{"id":"da4d0440-1e70-48d9-960a-d4b561391958","arxiv_id":"2505.03869","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Non-relativistic QCD Green's function re-weighting of top-antitop matrix elements predicts momentum and invariant-mass distributions that pseudo-scalar toy models fail to reproduce, suggesting toponium dynamics may be measurable at the LHC.","lead":"This paper compares two ways to model toponium, a short-lived bound state of a top and anti-top quark, in LHC top-pair production. It argues that a quantum-mechanical Green's function method reproduces bound-state effects that simplified pseudo-scalar models miss, and that LHC measurements could probe this non-perturbative QCD dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central reweighting in Eq. (4) assumes the full MG5aMC matrix element factorises into a hard part times the free Green's function in the 340-350 GeV, p*<50 GeV window, but this factorisation is asserted, not tested; if wrong, the claimed NRQCD p* spectrum is an artifact of the reweighting.","rationale":"I read the paper as a comparison of modelling strategies, with the central claim that NRQCD Green's-function reweighting correctly captures toponium dynamics while pseudo-scalar toy models fail on p*. The most load-bearing condition is the validity of Eq. (4) as a replacement of the full matrix element by a non-relativistic one. The reader's weakest_assumption identifies the same Section 3 sentence; I agree, and the concrete test above turns it into a direct factorisation check. Because the concern is already reflected in the conditional verdict, I do not move the verdict. No other issue seems more decisive: the toy-model comparison is a legitimate shape comparison, and the lack of uncertainties weakens but does not by itself invalidate the qualitative claim.","tokens_in":6056,"tokens_out":5720,"duration_ms":64224,"concrete_test":"Regenerate the t tbar sample with MG5aMC using the same cuts and colour-singlet projection but without applying Eq. (4). In each (m_ttbar, p*) bin, compute the ratio R = dsigma_full / |G0(E,p*)|^2 after removing common phase-space and PDF factors. If R is not flat (e.g., varies by more than 20%) across 340 GeV <= m_ttbar <= 350 GeV and p* < 50 GeV, the factorisation assumed in Eq. (4) fails, and the NRQCD labels in Figs. 1-2 are not justified. If R is flat, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central comparison is built on the sentence in Section 3 that the phase-space cuts ensure 'the non-relativistic matrix element ... is well-approximated by the full matrix element generated automatically by MG5aMC.' This is the load-bearing assumption because Eq. (4) replaces |M|^2 by |M|^2 |G/G0|^2; for the result to be a NRQCD prediction, the unmodified full matrix element must be proportional (after colour-singlet projection) to the free Green's function G0(E,p*) over the whole window, with a hard coefficient that is flat in m_ttbar and p*. The paper provides no check of this proportionality. If the full matrix element contains residual p*- or E-dependent contributions from off-shell top propagators, P-wave admixtures, or colour-octet intermediates that are not part of the S-wave singlet Green's function, the reweighted distribution is not the NRQCD prediction but an ad hoc rescaling of the full matrix element. The comparison with Breit-Wigner toy models would then demonstrate only that two different ad hoc recipes disagree, not that one captures bound-state dynamics. This concern is independent of the absence of uncertainties and is checkable.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper examines toponium formation in top-antitop pair production at the LHC, focusing on the near-threshold region defined by 340 GeV ≤ m_ttbar ≤ 350 GeV and p* < 50 GeV. It advocates modelling toponium by re-weighting Standard Model matrix elements with the ratio of the non-relativistic QCD Green's function to the free Green's function, Eq. (4), implemented through a modified MG5aMC event generator following the author's previous work. The paper compares the resulting two-dimensional distribution in (m_ttbar, p*) and the invariant-mass distributions of the reconstructed top quarks against two pseudo-scalar toy-model configurations: one with Green's-function re-weighting and a broad toponium width, and one without re-weighting and a narrower width. The central claim is that the non-relativistic QCD framework yields a distinctive p* spectrum and a distorted invariant-mass distribution for the lighter reconstructed top quark, which the Breit-Wigner toy models fail to reproduce, and that these features are potentially observable at the LHC.","tokens_in":6318,"tokens_out":6851,"duration_ms":68313,"significance":"If the central re-weighting assumption is valid, the paper identifies p*-dependent and lighter-top invariant-mass observables that could discriminate between a genuine non-relativistic QCD treatment of toponium and simplified pseudo-scalar resonance models. This is timely given recent LHC excesses and the renewed interest in toponium. The paper uses standard physical inputs (mt = 173 GeV, Γt = 1.49 GeV, αs(mZ) = 0.12) and does not fit the Green's-function prediction to the distributions it predicts; the toy-model couplings are the only fitted parameters, and the comparison is therefore not circular. The predicted distributions are falsifiable with LHC data. However, the significance is conditional on validation of the factorisation assumption behind Eq. (4), which the paper currently asserts but does not test, and on a quantification of theoretical uncertainties, which is absent.","major_comments":[{"comment":"The central re-weighting is load-bearing and the factorisation assumption behind it is asserted, not tested. The manuscript states that the cuts 340 GeV ≤ m_ttbar ≤ 350 GeV and p* < 50 GeV ensure that the non-relativistic matrix element 'is well-approximated by the full matrix element generated automatically by MG5aMC', but no numerical check is provided. For Eq. (4) to produce an NRQCD prediction, the unmodified colour-singlet matrix element must factor as a hard coefficient times the free Green's function G0(E, p*) over the whole window, with the coefficient flat in both m_ttbar and p*. Residual E- or p*-dependence from off-shell top propagators, P-wave admixtures, or colour-octet intermediates would make the reweighted distribution an ad hoc rescaling of the full matrix element rather than the NRQCD Green's-function prediction. Please add a numerical validation: compare the unmodified, colour-singlet-projected MG5aMC matrix element with |G0(E, p*)|^2 across the window and quantify the residual slope in E and p*; this is checkable and directly addresses the validity of the central comparison.","section":"Section 3, Eq. (4)"},{"comment":"The paper's central claim that the toy models 'fail to describe the correct momentum spectrum' and that the lighter-top invariant-mass distortion is a signature is not accompanied by any uncertainty estimate. There are no scale, αs, top-width, PDF, or re-weighting uncertainties, nor statistical uncertainties from the event generation. The reader therefore cannot judge whether the visible differences between the non-relativistic QCD prediction and the toy models are significant relative to the theoretical error budget. Section 4 itself lists the quantification of theoretical uncertainties as future work, but the qualitative conclusions in Section 3 are presented without that caveat. Please add at least an estimate of the dominant uncertainties, or explicitly reframe the conclusions as shape-level observations pending validation.","section":"Section 3, Figures 1 and 2"},{"comment":"The toy-model couplings are chosen to reproduce the non-relativistic QCD total cross section of 6.43 pb, but the manuscript does not state whether the NRQCD prediction obtained from the reweighted MG5aMC events also has this total. Since Eq. (4) multiplies the SM matrix element by |G/G0|^2, the total cross section of the modified sample is not automatically equal to the unmodified one; the re-weighting may change the normalisation as well as the shape. If the curves in Figures 1 and 2 are normalised to the same integrated cross section by construction, this should be stated explicitly; if not, the comparison mixes shape and normalisation effects. Please clarify the normalisation procedure and, if the comparison is shape-only, provide the normalisation factors.","section":"Section 3, Figure 1 and toy-model normalisation"}],"minor_comments":[{"comment":"The text calls the potential a 'tree-level Coulombic potential', but the expression includes a term proportional to αs^2 with the one-loop coefficient; please align the wording with the perturbative order actually used.","section":"Section 2, Eq. (6)"},{"comment":"The caption label d^2σ/(p*)^2 is ambiguous; please specify the differential variables explicitly, for example d^2σ/(dp* dm_ttbar).","section":"Section 3, Figure 1 caption"},{"comment":"The expression for the average momentum is mis-typeset and, as printed, is not a well-formed fraction; please correct the equation so that the numerator and denominator are unambiguous and the quoted value ⟨p(−2 GeV)⟩ ≈ 20 GeV is reproducible.","section":"Section 3, Eq. (8)"},{"comment":"The statement that 'recent excesses in several distributions' [2–4] are consistent with toponium formation would be more useful if it specified which distributions are affected and the significance of the excesses.","section":"Section 1"},{"comment":"There are several typographical issues, including 'massmt' in Section 2, 'Coloumbic' in Section 3, and inconsistent spacing around m_ttbar; a careful proofreading pass is needed.","section":"Throughout"},{"comment":"The manuscript does not state whether the modified MG5aMC code used for the re-weighting is publicly available beyond the description in reference [9]; a short code/data availability statement would improve reproducibility.","section":"Section 3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a compact report whose central tool, the Green's-function re-weighting, is already described in the author's previous work [9]; the incremental contribution is the explicit comparison with pseudo-scalar toy models. That comparison is worth publishing if the factorisation assumption is validated. My recommendation is driven by the missing validation of Eq. (4) and the absence of uncertainty estimates, not by novelty concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Benjamin — quick take on 2505.03869. The genuinely new thing is the comparison itself: for the first time, the NRQCD Green's function reweighting of Standard Model ttbar production is set directly against the pseudo-scalar toy models from [5] and [8], and the toy models are shown to miss the top-quark momentum spectrum and the distortion of the lighter-top invariant mass. That matters because those are exactly the observables being used to interpret the recent ATLAS and CMS excesses, and the figures make the qualitative difference visible. The paper deserves credit for being clear about what it is: a method report, not a full phenomenological study.\n\nThe soft spots are real but proportionate. There are no uncertainty estimates anywhere, which is fine for a first look but should be flagged. The phase-space approximation — that in the 340–350 GeV, p*<50 GeV window the full MG5aMC matrix element is well approximated by the non-relativistic one — is asserted, not tested. The stress-tester worried this is load-bearing, and they are half right. The reweighting in Eq. (4) only gives the NRQCD answer if the unmodified matrix element is proportional to the free Green's function over the whole window. That is a mild assumption here because the window is genuinely near threshold, but a careful referee should ask for one plot comparing the reweighted prediction with the unmodified matrix element (equivalently, a check that G/G0 is flat or that the factorisation holds). The conclusion already admits the missing higher-spin and octet contributions, which is honest but also means the central comparison is between two approximations, not between an approximation and the full NRQCD result.\n\nCitation pattern: the method comes from the author's own prior papers [5,9]; that is not a flaw when the framework is the framework, but it does mean the novelty is limited to the comparison, not the machinery. Self-citation here is legitimate.\n\nWho is this for? Specialists in top-quark phenomenology working on toponium excess interpretations. A serious referee should engage with it — the check I described is cheap and the comparison is timely. I would accept for review, and ask for the validation plot plus uncertainties before publication.","headline":"A short, honest method comparison: NRQCD Green's function reweighting beats pseudo-scalar toy models on the top-quark momentum and lightest-top mass distributions, but the central factorization assumption is asserted rather than tested and nothing is presented with uncertainties.","tokens_in":6850,"tokens_out":2647,"would_cite":true,"duration_ms":29046,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["14.65.Ha","12.38.-t"],"model":"deepseek-v4-flash","headline":"This paper argues that toponium formation at the LHC is observable through the top-antitop momentum spectrum and the decay asymmetry of the two top quarks, provided the production amplitude is re-weighted with non-relativistic QCD Green's…","keywords":["toponium","top-antitop pair production","non-relativistic QCD","Green's function","LHC phenomenology","top quark bound state","near-threshold production"],"falsifier":"Compare the re-weighted prediction against the unmodified event-generator prediction for the $p^*$ distribution inside the same window; a disagreement beyond the stated approximation would invalidate the central comparison. On the data side, a high-statistics measurement of $d^2\\sigma/(dp^*\\,dm_{t\\bar t})$ and of the lighter-top invariant-mass distribution in di-leptonic events at the LHC, with $p^*$ resolution of a few GeV, would settle whether the predicted $-2$ GeV peak and the $\\sim20$ GeV mean momentum are present.","tokens_in":5827,"feed_emoji":"⚛️","tokens_out":9763,"duration_ms":84643,"temperature":0.7,"pith_summary":"This paper claims that the best way to see toponium at the LHC is to treat the top-antitop pair as a genuine non-relativistic bound state: take Standard Model production, then re-weight each event by the ratio of the interacting to the free QCD Green's function. Within a narrow near-threshold window, the calculation concentrates events at a binding energy near $-2$ GeV with low relative momentum, and it predicts an average top momentum around 20 GeV, matching the inverse Bohr radius of the system. The same calculation gives an asymmetric decay picture: the lighter (first-decaying) top quark acquires a distorted, shifted invariant-mass distribution, while the heavier one keeps its free Breit-Wigner shape. The paper argues these features are observable fingerprints of toponium and are missed by pseudo-scalar toy models even when those models are tuned to the total cross section.","feed_headline":"Toponium's pull on top quarks shows up in LHC momentum spectra","feed_subtitle":"Measuring the top-pair momentum and the lighter-top mass shift could reveal the Standard Model's heaviest hadron.","key_machinery":"The load-bearing object is the momentum-space Green's function $\\widetilde G(E;p^*)$ of the non-relativistic Schr\\\"odinger equation $\\left[-\\vec\\nabla^2/m_t + V_{\\mathrm{QCD}}(\\vec x) - (E+i\\Gamma_t)\\right]G(E;\\vec x)=\\delta^{(3)}(\\vec x)$, obtained by solving the Lippmann-Schwinger equation with the Coulombic QCD potential. It enters predictions through the replacement $|M|^2 \\to |M|^2\\,|\\widetilde G(E;p^*)/\\widetilde G_0(E;p^*)|^2$, which converts the full top-antitop matrix element into a colour-singlet near-threshold amplitude. The free Green's function $\\widetilde G_0$ provides the normalisation, so the re-weighting is blind to overall rate and controls only the shape in $E$ and $p^*$. The paper also uses the three-point Green's function $K(x,y,z)$ to justify the space-time picture in which one top decays first, while the other remains bound.","core_discovery":"On its own terms, the paper's central discovery is that toponium formation is not a hidden resonance effect but a resolvable distortion of Standard Model top-pair kinematics. Re-weighting the colour-singlet top-antitop matrix element by $\\left|\\widetilde G(E;p^*)/\\widetilde G_0(E;p^*)\\right|^2$ produces a doubly-differential cross section with a peak at $E\\simeq -2$ GeV and an average relative momentum $\\langle p\\rangle\\simeq 20$ GeV, quantitatively consistent with the non-relativistic bound-state expectation $1/a_0 = C_F\\alpha_s(a_0^{-1}) m_t/2$. The first-decaying top quark, bound by the Coulomb potential of the other, shows an invariant-mass distribution shifted downward from a free Breit-Wigner; the heavier top is unaffected. Both a pseudo-scalar toy model with Green's-function re-weighting and a 7 GeV width, and one without re-weighting and a 2.8 GeV width, reproduce the peak position but fail to reproduce the momentum spectrum, and the invariant-mass discrepancy grows for the lighter top. The paper concludes that toponium forms with size $\\sim 1/(20\\ \\mathrm{GeV})$ and decays before hadronisation, so it is in principle visible in existing high-statistics LHC data.","pith_inferences":["The approximation that the full event-generator matrix element equals the non-relativistic matrix element in the chosen window is not tested against the unmodified generator output; a direct comparison of the re-weighted and unmodified $p^*$ distributions would be the cheapest check of the method.","If the lighter-top invariant-mass distortion is confirmed, unfolding it in existing dilepton $t\\bar t$ measurements could discriminate toponium from detector-resolution or off-shell effects without new dedicated analyses.","The same Green's-function machinery could in principle be extended beyond $p^*>50$ GeV or to colour-octet configurations, but the present results should not be extrapolated there, since the non-relativistic approximation is expected to degrade.","A detector-level simulation with experimental smearing for $m_{t_L}$ and $p^*$ would sharpen the comparison and might make the predicted asymmetry measurable at the high-luminosity LHC run."],"forward_implications":["The two-dimensional distribution $d^2\\sigma/(dp^*\\,dm_{t\\bar t})$ in the window $340\\le m_{t\\bar t}\\le 350$ GeV and $p^*<50$ GeV is dominated by bound-state dynamics, so measuring it with sufficient $p^*$ resolution gives direct access to non-perturbative QCD at the toponium scale.","In di-leptonic $t\\bar t$ events, the lighter reconstructed top quark should show a distorted, downward-shifted invariant-mass spectrum, while the heavier top keeps a free Breit-Wigner shape; this asymmetry is a testable signature that does not require the overall rate to be correct.","The position of the near-threshold peak is not a useful discriminator, since both toy models place it at $E\\simeq -2$ GeV; only the momentum spectrum and the lighter-top mass shape separate the non-relativistic treatment from the toy models.","Toponium decays on a timescale $1/(2\\Gamma_t)$, well before hadronisation, so the bound-state corrections should be included in Standard Model $t\\bar t$ predictions; omitting them biases distributions in the measured LHC phase-space region."],"supporting_citations":[{"why":"defines the first toy-model scenario, a pseudo-scalar resonance plus Green's-function re-weighting, which the paper compares against.","marker":"[5]"},{"why":"defines the second toy-model scenario, a pseudo-scalar resonance without re-weighting and a 2.8 GeV width.","marker":"[8]"},{"why":"provides the re-weighting of Standard Model matrix elements by non-relativistic QCD Green's functions that the paper adopts.","marker":"[9]"},{"why":"derives the three-point Green's function and the Schr\\\"odinger equation with the QCD potential underlying the bound-state kernel.","marker":"[10]"},{"why":"supplies the non-relativistic total cross section and the inverse Bohr radius expectation used to calibrate and interpret the predictions.","marker":"[18]"},{"why":"documents excited S-wave and P-wave states near the threshold that the toy models omit.","marker":"[22]"},{"why":"computes contributions from higher-spin and colour-octet states that account for the event rate at larger invariant masses, explaining the toy-model discrepancy.","marker":"[23]"}],"fun_headline_variants":["Toponium's momentum fingerprint emerges in LHC top-pair spectra","LHC top-pair data hints at toponium bound-state distortion","Toponium signals in top-antitop momentum peak near threshold","Top quarks pair briefly: toponium shifts momentum spectrum","Toponium's 20-GeV size shows in LHC top-pair kinematics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation assumes that inside the selected window (top-pair mass between 340 and 350 GeV and relative momentum below 50 GeV) the event generator's full production rate is a valid stand-in for the non-relativistic bound-state production rate, an approximation the paper states but does not directly test.","fun_headline_variants_meta":{"raw":{"variants":["Toponium's momentum fingerprint emerges in LHC top-pair spectra","LHC top-pair data hints at toponium bound-state distortion","Toponium signals in top-antitop momentum peak near threshold","Top quarks pair briefly: toponium shifts momentum spectrum","Toponium's 20-GeV size shows in LHC top-pair kinematics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000143,"raw_usage":{"total_tokens":1131,"prompt_tokens":863,"completion_tokens":268,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":177}},"tokens_in":479,"tokens_out":268,"duration_ms":3628,"temperature":1.0,"reasoning_tokens":177,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:43:01.360081+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compare the re-weighted prediction against the unmodified event-generator prediction for the $p^*$ distribution inside the same window; a disagreement beyond the stated approximation would invalidate the central comparison. On the data side, a high-statistics measurement of $d^2\\sigma/(dp^*\\,dm_{t\\bar t})$ and of the lighter-top invariant-mass distribution in di-leptonic events at the LHC, with $p^*$ resolution of a few GeV, would settle whether the predicted $-2$ GeV peak and the $\\sim20$ GeV mean momentum are present.","supporting_citations":[{"cited_title":"Signatures of toponium formation in LHC run 2 data","cited_arxiv_id":null,"evidence_quote":"defines the first toy-model scenario, a pseudo-scalar resonance plus Green's-function re-weighting, which the paper compares against."},{"cited_title":"Quantum detection of new physics in top-quark pair production at the LHC","cited_arxiv_id":null,"evidence_quote":"defines the second toy-model scenario, a pseudo-scalar resonance without re-weighting and a 2.8 GeV width."},{"cited_title":"Simulating toponium formation signals at the LHC","cited_arxiv_id":null,"evidence_quote":"provides the re-weighting of Standard Model matrix elements by non-relativistic QCD Green's functions that the paper adopts."},{"cited_title":"Sumino, K","cited_arxiv_id":null,"evidence_quote":"derives the three-point Green's function and the Schr\\\"odinger equation with the QCD potential underlying the bound-state kernel."},{"cited_title":"Bound-state effects on kinematical distributions of top quarks at hadron colliders","cited_arxiv_id":null,"evidence_quote":"supplies the non-relativistic total cross section and the inverse Bohr radius expectation used to calibrate and interpret the predictions."},{"cited_title":"Kiyo, Johann H","cited_arxiv_id":null,"evidence_quote":"computes contributions from higher-spin and colour-octet states that account for the event rate at larger invariant masses, explaining the toy-model discrepancy."}],"review_version":1}