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arxiv: 2605.15853 · v1 · pith:ULEMJ5KEnew · submitted 2026-05-15 · ✦ hep-ph · hep-ex· hep-th

Numerical Study of MRW-Type Unintegrated Double Parton Distribution Functions from Non-Factorized DPDFs

R. Kord Valeshabadi , S. Rezaie , K. Azizi This is my paper

Pith reviewed 2026-05-20 17:09 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-th
keywords double parton distribution functionsunintegrated distributionsMRW prescriptionnon-factorized DPDFsDGLAP evolutiondouble parton scatteringtransverse momentum dependence
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The pith

MRW-type prescriptions allow direct construction of unintegrated double parton distributions from non-factorized collinear DPDFs without piecewise splitting.

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

This paper numerically constructs unintegrated double parton distribution functions by adding transverse momentum dependence to non-factorized collinear DPDFs using MRW-inspired models. It employs the GS09 parametrization evolved via a double DGLAP framework validated by the momentum sum rule. The study examines three variants: double modified KMRW, double virtuality-ordered MRW, and a normalization-matched version of the latter. These can be applied to the entire DPDF including non-homogeneous terms, unlike conventional methods that require splitting into pieces. The results show preservation of normalization in one model, adjustments in others, and strong dependence on parton channels for the effects of non-factorized correlations.

Core claim

The authors demonstrate that MRW-inspired prescriptions for unintegrated double parton distributions can be applied directly to the full collinear double parton distribution functions from the GS09 parametrization, including their non-homogeneous components. This avoids the piecewise treatment needed in the standard leading-order MRW construction. Through numerical evolution to unequal scales and analysis of normalization, transverse momentum dependence, and flavor sensitivity, they find that the double modified KMRW model preserves normalization while the virtuality-ordered model requires matching to do so, with non-factorized effects manifesting differently across models and being most 1

What carries the argument

MRW-type prescriptions (DMKMRW, DVO-MRW, and matched DVO-MRW) for generating transverse-momentum dependence from non-factorized collinear DPDFs

If this is right

  • The DMKMRW prescription preserves the normalization of the underlying DPDF by construction.
  • The DVO-MRW model exhibits nontrivial normalization deviations that are eliminated in its normalization-matched version.
  • Non-factorized longitudinal correlations affect the transverse momentum shape in virtuality-ordered models but enter primarily through the longitudinal part in DMKMRW.
  • Largest deviations from factorized expectations appear near the double parton kinematic boundary and in channels with valence-number or quark-antiquark correlations.

Where Pith is reading between the lines

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

  • Such direct constructions could simplify calculations of double parton scattering cross sections at colliders by incorporating correlations more seamlessly.
  • The channel-dependent sensitivity highlights the need to model specific parton flavor combinations carefully in multi-parton interaction studies.
  • Extending this numerical approach to other parametrizations or evolution orders might reveal additional impacts on transverse momentum distributions.

Load-bearing premise

The GS09 parametrization provides an accurate enough description of non-factorized longitudinal correlations in double parton distributions to be directly extended to transverse momentum dependence using the MRW prescriptions.

What would settle it

A measurement of double parton scattering observables at the LHC showing significant normalization mismatch with the DVO-MRW predictions even after matching, or a failure of the evolved distributions to satisfy the momentum sum rule.

Figures

Figures reproduced from arXiv: 2605.15853 by K. Azizi, R. Kord Valeshabadi, S. Rezaie.

Figure 1
Figure 1. Figure 1: FIG. 1: Validation of the unequal scale momentum sum rule in [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Normalization test for the UDPDFs. The plotted quantity is the ratio of the transverse momentum [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Representative [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Contour maps of the non-factorized to factorized ratio for the virtuality ordered MRW construction, [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Contour maps of the non-factorized to factorized ratio for the matched virtuality ordered MRW [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Contour maps of the ratio [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
read the original abstract

Double parton scattering provides a sensitive probe of multi parton correlations inside hadrons. In this work we present a numerical study of unintegrated double parton distribution functions constructed from non-factorized collinear DPDFs. As input we use the GS09 DPDFs and evolve them to unequal scales with a numerical double DGLAP evolution framework, which is validated through the corresponding momentum sum rule. We investigate MRW-inspired prescriptions for generating transverse momentum dependence in the double parton case. In particular, we study the double modified KMRW approach (DMKMRW), the double virtuality ordered MRW (DVO-MRW) model, and a normalization-matched version of the latter. These prescriptions can be applied directly to the full collinear DPDF, including its non-homogeneous component, and avoid the piecewise treatment required in the conventional LO-MRW construction. We analyze the normalization, transverse momentum dependence, flavor dependence, and sensitivity to longitudinal DPDF correlations of the resulting distributions. The DMKMRW model is normalization preserving by construction, while the DVO-MRW model shows nontrivial normalization deviations that are removed in the matched version. Non-factorized effects are strongly channel dependent: in DMKMRW they enter mainly through longitudinal DPDF correlations, whereas in the virtuality ordered models they also modify the transverse momentum shape. The largest deviations occur near the double parton kinematic boundary and in channels affected by valence-number and quark-antiquark correlations.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a numerical study of unintegrated double parton distribution functions (uDPDFs) constructed from the non-factorized GS09 collinear DPDFs. The collinear DPDFs are evolved to unequal scales using a numerical double DGLAP framework validated by the momentum sum rule. Three MRW-inspired prescriptions (DMKMRW, DVO-MRW, and a normalization-matched DVO-MRW variant) are applied to generate transverse-momentum dependence. The central claim is that these prescriptions can be applied directly to the full collinear DPDF, including its non-homogeneous component, thereby avoiding the piecewise treatment required in conventional LO-MRW constructions. The study examines normalization, kT spectra, flavor dependence, and sensitivity to longitudinal correlations, reporting channel-dependent non-factorized effects that are largest near the double-parton kinematic boundary.

Significance. If the direct application of the MRW factors to the non-homogeneous term is justified, the work supplies useful numerical benchmarks for modeling double parton scattering, particularly the interplay between longitudinal correlations and transverse-momentum distributions. The momentum-sum-rule validation of the double DGLAP evolution is a positive technical check, and the explicit comparison of normalization-preserving versus virtuality-ordered prescriptions quantifies model differences in a reproducible way.

major comments (2)
  1. [momentum sum rule validation and numerical results on non-homogeneous component] The momentum-sum-rule validation is performed only at the collinear level and does not isolate whether the unintegrated non-homogeneous component recovers the input GS09 distribution after integration over transverse momentum. This check is required to confirm that the MRW-style Sudakov or virtuality-ordering factor acts consistently on the inhomogeneous term, whose evolution kernel differs from the homogeneous one.
  2. [analysis of transverse momentum dependence and non-factorized effects] The claim that the prescriptions can be applied directly to the full DPDF (including non-homogeneous part) without additional scale-dependent adjustment rests on the assumption that the transverse-momentum generating factor is uniform regardless of generation scale. The largest reported deviations occur precisely near the double-parton kinematic boundary, where this assumption is most likely to distort the kT spectrum; no explicit test isolating the non-homogeneous contribution is presented.
minor comments (2)
  1. [abstract and numerical results] The abstract states that the DMKMRW model is normalization preserving by construction while DVO-MRW shows deviations removed by matching, but no quantitative error estimates or cutoff sensitivity studies are provided for the reported normalization deviations.
  2. [prescription definitions] Notation for the three prescriptions (DMKMRW, DVO-MRW, matched version) should be defined once in a dedicated subsection with explicit formulas for the Sudakov factors and matching procedure to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions we plan to make.

read point-by-point responses

  1. Referee: The momentum-sum-rule validation is performed only at the collinear level and does not isolate whether the unintegrated non-homogeneous component recovers the input GS09 distribution after integration over transverse momentum. This check is required to confirm that the MRW-style Sudakov or virtuality-ordering factor acts consistently on the inhomogeneous term, whose evolution kernel differs from the homogeneous one.

    Authors: We agree with the referee that validating the integration of the unintegrated distributions back to the collinear level, specifically separating the non-homogeneous component, would strengthen the presentation. In our work, the double DGLAP evolution framework is validated using the momentum sum rule at the collinear level for the full DPDF. The MRW prescriptions are then applied to this evolved full DPDF. Since the MRW construction is such that the transverse momentum integration is designed to recover the collinear input by construction (via the Sudakov or ordering factors), we expect consistency. However, to explicitly address the concern for the inhomogeneous term, we will include in the revised manuscript a numerical demonstration that the kT-integrated uDPDFs reproduce the input GS09 DPDFs for both the homogeneous and non-homogeneous contributions. This will be added as an additional validation step. revision: yes

  2. Referee: The claim that the prescriptions can be applied directly to the full DPDF (including non-homogeneous part) without additional scale-dependent adjustment rests on the assumption that the transverse-momentum generating factor is uniform regardless of generation scale. The largest reported deviations occur precisely near the double-parton kinematic boundary, where this assumption is most likely to distort the kT spectrum; no explicit test isolating the non-homogeneous contribution is presented.

    Authors: The application of the MRW-type factors is performed on the collinear DPDF after it has been evolved to the final scales using the double DGLAP evolution. This evolution already accounts for the different kernels acting on the homogeneous and non-homogeneous terms up to the final scales. The transverse momentum dependence is then introduced via the MRW-inspired models at these final scales, consistent with how such models are applied in the single-parton case. We do not introduce additional scale-dependent adjustments because the prescriptions are scale-independent in their application once the collinear input is fixed. Regarding the deviations near the kinematic boundary, these arise from the longitudinal correlations in the GS09 DPDFs themselves. While we have not presented an explicit isolation of the non-homogeneous contribution to the kT spectra in the current version, the results for the full DPDF already incorporate all effects. In the revision, we will add a short discussion clarifying this point and, if space permits, a supplementary figure showing the kT spectra with and without the non-homogeneous term to quantify its impact. revision: partial

Circularity Check

0 steps flagged

No circularity: external GS09 input plus published MRW prescriptions produce numerical outputs

full rationale

The derivation chain begins with the external GS09 collinear DPDF parametrization as input, followed by numerical double DGLAP evolution validated against the momentum sum rule at the collinear level. Transverse-momentum dependence is then generated by applying published MRW-inspired prescriptions (DMKMRW, DVO-MRW and matched variant) directly to the full evolved DPDF, including its non-homogeneous term. No quantity is defined in terms of a fitted parameter that is subsequently relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled through prior author work. The central claim that the prescriptions avoid piecewise treatment is a direct numerical statement rather than an equivalence that reduces to the input by construction. The paper therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The study rests on the validity of the GS09 parametrization and the applicability of single-parton MRW constructions to the double-parton case; no new free parameters are introduced beyond those already present in the input DPDFs.

axioms (2)
  • domain assumption The GS09 DPDFs provide a reliable description of non-factorized longitudinal correlations inside the proton.
    Used directly as input for the evolution and MRW constructions.
  • standard math The double DGLAP evolution equations can be solved numerically to unequal scales while preserving the momentum sum rule.
    Invoked to justify the scale evolution step.

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discussion (0)

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Reference graph

Works this paper leans on

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