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Updated predictions for toponium production at the LHC

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Standard Model QCD, including toponium bound-state effects, predicts a top-pair background near 345 GeV of 12 pb, about 40% above the current Monte Carlo estimate.

desk verdict Solid computational update with useful new numbers, but the 'lower bound' label is overclaimed and should be either verified or softened before the paper is used as an experimental input. read the letter →

arxiv 2412.16685 v2 pith:ERF4RDOG submitted 2024-12-21 hep-ph hep-ex

M.V. Garzelli , G. Limatola , S.-O. Moch , M. Steinhauser , O. Zenaiev This is my paper · ORCID
classification hep-phhep-ex
keywords top-quarkpairproductiontoponiumthresholdresummationCoulombNRQCDinvariantmassdistributionLHCheavyHiggssearch
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

This paper updates the Standard Model prediction for top-quark pair production near the threshold where the pair's invariant mass is about twice the top-quark mass, matching the Run 2 kinematics and settings of the LHC search for heavy Higgs bosons decaying to top pairs. It includes Coulomb resummation for the quasi-bound 'toponium' state formed by gluon exchange between top and anti-top, combined with threshold resummation of soft and collinear gluon emissions at next-to-leading-logarithmic accuracy. The central result is that the Standard Model background in the 340–350 GeV bin is predicted to be about 12 pb, compared with about 8.4 pb from the Monte Carlo generator predictions used by the experimental analysis. If correct, this means the excess seen near a top-pair mass of 345 GeV sits on a larger and differently shaped Standard Model background than previously assumed, and the paper argues its prediction should be used as a conservative input for background modelling and new-physics searches.

What carries the argument

The central object is the NRQCD factorization formula $M_{t\bar{t}}\,d\hat{\sigma}_{ij\to T}/dM_{t\bar{t}} = F_{ij\to T}(\hat{s},M_{t\bar{t}}^2,\mu_f^2)\,\frac{1}{m_t^2}\mathrm{Im}\,G^{[1,8]}(M_{t\bar{t}}+i\Gamma_t)$, which separates the cross section into a perturbative hard function $F$ and the imaginary part of the non-relativistic Green's function $G$, evaluated at zero distance. The Green's functions solve a Schrödinger equation with the next-to-leading-order QCD potential; the attractive color-singlet solution generates the toponium enhancement, and the repulsive color-octet solution shapes the steeply rising background just above threshold. Threshold logarithms from soft and collinear gluons are resummed in Mellin space to NLL accuracy and matched to NLO fixed order, following the framework of Ref. [8]. The result is a parameter-free Standard Model prediction for $d\sigma/dM_{t\bar{t}}$ in the window roughly $|M_{t\bar{t}}-2m_t|\lesssim5$ GeV, which the paper proposes to combine additively with fixed-order NNLO predictions above $M_{t\bar{t}}\gtrsim350$ GeV.

What would settle it

The transfer of the LP/NLP bounding property can be tested directly: compute the exact NLO (or NNLO) threshold corrections including next-to-leading-power terms for the three channels $gg\to {}^1S_0^{[1]}$, $gg\to {}^1S_0^{[8]}$, and $q\bar q\to {}^3S_1^{[8]}$ in Mellin space and check whether the NLO+NLL resummed result lies below them in every channel. A single channel where it lies above would falsify the lower-bound claim; alternatively, a measurement of the 340–350 GeV bin at the LHC with total uncertainty below about 20% would discriminate 12 pb from 8.4 pb directly.

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Extended reading notes

Core claim

The paper's central claim is that QCD bound-state and threshold effects, both part of the Standard Model, produce a clear enhancement in $d\sigma/dM_{t\bar{t}}$ for $M_{t\bar{t}}\simeq 2m_t$ that is missing from the fixed-order and shower Monte Carlo calculations used in current experimental analyses. Working in non-relativistic QCD, the authors factor the partonic cross section into a hard function times the imaginary part of a non-relativistic Green's function, $F_{ij\to T}\times \frac{1}{m_t^2}\,\mathrm{Im}\,G^{[1,8]}(M_{t\bar{t}}+i\Gamma_t)$. The color-singlet Green's function feels an attractive potential and produces the broad 'toponium' resonance near $M_{t\bar{t}}\sim 345$ GeV, while the color-octet channels are repulsive and dominate above about 350 GeV. With the hard function resummed to NLL accuracy and matched to NLO fixed order, the integrated cross section in the bin $M_{t\bar{t}}\in[340,350]$ GeV is 12 pb for the central scale choice, versus 8.4 pb from the Monte Carlo prediction used in the experimental analysis. The paper further argues that because the missing NNLO and subleading-power corrections are expected to be positive and of order 10%, the NLO+NLL result acts as a lower bound on the exact Standard Model prediction in the threshold region.

Load-bearing premise

The load-bearing premise is that the leading-power/next-to-leading-power bounding property, proven to N4LL accuracy for deep-inelastic scattering and Higgs production in Refs. [23, 24], carries over to top-quark-pair hadroproduction in the three channels considered; if that transfer fails, the paper's statement that the NLO+NLL result is a lower bound on $d\sigma/dM_{t\bar{t}}$ is unsupported even though the NLL calculation itself might still be correct.

Editorial extensions

If this is right

  • The Standard Model background in the $M_{t\bar{t}}\in[340,350]$ GeV window is about 12 pb at NLO+NLL, roughly 40% above the 8.4 pb Monte Carlo estimate, so experimental excesses in this bin must be reinterpreted against the larger background.
  • The prediction provides a lower bound on $d\sigma/dM_{t\bar{t}}$ near threshold, because missing NNLO corrections to the Green's functions and hard functions, and the omitted next-to-leading-power terms, are both estimated to increase the cross section by about 10%.
  • Above $M_{t\bar{t}}\simeq350$ GeV the NRQCD approach ceases to be valid, so fixed-order NNLO and the Monte Carlo description remain the appropriate tools there; the paper recommends a transition region beginning around that mass.
  • Models that add a toponium-like pseudoscalar resonance to Monte Carlo samples, as used by the experimental analysis, give different peak shape, width, and height; the full NRQCD prediction is a more complete Standard Model template for the bound-state contribution.

Reading between the lines

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

  • If the lower-bound property survives a direct test in top-pair production, the statistical significance of the excess near $M_{t\bar{t}}\sim345$ GeV would likely drop, because part of the apparent signal would be absorbed into the larger Standard Model background; re-running the experimental fit with this background shape would quantify the shift.
  • The same machinery could be applied to the spin-correlation and angular observables used to characterise the excess; the predicted background angular distribution would differ from the current Monte Carlo one, giving an additional discriminating test.
  • A direct verification of the bounding property needs no new data: computing the exact next-to-leading-power threshold terms at NLO in the $gg$ and $q\bar q$ channels and comparing them with the NLL-resummed result would settle whether the 'lower bound' wording is justified in every channel.
  • With more LHC data, the 340–350 GeV bin will be populated densely enough that the 12 pb versus 8.4 pb difference may become visible as a pure rate measurement, provided systematic uncertainties on the background shape can be controlled below about 20%.
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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

4 major / 5 minor

Summary. The manuscript presents updated NRQCD-based predictions for the top-quark pair invariant mass distribution near threshold at the LHC (13 TeV). Using the framework of Ref. [8], the authors compute the dominant channels gg -> 1S0[1], gg -> 1S0[8], and q qbar -> 3S1[8], combining Coulomb resummation through non-relativistic Green's functions with NLL threshold resummation of the hard functions in Mellin space. They compare the NLO+NLL results with NLO, fixed-order NNLO, and POWHEG predictions, finding an enhancement in the threshold region (12 pb versus 8.4 pb in the [340,350] GeV bin), and they propose using their results as a conservative SM background input for experimental analyses. They also argue that their prediction constitutes a lower bound on the cross section, based on LP/NLP considerations and on estimates of missing NNLO corrections.

Significance. If the calculation is correct, this is a valuable theory input for CMS and ATLAS searches in the top-pair threshold region, with a concrete numerical difference from the currently used POWHEG-based background. The paper is transparent about its framework and provides an ancillary file with the NLO+NLL predictions, which is a strength. The central quantitative claim is a direct calculation and is not fitted to the CMS excess. However, the 'conservative lower bound' interpretation is not yet established, and the scale-stability statement is not supported by the tables for all channels. The paper would be strengthened by either removing or properly qualifying the lower-bound claim, and by clarifying the channel content of the quoted integrated cross section.

major comments (4)
  1. [Paragraph beginning 'In passing let us also address concerns raised in Ref. [10]'] The claim that 'the LP and NLP terms provide a lower and upper bound on the exact result' is supported only by references to DIS and Higgs boson production in gluon-gluon fusion (Refs. [23,24]), and is then applied without proof to top-quark pair hadroproduction. The channels used here include gg -> 1S0[8] and q qbar -> 3S1[8], which involve color-octet final states with a repulsive potential and a q qbar initial state that has no direct analogue in the cited processes. Moreover, footnote 1 concedes that the Mellin N-space power expansion used here differs from the z-space expansion of Ref. [10] by 'numerically important sub-leading terms', so the sign pattern of LP/NLP terms found in Refs. [23,24] cannot simply be assumed to transfer. This matters because the statement 'the results presented here can be considered as a lower bound on the cross section' is used to characterize the 12 pb integrated prediction in [340,350] GeV as conservative relative to the 8.4 pb POWHEG result. Please either remove the lower-bound claim or verify it numerically for each of the three channels, for example by comparing the exact NLO result with LP-only and LP+NLP truncations in this process.
  2. [Tables I and II and paragraph following Table II] The text states that 'the scale stability of the resummed results is somewhat improved', but Tables I and II show the opposite for the channel that produces the toponium resonance, gg -> 1S0[1]. For M_ttbar = 2 m_t, the NLO values across mu_r = mu_f = m_t, 2 m_t, 4 m_t are 18.2, 18.7, 18.3 (range 0.5) while the resummed values are 19.4, 20.5, 21.1 (range 1.7). For M_ttbar = 2 m_t - 5 GeV, the corresponding ranges are 0.4 and 1.9. Thus the resummation does not improve scale stability in this channel, and may worsen it. Please qualify the claim or explain why this is not a concern for the stated improvement.
  3. [Final paragraph before 'The impact of different parameter settings'] The sentence 'Both these effects are expected to enhance the cross section, so that our current NRQCD prediction can be considered as an estimate for the lower bound' combines two estimates (NNLO corrections to the Green's function and to the hard functions, each O(10%)) with the LP/NLP bound. These are estimates based on e+e- studies and on expected higher-order behavior, not demonstrated bounds for this process. As written, the 'lower bound' conclusion is a conjecture rather than a proven statement. If the authors intend to keep this phrasing, they should provide a concrete numerical check or clearly label the statement as an expectation rather than a bound.
  4. [Paragraph beginning 'In a next step, we compute the full partonic and hadronic cross sections' and paragraph with the…] The quoted integrated cross section of 12 pb in [340,350] GeV and the curves labeled 'sum' in Figures 1-4 include only the three channels of Table I. The text states that other S-wave channels (gg -> 3S1[1,8], q qbar -> 1S0[1,8], gq -> 1S0[1,8] and gq -> 3S1[8]) contribute an additional O(5%). Since the POWHEG comparison includes all channels, the 12 pb versus 8.4 pb comparison is not fully inclusive. Please either include the missing channels in the plotted sum and in the integrated bin, or explicitly state that the quoted numbers exclude them and estimate the impact on the comparison.
minor comments (5)
  1. [Equation (5)] The inverse Mellin integrand in Eq. (5) uses x^{-N}, whereas the forward transform in Eq. (4) is defined with z^{N-1} and the natural inverse is z^{-N}; please make the variable consistent and define x if it is not z.
  2. [Discussion around e+e- annihilation] Typo: 'preocess' should be 'process' in the sentence 'From studies of the related preocess of e+e- annihilation'.
  3. [Figure 4 and matching proposal] Figure 4 shows a 'recommended transition' between NRQCD and POWHEG, but the text does not provide a concrete formula or algorithm for the additive matching; please state how the transition is implemented (for example, a switching function or a hard cutoff) so that the proposal is reproducible.
  4. [Footnote 2] The statement that the dynamical scale H_T/2 is close to m_t near threshold is plausible but not quantified; a short numerical estimate of H_T/2 in the threshold region would be helpful.
  5. [Ancillary file] Please state explicitly in the ancillary file or its documentation whether the tabulated NLO+NLL predictions include only the three dominant channels, and which scale choices and PDF set are used.

Circularity Check

1 steps flagged · score 4.0 of 10

The enhanced dσ/dM prediction is independently computed, but the 'conservative lower bound' label rests on a self-cited LP/NLP bounding result asserted to transfer from DIS/Higgs to top-pair hadroproduction without proof.

  1. self citation load bearing [Page 4, unnumbered paragraph beginning 'In passing let us also address concerns raised in Ref. [10]' (after Tables I–II, before Fig. 1)]
    "It turns out that the LP and NLP terms provide a lower and upper bound on the exact result. This is well documented to much higher accuracy, i.e. (next-to)4-leading logarithmic (N4LL) accuracy in Refs. [23, 24] for deep-inelastic scattering and the Higgs boson production in gluon-gluon fusion, which is directly related to the toponium channel gg→ 1S [1]0 . With the resummation of the LP contributions, the results presented here can be considered as a lower bound on the cross section."

    The 'lower bound' property is the load-bearing reason for calling the NLO+NLL NRQCD prediction a conservative SM-background input (12 pb vs 8.4 pb in [340,350] GeV). Its only cited support, Refs. [23,24], is the authors' own four-loop LP/NLP study of DIS and gg→H, not of top-pair hadroproduction. The paper does not prove that the LP/NLP bounding pattern transfers to the three channels actually used (gg→1S0[1], gg→1S0[8], qqbar→3S1[8]); the octet channels have repulsive Coulomb potentials and the qqbar channel has no analogue in the cited processes. Footnote 1 even concedes that the Mellin-N power expansion here differs from the z-space expansion of Ref. [10] by numerically important sub-leading terms, so the sign pattern is not guaranteed.

full rationale

The core numerical content of the paper is not circular. dσ/dM_tbar-tbar is obtained from a standard NRQCD factorization (Eqs. 1–3) with external inputs: NNPDF3.1 PDFs, m_t=172.5 GeV, Γ_t=1.4 GeV, α_s(MZ)=0.1180, NLO hard functions from Petrelli et al., and Schrödinger-equation Green's functions with an NLO QCD potential. No parameter is fitted to the CMS excess or to the POWHEG curve; the 12 pb vs 8.4 pb comparison is a direct calculation. The matching/resummation setup is adopted from the authors' earlier Ref. [8], but that is a documented, reproducible framework rather than a definitional identification of the prediction with its input. The one load-bearing self-citation is the lower-bound argument: the claim that LP/NLP terms bound the exact result is cited only to the same authors' four-loop DIS/Higgs papers and then asserted to transfer to top-pair hadroproduction, with the octet and qqbar channels unsupported and footnote 1 undermining the expansion equivalence. Because the 'conservative lower bound' recommendation depends on this unproven transfer, the paper earns a moderate circularity score; the central enhancement prediction itself has independent content.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The calculation has no fitted parameters tuned to the CMS data; the central enhancement is produced by an ab initio NLO+NLL NRQCD calculation. The only hand-chosen scale is the soft scale for the Green's functions, which is not varied and for which the authors state that NLO variation would underestimate the uncertainty. The main axiomatic load is the NRQCD factorization and the transfer of the LP/NLP bound from other processes. No new particles or mediators are introduced; 'toponium' is an existing Standard Model quasi-bound-state concept, not a new invented entity.

free parameters (1)
  • soft renormalization scale mu_s = ~32 GeV (fixed, not varied)
    Chosen by hand for the NRQCD Green's functions. The authors state that soft-scale variations at NLO underestimate the uncertainty, and the paper does not vary this scale.
assumptions (5)
  • domain assumption NRQCD factorization of the partonic cross section into a hard function times the imaginary part of the zero-distance non-relativistic Green's function (Eq. 3).
    Underlies all predictions; valid only for small top-quark velocity beta_t, restricting the computation to |M_ttbar - 2m_t| less than about 5 GeV as stated in the text.
  • domain assumption The Schroedinger equation with the QCD potential at NLO accuracy (Refs. [15,16]) describes the color-singlet and color-octet Green's functions.
    Used to compute Im G^[1,8] with fixed soft scale mu_s ~ 32 GeV; higher-order corrections are not included.
  • domain assumption The Mellin-space NLL threshold resummation with minimal prescription and NLO matching (Eq. 5) correctly resums the leading-power soft and collinear logarithms in the three dominant channels.
    Technical basis of the NLO+NLL tables; implementation details are inherited from Ref. [8] and not fully reproduced here.
  • ad hoc to paper The LP/NLP bounding property established for DIS and Higgs production in Refs. [23,24] also holds for top-quark pair hadroproduction.
    Used to justify the statement that the NLO+NLL result is a lower bound on dsigma/dM_ttbar; no direct proof or numerical check for this process is provided.
  • domain assumption External inputs: NNPDF3.1 NNLO PDFs, m_t=172.5 GeV, Gamma_t=1.4 GeV, alpha_s(M_Z)=0.1180, taken from prior literature.
    Inputs to the calculation; the paper does not assess their uncertainties.

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Cite this review

Pith. "Pith review of Updated predictions for toponium production at the LHC." pith.science (2026). https://pith.science/paper/ERF4RDOG

@misc{pith2026241216685,
  author       = {Pith},
  title        = {Pith review of: Updated predictions for toponium production at the LHC},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ERF4RDOG}},
  note         = {Machine review of arXiv:2412.16685}
}
abstract

We provide an update on QCD predictions for top-quark pair production close to threshold including bound state effects at the Large Hadron Collider. We compute the top-quark pair invariant mass distribution $d\sigma/dM_{t\bar{t}}$, including Coulomb resummation for bound-state effects, as well as threshold resummation for emissions of soft and collinear gluons. We discuss uncertainty estimates and present a proposal for the use of these predictions in experimental analyses.

Figures

Figures reproduced from arXiv: 2412.16685 by the authors.

Figure 1
Figure 1. FIG. 1. Invariant mass distribution [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Invariant mass distribution [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 10 Pith papers

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  3. Top-Antitop Production and Decay at Threshold at the LHC in QCD Perturbation Theory

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    NLO+PS generators with resummed Coulomb threshold corrections show that N3LO already captures most of the ATLAS-bin enhancement, finite-width effects are small, and the loosely defined toponium piece is only a few picobarns.

  4. Extracting a Toponium Signal at the LHC with Spin and Quantum Information Tools

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    Consistently including toponium bound-state effects inside the new-physics amplitude, not just the Standard Model one, reshapes the allowed mass–coupling region for a top-philic pseudoscalar near the top-antitop threshold.

  6. Toponia at the HL-LHC and FCC-ee

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    Toponium ηt and ψt states have promising discovery channels at HL-LHC (ηt b bbar, about 6σ) and FCC-ee (ψt interference in b bbar, about 14σ), while P-wave states remain out of reach.

  7. Phenomenology of Hypothetical Single-Top Hadronic States

    hep-ph 2026-05 unverdicted novelty 5.0 of 10

    QCD sum-rule calculations yield single-top baryon and meson masses near the top-quark mass, with a few channels slightly below the naive quark-sum threshold.

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  10. Top-quark pole mass extraction at NNLO accuracy

    hep-ph 2024-12 conditional novelty 2.0 of 10

    Top-quark pole mass values extracted from NNLO fits to ttbar cross-section data are consistent across four PDF sets and with the PDG 2024 average.

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