alpha_s=0.5 in the numerical diagonalization.
    Standard QCD input plus an ad hoc numerical guard; the paper says the guard should not affect the lowest states.

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Cite this review

Pith. "Pith review of Ensuring that toponium is glued, not nailed." pith.science (2026). https://pith.science/paper/JLYV6B6C

@misc{pith2026241119180,
  author       = {Pith},
  title        = {Pith review of: Ensuring that toponium is glued, not nailed},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JLYV6B6C}},
  note         = {Machine review of arXiv:2411.19180}
}
read the original abstract

Hints of toponium might be incipient in LHC data, as given the vast numbers of t quarks produced, some survive on the exponential-decay tail long enough to fasten ttbar together. I here discuss a few differences between the standard Quantum Chromodynamics (QCD) binding (the ``glue'') and exotic short-range binding (the ``nail''). If the binding energy below threshold reaches the 3 GeV range the peak of the eta_t is distinct enough that a cross-section dip should be apparent in the line shape, should there only be one isolated resonance, but is filled by the excited QCD states adding about a pbarn to the cross section of ttbar production. Their effect for smaller binding energies is a tenuous increase in the cross section. A new-physics short-range interaction, on the other hand, yields a larger cross-section for equal binding energy (or hardly a visible bound state for similar cross section). This is due to its larger ttbar relative wavefunction at small distances. Finally, assuming that standard QCD plays out, I comment on what size of constraints on new-physics coefficients one can expect at given precision.

Figures

Figures reproduced from arXiv: 2411.19180 by the authors.

Figure 1
Figure 1. ηt (ground state toponium) production from two hard gluons at the LHC. Left: tt¯ with binding energy BE from soft gluons with µ ∼ mtαs , (“glued”). Right: tt¯ bound by BSM contact interactions (“nailed”). in [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Upwards from bottom: Phase-space factor ( [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Factor substituting the βt phase-space one for the color-singlet tt¯ channel, including the Sommerfeld enhancement and either one or five toponia states (since they cluster near threshold, this eliminates any dip in the cross￾section which appears if the binding energy approaches 3 GeV). The Bohr spectrum of a Coulomb potential does include the excited s-wave states η n t with decreasing binding energy BE/n 2 and de… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: with the ηt barely separating from the nominal threshold (here at 344 GeV). Interestingly, the decay width Γtt¯ does not [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]
Figure 6
Figure 6. Figure 6: Upper plot: a quasicontact regulated potential with scale [PITH_FULL_IMAGE:figures/full_fig_p003_6.png]
Figure 5
Figure 5. Figure 5: Gross estimate of the pp differential cross-sections as functions of m 2 tt¯ . From bottom to top, the would-be production of tt¯with no rescattering; the effect of the Sommerfeld enhancement; the addition of one ηt toponium under the nominal threshold; and the additio…
Figure 7
Figure 7. Figure 7: The resonant glued-tt¯lineshape (red online) with a soft renormaliza￾tion scale so that BE ≃ 2 GeV, compared with short range potentials. Upper plot: if the new physics interaction produces one clear bound state very near threshold, akin to the ηt , in exchange the cro…

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Forward citations

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