{"id":"1528f1e0-f74b-4459-96ee-da76dd346b09","arxiv_id":"2506.04846","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"The tensor-pomeron model predicts 0.3 to 0.7 microbarn cross sections for central eta-prime production at the LHC, and shows that scalar-pomeron models forbid eta, eta-prime, and f1 production entirely.","lead":"This paper calculates how often LHC proton collisions should produce an eta or eta-prime meson in the middle while both protons stay intact, using a model in which the pomeron is a spinning object. It predicts measurable rates and argues that detecting these mesons would rule out the simpler picture of a spinless pomeron.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The WA102 fit does not uniquely determine the PPη and PPη′ couplings, so the LHC cross-section predictions and their 'upper limit' status inherit a factor-2–4 ambiguity; the scalar-pomeron no-go itself is sound.","rationale":"The strongest part of the paper is the scalar-pomeron no-go in Sec. II. With only two independent momenta q1 and q2 (and k = q1 + q2) available at the vertex, there is no parity-odd Lorentz invariant that could couple two scalar pomerons to a pseudoscalar meson, and the axial-vector vertex is forced to vanish by the same parity argument; Eqs. (2.2)–(2.10) are correct. The tensor-pomeron formalism is well established, and the inclusion of absorption at the amplitude level is a genuine improvement. The load-bearing weak point is the calibration of the PPη and PPη′ couplings. Table I and the discussion following Eq. (3.4) make the degeneracy explicit: the fit cannot cleanly separate PP from P f2R, f2RP, and f2R f2R exchanges at WA102 energies, and the only parameter sets reproducing the WA76/WA102 energy ratio require an unrealistically large, explicitly effective f2R f2Rη′ coupling of about ±25. Without a fit-quality statistic or parameter uncertainties, the quoted LHC cross sections are not uniquely determined; the spread among the published sets is a factor of 2–4, and the 'upper limit' label is an assumption, not a rigorous bound. This matches the reader's weakest-assumption analysis and supports the conditional verdict. The paper is honest about the limitation, stating in footnote 1 that the reduction could be up to a factor of 4, but the central numerical claim remains dependent on an unquantified model-selection ambiguity. I therefore see no reason to change the verdict.","tokens_in":18681,"tokens_out":8424,"duration_ms":114376,"concrete_test":"Perform a profile-likelihood scan over the PPη and PPη′ couplings: for each (g′_PPη, g″_PPη, g′_PPη′, g″_PPη′) on a grid, refit the subleading couplings and the form-factor cutoff to the WA102 total cross sections (3.1)–(3.2), the differential shapes in Figs. 2–3, and the energy ratio (3.4), using a proper χ² with experimental covariance and allowing the ±25 effective f2R f2R values. Then propagate the resulting allowed PP-coupling region to √s = 13 TeV and compute the envelope of σ(pp→ppη) and σ(pp→ppη′) for |η_M| < 1 and 2 < η_M < 5. If the envelope extends above 2.5 μb for η or above 0.7 μb for η′, the 'upper limit' and predicted-range statements in the abstract are not established by the current fits.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim — LHC σ(pp→ppη) ≤ 2.5 μb and σ(pp→ppη′) = 0.3–0.7 μb for |η_M| < 1 — rests on fixing the PPη and PPη′ couplings from WA102 data at √s = 29.1 GeV. Section III A and Table I show that several parameter sets describe the same WA102 distributions with very different shares of PP versus P f2R and f2R f2R contributions. The energy ratio (3.4), σ(29.1 GeV)/σ(12.7 GeV) = 0.72 ± 0.16, is not reproduced by fits 1–6, which give ratios 1.19–1.47; only by inflating g′_{f2R f2R η′} to about ±25 (sets 7–8) is the ratio matched, and the authors themselves state that this value is effective, replacing non-included channels. No χ², likelihood, or parameter uncertainties are reported, so the extracted PP couplings are not statistically characterized. Since subleading exchanges are suppressed at 13 TeV, the LHC rate is essentially proportional to |g_PP M|²; across the published sets the predictions differ by a factor of about 2 for η′ and about 4 for η. The paper labels the larger results as 'upper limits', but this is an interpretive assumption rather than a demonstrated bound: a scan allowing larger PP couplings with compensating destructive interference from subleading exchanges could in principle yield even larger LHC cross sections while still fitting WA102. Thus the numerical LHC predictions are conditional on an unquantified model-selection ambiguity, even though the scalar-pomeron no-go of Eqs. (2.6) and (2.10) is rigorous.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies central exclusive production (CEP) of η and η′ mesons in proton-proton collisions at the LHC within the tensor-pomeron approach. It first proves that a scalar pomeron cannot mediate CEP of pseudoscalar mesons (η, η′) or of the axial-vector meson f1(1285), using Lorentz invariance and parity (Eqs. (2.2)–(2.10)). The authors then construct amplitudes with pomeron-pomeron, pomeron-f2R, and f2R-f2R exchanges, including absorption corrections, fit the model parameters to WA102 data at √s = 29.1 GeV, and extrapolate to √s = 13 TeV. They present integrated cross sections and differential distributions, reporting upper limits of 2.5 μb for η (|η_M|<1) and 5.6 μb for η (2<η_M<5), and ranges 0.3–0.7 μb and 0.9–2.1 μb for η′. The paper also discusses SU(3)-flavor arguments and concludes that they do not forbid a PPη coupling.","tokens_in":19239,"tokens_out":5321,"duration_ms":64315,"significance":"The scalar-pomeron no-go result is a clean, rigorous analytic statement with a clear experimental consequence: observation of CEP of η, η′, or f1(1285) would disfavor a scalar pomeron. The tensor-pomeron formalism, the detailed treatment of subleading reggeon exchanges, and the inclusion of absorption effects are valuable technical contributions. The authors are also transparent about the model dependence of their predictions. However, the central quantitative LHC predictions are not yet on firm ground: they rely on WA102 fits with no reported fit quality, with multiple equally plausible parameter sets giving cross sections that differ by factors of two to four. The paper is therefore significant as a framework and a no-go theorem, but its headline cross-section numbers require either a demonstrated bound or a reframing as conditional estimates.","major_comments":[{"comment":"The parameter sets used for the LHC predictions (sets 1–6 for η′ and A–D for η) do not reproduce the measured energy dependence of the η′ cross section. The paper reports that fits 1–6 give σ(29.1 GeV)/σ(12.7 GeV) ratios of 1.19–1.47, whereas the WA76/WA102 result is 0.72 ± 0.16. Only sets 7 and 8, which require an effective g′_{f2R f2R η′} ≈ ±25, reproduce the ratio, but these sets are not carried into the LHC predictions in Sec. III C. Thus the couplings that feed the LHC extrapolation are not validated by the available energy-dependent data, which is a load-bearing gap for the central quantitative claim.","section":"Sec. III A, Table I, Eqs. (3.1)–(3.4)"},{"comment":"The WA102 fit does not uniquely determine the PPη and PPη′ couplings: no χ², likelihood, or parameter uncertainties are reported, and the relative PP versus P f2R and f2R f2R shares vary strongly across the equally motivated parameter sets. This ambiguity propagates directly into the LHC cross sections in Table II, where the η′ predictions differ by about a factor of 2 and the η predictions by about a factor of 4. The paper's characterization of the larger results as 'upper limits' is an interpretive assumption, not a derived bound, because no systematic scan over the full parameter space (including the possibility of larger PP couplings with compensating destructive interference from subleading exchanges) is performed.","section":"Sec. III A, Table I vs. Table II"},{"comment":"The abstract prominently presents 'upper limits' of 2.5 μb and 5.6 μb for pp→ppη, but the paper itself states in footnote 1 that if subleading reggeon contributions are important at WA102 energies, the LHC cross sections could be smaller by up to a factor of 4. This undercuts the upper-limit wording: the quoted numbers are upper limits only under a specific model-selection assumption. The predictions should either be reframed as model-dependent estimates with an explicit uncertainty band, or the upper-limit claim should be supported by a maximization over the parameter space consistent with the WA102 data.","section":"Abstract, Sec. III C, and footnote 1"}],"minor_comments":[{"comment":"The sentence 'We can see from Table I how the choice of the type of the form factor F(t1,t2) in (2.17) and the cutoff parameter affect the strength of the different coupling constants' has a subject-verb agreement error: 'the choice ... affect' should be 'the choice ... affects'.","section":"Sec. III A, near Table I"},{"comment":"In Fig. 6, the P f2R and f2RP contributions are multiplied by a factor of 10 for visualization, and the caption states this. However, the unscaled contributions are not shown or tabulated, so a reader cannot immediately judge their actual magnitude; consider adding the unscaled curves or a numerical statement of their relative contribution.","section":"Sec. III C, Fig. 6"},{"comment":"Sets 7 and 8, which reproduce the energy ratio (3.4) with an effective g′_{f2R f2R η′} ≈ ±25, are not used for the LHC predictions in Table II. This choice is not explained in the text; a remark clarifying why sets 7 and 8 are excluded from the LHC extrapolation would help the reader.","section":"Table I and Sec. III A, sets 7 and 8"},{"comment":"The abstract quotes the η′ range for |η_M|<1 as 0.3–0.7 μb, while Table II lists values from 0.30 to 0.72 μb. Quoting the exact Table II range, or noting the rounding, would remove a small inconsistency.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper deserves a major-revision opportunity rather than rejection. The scalar-pomeron no-go result is rigorous and publishable on its own, and the tensor-pomeron framework is competent. The load-bearing problem is the unquantified model-selection ambiguity in the LHC predictions, which can be addressed by a more systematic fit with uncertainties, a demonstrated upper-limit scan, or a softened presentation of the numerical claims. The fit-to-data validation chain also needs to confront the failure to reproduce the WA76/WA102 energy ratio with the parameter sets used for extrapolation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The scalar-pomeron no-go in Sec. II is clean and rigorous: parity plus Lorentz invariance forces Gamma(PS PS -> pseudoscalar) = 0, and the same argument kills axial-vector f1 production. That result stands on its own. The LHC cross sections are honest model predictions, but the 'upper limit' label is an interpretation, not a demonstrated bound.\n\nWhat is new: the authors add absorption corrections to the earlier Born amplitudes, refit the PP, P f2R, and f2R f2R couplings to WA102 data, and give LHC differential distributions and integrated cross sections in Table II. They are transparent about limitations — footnote 1 says subleading exchanges could reduce LHC rates by up to a factor of 4, and Table I shows several parameter sets with very different coupling shares that all describe the same WA102 distributions.\n\nThe soft spot is the one the stress test identifies. No objective fit statistic is reported, and no parameter uncertainties. Sets 1–6 fail to reproduce the WA76/WA102 energy ratio in Eq. (3.4); only sets 7–8 match it, by inflating g'_f2R f2R eta' to about +-25, which the authors call effective. So the PP eta and PP eta' couplings are not pinned down by the fit. The LHC cross sections scale as |g_PP M|^2, and across the published sets they differ by a factor of about 2 for eta' and 4 for eta. The largest values are labeled upper limits, but one could imagine larger PP couplings compensated by destructive interference from subleading terms that still fit the lower-energy data. The conditionality is stated, but the spread is real.\n\nThe SU(3) discussion in Sec. IV is qualitative and does not affect the numerics. The citation pattern leans on the authors' own tensor-pomeron program, which is appropriate since the Born amplitudes are theirs.\n\nFor whom: the soft-diffraction subfield and LHC experiments that could search for central eta, eta', f1. The no-go alone is worth a referee. The LHC predictions give a first target, with the factor-2–4 caveat attached.\n\nI would send this to a serious referee. The central argument is sound, the model is clearly stated, and the honesty about limitations is a credit.","headline":"Sound scalar-pomeron no-go plus honest but fit-ambiguous LHC predictions; the upper-limit label is an interpretation, not a bound.","tokens_in":19718,"tokens_out":3010,"would_cite":true,"duration_ms":35635,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper argues that observing central exclusive production of η, η′ or f1(1285) at the LHC would rule out a scalar pomeron, and it predicts tensor-pomeron cross sections large enough to measure.","keywords":["central exclusive production","tensor pomeron","eta meson","eta-prime meson","diffractive proton-proton collisions","scalar pomeron","pomeron spin","LHC cross-section predictions"],"falsifier":"A dedicated measurement of $pp\\to pp\\,\\eta$ and $pp\\to pp\\,\\eta'$ at $\\sqrt{s}=13$ TeV with forward protons tagged or a central rapidity gap. If either channel is observed at non-zero rate, the scalar-pomeron prediction of zero from pomeron fusion is falsified; if the $\\eta$ rate at $|\\eta_M|<1$ comes in below roughly 0.6 $\\mu$b, the fitted pomeron couplings were inflated by sub-leading exchanges at 29.1 GeV, and the tensor-pomeron parameters would need revision.","tokens_in":18468,"feed_emoji":"⚛️","tokens_out":9204,"duration_ms":102440,"temperature":0.7,"pith_summary":"At issue is the spin of the pomeron, the object that governs high-energy diffractive scattering. The paper shows that a scalar pomeron predicts zero central exclusive production of $\\eta$, $\\eta'$, and $f_1(1285)$, while a tensor pomeron naturally allows it. After fitting its couplings to 29.1 GeV fixed-target data and adding absorption, the model predicts LHC cross sections at the level of a few microbarns for $\\eta$ and 0.3–2.1 $\\mu$b for $\\eta'$ — rates that should be measurable. Measurement of any of these channels would therefore be a direct test of pomeron spin, and the size of the rates would fix the pomeron-pomeron-meson couplings that low-energy data cannot determine uniquely.","feed_headline":"LHC observation of eta mesons would rule out a scalar pomeron","feed_subtitle":"Tensor-pomeron model predicts up to 2.5 microbarns for pp→pp eta and 0.3–0.7 for eta' at 13 TeV.","key_machinery":"The load-bearing object is the effective $\\mathbb{P}\\mathbb{P}M$ vertex $\\Gamma^{\\mu\\nu,\\kappa\\lambda}_{\\mathbb{P}\\mathbb{P}M}(q_1,q_2)$: two antisymmetric Levi-Civita tensor structures, corresponding to the $(l,S)=(1,1)$ and $(3,3)$ partial waves of two spin-2 pomerons, contracted with the pomeron momenta and multiplied by a form factor $F(t_1,t_2)$. The same vertex forms, with different couplings, also describe $\\mathbb{P} f_{2\\mathbb{R}}M$ and $f_{2\\mathbb{R}}f_{2\\mathbb{R}}M$ fusion. The other essential piece is the absorption correction, an integral over the elastic $pp$ amplitude that converts the Born amplitude into the physical one; at LHC energies it cuts the cross section by roughly 60% and shifts the azimuthal-angle distribution.","core_discovery":"The central claim is that the spin of the pomeron is directly readable in central exclusive production. In a theory with a scalar pomeron, the vertex $\\Gamma^{(\\mathrm{PS\\,PS}\\to M)}(q_1^2,q_2^2,k^2)$ is a scalar function of invariant momentum transfers; parity reverses its sign while leaving the arguments unchanged, so it must vanish, and the same argument forces the $f_1(1285)$ vertex to vanish. Thus scalar pomerons predict exactly zero CEP of $\\eta$, $\\eta'$, and $f_1(1285)$ via pomeron fusion. In the tensor-pomeron model, by contrast, the pomeron is a rank-2 tensor exchange and the $\\mathbb{P}\\mathbb{P}M$ vertex has two independent Lorentz structures, labelled by $(l,S)=(1,1)$ and $(3,3)$, with couplings $g'_{\\mathbb{P}\\mathbb{P}M}$ and $g''_{\\mathbb{P}\\mathbb{P}M}$. The paper fits these couplings, together with reggeon-pomeron and reggeon-reggeon contributions and cutoff parameters, to 29.1 GeV central-production data, applies absorption corrections, and extrapolates to 13 TeV. The resulting upper limits are 2.5 $\\mu$b for $\\eta$ at $|\\eta_M|<1$, 5.6 $\\mu$b for $\\eta$ at $2<\\eta_M<5$, 0.3–0.7 $\\mu$b for $\\eta'$ at $|\\eta_M|<1$, and 0.9–2.1 $\\mu$b for $\\eta'$ at $2<\\eta_M<5$.","pith_inferences":["Beyond the paper, the scalar-pomeron null result is a clean diagnostic: a rapidity-gap search for $\\eta\\to\\pi^+\\pi^-\\pi^0$ in the same sample as $\\omega$ production could distinguish pomeron fusion from photon-pomeron background and give a quick verdict on pomeron spin.","Beyond the paper, if LHC data come in near the lower edge of the predicted ranges, the likely lesson is that the fitted pomeron couplings at 29.1 GeV were inflated by sub-leading exchanges; that would not overturn the tensor-pomeron framework but would require a re-fit with running energy dependence.","Beyond the paper, comparing $\\eta'$ production in the forward region $2<\\eta_M<5$ with midrapidity would give a direct handle on sub-leading exchanges, since reggeon contributions are enhanced at forward and backward meson rapidity."],"forward_implications":["Any non-zero observation of central exclusive $\\eta$, $\\eta'$, or $f_1(1285)$ production at the LHC would contradict the scalar-pomeron prediction that these vertices vanish by parity.","The predicted cross sections at $\\sqrt{s}=13$ TeV, up to 2.5 $\\mu$b for $\\eta$ and 0.3–0.7 $\\mu$b for $\\eta'$ at $|\\eta_M|<1$, are large enough to be measured with rapidity-gap or forward-proton selections, so the pomeron-pomeron-meson couplings can be extracted.","Because reggeon-pomeron and reggeon-reggeon fusion die off with energy, LHC data will separate the clean pomeron-pomeron signal from the sub-leading exchanges that contaminate the 29.1 GeV fits.","Comparing the $\\eta$ and $\\eta'$ rates will test whether the pomeron couples to the mesons' extended gluonic string rather than to their flavor quantum numbers; a relatively large $\\mathbb{P}\\mathbb{P}\\eta$ coupling favours the string-extension picture."],"supporting_citations":[{"why":"Derives the two independent pomeron-pomeron-meson couplings and the amplitude formalism that this paper extends.","marker":"[1]"},{"why":"Defines the tensor-pomeron model, including the proton vertex, propagator, and elastic amplitude used for absorption corrections.","marker":"[2]"},{"why":"Provides the 29.1 GeV central-production distributions that are fitted to fix coupling constants and cutoff parameters.","marker":"[3]"},{"why":"Supplies the measured total cross sections for central eta and eta-prime production used as fit targets.","marker":"[4]"},{"why":"Establishes the absorption treatment and the upper-limit interpretation for central f1 production that this analysis adopts.","marker":"[5]"},{"why":"Gives the measured ratio of cross sections at 29.1 and 12.7 GeV used to constrain the size of sub-leading exchange contributions.","marker":"[16]"},{"why":"Provides an independent derivation of the pomeron-pomeron-meson vertex whose coupling ratio is compared with the fitted tensor-pomeron couplings.","marker":"[23]"}],"fun_headline_variants":["Observing eta at LHC would falsify scalar pomeron","Scalar pomeron forbids exclusive eta production","Tensor pomeron predicts eta cross section at LHC","Eta meson production tests pomeron spin nature","If LHC sees eta, pomeron is not scalar"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the pomeron-pomeron-meson couplings extracted from the 29.1 GeV data are not drastically smaller than fitted, because sub-leading exchanges could be absorbing part of the measured rate; if those exchanges dominate, the LHC cross sections could be up to four times lower than quoted.","fun_headline_variants_meta":{"raw":{"variants":["Observing eta at LHC would falsify scalar pomeron","Scalar pomeron forbids exclusive eta production","Tensor pomeron predicts eta cross section at LHC","Eta meson production tests pomeron spin nature","If LHC sees eta, pomeron is not scalar"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000257,"raw_usage":{"total_tokens":1789,"prompt_tokens":1367,"completion_tokens":422,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":983,"completion_tokens_details":{"reasoning_tokens":339}},"tokens_in":983,"tokens_out":422,"duration_ms":5422,"temperature":1.0,"reasoning_tokens":339,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:32:51.223444+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A dedicated measurement of $pp\\to pp\\,\\eta$ and $pp\\to pp\\,\\eta'$ at $\\sqrt{s}=13$ TeV with forward protons tagged or a central rapidity gap. If either channel is observed at non-zero rate, the scalar-pomeron prediction of zero from pomeron fusion is falsified; if the $\\eta$ rate at $|\\eta_M|<1$ comes in below roughly 0.6 $\\mu$b, the fitted pomeron couplings were inflated by sub-leading exchanges at 29.1 GeV, and the tensor-pomeron parameters would need revision.","supporting_citations":[{"cited_title":"Exclusive central diffractive production of scalar and pseudoscalar mesons; tensorial vs. vectorial pomeron","cited_arxiv_id":"1309.3913","evidence_quote":"Derives the two independent pomeron-pomeron-meson couplings and the amplitude formalism that this paper extends."},{"cited_title":"(2.10) CEP of an axial-vector meson ~M is not possible with a scalar pomeron","cited_arxiv_id":null,"evidence_quote":"Defines the tensor-pomeron model, including the proton vertex, propagator, and elastic amplitude used for absorption corrections."},{"cited_title":"glueball-filter variable","cited_arxiv_id":null,"evidence_quote":"Provides the 29.1 GeV central-production distributions that are fitted to fix coupling constants and cutoff parameters."},{"cited_title":"A study of pseudoscalar states produced centrally in pp interactions at 450 GeV/c","cited_arxiv_id":"hep-ex/9803029","evidence_quote":"Establishes the absorption treatment and the upper-limit interpretation for central f1 production that this analysis adopts."},{"cited_title":"Central production of mesons: Exotic states versus Pomeron structure","cited_arxiv_id":"hep-ph/9902243","evidence_quote":"Gives the measured ratio of cross sections at 29.1 and 12.7 GeV used to constrain the size of sub-leading exchange contributions."},{"cited_title":"Searching for the odderon in $pp \\to pp K^{+}K^{-}$ and $pp \\to pp \\mu^{+}\\mu^{-}$ reactions in the $\\phi(1020)$ resonance region at the LHC","cited_arxiv_id":"1911.01909","evidence_quote":"Provides an independent derivation of the pomeron-pomeron-meson vertex whose coupling ratio is compared with the fitted tensor-pomeron couplings."}],"review_version":1}