arxiv: 2510.11809 · v3 · submitted 2025-10-13 · ✦ hep-ph

Modeling the TMD shape function in J/psi electroproduction

Miguel G. Echevarria , Raj Kishore , Samuel F. Romera This is my paper

Pith reviewed 2026-05-18 07:06 UTC · model grok-4.3

classification ✦ hep-ph

keywords TMD factorizationJ/ψ electroproductionhard functionshape functionElectron-Ion Collidertransverse momentumquarkonium productiongluon distribution

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The pith

The next-to-leading order hard function for J/ψ electroproduction is calculated in TMD factorization for the low transverse momentum regime.

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

This paper computes the next-to-leading order hard function for quarkonium electroproduction within transverse-momentum-dependent factorization when the transverse momentum is small. It defines the TMD shape function at the operator level and examines how it convolves with the unpolarized TMD gluon distribution to determine the cross section. Predictions for the unpolarized differential cross section of J/ψ electroproduction are then given for the future Electron-Ion Collider. A sympathetic reader would care because the work supplies a concrete way to connect gluon distributions inside the proton to observable production rates at an upcoming collider facility.

Core claim

The next-to-leading order hard function for quarkonium electroproduction is calculated within the framework of transverse-momentum-dependent factorization in the low-transverse-momentum regime. The structure of the TMD shape function in quarkonium leptoproduction is analyzed through its operator-level definition. Particular attention is given to the convolution of the unpolarized TMD gluon distribution with the TMD shape function, thereby illustrating the latter's phenomenological role. Building on this framework, predictions are provided for the unpolarized differential cross-section of J/ψ electroproduction at the future Electron-Ion Collider in the region of small transverse momentum.

What carries the argument

The TMD shape function, which is defined through an operator matrix element that describes the nonperturbative transition of a heavy quark-antiquark pair into the observed quarkonium and enters the cross section via convolution with the TMD gluon distribution.

If this is right

The unpolarized differential cross-section of J/ψ electroproduction can be predicted at small transverse momentum using the calculated hard function.
The TMD shape function acquires a concrete phenomenological role through its convolution with the TMD gluon distribution.
The operator definition of the shape function provides a starting point for modeling its nonperturbative content in quarkonium leptoproduction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same framework could be applied to other vector quarkonia or to polarized observables to test the generality of the TMD approach.
Future EIC data in the low-pT region might be used to constrain the parameters inside the TMD shape function.
Inclusion of higher-order corrections or explicit power-suppressed terms could extend the kinematic reach of the predictions.

Load-bearing premise

TMD factorization applies reliably in the low-transverse-momentum regime for quarkonium electroproduction, allowing the hard function and shape function to be defined and convoluted with the TMD gluon distribution without significant power corrections.

What would settle it

A measurement of the J/ψ electroproduction differential cross section at the Electron-Ion Collider in the small transverse momentum region that deviates from the calculated prediction by more than the combined theoretical and experimental uncertainties.

read the original abstract

The next-to-leading order hard function for quarkonium electroproduction is calculated within the framework of transverse-momentum-dependent (TMD) factorization in the low-transverse-momentum regime. The structure of the TMD shape function in quarkonium leptoproduction is analyzed through its operator-level definition. Particular attention is given to the convolution of the unpolarized TMD gluon distribution with the TMD shape function, thereby illustrating the latter's phenomenological role. Building on this framework, we provide predictions for the unpolarized differential cross-section of $J/\psi$ electroproduction at the future Electron-Ion Collider in the region of small transverse momentum.

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

The paper delivers a new NLO hard function for TMD J/ψ electroproduction plus EIC predictions, but leaves power corrections unquantified.

read the letter

This paper computes the next-to-leading order hard function for J/ψ electroproduction in TMD factorization and gives predictions for the unpolarized differential cross section at the EIC in the small-pT region. The operator definition and analysis of the TMD shape function are handled directly, and the convolution with the unpolarized TMD gluon distribution is shown explicitly. That part is useful and extends standard TMD methods with the specific perturbative input for this quarkonium process. The NLO calculation supplies concrete technical progress that was missing before. The main soft spot is the reliance on TMD factorization without a numerical bound on power corrections. Multiple scales (charm mass, Q, pT) are present, yet the paper does not supply an explicit estimate or suppression factor for the EIC kinematics used in the predictions. If those corrections are not small, the low-pT results shift. The framework itself is coherent and follows established TMD definitions without obvious circularity or invented parameters. This work is aimed at TMD phenomenologists and people preparing EIC analyses of gluon distributions through quarkonia. Readers who track factorization for heavy quarks will find the operator structure and hard-function details worth checking. The thinking is clear and engages the relevant literature on its own terms. It deserves a serious referee. Send it for review and ask the referees to examine the power-counting estimates and any cross-checks against existing data or limits.

Referee Report

1 major / 2 minor

Summary. The paper calculates the next-to-leading order hard function for quarkonium electroproduction within TMD factorization in the low transverse-momentum regime. It analyzes the structure of the TMD shape function via its operator definition, illustrates the convolution of the unpolarized TMD gluon distribution with this shape function, and provides predictions for the unpolarized differential cross-section of J/ψ electroproduction at the future Electron-Ion Collider in the small-pT region.

Significance. If the TMD factorization framework holds with controlled power corrections, the NLO hard function and explicit shape-function modeling would supply useful theoretical input for gluon TMD studies at the EIC. The work supplies concrete perturbative ingredients and a phenomenological illustration that could be directly compared with future data.

major comments (1)

[Predictions for EIC kinematics] The central predictions for the EIC differential cross section in the low-pT regime rest on the assumption that power-suppressed corrections (involving the scales m_c, Q, and pT) remain small. The manuscript provides no explicit power-counting estimate or numerical bound on these corrections for the kinematics used in the predictions, leaving the reliability of the results unquantified.

minor comments (2)

The abstract states that the NLO hard function is calculated but does not quote its explicit form or key features; adding a brief statement of the result would improve clarity.
Notation for the TMD shape function and its operator definition should be checked for consistency between the analysis section and the phenomenological convolution discussion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for highlighting an important aspect regarding the reliability of our predictions. We address the comment as follows.

read point-by-point responses

Referee: The central predictions for the EIC differential cross section in the low-pT regime rest on the assumption that power-suppressed corrections (involving the scales m_c, Q, and pT) remain small. The manuscript provides no explicit power-counting estimate or numerical bound on these corrections for the kinematics used in the predictions, leaving the reliability of the results unquantified.

Authors: We agree with the referee that an explicit power-counting estimate would strengthen the manuscript by quantifying the applicability of the TMD factorization in the chosen EIC kinematics. Although the primary focus of the paper is the calculation of the NLO hard function and the analysis of the TMD shape function, we recognize the value of this addition for the phenomenological section. In the revised version, we will incorporate a brief discussion of the power corrections, including a power-counting argument and a numerical estimate of their relative size for the kinematics used in our cross-section predictions (e.g., for Q values of several GeV and small p_T). This will be added without altering the main results or conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in TMD factorization derivation or predictions

full rationale

The paper calculates the NLO hard function for quarkonium electroproduction and analyzes the TMD shape function via its explicit operator-level definition, then convolves it with the unpolarized TMD gluon distribution to produce predictions for the EIC cross section. No steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the central results consist of new perturbative input and operator analysis that remain independent of the final numerical predictions. The framework is self-contained against external benchmarks such as standard TMD factorization theorems and does not rename known results or smuggle ansatze through citations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of TMD factorization at low pT and the existence of a well-defined TMD shape function for quarkonium; no free parameters or new entities are explicitly introduced in the abstract.

axioms (1)

domain assumption TMD factorization holds in the low-transverse-momentum regime for quarkonium electroproduction.
Invoked to justify the hard function calculation and convolution with the TMD gluon distribution.

pith-pipeline@v0.9.0 · 5634 in / 1202 out tokens · 30213 ms · 2026-05-18T07:06:58.272567+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

W μν = … convolution of unpolarized TMD gluon distribution G g/N with TMD shape function S [n]→J/ψ (Eq. 2.6); NLO hard function H[n] obtained from one-loop virtual corrections (Eq. 2.9–2.10)
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

TMDShF modeled with Gaussian NP term B_S b_T² and perturbative matching coefficients C[n][m] (Eq. 3.5, 3.7, A.11)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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