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REVIEW 4 major objections 4 minor 72 references

A new analysis shows double parton scattering in pA and AA collisions can probe the spatial separation of partons in the proton, the bound nucleon, and the nucleus.

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2026-08-02 18:48 UTC pith:TD2JHHMW

load-bearing objection A useful phenomenological extension of DPS to pA/AA with testable predictions, but the pPb 'agreement' is weaker than claimed because the widening term was chosen after the fact and the data are lower limits. the 4 major comments →

arxiv 2603.04510 v2 pith:TD2JHHMW submitted 2026-03-04 hep-ph

Momentum fraction and hard scale dependence of double parton scattering in heavy-ion collisions

Joao Vitor C. Lovato , Edgar Huayra , Emmanuel G. de Oliveira This is my paper
classification hep-ph
keywords double parton scatteringheavy-ion collisionseffective cross sectiontransverse parton distributionsnuclear shadowingantishadowingpA collisionsAA collisions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper extends a model of double parton scattering (DPS) from proton–proton to proton–nucleus and nucleus–nucleus collisions, aiming to show that the DPS effective cross section depends on the final state in a way that carries geometric information. It argues that in proton–lead collisions the variation across final states and rapidities—about 50%—is driven by the transverse distance between two partons in a single nucleon, while in lead–lead collisions the variation—about 10%—reflects how nuclear shadowing and antishadowing reshape the transverse parton profile of the nucleus. To describe the available pPb data, the authors hypothesize that partons inside a bound nucleon are more widely separated than in a free proton; with that hypothesis, the model agrees with all five existing data points. A sympathetic reader would care because DPS measurements, which are already being made, could become a direct probe of the transverse spatial structure of hadrons and nuclei, a quantity that is otherwise very hard to access.

Core claim

The paper's central claim is that the effective DPS cross section in heavy-ion collisions is not a universal constant but a function of the chosen final state and rapidity, and that this dependence is governed by the transverse separation of the two partons inside the projectile and target. In pA collisions, the authors decompose the scale factor into a 1x1 term (both nuclear partons from one nucleon) and a 1x2 term (from two nucleons); the 1x1 term, although comparable in size, produces the dominant variation. They show that the observed forward/backward asymmetry in pPb data can only be reproduced if partons in a bound nucleon are more widely separated at small x than in a free proton, imp

What carries the argument

The load-bearing object is a momentum-fraction- and hard-scale-dependent Gaussian profile for the transverse distance r between two partons, with variance B(x, mu). The DPS scale factor Theta is the overlap integral of two such profiles, and it enters the denominator of the effective cross section. For heavy-ion collisions, the paper adds a gamma_A term that widens the profile for bound nucleons at small x, and a nuclear profile built from the gluon nuclear modification factor R_g, normalized without free parameters; in AA the dominant 2x2 term is the convolution of two such nuclear profiles. The work these objects do is to turn the final-state dependence of sigma_eff into a map of the trans

Load-bearing premise

The whole picture rests on the hypothesis, proposed after seeing the data, that partons inside a bound nucleon are more widely separated than free-proton partons, with the size of that spreading (gamma_A = 1 mb, x_A = 5e-3) chosen by hand rather than fitted.

What would settle it

A measurement of the forward and backward double-J/psi or J/psi-D0 effective cross section in pPb collisions at 8.16 TeV that matches the gamma_A = 0 prediction—forward values systematically lower than backward—would falsify the bound-nucleon widening hypothesis; conversely, a precision pp measurement that already reproduces the pPb asymmetry through nuclear PDFs alone would remove the need for the gamma_A term.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the model is correct, the DPS effective cross section in pPb collisions should vary by about 50% between final states and rapidities, and that variation directly encodes the transverse separation of partons in a single nucleon.
  • In PbPb collisions, sigma_eff should vary by about 10%, dominated by the nuclear transverse profile set by shadowing and antishadowing, so DPS becomes a nuclear-structure probe.
  • The five existing pPb data points are consistent with the hypothesis of wider parton separation in bound nucleons, which provides a target for future higher-precision measurements to confirm or exclude.
  • Future measurements of forward and backward rapidities in pA collisions can constrain the strength of the bound-nucleon broadening and separate it from free-proton structure.
  • The model predicts that heavier final states probe larger x and thus see weaker shadowing, which translates into systematically different sigma_eff across D0, J/psi, Upsilon, W, and Z channels.

Where Pith is reading between the lines

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

  • By extension, if the bound-nucleon broadening is real, a similar effect should appear in other small-x two-parton observables—for example, double charm production in pA collisions—where the same gamma_A term would shift rates.
  • The nuclear profile construction suggests an inversion scheme the paper does not spell out: by measuring sigma_eff,AA as a function of final state, one could extract an effective x-dependent transverse radius of the nucleus, effectively using DPS as a tomographic probe.
  • The model's use of the gluon modification factor for all parton species is an approximation that could be tested by comparing channels dominated by quark-initiated versus gluon-initiated processes; a deviation would indicate flavor-dependent transverse nuclear structure, a scenario beyond the current paper.
  • Since the gamma_A parameters are fixed by hand, fitting them to the full pPb dataset would turn the model into a quantitative extraction tool; the reported 0.5–1.5 mb range suggests the current data already provide some discriminating power.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper extends a previous phenomenological model of double parton scattering (DPS) in proton-proton collisions [28] to proton-nucleus (pA) and nucleus-nucleus (AA) collisions. The DPS effective cross section is computed from PYTHIA-determined single-parton-scattering inputs convoluted with a scale factor Θ that encodes transverse parton-parton correlations. For pA, the paper separates the 1x1 and 1x2 contributions and introduces a new hypothesis that partons inside a bound nucleon are more widely separated at small x, implemented by an extra term γ_A H(x_A - x'_CD) ln(x_A/x'_CD) in Eq. (2.7) with γ_A = 1 mb and x_A = 5×10^-3. For the 1x2 and AA 2x2 contributions, a new nuclear transverse profile is proposed in Eq. (2.13), depending on the gluon nuclear modification factor R_g. The model is compared to five pPb data points, all lower limits, and predictions are made for pPb and PbPb final states. The central claim is that the final-state and rapidity dependence of σ_eff in pA is primarily sensitive to the transverse separation of partons inside the bound nucleon, while in AA it probes the nuclear transverse profile.

Significance. If the framework is correct, this is a valuable step toward using DPS measurements to access transverse structure in nuclear collisions. The paper is transparent about its ingredients, propagates uncertainties from nNNPDF replicas, cross-checks with EPPS21, and provides concrete predictions for many final states in pPb and PbPb collisions. These strengths make the paper potentially useful even if the specific bound-nucleon widening hypothesis is ultimately not confirmed. However, the empirical anchor for the central pA claim is weak: all available pPb data points are lower limits, and the key widening term was introduced after the γ_A=0 model failed to reproduce the LHCb trend, with parameters chosen a posteriori. The significance of the main physics claim is therefore conditional on further scrutiny of this circularity and on the statistical significance of the trend.

major comments (4)
  1. [§II, Eq. (2.7) and §III, Fig. 3] The bound-nucleon widening term is the load-bearing element for the pA part of the central claim, yet it is introduced only after the γ_A=0 model fails to reproduce the LHCb forward/backward trend, and the values γ_A=1 mb and x_A=5×10^-3 are explicitly 'justified a posteriori'. All five pPb data points are lower limits, and no statistical test is provided for whether the opposite trend seen with γ_A=0 is actually excluded. Since the γ_A term selectively raises forward σ_eff while leaving backward values essentially unchanged, the agreement in Fig. 3 is substantially built into the model. The conclusion that 'This agreement supports the hypothesis of more widely separated partons inside the same nucleon' is therefore circular to a significant degree. Please quantify the compatibility of γ_A=0 with the lower-limit data (e.g., a chi-square or a coverage test) and reframe the widening hypoth
  2. [§III, Fig. 3] The paper states that for γ_A=0 the backward-rapidity σ_eff values are 'significantly larger' than forward ones and that the LHCb trend goes in the opposite direction. But because each experimental point is a lower limit without an upper uncertainty, the sign of the experimental 'trend' may not be statistically meaningful. If the forward and backward lower limits are mutually compatible with a flat or increasing σ_eff, the motivation for the γ_A term is not established. The authors should display the numerical values, uncertainties, and lower-limit nature of the five LHCb/CMS points explicitly in the text and perform a simple significance test of the forward/backward asymmetry before using this as motivation.
  3. [§II, Eqs. (2.13)–(2.14) and §III, Figs. 6–7] The AA predictions depend dominantly on the 2x2 contribution, which uses only the new nuclear profile ρ(x;r) built from R_g as a proxy for all parton species, with R_g≥0.1 imposed ad hoc. This profile has no free parameters but is also not derived or validated. The claim that σ_eff,AA varies by about 10% across final states and thereby probes the nuclear transverse structure is only as credible as this ansatz. Please discuss the sensitivity of the AA predictions to the choice of using R_g for all flavors and to the lower bound R_g≥0.1; a comparison with an alternative ansatz, or with EPPS21-based R_g, would help.
  4. [§II, Eq. (2.1)] The statement that 'no approximation other than working within collinear factorization has been made' is misleading because the scale factor Θ is left completely unspecified and in practice is modeled through Gaussian profiles and the new nuclear/rg-dependent ansätze. The factorization in Eq. (2.1) absorbs all dynamical information into Θ, so it is not an assumption-free starting point. The sentence should be revised to avoid overstating the rigor of the framework.
minor comments (4)
  1. [§II, Eq. (2.13)] The profile ρ(x;r) is defined with r as a transverse coordinate, but the Woods-Saxon function in Eq. (2.10) is written in three-dimensional form. Please clarify the notation and the reduction to two dimensions, including the normalization convention in Eq. (2.14).
  2. [§III, Fig. 3] The caption labels the experimental points as 'exp' but does not state the lower-limit nature explicitly; that is mentioned only in the text. Please add 'lower limits' to the figure legend or caption for clarity.
  3. [§III, paragraph after Fig. 3] The sentence 'Variations of the parameter x_A lead to correlated effects and can be partially compensated by corresponding changes in γ_A' is not demonstrated. A small plot or table showing this degeneracy would be helpful for interpreting the dashed uncertainty bars.
  4. [Abstract/Conclusions] The phrase 'The observed dependence of our predictions on the final state indicates that DPS... can be used to probe...' overstates what are model-dependent predictions, not measurements. Consider replacing 'observed dependence' with 'predicted dependence' or similar.

Circularity Check

1 steps flagged

pPb validation of the bound-nucleon widening hypothesis reduces to a post-hoc γ_A term tuned to the same lower-limit data points it is said to support.

specific steps
  1. fitted input called prediction [Sec. III, Fig. 3 and Eq. (2.7); Conclusions]
    ""This failure constitutes our main motivation to hypothesize that partons inside a nucleon are more spread out than in a free proton. Consequently, the additional γ A = 1 mb term introduced at small x in Eq. 2.7 leaves the backward results essentially unchanged while increasing the forward effective cross section." ... "This agreement supports the hypothesis of more widely separated partons inside the same nucleon.""

    The γ_A-dependent term in Eq. (2.7) was introduced only after the γ_A=0 model failed to reproduce the same five LHCb data points that are later cited as support. The paper itself states that the parameters γ_A=1 mb and x_A=5e-3 are "justified a posteriori." The term is designed to raise forward σ_eff while leaving backward points essentially unchanged, exactly inverting the γ_A=0 trend. Since all five pPb measurements are lower limits, the discriminating power is limited even as a test. The subsequent "agreement" is therefore not an independent confirmation of the widening hypothesis; it is a restatement of the hand-chosen term. Because this agreement is the main evidence for the central pA probe claim, that part of the conclusion is partially built into the fit.

full rationale

The pp DPS model imported from Ref. [28] is anchored to independent pp data, including a postdiction of ATLAS WW, and is not circular here. The nuclear profile in Eq. (2.13) is parameter-free and the AA predictions are dominated by the 2x2 term, so those parts do not reduce to their inputs. The circularity is confined to the pPb validation of the bound-nucleon widening hypothesis: the γ_A term in Eq. (2.7) is a post-hoc adjustment made after γ_A=0 failed to match the LHCb forward/backward trend, and the same data are then invoked as supporting the hypothesis. This makes the central pA probe claim partially built-in, although the model still has predictive content for unmeasured final states and the AA predictions are independent. The score reflects this partial, load-bearing circularity rather than a fully forced derivation.

Axiom & Free-Parameter Ledger

4 free parameters · 8 axioms · 0 invented entities

The central predictions import four fitted parameters from the authors' pp analysis, add two hand-chosen bound-nucleon parameters set to match pPb data, and adopt a normalization-fixed but otherwise unvalidated profile ansatz and kinematic cutoffs. No new physical entities are postulated.

free parameters (4)
  • Free-proton Gaussian variance parameters (beta, gamma1, gamma2, kappa) = beta=0.067+-0.068 mb; gamma1=1.68+-0.48; gamma2=0.85+-0.16; kappa=0.087+-0.036
    Fitted to pp DPS data in the authors' previous paper [28]; used as fixed inputs here.
  • gamma_A (bound-nucleon widening coefficient) = 1 mb (0.5-1.5 mb explored as uncertainty band)
    Hand-chosen so pPb predictions are compatible with LHCb data; explicitly not a statistical fit due to insufficient data (Sec. II and III).
  • x_A (small-x threshold for bound-nucleon widening) = 5e-3
    Small-x threshold for the widening term, chosen alongside gamma_A; the text notes variations in x_A can be compensated by gamma_A.
  • Nuclear profile kinematic bounds (R_g floor, x range) = R_g >= 0.1; x < 0.3
    Ad hoc numerical limits imposed to keep the profile and nPDF values physical; no data basis is provided.
axioms (8)
  • domain assumption DPS cross section factorizes as a product of two SPS cross sections times an x- and scale-dependent overlap factor Theta (Eq. 2.1).
    Standard DPS pocket-formula extension, assumed without derivation; Theta is left unspecified and carries all transverse information.
  • ad hoc to paper The two-parton transverse distribution in a free proton is a normalized Gaussian with variance B(x,mu) given by Eq. 2.4.
    Phenomenological model from Ref. [28], fitted to pp DPS data; not derived from QCD.
  • domain assumption The overlap factor Theta is independent of parton flavor.
    Explicitly assumed in Sec. II for all cases considered.
  • ad hoc to paper Partons inside a bound nucleon are more widely separated at small x, implemented via gamma_A term in Eq. 2.7 with gamma_A=1 mb and x_A=5e-3.
    Hypothesis introduced after the gamma_A=0 model failed to match the pPb trend; parameters are justified a posteriori.
  • ad hoc to paper Nuclear transverse profile is rho(x;r) = (exp(Delta_sigma rho_WS(r))-1)/(Delta_sigma R_g(x|mu)), with Delta_sigma fixed by normalization (Eq. 2.13-2.14).
    Purely phenomenological ansatz with no derivation; it determines the 1x2 and 2x2 contributions.
  • ad hoc to paper The gluon nuclear modification factor R_g is used as a proxy for the transverse modification of all parton species, for x < 0.3, with R_g >= 0.1.
    Motivated by gluon and sea-quark dominance, but not derived; the kinematic bounds are arbitrary.
  • domain assumption Nucleon distribution in the nucleus follows the Woods-Saxon form with R_A=6.62 fm and delta=0.546 fm.
    Standard nuclear geometry input taken from Ref. [67].
  • domain assumption PYTHIA 8.3 default settings with NNPDF2.3 LO proton PDFs and nNNPDF3.0 NLO lead PDFs, omitting MPI and Angantyr nuclear rescattering.
    Computational setup assumed to describe SPS cross sections; neglected effects could alter DPS normalizations.

pith-pipeline@v1.3.0-alltime-deepseek · 11929 in / 16557 out tokens · 144198 ms · 2026-08-02T18:48:27.598058+00:00 · methodology

0 comments
read the original abstract

In a previous work, we studied the momentum fraction and hard--scale dependence of double parton scattering (DPS) in proton--proton collisions and the resulting dependence of the effective cross section on the final--state observables. In this paper, we extend those results to heavy--ion ($pA$ and $AA$) collisions, accounting for nuclear effects in the relevant kinematic region, namely shadowing and antishadowing. In addition to modifying the longitudinal parton distributions, these effects also alter the transverse parton distribution of the nucleus, for which we propose a simple new nuclear profile. We further hypothesize that partons inside a bound nucleon are more widely separated than in a free proton. We compute the effective cross section for the available $p$Pb data, obtaining reasonable agreement, and provide predictions for future measurements at the LHC. The observed dependence of our predictions on the final state indicates that DPS in heavy--ion collisions can be used to probe the transverse profile of the free proton and the bound nucleon, primarily in $pA$ collisions, as well as the transverse structure of the nucleus, mainly in $AA$ collisions.

Figures

Figures reproduced from arXiv: 2603.04510 by Edgar Huayra, Emmanuel G. de Oliveira, Joao Vitor C. Lovato.

Figure 1
Figure 1. Figure 1: Schematic diagrams of double parton scattering (DPS) contributions in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Our proposed two-dimensional nuclear profile for different values of the nuclear gluon [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between available experimental results and theoretical predictions for the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Theoretical predictions for the DPS effective cross section in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Theoretical predictions for the DPS effective cross section in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Theoretical predictions for the DPS effective cross section in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Theoretical predictions for the DPS effective cross section in [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

72 extracted references · 53 linked inside Pith

  1. [1]

    take into account that nuclear effects modify thexdistribution of partons in a bound nucleon and, consequently, reweight thex–dependent transverse profile

  2. [2]

    introduce a transverse nuclear profile that phenomenologically incorporates thexde- pendent shadowing and antishadowing effects without introducing free parameters; 2

  3. [3]

    ce”) doubleJ/ψresult [56]. The LHCb results correspond toJ/ψ+D 0 and doubleD 0 production in the forward (“fw

    further hypothesize that partons inside a bound nucleon are more widely separated than in a free proton. This framework leads to agreement with the available experimental measurements. We also provide predictions for DPS effective cross sections across a wide range of final states inpPb and PbPb collisions at nucleon–nucleon center–of–mass (c.o.m.) energi...

  4. [4]

    Paver and D

    N. Paver and D. Treleani, Nuovo Cim. A70, 215 (1982)

    1982

  5. [5]

    Mekhfi, Phys

    M. Mekhfi, Phys. Rev. D32, 2371 (1985)

    1985

  6. [6]

    Sjostrand and M

    T. Sjostrand and M. van Zijl, Phys. Rev. D36, 2019 (1987)

    2019

  7. [7]

    Bartaliniet al.[arXiv:1111.0469 [hep-ph]]

    P. Bartaliniet al.[arXiv:1111.0469 [hep-ph]]

    Pith/arXiv arXiv

  8. [8]

    Bansalet al.[arXiv:1410.6664 [hep-ph]]

    S. Bansalet al.[arXiv:1410.6664 [hep-ph]]

    Pith/arXiv arXiv

  9. [9]

    Szczurek, Acta Phys

    A. Szczurek, Acta Phys. Polon. Supp.8, no.2, 483 (2015) [arXiv:1505.04067 [hep-ph]]

    Pith/arXiv arXiv 2015

  10. [10]

    Diehl and J

    M. Diehl and J. R. Gaunt, Adv. Ser. Direct. High Energy Phys.29, 07 (2018) [arXiv:1710.04408 [hep-ph]]

    Pith/arXiv arXiv 2018

  11. [11]

    Cattaruzza, A

    E. Cattaruzza, A. Del Fabbro, and D. Treleani, Phys. Rev. D72, 034022 (2005) [arXiv:hep- ph/0507052 [hep-ph]]

    arXiv 2005

  12. [12]

    A. M. Snigirev, Phys. Rev. D81, 065014 (2010) [arXiv:1001.0104 [hep-ph]]

    Pith/arXiv arXiv 2010

  13. [13]

    J. R. Gaunt and W. J. Stirling, JHEP03, 005 (2010) [arXiv:0910.4347 [hep-ph]]

    Pith/arXiv arXiv 2010

  14. [14]

    Diehl, D

    M. Diehl, D. Ostermeier, and A. Schafer, JHEP03, 089 (2012) [arXiv:1111.0910 [hep-ph]]

    Pith/arXiv arXiv 2012

  15. [15]

    B. Blok, Y. Dokshitser, L. Frankfurt, and M. Strikman, Eur. Phys. J. C72, 1963 (2012) [arXiv:1106.5533 [hep-ph]]

    Pith/arXiv arXiv 1963

  16. [16]

    H. M. Chang, A. V. Manohar, and W. J. Waalewijn, Phys. Rev. D87, 034009 (2013) [arXiv:1211.3132 [hep-ph]]

    Pith/arXiv arXiv 2013

  17. [17]

    B. Blok, M. Strikman, and U. A. Wiedemann, Eur. Phys. J. C73, 2433 (2013) [arXiv:1210.1477 [hep-ph]]

    Pith/arXiv arXiv 2013

  18. [18]

    B. Blok, Y. Dokshitzer, L. Frankfurt, and M. Strikman, Eur. Phys. J. C74, 2926 (2014) [arXiv:1306.3763 [hep-ph]]

    Pith/arXiv arXiv 2014

  19. [19]

    Salvini, D

    S. Salvini, D. Treleani, and G. Calucci, Phys. Rev. D89, 016020 (2014) [arXiv:1309.6201 [hep-ph]]

    Pith/arXiv arXiv 2014

  20. [20]

    Strikman and D

    M. Strikman and D. Treleani, Phys. Rev. Lett.88, 031801 (2002) [arXiv:hep-ph/0111468 [hep-ph]]

    Pith/arXiv arXiv 2002

  21. [21]

    d’Enterria and A

    D. d’Enterria and A. M. Snigirev, Eur. Phys. J. C78, no.5, 359 (2018) [arXiv:1612.08112 [hep-ph]]. 14

    Pith/arXiv arXiv 2018

  22. [22]

    d’Enterria and A

    D. d’Enterria and A. M. Snigirev, Adv. Ser. Direct. High Energy Phys.29, 159-187 (2018) [arXiv:1708.07519 [hep-ph]]

    Pith/arXiv arXiv 2018

  23. [23]

    M. G. A. Buffing, M. Diehl, and T. Kasemets, JHEP01, 044 (2018) [arXiv:1708.03528 [hep- ph]]

    Pith/arXiv arXiv 2018

  24. [24]

    J. R. Gaunt and T. Kasemets, Adv. High Energy Phys.2019, 3797394 (2019) [arXiv:1812.09099 [hep-ph]]

    Pith/arXiv arXiv 2019

  25. [25]

    Huayra, E

    E. Huayra, E. G. de Oliveira, and R. Pasechnik, Eur. Phys. J. C79, 880 (2019) [arXiv:1905.03294 [hep-ph]]

    Pith/arXiv arXiv 2019

  26. [26]

    Huayra, E

    E. Huayra, E. G. de Oliveira, and R. Pasechnik, Eur. Phys. J. C80, 772 (2020) [arXiv:2003.06412 [hep-ph]]

    Pith/arXiv arXiv 2020

  27. [27]

    Huayra, E

    E. Huayra, E. G. de Oliveira, R. Pasechnik, and B. O. Stahlh¨ ofer, Phys. Rev. D104, 096003 (2021) [arXiv:2106.11465 [hep-ph]]

    Pith/arXiv arXiv 2021

  28. [28]

    B. Blok, R. Segev, and M. Strikman, Eur. Phys. J. C83, 415 (2023) [arXiv:2212.08848 [hep- ph]]

    Pith/arXiv arXiv 2023

  29. [29]

    Huayra, J

    E. Huayra, J. V. C. Lovato, and E. G. de Oliveira, JHEP09, 177 (2023) [arXiv:2305.11106 [hep-ph]]

    Pith/arXiv arXiv 2023

  30. [30]

    B. O. Stahlh¨ ofer, E. Huayra, and E. G. de Oliveira, [arXiv:2510.05420 [hep-ph]]

    Pith/arXiv arXiv

  31. [31]

    J. V. C. Lovato, E. Huayra, and E. G. de Oliveira, [arXiv:2506.05337 [hep-ph]]

    Pith/arXiv arXiv

  32. [32]

    ˚Akessonet al.(AFS Collaboration), Z

    T. ˚Akessonet al.(AFS Collaboration), Z. Phys. C34, 163 (1987)

    1987

  33. [33]

    Abeet al.(CDF Collaboration), Phys

    F. Abeet al.(CDF Collaboration), Phys. Rev. D47, 4857 (1993)

    1993

  34. [34]

    Abeet al.(CDF Collaboration), Phys

    F. Abeet al.(CDF Collaboration), Phys. Rev. D56, 3811 (1997)

    1997

  35. [35]

    Aadet al.(ATLAS Collaboration), New J

    G. Aadet al.(ATLAS Collaboration), New J. Phys.15, 033038 (2013) [arXiv:1301.6872 [hep- ex]]

    Pith/arXiv arXiv 2013

  36. [36]

    V. M. Abazovet al.(D0 Collaboration), Phys. Rev. D89, 072006 (2014) [arXiv:1402.1550 [hep-ex]]

    Pith/arXiv arXiv 2014

  37. [37]

    V. M. Abazovet al.(D0 Collaboration), Phys. Rev. D90, 111101 (2014) [arXiv:1406.2380 [hep-ex]]

    Pith/arXiv arXiv 2014

  38. [38]

    Chatrchyanet al.(CMS Collaboration), JHEP03, 032 (2014) [arXiv:1312.5729 [hep-ex]]

    S. Chatrchyanet al.(CMS Collaboration), JHEP03, 032 (2014) [arXiv:1312.5729 [hep-ex]]

    Pith/arXiv arXiv 2014

  39. [39]

    V. M. Abazovet al.(D0 Collaboration), Phys. Rev. D93, 052008 (2016) [arXiv:1512.05291 [hep-ex]]

    Pith/arXiv arXiv 2016

  40. [40]

    Khachatryanet al.(CMS Collaboration), Eur

    V. Khachatryanet al.(CMS Collaboration), Eur. Phys. J. C76, 155 (2016) [arXiv:1512.00815 [hep-ex]]

    Pith/arXiv arXiv 2016

  41. [41]

    Aaboudet al.(ATLAS Collaboration), JHEP11, 110 (2016) [arXiv:1608.01857 [hep-ex]]

    M. Aaboudet al.(ATLAS Collaboration), JHEP11, 110 (2016) [arXiv:1608.01857 [hep-ex]]

    Pith/arXiv arXiv 2016

  42. [42]

    Aaijet al.(LHCb Collaboration), JHEP07, 052 (2016) [arXiv:1510.05949 [hep-ex]]

    R. Aaijet al.(LHCb Collaboration), JHEP07, 052 (2016) [arXiv:1510.05949 [hep-ex]]

    Pith/arXiv arXiv 2016

  43. [43]

    Aaboudet al.(ATLAS Collaboration), Eur

    M. Aaboudet al.(ATLAS Collaboration), Eur. Phys. J. C77, 76 (2017) [arXiv:1612.02950 [hep-ex]]

    Pith/arXiv arXiv 2017

  44. [44]

    Aaijet al.(LHCb Collaboration), JHEP06, 047 (2017) [arXiv:1612.07451 [hep-ex]]

    R. Aaijet al.(LHCb Collaboration), JHEP06, 047 (2017) [arXiv:1612.07451 [hep-ex]]

    Pith/arXiv arXiv 2017

  45. [45]

    J. P. Lansberg, H. S. Shao, and N. Yamanaka, Phys. Lett. B781, 485 (2018) [arXiv:1707.04350 [hep-ph]]

    Pith/arXiv arXiv 2018

  46. [46]

    J. P. Lansberg, Phys. Rept.889, 01 (2020) [arXiv:1903.09185 [hep-ph]]

    Pith/arXiv arXiv 2020

  47. [47]

    A. M. Sirunyanet al.(CMS Collaboration), Eur. Phys. J. C80, 41 (2020) [arXiv:1909.06265 [hep-ex]]

    Pith/arXiv arXiv 2020

  48. [48]

    Tumasyanet al.(CMS Collaboration), JHEP01, 177 (2022) [arXiv:2109.13822 [hep-ex]]

    A. Tumasyanet al.(CMS Collaboration), JHEP01, 177 (2022) [arXiv:2109.13822 [hep-ex]]

    Pith/arXiv arXiv 2022

  49. [49]

    Tumasyanet al.(CMS Collaboration), Phys

    A. Tumasyanet al.(CMS Collaboration), Phys. Rev. Lett.131, 091803 (2023) [arXiv:2206.02681 [hep-ex]]

    Pith/arXiv arXiv 2023

  50. [50]

    Acharyaet al.(ALICE Collaboration), Phys

    S. Acharyaet al.(ALICE Collaboration), Phys. Rev. C108, 045203 (2023) [arXiv:2303.13431 15 [hep-ex]]

    Pith/arXiv arXiv 2023

  51. [51]

    Tumasyanet al.(CMS Collaboration), Nature Phys.19, 338 (2023) [arXiv:2111.05370 [hep-ex]]

    A. Tumasyanet al.(CMS Collaboration), Nature Phys.19, 338 (2023) [arXiv:2111.05370 [hep-ex]]

    Pith/arXiv arXiv 2023

  52. [52]

    Aaijet al.(LHCb Collaboration), JHEP08, 093 (2023) [arXiv:2305.15580 [hep-ex]]

    R. Aaijet al.(LHCb Collaboration), JHEP08, 093 (2023) [arXiv:2305.15580 [hep-ex]]

    arXiv 2023

  53. [53]

    H. S. Shao and Y. J. Zhang, Phys. Rev. Lett.117, 062001 (2016) [arXiv:1605.03061 [hep-ph]]

    Pith/arXiv arXiv 2016

  54. [54]

    J. P. Lansberg and H. S. Shao, Nucl. Phys. B916, 132 (2017) [arXiv:1611.09303 [hep-ph]]

    Pith/arXiv arXiv 2017

  55. [55]

    Alittiet al.[UA2], Phys

    J. Alittiet al.[UA2], Phys. Lett. B268, 145-154 (1991)

    1991

  56. [56]

    CMS Collaboration, CMS-PAS-BPH-23-007

  57. [57]

    Aaijet al.(LHCb Collaboration), JHEP03, 088 (2024) [arXiv:2311.14085 [hep-ex]]

    R. Aaijet al.(LHCb Collaboration), JHEP03, 088 (2024) [arXiv:2311.14085 [hep-ex]]

    arXiv 2024

  58. [58]

    Aadet al.[ATLAS], Phys

    G. Aadet al.[ATLAS], Phys. Lett. B870, 139892 (2025) [arXiv:2505.08313 [hep-ex]]

    arXiv 2025

  59. [59]

    Hayrapetyanet al.[CMS], Phys

    A. Hayrapetyanet al.[CMS], Phys. Rev. D110, 092002 (2024) [arXiv:2407.03223 [nucl-ex]]

    Pith/arXiv arXiv 2024

  60. [60]

    Aaijet al.[LHCb], Phys

    R. Aaijet al.[LHCb], Phys. Rev. Lett.125, 212001 (2020) [arXiv:2007.06945 [hep-ex]]

    arXiv 2020

  61. [61]

    Klasen and H

    M. Klasen and H. Paukkunen, Ann. Rev. Nucl. Part. Sci.74, 49-87 (2024) [arXiv:2311.00450 [hep-ph]]

    Pith/arXiv arXiv 2024

  62. [62]

    M. L. Miller, K. Reygers, S. J. Sanders, and P. Steinberg, Ann. Rev. Nucl. Part. Sci.57, 205-243 (2007) [arXiv:nucl-ex/0701025 [nucl-ex]]

    Pith/arXiv arXiv 2007

  63. [63]

    d’Enterria and C

    D. d’Enterria and C. Loizides, Ann. Rev. Nucl. Part. Sci.71, 315-344 (2021) [arXiv:2011.14909 [hep-ph]]

    Pith/arXiv arXiv 2021

  64. [64]

    H. S. Shao, Phys. Rev. D101, 054036 (2020) [arXiv:2001.04256 [hep-ph]]

    Pith/arXiv arXiv 2020

  65. [65]

    Bierlichet al.SciPost Phys

    C. Bierlichet al.SciPost Phys. Codeb.2022, 08 (2022) [arXiv:2203.11601 [hep-ph]]

    Pith/arXiv arXiv 2022

  66. [66]

    R. D. Ballet al.(NNPDF Collaboration), Eur. Phys. J. C77, 663 (2017) [arXiv:1706.00428 [hep-ph]]

    Pith/arXiv arXiv 2017

  67. [67]

    Abdul Khalek, R

    R. Abdul Khalek, R. Gauld, T. Giani, E. R. Nocera, T. R. Rabemananjara, and J. Rojo, Eur. Phys. J. C82, 507 (2022) [arXiv:2201.12363 [hep-ph]]

    Pith/arXiv arXiv 2022

  68. [68]

    Bierlich, G

    C. Bierlich, G. Gustafson, L. L¨ onnblad, and H. Shah, JHEP10, 134 (2018) [arXiv:1806.10820 [hep-ph]]

    Pith/arXiv arXiv 2018

  69. [69]

    R. D. Woods and D. S. Saxon, Phys. Rev.95, 577-578 (1954)

    1954

  70. [70]

    De Vries, C

    H. De Vries, C. W. De Jager, and C. De Vries, Atom. Data Nucl. Data Tabl.36, 495-536 (1987)

    1987

  71. [71]

    James and M

    F. James and M. Winkler,MINUIT User’s Guide(CERN, Geneva, 2004)

    2004

  72. [72]

    K. J. Eskola, P. Paakkinen, H. Paukkunen, and C. A. Salgado, Eur. Phys. J. C82, no.5, 413 (2022) [arXiv:2112.12462 [hep-ph]]. 16

    Pith/arXiv arXiv 2022