arxiv: 2605.12373 · v1 · submitted 2026-05-12 · ✦ hep-ph

Recognition: no theorem link

Quasi Parton Distribution Functions in Covariant Quark Models

Fatma Aslan , Asli Tandogan , Peter Schweitzer

Authors on Pith no claims yet

Pith reviewed 2026-05-13 03:35 UTC · model grok-4.3

classification ✦ hep-ph

keywords quasi parton distribution functionscovariant quark modelssum rulesconvergenceenergy-momentum tensorWandzura-Wilczek approximationparton distribution functions

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The pith

In gauge-free quark models, quasi-PDFs converge to standard PDFs and obey sum rules for both gamma0 and gamma3 choices.

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

The paper proves that in a broad class of quark models without gauge fields, the unpolarized quasi parton distribution functions for quarks and antiquarks converge to the usual PDFs as the hadron velocity approaches the speed of light. It shows this convergence holds along with the expected sum rules, for both choices of the Dirac matrix gamma0 and gamma3. Using the covariant parton model as an example, it derives exact expressions for the behavior at small momentum fraction and links the results to an approximation for the energy-momentum tensor form factor. A sympathetic reader would care because these proofs provide a controlled setting to understand how lattice-accessible quasi-PDFs relate to the parton distributions that describe high-energy scattering.

Core claim

In the absence of gauge degrees of freedom, quasi parton distribution functions defined in terms of QCD fields at spacelike separations in matrix elements of hadrons moving with velocity v converge to the standard parton distribution functions when v approaches the speed of light. General proofs establish that the unpolarized quark and antiquark QPDFs satisfy sum rules for both choices of γ⁰ and γ³. In the covariant parton model, analytical results are obtained for the small-x_v behavior of the QPDFs and for the energy-momentum tensor form factor c̄^q(t) at zero momentum transfer, which correspond to a Wandzura-Wilczek-type approximation.

What carries the argument

Quasi parton distribution functions (QPDFs) in covariant quark models without gauge degrees of freedom; they allow general proofs of convergence to PDFs and satisfaction of sum rules independent of the Dirac structure choice.

Load-bearing premise

The models completely lack gauge degrees of freedom, so that if gauge-field dynamics in QCD alter convergence or sum-rule properties, the general proofs would not apply.

What would settle it

A direct computation of QPDF sum rules in a lattice QCD simulation or in a model that includes dynamical gauge fields, checking if they match the expected values independent of velocity.

Figures

Figures reproduced from arXiv: 2605.12373 by Asli Tandogan, Fatma Aslan, Peter Schweitzer.

**Figure 1.** Figure 1: b. In our case (which is realistic for that matter) at lower velocities the cancellation in the negative xv-region is so strong that it reverses the sign of the QPDF. Whether or not in the case of Γ = γ 0 the leaking of xvD q +(xv, γ0 , v) into the negative xv-region is strong enough for xvD q +(xv, γ0 , v) to dominate over xvD q −(xv, γ0 , v) and reverse the sign of the total QPDF, this depends on the inp… view at source ↗

**Figure 2.** Figure 2: FIG. 2: The QPDFs [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: The QPDFs [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: The QPDFs [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

**Figure 5.** Figure 5: The pertinent asymptotics from respectively Eqs. (64, 66, 67) [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: (a) Unpolarized isovector QPDF ( [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

read the original abstract

Quasi parton distribution functions (QPDFs) are defined in terms of QCD fields at spacelike separations evaluated in matrix elements of hadrons moving with velocity $v$. These objects can be studied in lattice QCD. In the limit when $v$ approaches the speed of light, QPDFs converge to PDFs. It is insightful to study QPDFs and their convergence in models. In this work, we first study the QPDFs in a broad class of quark models characterized by one common feature, namely the absence of gauge degrees of freedom. We provide general proofs for the convergence and sum rules of the unpolarized quark and antiquark QPDFs for both choices $\gamma^0$ and $\gamma^3$. We choose the Covariant Parton Model (CPM) as an illustration. We derive analytical results for the small-$x_v$ behavior of QPDFs and the energy-momentum tensor form factor $\bar{c}^q(t)$ at zero momentum transfer. These results are of interest as they correspond to a Wandzura-Wilczek-type approximation.

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 paper proves convergence and sum rules for QPDFs in gauge-free quark models and gives exact small-x results in the Covariant Parton Model.

read the letter

The main thing to know is that the authors prove convergence of quasi-PDFs to ordinary PDFs plus the associated sum rules for unpolarized quarks and antiquarks, and they do it for both the gamma-zero and gamma-three definitions. They then work out closed-form small-x_v expressions and the zero-momentum energy-momentum tensor form factor inside the Covariant Parton Model, where these quantities line up with a Wandzura-Wilczek-type relation that emerges automatically rather than being imposed by hand. That analytic control is the useful part. Having exact expressions across an entire class of models without gauge fields supplies clean reference points that lattice groups can use when they check their own extrapolations to the light-cone limit. The proofs rest only on the Lorentz properties and the model definitions, so they are internally consistent within the stated framework. The authors are explicit that they are not claiming the same behavior holds once gluons are restored, which keeps the scope honest. The obvious limitation is that the absence of gauge degrees of freedom is built into the whole class of models, so any gluon-driven corrections that matter in real QCD are simply not present. That is not a flaw in the paper, just a boundary on what the results can tell us. The work is aimed at people who already run lattice calculations of quasi-PDFs or who build simple quark models to test ideas about nucleon structure. The analytic benchmarks and the general proofs are the pieces that could actually get referenced. I would send it to peer review. The claims are narrow enough and the derivations are presented as exact within the model class, so referees can check the algebra and decide whether the CPM results add enough new material to justify publication.

Referee Report

0 major / 3 minor

Summary. The paper claims to establish general proofs of convergence to PDFs and associated sum rules for unpolarized quark and antiquark QPDFs within a broad class of covariant quark models that contain no gauge degrees of freedom; the proofs are stated to hold for both the γ⁰ and γ³ choices of the Dirac structure. The Covariant Parton Model is used as a concrete illustration, where closed-form expressions are derived for the small-x_v behavior of the QPDFs and for the energy-momentum tensor form factor c̄^q(t) evaluated at zero momentum transfer; these expressions are identified with a Wandzura-Wilczek-type approximation.

Significance. Within the stated model class the results supply an exactly solvable laboratory in which the approach of QPDFs to light-cone PDFs, the validity of sum rules, and the emergence of Wandzura-Wilczek relations can be examined analytically. Such controlled benchmarks are useful for interpreting lattice-QCD extractions of QPDFs and for testing numerical methods that aim to recover PDFs from finite-velocity matrix elements.

minor comments (3)

§2: the precise definition of the velocity parameter v and the relation between the quasi-momentum fraction x_v and the light-cone fraction x should be stated explicitly before the general proofs are presented, to avoid ambiguity when the two γ choices are compared.
§4 (CPM illustration): the analytic small-x_v expressions are given, but the range of validity (e.g., the regime of x_v where higher-order corrections in the model remain negligible) is not quantified; a brief statement or plot would strengthen the claim that the results correspond to a Wandzura-Wilczek approximation.
References: the discussion of the Wandzura-Wilczek approximation would benefit from citing the original Wandzura-Wilczek paper and at least one recent lattice-QCD study that employs the same approximation for comparison.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the accurate summary of the claims, and the recommendation for minor revision. The report correctly identifies the scope of the general proofs for QPDF convergence and sum rules in gauge-free covariant quark models, as well as the analytic results obtained in the Covariant Parton Model. Since the report contains no specific major comments or criticisms, we provide no point-by-point rebuttals below. We will implement minor editorial improvements in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained within model definitions

full rationale

The paper defines a broad class of quark models by the explicit absence of gauge degrees of freedom and derives general proofs of convergence and sum rules for unpolarized QPDFs (both quark and antiquark, for γ⁰ and γ³ choices) directly from Lorentz invariance and the model kinematics. The CPM is introduced only as an illustration, with analytical small-x_v behavior and the EMT form factor c̄^q(0) obtained as exact consequences inside that model; the Wandzura-Wilczek-type correspondence is stated as an emergent outcome rather than an input ansatz. No load-bearing step reduces to a fitted parameter renamed as prediction, a self-citation chain, or a self-definitional loop. The central claims remain internal to the stated assumptions and do not rely on external data fits or prior author results for their validity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the definition of QPDFs at spacelike separations and on the restriction to quark models that contain no gauge fields; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Quark models are defined by the complete absence of gauge degrees of freedom
The paper states it studies a broad class of models characterized by this single common feature.

pith-pipeline@v0.9.0 · 5491 in / 1320 out tokens · 107866 ms · 2026-05-13T03:35:55.293793+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

116 extracted references · 116 canonical work pages · 1 internal anchor

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it is better to get rid of it

=N q which coincides with the above-encountered integral expressions and completes the proof. 6 F. Momentum sum rule Next we investigate the second Mellin moment sum rule. For PDFs it corresponds to the momentum sum rule. For QPDFs there is no direct correspondence to momentum carried by partons, but we shall nevertheless loosely refer to this sum rule by...

work page
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= 2P 3 Z ∞ −∞ dxv eiνxv Dq(xv, γ3, v) (41) which can be inverted and simplified usingz 3 = ν P 3 as follows v Dq(xv, γ0, v) = Z ∞ −∞ dν 2π e−iνxv Mq(ν,−z 2 3) Dq(xv, γ3, v) = Z ∞ −∞ dν 2π e−iνxv Mq(ν,−z 2

work page
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+ M2 P 2 3 νJ q(ν,−z 2 3) (42) where we have included for comparison the result forD q(xv, γ0, v) discussed already in Sec. II G. SinceD q(xv, γ3, v) contains the power-suppressed term M 2 P 2 3 νJ q(ν,−z 2 3), the choice Γ =γ 0 appears preferable because inD q(xv, γ0, v) this power-suppressed term is absent to begin with [34, 35]. To investigate this que...

work page
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interaction dependent

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= 2 Z d4k eiν k3 P 3 Aq 2 + k3 P 3 Aq 3 ,(45c) which is rather remarkable. Inserting the results from Eq. (45) in Eq. (42) and taking theν-integrals we obtain Dq(xv, γ0, v) = 2 Z d4k δ xv − k3 P 3 Aq 2 + xv + P·k P 2 3 1 + M 2 P 2 3 Aq 3 q 1 + M 2 P 2 3 (46a) Dq(xv, γ3, v) = 2 Z d4k δ xv − k3 P 3 Aq 2 +x v Aq 3 (46b) The expressions in Eq. (46) coincide w...

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