Central exclusive production of $\eta$ and $\eta'$ mesons in diffractive proton-proton collisions at the LHC within the tensor-pomeron approach

Antoni Szczurek; Otto Nachtmann; Piotr Lebiedowicz

REVIEW 3 major objections 4 minor 40 references

Central exclusive production of $\eta$ and $\eta'$ mesons in diffractive proton-proton collisions at the LHC within the tensor-pomeron approach

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Sound scalar-pomeron no-go plus honest but fit-ambiguous LHC predictions; the upper-limit label is an interpretation, not a bound. read the letter →

arxiv 2506.04846 v2 pith:WCJ7RURX submitted 2025-06-05 hep-ph hep-ex

Piotr Lebiedowicz , Otto Nachtmann , Antoni Szczurek This is my paper · ORCID

classification hep-phhep-ex

keywords centralexclusiveproductiontensorpomeronetamesoneta-primediffractiveproton-protoncollisionsscalarspinLHCcross-sectionpredictions

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $2<\eta_M<5$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What 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.

The 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.

The 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.

For 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.

I 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[Sec. III A, Table I, Eqs. (3.1)–(3.4)] 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.
[Sec. III A, Table I vs. Table II] 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.
[Abstract, Sec. III C, and footnote 1] 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.

minor comments (4)

[Sec. III A, near Table I] 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'.
[Sec. III C, Fig. 6] 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.
[Table I and Sec. III A, sets 7 and 8] 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.
[Abstract] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the LHC predictions are energy extrapolations of couplings fitted to WA102 data, and the scalar-pomeron no-go is derived independently from Lorentz and parity invariance.

full rationale

The paper's central quantitative predictions are obtained by fitting the PPη, PPη′, and related reggeon couplings to WA102 data at √s = 29.1 GeV and then extrapolating these same couplings to LHC energies where subleading reggeon exchanges are suppressed. This is calibration followed by extrapolation, not a prediction that reduces by construction to its input: the LHC cross sections are not identical to the fitted WA102 quantities, and the paper explicitly shows that several parameter sets with different PP/reggeon shares describe the WA102 data, leading to a factor-of-2–4 spread in the LHC predictions. The 'upper limit' language is an interpretive label attached to the parameter set with the largest PP contribution, not a derived bound, so it is a model-ambiguity concern rather than circularity. The scalar-pomeron no-go result for η, η′, and f1(1285) is derived directly from Lorentz invariance and parity, Eqs. (2.2)–(2.10), with no fitted parameters and no reliance on self-citation. The tensor-pomeron framework is imported from the authors' earlier work, but it is supported by external data (STAR spin asymmetries, DIS analyses) and by the WA102 comparisons made here; the self-citations are contextual, not load-bearing. No step in the derivation chain is equivalent to its own input by definition.

Assumptions & free parameters 8 free parameters · 5 assumptions · 0 invented entities

The central LHC numbers rest on seven fitted coupling groups and a model form factor (Table I), plus trajectory parameters taken from Refs. [1] and [2]. The scalar-pomeron no-go, by contrast, uses only Lorentz invariance and parity and is independent of those fits. No new particles or new fundamental entities are introduced.

free parameters (8)

g'_PP eta and g''_PP eta = Set A: 1.8, 1.8; Sets B-D: 1.0, 1.0
Fitted to WA102 differential distributions for pp -> pp eta; directly control the central LHC eta prediction.
g'_P f2R eta and g''_P f2R eta = Set B: 1.1, 3.0; Set C: 1.1, 3.0; Set D: 1.7, 4.0
Subleading pomeron-reggeon fusion couplings fitted at WA102; strongly affect the decomposition of the total cross section.
g'_f2R f2R eta and g''_f2R f2R eta = Set A: 10, 0; Set B: 4, 0; Set C: -4, 0; Set D: -4, 0
Reggeon-reggeon couplings; their interference terms are significant even though the direct contribution is small.
g'_PP eta-prime and g''_PP eta-prime = Fits 1-3: 2.7 or 2.05, 1.2; Fits 4-6: 1.6 or 1.4, 1.2 or 1.4
Fitted to WA102 eta-prime data; the ratio g''/g' controls the shape of the phi_pp distribution.
g'_P f2R eta-prime and g''_P f2R eta-prime = Fits 4-6: 1.8 to 2.5, 0
Secondary exchange couplings; set to zero in Fits 1-3 and used in Fits 4-6 to estimate subleading effects.
g'_f2R f2R eta-prime and g''_f2R f2R eta-prime = Fits 1-5: -10, 0; Fit 6: 10, 0; Sets 7-8: +-25, 0
Only a large effective value around +-25 reproduces the WA76/WA102 energy ratio in Eq. (3.4); the authors state this value is effective and replaces omitted processes.
Central vertex cutoff Lambda = Lambda_0^2 = 0.5 GeV^2 or Lambda_E = 0.8 or 1.0 GeV
Two functional forms are tried and the cutoff is fitted by hand; it changes the fitted couplings and the LHC cross sections.
Pomeron and f2R trajectory parameters = alpha_P(0) = 1.0808, alpha'_P = 0.25 GeV^-2, beta_PNN = 1.87 GeV^-1; alpha_R+(0) = 0.5475, alpha'_R+ = 0.9 GeV^-2…
Taken from prior tensor-pomeron fits in Ref. [2]; they are load-bearing for every amplitude in this paper but are not fitted here.

assumptions (5)

domain assumption The pomeron is an effective rank-two tensor exchange with trajectory alpha_P(t) = 1 + epsilon_P + alpha'_P t, epsilon_P = 0.0808, alpha'_P = 0.25 GeV^-2, and proton coupling beta_PNN = 1.87 GeV^-1 (Eqs. 2.13 to 2.16, from Ref. [2]).
This is the defining assumption of the tensor-pomeron model; all cross sections follow from it. It is not derived in this paper.
domain assumption The f2R reggeon is also a tensor exchange with g_f2R pp / M0 = 11.04, alpha_R+(0) = 0.5475, alpha'_R+ = 0.9 GeV^-2 (Eqs. 2.23 and 2.24, from Ref. [2]).
Used for P f2R, f2R P, and f2R f2R fusion contributions; parameters are inherited from prior fits.
domain assumption The PP M and P f2R M vertices have only the (l, S) = (1, 1) and (3, 3) tensor couplings with the Lagrangians in Eqs. (2.17) to (2.19) and Appendix A.
The restriction to two couplings follows from angular momentum and parity, but the specific Lorentz structures and form factors are model choices from the authors' earlier work.
ad hoc to paper The central form factor is either 1/((1 - t1/Lambda_0^2)(1 - t2/Lambda_0^2)) with Lambda_0^2 = 0.5 GeV^2 or exp((t1 + t2)/Lambda_E^2) with Lambda_E = 0.8 or 1.0 GeV (Eqs. 2.20 and 2.21).
The functional form is chosen for convenience and the cutoff is fitted to WA102 data; the central predictions depend on this choice.
standard math Lorentz invariance and parity conservation force the scalar-pomeron vertex Gamma(PS PS -> M) = 0 (Eqs. 2.2 to 2.6).
This is the basis for the no-go statement against a scalar pomeron; it does not depend on fitted parameters.

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Cite this review

Pith. "Pith review of Central exclusive production of $\eta$ and $\eta'$ mesons in diffractive proton-proton collisions at the LHC within the tensor-pomeron approach." pith.science (2026). https://pith.science/paper/WCJ7RURX

@misc{pith2026250604846,
  author       = {Pith},
  title        = {Pith review of: Central exclusive production of $\eta$ and $\eta'$ mesons in diffractive proton-proton collisions at the LHC within the tensor-pomeron approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WCJ7RURX}},
  note         = {Machine review of arXiv:2506.04846}
}

abstract

We present a study of the central exclusive production (CEP) of $\eta$ and $\eta'(958)$ mesons in diffractive proton-proton collisions at high energies. The amplitudes, including pomeron and $f_{2 \mathbb{R}}$ reggeon exchanges, are calculated within the tensor-pomeron model. Absorption effects are also taken into account at the amplitude level. We fit some undetermined model parameters (coupling constants and cutoff parameters in form factors) to the WA102 experimental data and then make predictions for the LHC energy $\sqrt{s} = 13$ TeV. Both, total cross sections and several differential distributions are presented. For $pp \to pp \eta$, we find an upper limit for the total cross section of 2.5 $\mu$b for pseudorapidity of the $\eta$ meson $|\eta_{M}| < 1$ and 5.6 $\mu$b for $2 < \eta_{M} < 5$. For $pp \to pp \eta'$, we predict the cross section to be in the range of 0.3-0.7 $\mu$b for pseudorapidity of the $\eta'$ meson $|\eta_{M}| < 1$ and 0.9-2.1 $\mu$b for $2 < \eta_{M} < 5$. This opens the possibility to study diffractive production of pseudoscalar mesons in experiments at the LHC. We discuss if there are arguments from SU(3)-flavor symmetry which would forbid pomeron-pomeron fusion giving an $\eta$ meson. In our opinion such arguments do not exist. We also consider CEP of the pseudoscalars $\eta$ and $\eta'(958)$ and the pseudovector meson $f_{1}(1285)$ in diffractive proton-proton collisions in a theory with a scalar pomeron. We show that none of these particles can be produced in this way in the scalar-pomeron theory. Thus, experimental observation of any of these particles in the above CEP processes at the LHC would give striking evidence against a scalar character of the pomeron.

Figures

Figures reproduced from arXiv: 2506.04846 by the authors.

**Figure 2.** FIG. 2. Fits to the WA102 data [3] for [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. Fits to the WA102 data [3] for [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. The two-dimensional distributions in ( [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5. The differential cross sections for the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. The distributions in rapidity (y [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: FIG. 7. The differential cross sections for the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: FIG. 8. The distributions in ( [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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(2.10) CEP of an axial-vector meson ~M is not possible with a scalar pomeron

= −P µ νΓ ν (PSPS→~M)(q1, q2) , (2.9) which implies A(q2 1, q2 2, k2) = B(q2 1, q2 2, k2) = 0 . (2.10) CEP of an axial-vector meson ~M is not possible with a scalar pomeron. A prominent axial-vec tor meson is the f1(1285). In our paper we treat the reactions (2.1) in the tensor-pomer on approach as introduced in [2]. The pomeron ( P) with the charge conju...

work page

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, (2.17) iΓ ′(PP M) µν,κλ (q1, q2) |bare = i g′ PP M 2M0 ( gµκενλρσ + gνκεµλρσ + gµλενκρσ + gνλεµκρσ ) (q1 − q2)ρ(q1 + q2)σ , (2.18) iΓ ′′(PP M) µν,κλ (q1, q2) |bare = i g′′ PP M M3 0 { ενλρσ [ q1κq2µ − (q1 · q2)gµκ ] + εµλρσ [q1κq2ν − (q1 · q2)gνκ] +ενκρσ [ q1λq2µ − (q1 · q2)gµλ ] + εµκρσ [q1λq2ν − (q1 · q2)gνλ] } (q1 − q2)ρ(q1 + q2)σ . (2.19) In (2.18) ...

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