Target-Mass Corrections in the OPE Sum-Rule Approach to Quarkonium-Nucleon Interactions with Global-Fit PDFs: an $x$-Resolved Analysis

Arkadiy I. Syamtomov

arxiv: 2604.20217 · v3 · pith:YA5DPU76new · submitted 2026-04-22 · ✦ hep-ph

Target-Mass Corrections in the OPE Sum-Rule Approach to Quarkonium-Nucleon Interactions with Global-Fit PDFs: an x-Resolved Analysis

Arkadiy I. Syamtomov This is my paper

Pith reviewed 2026-05-10 00:38 UTC · model grok-4.3

classification ✦ hep-ph

keywords target-mass correctionsOPE sum rulesquarkonium-nucleon interactionsglobal-fit PDFsMellin momentsx-resolved analysispartonic convolutiongluon distribution

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The pith

Target-mass corrections in quarkonium-nucleon OPE sum rules depend on both kinematic weights and PDF x-support redistribution

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

This paper aims to show that the size of target-mass corrections in the operator product expansion sum-rule method for quarkonium-nucleon scattering is influenced not just by a standard kinematic factor but also by how different modern global-fit parton distribution functions place the weight of their Mellin moments at small, medium, and large x. Through a detailed breakdown of the moment densities by momentum fraction regions, it traces how the gluon distribution contributes to each sum rule and how mass effects suppress those contributions differently for each PDF set. The cross section is then obtained straight from the partonic convolution instead of relying on a two-parameter threshold model that can produce unrealistic behavior with current PDFs. Readers interested in high-energy QCD phenomenology would value this because it makes the connection from PDF data to physical cross-section predictions more explicit and reliable.

Core claim

We revisit the operator-product-expansion sum-rule approach to inelastic quarkonium-nucleon interactions using global-fit parton distribution functions ABMP16, MSHT20, CT18 and NNPDF4.0. In contrast to previous work, we trace the full chain from the gluon distribution g(x,Q) through the Mellin moments A_n(Q) and modified sum rules to the cross section σ_ΦN(s). An x-resolved analysis of the weighted moment densities and their decomposition into different x-regions demonstrates that the magnitude of the target-mass correction effect is controlled not only by the universal kinematic weight in the modified sum rules but also by the way in which a given global-fit PDF redistributes the support of

What carries the argument

The x-resolved decomposition of weighted moment densities into contributions from different x regions in the gluon PDF, which determines how target mass suppresses each moment, together with the direct partonic convolution integral used to obtain the cross section.

Load-bearing premise

The operator-product-expansion sum-rule framework correctly captures the physics of inelastic quarkonium-nucleon interactions and the global-fit PDFs accurately describe the gluon distribution in the relevant kinematic regions.

What would settle it

A calculation showing that the target-mass correction magnitude remains identical across all PDFs despite their different redistributions of moment support in the small-, intermediate- and large-x regions would falsify the claim that PDF redistribution controls the TMC effect size.

Figures

Figures reproduced from arXiv: 2604.20217 by Arkadiy I. Syamtomov.

**Figure 1.** Figure 1: Normalised log-x moment density ρn(x; Q) = x n−2 xg(x, Q)/An(Q) for n = 2, 4, 6, 8 at Q = 10 GeV, shown for ABMP16, MSHT20, CT18 and NNPDF4.0. This quantity is the integrand of An per dln x and integrates to unity, making it the natural indicator of where each Mellin moment receives its contribution. The support shifts systematically from small x (n = 2, blue) to large x (n = 8, red) as n increases, reflec… view at source ↗

**Figure 2.** Figure 2: Local target-mass correction factor rn(x) = Tn(x) for representative moments n = 2, 4, 6, 8 with τ ≈ 8.59. Tn(x) is the 3F2 hypergeometric function of Eq. (12), arising from the trace corrections with explicit n-dependent Wilson coefficients. The suppression is weak at small x, grows rapidly towards large x, and is stronger for larger n. The n = 2 curve is shown for context only; A2 is excluded from the su… view at source ↗

**Figure 3.** Figure 3: Partial TMC ratios R (k) n (Q) = Ae(k) n (Q)/A(k) n (Q) for the four x-windows [10−4 , 0.01], [0.01, 0.1], [0.1, 0.5], [0.5, 0.99] at Q = 10 GeV and for the PDF sets ABMP16, MSHT20, CT18 and NNPDF4.0. The strongest suppression is generated in the largest-x interval, while the smallest-x window remains only weakly affected. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** Figure 4: Inelastic quarkonium–nucleon cross sections σΦN ( √ s ) computed via the direct OPE convolution (22) for ABMP16, MSHT20, CT18 and NNPDF4.0, shown with (dashed, TMC kernel (25)) and without (solid) target-mass corrections, normalised to the respective no-TMC peak. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

**Figure 5.** Figure 5: Energy-dependent target-mass correction ratio RTMC( √ s ; PDF) = σ TMC ΦN /σ(0) ΦN for ABMP16, MSHT20, CT18 and NNPDF4.0, computed as the ratio of the same truncated dn series with and without the Tn(x) weights. Near threshold RTMC ≈ 0.60–0.63; the ratio approaches unity at high energies where the integration region is dominated by small x and Tn(x) → 1. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗

**Figure 6.** Figure 6: Cross-section ratios between global-fit PDF families (with TMC, Q = 10 GeV), with CT18 taken as the reference set. The cross sections are computed via the TMC convolution (24). These ratios isolate the spread induced by different gluon inputs after full propagation through the OPE convolution integral. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗

read the original abstract

We present a controlled $x$-resolved numerical diagnosis of target-mass corrections (TMC) in the operator-product-expansion sum-rule approach to inelastic quarkonium--nucleon interactions, using global-fit gluon PDFs ABMP16, MSHT20, CT18 and NNPDF4.0. Applying the known hypergeometric TMC weight to native LHAPDF grids, we decompose the Mellin-moment suppression into finite $x$-windows and identify which parts of the gluon distribution drive the near-threshold cross-section reduction. PDF uncertainties are propagated to the integrated ratios $R_n$ and to the energy-dependent ratio $R_{\rm TMC}(\sqrt{s}\,)$; the factorisation-scale dependence and the sensitivity to the binding-energy parameter $\epsilon_0$ are quantified explicitly. The cross section is computed via direct partonic convolution, bypassing the finite-moment threshold ansatz that produces unphysical exponents ($a\approx 8$-$11$ without and $28$-$32$ with TMC weights) with modern small-$x$-singular gluons.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a careful technical update that clarifies PDF-driven variations in target-mass corrections for quarkonium sum rules and replaces the threshold ansatz with direct convolution, but the link between the two routes to the cross section needs explicit checking.

read the letter

The paper's core contribution is an x-resolved breakdown of how modern global PDFs redistribute moment support across small, intermediate, and large x, which in turn sets the size of target-mass corrections in the modified OPE sum rules for inelastic quarkonium-nucleon scattering. It also computes the cross section directly from partonic convolution instead of the two-parameter threshold form that becomes unphysical with current fits. That shift is practical and removes one source of arbitrariness. The use of four recent PDF sets (ABMP16, MSHT20, CT18, NNPDF4.0) lets the reader see the spread coming from the input distributions rather than from the sum-rule machinery itself. The decomposition into weighted moment densities is transparent and shows which x regions dominate each Mellin moment and how finite target mass suppresses them differently. This is the kind of incremental clarification that helps when people actually run these calculations for phenomenology. The chain from g(x,Q) through A_n(Q) to the sum rules is laid out without unnecessary layers. On the downside, the stress-test concern lands: the paper does not appear to demonstrate numerical equivalence between the direct convolution result and the cross section one would extract from the hadronic spectral density of the same modified sum rules. If those two numbers differ for identical A_n(Q), then the TMC effect shown on the sum-rule side does not automatically control the reported sigma. The validity of the OPE sum-rule framework for inelastic processes is taken as given, which is reasonable for this community but still carries the usual uncertainties from higher twists and continuum modeling. No direct comparison to data is presented in the abstract, so the practical improvement remains internal to the method. This work is for readers already using or extending quarkonium-nucleon sum rules, especially those who need to update older results with current PDFs. It is not aimed at a broad audience. The technical steps are reproducible enough that a serious referee could check the numerics and the consistency between the two cross-section routes. I would send it to peer review rather than desk-reject.

Referee Report

1 major / 1 minor

Summary. The manuscript revisits the operator-product-expansion sum-rule approach to inelastic quarkonium-nucleon interactions using global-fit PDFs (ABMP16, MSHT20, CT18, NNPDF4.0). It performs an x-resolved analysis of target-mass corrections by decomposing weighted moment densities into small-, intermediate- and large-x regions, showing that TMC magnitude is controlled by both the universal kinematic weight in the modified sum rules and the PDF-dependent redistribution of moment support. The cross section σ_ΦN(s) is computed directly via the partonic convolution integral, bypassing the two-parameter threshold ansatz.

Significance. If the results hold, the work provides a transparent reanalysis clarifying the role of modern PDF information in σ_ΦN(s) predictions. The x-resolved decomposition of moments and use of multiple global-fit PDFs are strengths, as is the direct convolution method that avoids unphysical exponents in the threshold ansatz. These elements could improve reliability of sum-rule predictions for quarkonium-nucleon interactions.

major comments (1)

[Cross-section computation section] The central claim states that TMC effects demonstrated via modified sum rules and x-resolved moment analysis control the reported cross sections. However, σ_ΦN(s) is obtained by direct partonic convolution rather than extraction from the hadronic spectral density of the modified OPE sum rules for the same A_n(Q). The manuscript does not demonstrate numerical consistency between these two routes to the cross section (e.g., via benchmark comparisons for fixed moments), which is load-bearing for linking the TMC analysis to the final predictions.

minor comments (1)

The abstract notes that the threshold ansatz develops unphysically large exponents with modern PDFs; a short quantitative illustration of this issue (e.g., exponent values for one of the PDFs) would help readers appreciate the motivation for the direct-convolution approach.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment on the cross-section computation. We address the point below and will revise the manuscript to strengthen the explicit link between the TMC analysis and the reported cross sections.

read point-by-point responses

Referee: [Cross-section computation section] The central claim states that TMC effects demonstrated via modified sum rules and x-resolved moment analysis control the reported cross sections. However, σ_ΦN(s) is obtained by direct partonic convolution rather than extraction from the hadronic spectral density of the modified OPE sum rules for the same A_n(Q). The manuscript does not demonstrate numerical consistency between these two routes to the cross section (e.g., via benchmark comparisons for fixed moments), which is load-bearing for linking the TMC analysis to the final predictions.

Authors: We agree that the manuscript would benefit from an explicit demonstration of consistency between the two routes to σ_ΦN(s). The TMC analysis is performed on the Mellin moments A_n(Q) within the modified OPE sum rules, with the x-resolved decomposition showing how kinematic weights and PDF support in different x-regions control the suppression. The cross section itself is evaluated via direct partonic convolution to avoid the unphysical exponents that arise in the two-parameter threshold ansatz when modern global-fit PDFs are used. Because the convolution integral is the fundamental definition of σ_ΦN(s) from the gluon distribution, the TMC-induced changes in the moments translate directly into changes in the predicted cross section. To make this link fully transparent and address the referee's concern, we will add a benchmark subsection in the revised manuscript. For a representative set of fixed moments A_n(Q), we will compare the cross section obtained from direct convolution against the value extracted from the hadronic spectral density of the modified sum rules (using the threshold ansatz for this limited benchmark only). This will quantify the numerical agreement and confirm that the TMC effects identified in the moment analysis propagate consistently to the final predictions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external PDFs and independent convolution

full rationale

The paper takes external global-fit PDFs (ABMP16, MSHT20, CT18, NNPDF4.0) as given inputs, constructs Mellin moments A_n(Q) from g(x,Q), applies modified sum rules for TMC, performs x-resolved decomposition of moment support, and computes σ_ΦN(s) via direct partonic convolution (explicitly bypassing the threshold ansatz). No equation or step reduces by construction to the paper's own fitted outputs, self-citations, or redefinitions. The TMC magnitude claim follows from the kinematic weights and PDF redistribution, both traceable to the external inputs without circular closure. This matches the default case of a self-contained analysis against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on standard QCD tools and external PDF fits without introducing new free parameters or entities in this work.

axioms (2)

domain assumption Validity of the operator product expansion for the relevant correlation functions in QCD.
The sum-rule approach relies on this standard assumption in QCD phenomenology.
domain assumption Accuracy of global-fit PDFs for the gluon distribution at the scales used.
The analysis depends on the reliability of these fitted distributions.

pith-pipeline@v0.9.0 · 5570 in / 1466 out tokens · 29662 ms · 2026-05-10T00:38:55.463711+00:00 · methodology

discussion (0)

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