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Distribution Functions of a Radially Excited Pion

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Pith's one-line read This paper predicts that the pion's first radial excitation has a three-peak valence quark distribution at the hadron scale, with zeros near x≈0.2 and x≈0.8.

desk verdict First hadron-scale valence DF for the pion's first radial excitation, with a novel three-peak structure that is intriguing but rests on SPM-extrapolated high moments whose stability is not demonstrated. read the letter →

arxiv 2501.13243 v1 pith:I57XUSUJ submitted 2025-01-22 hep-ph hep-exhep-latnucl-exnucl-th

Z.-N. Xu , Z.-Q. Yao , D. Binosi , M. Ding , C. D. Roberts , J. Rodríguez-Quintero This is my paper · ORCID
classification hep-phhep-exhep-latnucl-exnucl-th
keywords piondistributionfunctionsradialexcitationMellinmomentspartondistributionsBethe-Salpeterequationdynamicalchiralsymmetrybreakingemergenthadronmassthree-peakstructure
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Working from a symmetry-preserving approximation to QCD's bound-state equations, this paper predicts the parton distribution functions of the ground-state pion and its first radial excitation. Its central result is that the excited pion's valence quark distribution at the hadron scale is not a single bell-shaped curve: it has three peaks, with a central maximum near $x=1/2$, zeros near $x\approx0.2$ and $x\approx0.8$, and secondary peaks near $x\approx0.1$ and $x\approx0.9$. This shape follows from the pattern of chiral symmetry breaking, and it is the kind of structural fact that distinguishes a radially excited bound state from its ground state. If the prediction holds, it gives other nonperturbative approaches a sharp target: the same moments, the same zeros, the same peak locations.

What carries the argument

The carrier of the argument is the Mellin-moment representation of the hadron-scale valence quark distribution, computed from the light-front projection of the Bethe-Salpeter amplitude and the dressed quark propagator. Symmetry fixes the $m=0,1$ moments; the paper computes moments up to $m=5$ (or $6$) directly, extends the sequence to $m=10$ using the Schlessinger point method, and reconstructs the pointwise distribution from an ansatz that respects the QCD endpoint behaviour $(1-x)^2$. For the excited state, the load-bearing physical ingredient is the Bethe-Salpeter kernel built from a process-independent effective charge and a dressed-quark anomalous chromomagnetic moment; this kernel is what fixes the failure of rainbow-ladder truncation, which violates the Cauchy-Schwarz stability condition for a non-negative distribution. Evolution to $\zeta=3.2\,$GeV is performed with an all-orders scheme, which also produces the glue and sea distributions.

What would settle it

An independent determination of the hadron-scale $\pi_1$ valence Mellin moments—for instance from a lattice or light-front calculation—that finds the $m\ge8$ moments below the scale-free values, or a pointwise reconstruction without zeros near $x\approx0.2$ and $x\approx0.8$, would falsify the three-peak claim. In the paper's Table 2, the reconstruction needs $\langle x^8\rangle$ at least near the scale-free value $0.0350$.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the hadron-scale valence distribution function of the pion's first radial excitation, $q_{\pi_1}(x;\zeta_H)$, has three peaks. The $m=0$ and $m=1$ Mellin moments of $\pi_0$ and $\pi_1$ are forced equal by baryon-number and momentum conservation, while every moment with $m\ge2$ is larger for the ground state. The computed low-order moments are followed by a Schlessinger-point extension to $m\le10$ and a pointwise reconstruction, which produces a central peak at $x=1/2$ that is taller and narrower than the scale-free profile $30x^2(1-x)^2$, zeros at $x\approx0.2$ and $x\approx0.8$, and secondary peaks near $x\approx0.1$ and $x\approx0.9$. The paper interprets this as the momentum-space analogue of a first radial excitation: central momentum is favoured, the extreme all-or-nothing fractions are suppressed, and the support that would sit near $x\approx0.2,0.8$ is pushed outward to the endpoint shoulders.

Load-bearing premise

The load-bearing premise is that the analytic extension from the five directly computed Mellin moments to the tenth is reliable; the outer peaks only appear if the extrapolated high-order moments stay at or above the scale-free baseline.

Editorial extensions

If this is right

  • A radially excited pseudoscalar meson can have a valence distribution that is not bell-shaped: the $\pi_1$ DF possesses three peaks, with the outer peaks separated from the central maximum by zeros near $x\approx0.2$ and $x\approx0.8$.
  • The moment ordering is robust under evolution: $m=0,1$ moments remain equal and every $m\ge2$ moment of $\pi_0$ stays larger than the $\pi_1$ partner at $\zeta=3.2\,$GeV, so the valence difference survives to measurable scales.
  • Glue and sea distributions of $\pi_0$ and $\pi_1$ at $3.2\,$GeV are nearly identical, so the radial excitation's fingerprint is carried almost entirely by its valence distribution.
  • The relation $q(x)\propto|\varphi(x)|^2$, already known for the ground-state pion, holds at least qualitatively for the radial excitation: the zeros of the $\pi_1$ distribution amplitude line up with the zeros of its valence DF.
  • Because the reconstruction respects non-negativity and the $(1-x)^2$ endpoint behaviour, the three-peak shape, if correct, is not an artefact of the polynomial expansion but a consequence of the high-order moments lying above the scale-free values.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If confirmed, the three-peak structure would give searches a cleaner discriminator between ground-state and excited pion structure than mass spectra alone, since two Hamiltonians with similar spectra can have very different momentum-space wave functions.
  • The endpoint peaks are controlled by the extrapolated moments $m\ge8$; a future calculation that resolves these moments directly would settle whether the peaks are real or an artifact of the analytic continuation.
  • A similar three-peak pattern could be expected for other radially excited pseudoscalar mesons, because the mechanism is the node structure of the excited-state wave function combined with the symmetries of the pion-like system.
  • The paper's argument implies that the $\pi_1$ valence distribution at the hadron scale is effectively the modulus-squared of its light-front amplitude; if one measures or computes the amplitude's zero positions accurately, the DF zeros should coincide with them.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This manuscript computes the hadron-scale valence quark distribution functions of the pion ground state (π0) and its first radial excitation (π1) using a symmetry-preserving continuum Schwinger function framework. The valence Mellin moments are obtained from dressed quark propagators and Bethe-Salpeter amplitudes, the pointwise DFs are reconstructed using a parametrized form, and all DFs (valence, glue, sea) are evolved to 3.2 GeV with an all-orders evolution scheme. The central prediction is that the π1 valence DF has a three-peak structure at the hadron scale, with zeros near x≈0.2 and x≈0.8 and secondary peaks near x≈0.1 and x≈0.9, in contrast to the single-peak ground-state pion DF.

Significance. If the three-peak prediction is robust, it is a novel and testable statement about the structure of excited pseudoscalar mesons, and the paper provides useful benchmarks for other nonperturbative frameworks and for future lattice QCD studies. The analysis has several credible ingredients: an EHM-improved Bethe-Salpeter kernel that reproduces the meson spectrum, a reconstruction of the ground-state pion DF consistent with existing results, and an all-orders evolution scheme that is applied to all parton species. The main weakness is that the headline feature rests on Schlessinger-point extrapolation of Mellin moments, and the paper does not demonstrate that this extrapolation is stable for the specific five-point input set used for the π1.

major comments (3)
  1. [Section 4, Table 2] The three-peak structure of the π1 valence DF in Fig. 3 is driven by the m≥6 (especially m≥8) π1 moments being larger than the scale-free moments. These high moments are obtained by SPM extrapolation from the m≤5 computed moments. The text states that standard algorithms provide reliable access to all m≤6 moments for the π1, yet neither the π(a)1 nor the π(b)1 column in Table 2 uses the m=6 moment as an input; the m=6 entry is itself an extrapolated value. The validation sentence in Section 4 ('in nontrivial test cases, M(z) returns moments out to m=10 whose relative error is <0.1% in magnitude') is a test of the method on other functions, not on this particular five-point input set. Since the endpoint peaks in the reconstructed DF are the central claim, the authors should provide a direct stability test: include the actually computed m=6 moment as a check, vary the input set used for the SPM, compare the SPM predictions with the computed moments, and propagate the resulting uncertainty into the reconstructed DF.
  2. [Section 5, Table 3] The two SPM variants (columns π(a)1 and π(b)1) yield reconstruction parameters a6 = -0.003 and a6 = +0.0684, a substantial relative difference in the Gegenbauer coefficient that controls the oscillatory structure in Eq. (19). The paper states that the resulting DFs are 'qualitatively identical,' but this is not quantified, and the physical claim is precisely about the qualitative features: the zeros near x≈0.2, 0.8 and the secondary peaks near x≈0.1, 0.9. Without an uncertainty envelope for the reconstructed DF, or a demonstration that all plausible reconstructions preserve the three-peak structure, the robustness of the headline prediction is not established. Please provide the range of reconstructed DFs obtained from the two moment sets and state explicitly whether the three peaks and the zero locations are stable under this variation.
  3. [Section 6, paragraph after Fig. 4] The discussion that dismisses the lattice QCD results of Ref. [81] relies on the ordering of the hadron-scale moments in Table 2: the paper argues that the lattice ordering ⟨x^m⟩_{qπ1} > ⟨x^m⟩_{qπ0} at ζ3 is incompatible with the moments in Table 2 and with DGLAP evolution. This conclusion is contingent on the hadron-scale moments themselves, which for the π1 depend on the SPM extrapolation. If the SPM overestimates the higher π1 moments, the ordering of π0 vs π1 moments at ζ3 could be reversed. The paper should either treat the lattice comparison as an indication rather than a definitive incompatibility, or show explicitly that the SPM uncertainty is too small to affect the ordering conclusion.
minor comments (6)
  1. [Section 4] Please clarify why the reliable m=6 moment for the π1 is not used as an input to the SPM, given the statement that standard algorithms provide access to all m≤6 moments; this would strengthen the credibility of the extrapolation.
  2. [Section 7 and abstract] The summary says 'we computed eleven Mellin moments of the π0,1 valence DFs.' For the π1, moments m=6,...,10 are extrapolated rather than directly computed; 'obtained' or 'determined' would be more precise.
  3. [Table 4 caption] The caption contains a placeholder '[?]' for the G-parity reference; please supply the full citation.
  4. [Figure 3] The two π1 reconstructions are very close; consider adding a magnified inset near the endpoint peaks (x≈0.1 and x≈0.9) so that the secondary peaks are clearly distinguishable.
  5. [Section 5, Eq. (20)] The relation qπ0 ∝ |φπ0|^2 is cited but not derived; for the excited state the qualitative use is reasonable, but the sentence locating the π1 DF zeros 'in the vicinity of' the DA zeros should be supported by a quantitative comparison of the zero positions.
  6. [Section 3, Eqs. (11)-(13)] The parameters ω, D, η, and the current masses are model inputs; the paper would benefit from a statement on the sensitivity of the DF predictions to their variation, or a reference to the earlier study where these values were determined.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; the three-peak π1 valence DF is an inference from model-computed Mellin moments, with the SPM extrapolation as a robustness caveat rather than a circular step.

full rationale

The derivation chain is: (1) the gap and Bethe-Salpeter equations are solved with kernels fixed by the meson spectrum and decay constants (Table 1); (2) Mellin moments of the valence DFs are computed directly from Eq. (14) for m≤5, with m≤10 obtained by the Schlessinger point method (Section 4, Table 2); (3) the pointwise DF is reconstructed by fitting the ansatz Eq. (19) to those even moments (Section 5, Table 3). At no point is the target three-peak shape or any external DF dataset used as input. The reconstruction parameters (ρ, a2, a4, a6) are fitted to the computed moments, but this is a standard inversion of moments into a pointwise function, not a circular prediction: the three-peak structure is not imposed by the ansatz alone but emerges from the relative sizes of the low- and high-order moments, as the paper itself states: 'the low-m π1 moments are smaller than those of q sf; so, the π1 DF should be narrower and taller on x≃1/2. Yet, the large-m π1 moments are greater than those of q sf. Consequently, the π1 DF should exhibit greater support on the endpoint domains.' The main caveat is that the m≥6 moments, and especially the m≥8 moments that lie above the scale-free DF, come from the SPM extrapolation based on the m≤5 moments. The two SPM variants in Table 2 give materially different a6 values (Table 3), showing sensitivity in the extrapolated tail. However, this is an accuracy/robustness concern, not circularity: the extrapolated moments are not defined to equal the final DF, nor are they fitted to the three-peak structure. Self-citations to Refs. [23,29,33,34,79] provide methodology and prior π0 validation, but the central π1 claim is not justified solely by those citations; it is derived from the calculations reported here. Thus no specific step reduces by construction to its own input.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The calculations rest on a phenomenological interaction model with parameters fitted to the meson spectrum, an all-orders evolution scheme developed largely by the same group, and a moment reconstruction using SPM extrapolation and a chosen functional ansatz. The three-peak claim depends on the SPM-extrapolated high moments and the reconstruction parameters.

free parameters (7)
  • D = (0.68 GeV)^2
    Interaction strength in the effective charge, Eq. (11b), adjusted to reproduce the meson spectrum in Table 1.
  • omega = 0.8 GeV
    Width parameter of the effective interaction, Eq. (11b), chosen as the contemporary standard from Ref. [23].
  • eta = 1.1
    Strength of the dressed-quark anomalous chromomagnetic moment in the vertex, Eq. (12), fitted to improve agreement with excited-state masses and decay constants.
  • current quark masses = m_u = m_d = 4.07 MeV, m_s = 110 MeV (renormalization-point invariant)
    Set so that one-loop masses at ζ=2 GeV match empirical estimates, Eq. (13).
  • rho_pi0 = 0.0750
    Reconstruction parameter for the π0 valence DF in Eq. (18), fitted to the even Mellin moments via least squares.
  • rho_pi1, a2, a4, a6 = π1(a): 0.743, 0.116, 0.441, -0.003; π1(b): 0.663, 0.191, 0.487, 0.0684
    Parameters of the π1 reconstruction, Eq. (19), fitted to the moments in Table 2 with a non-negativity constraint. These directly control the three-peak shape.
  • zeta_H = 0.331(2) GeV
    Hadron scale obtained from the process-independent effective charge [48]; the starting point for AO evolution in Section 6.
assumptions (6)
  • standard math The axialvector Ward-Green-Takahashi identity connects the gap and Bethe-Salpeter kernels, so the truncation preserves chiral symmetry.
    Used in Section 3 to justify the kernel construction; this is a standard QCD identity.
  • domain assumption The AO evolution scheme is all-orders exact and can evolve DFs from ζH to ζ3.
    Adopted from Refs. [33,34,38] (overlapping authorship); not independently proved here.
  • ad hoc to paper The effective interaction in Eq. (11) accurately represents QCD's process-independent charge on the nonperturbative domain.
    Eq. (11) is a phenomenological fit; its use for excited states is an assumption.
  • ad hoc to paper SPM extrapolation gives correct moments for m=6,...,10 from the computed m≤5 moments.
    Section 4 claims test-case errors <0.1%, but the accuracy for the pion moments is assumed.
  • ad hoc to paper The functional forms in Eqs. (18) and (19) are flexible enough to represent the true valence DFs.
    Section 5 uses these ansätze; no proof that the DFs lie in this family.
  • domain assumption Non-negativity of the DF is enforced in the π1 parameter search.
    Imposed in Section 5 because a DF must be non-negative; this constraint can bias the reconstruction.

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Cite this review

Pith. "Pith review of Distribution Functions of a Radially Excited Pion." pith.science (2026). https://pith.science/paper/I57XUSUJ

@misc{pith2026250113243,
  author       = {Pith},
  title        = {Pith review of: Distribution Functions of a Radially Excited Pion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I57XUSUJ}},
  note         = {Machine review of arXiv:2501.13243}
}
abstract

A nonperturbatively-improved, symmetry-preserving approximation to the quantum field equations relevant in calculations of meson masses and interactions is used to deliver predictions for all distribution functions (DFs) of the ground state pion, $\pi_0$, and its first radial excitation, $\pi_1$, viz. valence, glue, and sea. Regarding Mellin moments of the valence DFs, the $m=0,1$ moments in both states are identical; but for each $m\geq 2$, that in the $\pi_0$ is greater than its partner in the $\pi_1$. Working with such information, pointwise reconstructions of the hadron-scale $\pi_{0,1}$ valence DFs are developed. The predicted $\pi_0$ valence DF is consistent with extant results. The $\pi_1$ valence DF is novel: it possesses three-peaks, with the central maximum partnered by secondary peaks on either side, each separated from the centre by a zero: the zeroes lie at $x\approx 0.2,0.8$ and the secondary peaks at $x\approx 0.1,0.9$. Evolution to $\zeta =3.2\,$GeV, a typical scale for nonperturbative calculations, is accomplished using an evolution scheme for parton DFs that is all-orders exact. At this higher scale, differences between the $\pi_{0,1}$ valence DFs remain significant, but analogous differences between glue and sea DFs are far smaller. This analysis shows that, owing to constraints imposed by chiral symmetry and the pattern by which it is broken in Nature, there are noticeable differences between the structural properties of the pion ground state and its radial excitations.

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Reference graph

Works this paper leans on

81 extracted references · 58 canonical work pages · cited by 2 Pith papers

  1. [81]

    X. Gao, N. Karthik, S. Mukherjee, P. Petreczky, S. Syrit- syn, Y. Zhao, Towards studying the structural differences between the pion and its radial excitation, Phys. Rev. D 103 (9) (2021) 094510

    work page 2021

  2. [1]

    Maris, C

    P. Maris, C. D. Roberts, P. C. Tandy, Pion mass and decay constant, Phys. Lett. B 420 (1998) 267–273

    1998

  3. [2]

    S. J. Brodsky, C. D. Roberts, R. Shrock, P. C. Tandy, Confinement contains condensates, Phys. Rev. C 85 (2012) 065202

    work page 2012

  4. [3]

    S.-X. Qin, C. D. Roberts, S. M. Schmidt, Ward-Green- Takahashi identities and the axial-vector vertex, Phys. Lett. B 733 (2014) 202–208

    work page 2014

  5. [4]

    T. Horn, C. D. Roberts, The pion: an enigma within the Standard Model, J. Phys. G. 43 (2016) 073001

    2016

  6. [5]

    C. D. Roberts, D. G. Richards, T. Horn, L. Chang, In- sights into the emergence of mass from studies of pion and kaon structure, Prog. Part. Nucl. Phys. 120 (2021) 103883

    2021

  7. [6]

    Binosi, Emergent Hadron Mass in Strong Dynamics, Few Body Syst

    D. Binosi, Emergent Hadron Mass in Strong Dynamics, Few Body Syst. 63 (2) (2022) 42

    2022

  8. [7]

    M. Ding, C. D. Roberts, S. M. Schmidt, Emergence of Hadron Mass and Structure, Particles 6 (1) (2023) 57– 120

    work page 2023

Show all 81 references
  1. [8]

    C. D. Roberts, Origin of the Proton Mass, EPJ Web Conf. 282 (2023) 01006

    2023

  2. [9]

    M. N. Ferreira, J. Papavassiliou, Gauge Sector Dynamics in QCD, Particles 6 (1) (2023) 312–363

    2023

  3. [10]

    D. S. Carman, R. W. Gothe, V. I. Mokeev, C. D. Ro- berts, Nucleon Resonance Electroexcitation Amplitudes and Emergent Hadron Mass, Particles 6 (1) (2023) 416– 439

    2023

  4. [11]

    K. Raya, A. Bashir, D. Binosi, C. D. Roberts, J. Rodr ´ ıguez-Quintero, Pseudoscalar Mesons and Emer- gent Mass, Few Body Syst. 65 (2) (2024) 60

    2024

  5. [12]

    Gell-Mann, R

    M. Gell-Mann, R. J. Oakes, B. Renner, Behavior of cur- rent divergences under SU(3) x SU(3), Phys. Rev. 175 (1968) 2195–2199

    1968

  6. [13]

    Gasser, H

    J. Gasser, H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158 (1984) 142

    1984

  7. [14]

    C. H. Llewellyn-Smith, A relativistic formulation for the quark model for mesons, Annals Phys. 53 (1969) 521–558

    1969

  8. [15]

    C. A. Dominguez, Extended Partially Conserved Axial- Vector Current Hypothesis and Chiral Symmetry Break- ing, Phys. Rev. D 15 (1977) 1350–1360

    1977

  9. [16]

    H¨ oll, A

    A. H¨ oll, A. Krassnigg, C. D. Roberts, Pseudoscalar meson radial excitations, Phys. Rev. C 70 (2004) 042203(R)

    2004

  10. [17]

    H¨ oll, A

    A. H¨ oll, A. Krassnigg, P. Maris, C. D. Roberts, S. V. Wright, Electromagnetic properties of ground and ex- cited state pseudoscalar mesons, Phys. Rev. C 71 (2005) 065204

    2005

  11. [18]

    M. S. Bhagwat, A. Krassnigg, P. Maris, C. D. Roberts, Mind the gap, Eur. Phys. J. A 31 (2007) 630–637

    2007

  12. [19]

    Ballon-Bayona, G

    A. Ballon-Bayona, G. Krein, C. Miller, Decay constants of the pion and its excitations in holographic QCD, Phys. Rev. D 91 (2015) 065024

    2015

  13. [20]

    Jiang, S.-L

    J.-F. Jiang, S.-L. Zhu, Radial excitations of mesons and nucleons from QCD sum rules, Phys. Rev. D 92 (7) (2015) 074002

    2015

  14. [21]

    Diehl, G

    M. Diehl, G. Hiller, New ways to explore factorization in b decays, JHEP 06 (2001) 067

    2001

  15. [22]

    McNeile, C

    C. McNeile, C. Michael, The decay constant of the first excited pion from lattice QCD, Phys. Lett. B 642 (2006) 244–247

    2006

  16. [23]

    Xu, Z.-Q

    Z.-N. Xu, Z.-Q. Yao, S.-X. Qin, Z.-F. Cui, C. D. Ro- berts, Bethe-Salpeter kernel and properties of strange- quark mesons, Eur. Phys. J. A 59 (3) (2023) 39

    2023

  17. [24]

    B.-L. Li, L. Chang, F. Gao, C. D. Roberts, S. M. Schmidt, H.-S. Zong, Distribution amplitudes of radially-excited π and K mesons, Phys. Rev. D 93 (11) (2016) 114033

    2016

  18. [25]

    Hua, et al., Pion and Kaon Distribution Amplitudes from Lattice QCD, Phys

    J. Hua, et al., Pion and Kaon Distribution Amplitudes from Lattice QCD, Phys. Rev. Lett. 129 (13) (2022) 132001

    2022

  19. [26]

    Li, Dispersive derivation of the pion distribution amplitude, Phys

    H.-N. Li, Dispersive derivation of the pion distribution amplitude, Phys. Rev. D 106 (3) (2022) 034015

    2022

  20. [27]

    X. Gao, A. D. Hanlon, N. Karthik, S. Mukherjee, P. Pe- treczky, P. Scior, S. Syritsyn, Y. Zhao, Pion distribution amplitude at the physical point using the leading-twist expansion of the quasi-distribution-amplitude matrix el- ement, Phys. Rev. D 106 (7) (2022) 074505

    2022

  21. [28]

    H. Y. Xing, M. Ding, Z. F. Cui, A. V. Pimikov, C. D. Ro- berts, S. M. Schmidt, Constraining the pion distribution amplitude using Drell-Yan reactions on a proton, Phys. Lett. B 849 (2024) 138462

    2024

  22. [29]

    Lu, Y.-Z

    Y. Lu, Y.-Z. Xu, K. Raya, C. D. Roberts, J. Rodr ´ ıguez- Quintero, Pion distribution functions from low-order Mellin moments, Phys. Lett. B 850 (2024) 138534

    2024

  23. [30]

    Alexandrou, et al., Quark and gluon momentum frac- tions in the pion and in the kaon – arXiv:2405.08529 [hep- lat]

    C. Alexandrou, et al., Quark and gluon momentum frac- tions in the pion and in the kaon – arXiv:2405.08529 [hep- lat]

    arXiv

  24. [31]

    Chang, I

    L. Chang, I. C. Cloet, J. J. Cobos-Martinez, C. D. Ro- berts, S. M. Schmidt, P. C. Tandy, Imaging dynamical chiral symmetry breaking: pion wave function on the light front, Phys. Rev. Lett. 110 (2013) 132001

    2013

  25. [32]

    Chang, C

    L. Chang, C. Mezrag, H. Moutarde, C. D. Roberts, J. Rodr ´ ıguez-Quintero, P. C. Tandy, Basic features of the pion valence-quark distribution function, Phys. Lett. B 737 (2014) 23–29

    2014

  26. [33]

    M. Ding, K. Raya, D. Binosi, L. Chang, C. D. Roberts, S. M. Schmidt, Drawing insights from pion parton distri- butions, Chin. Phys. C (Lett.) 44 (2020) 031002

    2020

  27. [34]

    M. Ding, K. Raya, D. Binosi, L. Chang, C. D. Roberts, S. M. Schmidt, Symmetry, symmetry breaking, and pion parton distributions, Phys. Rev. D 101 (5) (2020) 054014

    2020

  28. [35]

    Diehl, Generalized parton distributions, Phys

    M. Diehl, Generalized parton distributions, Phys. Rept. 388 (2003) 41–277

    2003

  29. [36]

    Nakanishi, A General survey of the theory of the Bethe-Salpeter equation, Prog

    N. Nakanishi, A General survey of the theory of the Bethe-Salpeter equation, Prog. Theor. Phys. Suppl. 43 (1969) 1–81

    1969

  30. [37]

    R. K. Ellis, W. J. Stirling, B. R. Webber, QCD and collider physics, Cambridge University Press, Cambridge, UK, 1991

    1991

  31. [38]

    Yin, Y.-Z

    P.-L. Yin, Y.-Z. Xu, Z.-F. Cui, C. D. Roberts, J. Rodr ´ ıguez-Quintero, All-Orders Evolution of Parton Distributions: Principle, Practice, and Predictions, Chin. Phys. Lett. Express 40 (9) (2023) 091201

    2023

  32. [39]

    Grunberg, Renormalization Group Improved Pertur- bative QCD, Phys

    G. Grunberg, Renormalization Group Improved Pertur- bative QCD, Phys. Lett. B 95 (1980) 70, [Erratum: Phys. Lett. B 110, 501 (1982)]

    1980

  33. [40]

    Grunberg, Renormalization Scheme Independent QCD and QED: The Method of Effective Charges, Phys

    G. Grunberg, Renormalization Scheme Independent QCD and QED: The Method of Effective Charges, Phys. Rev. D 29 (1984) 2315

    1984

  34. [41]

    A. Deur, S. J. Brodsky, C. D. Roberts, QCD Running Couplings and Effective Charges, Prog. Part. Nucl. Phys. 134 (2024) 104081

    2024

  35. [42]

    Y. L. Dokshitzer, Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e− Annihilation by 11 Perturbation Theory in Quantum Chromodynamics. (In Russian), Sov. Phys. JETP 46 (1977) 641–653

    1977

  36. [43]

    V. N. Gribov, L. N. Lipatov, Deep inelastic electron scat- tering in perturbation theory, Phys. Lett. B 37 (1971) 78–80

    1971

  37. [44]

    L. N. Lipatov, The parton model and perturbation the- ory, Sov. J. Nucl. Phys. 20 (1975) 94–102

    1975

  38. [45]

    Altarelli, G

    G. Altarelli, G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) 298–318

    1977

  39. [46]

    Binosi, L

    D. Binosi, L. Chang, J. Papavassiliou, C. D. Roberts, Bridging a gap between continuum-QCD and ab ini- tio predictions of hadron observables, Phys. Lett. B 742 (2015) 183–188

    2015

  40. [47]

    Binosi, C

    D. Binosi, C. Mezrag, J. Papavassiliou, C. D. Roberts, J. Rodr ´ ıguez-Quintero, Process-independent strong run- ning coupling, Phys. Rev. D 96 (2017) 054026

    2017

  41. [48]

    Cui, J.-L

    Z.-F. Cui, J.-L. Zhang, D. Binosi, F. de Soto, C. Mezrag, J. Papavassiliou, C. D. Roberts, J. Rodr ´ ıguez-Quintero, J. Segovia, S. Zafeiropoulos, Effective charge from lattice QCD, Chin. Phys. C 44 (2020) 083102

    2020

  42. [49]

    H. Y. Xing, Z. Q. Yao, B. L. Li, D. Binosi, Z. F. Cui, C. D. Roberts, Developing predictions for pion fragmentation functions, Eur. Phys. J. C 84 (1) (2024) 82

    2024

  43. [50]

    Z.-F. Cui, M. Ding, F. Gao, K. Raya, D. Binosi, L. Chang, C. D. Roberts, J. Rodr ´ ıguez-Quintero, S. M. Schmidt, Kaon and pion parton distributions, Eur. Phys. J. C 80 (2020) 1064

    2020

  44. [51]

    Chang, F

    L. Chang, F. Gao, C. D. Roberts, Parton distributions of light quarks and antiquarks in the proton, Phys. Lett. B 829 (2022) 137078

    2022

  45. [52]

    Y. Lu, L. Chang, K. Raya, C. D. Roberts, J. Rodr ´ ıguez- Quintero, Proton and pion distribution functions in coun- terpoint, Phys. Lett. B 830 (2022) 137130

    2022

  46. [53]

    Cheng, Y

    P. Cheng, Y. Yu, H.-Y. Xing, C. Chen, Z.-F. Cui, C. D. Roberts, Perspective on polarised parton distribution functions and proton spin, Phys. Lett. B 844 (2023) 138074

    2023

  47. [54]

    Y. Yu, P. Cheng, H.-Y. Xing, F. Gao, C. D. Roberts, Contact interaction study of proton parton distributions, Eur. Phys. J. C 84 (7) (2024) 739

    2024

  48. [55]

    Z. Q. Yao, Y. Z. Xu, D. Binosi, Z. F. Cui, M. Ding, K. Raya, C. D. Roberts, J. Rodr ´ ıguez-Quintero, S. M. Schmidt, Nucleon Gravitational Form Factors – arXiv:2409.15547 [hep-ph]

    arXiv

  49. [56]

    Y. Yu, C. D. Roberts, Impressions of Parton Distribution Functions, Chin. Phys. Lett. 41 (2024) 121202

    2024

  50. [57]

    C. D. Roberts, A. G. Williams, Dyson-Schwinger equa- tions and their application to hadronic physics, Prog. Part. Nucl. Phys. 33 (1994) 477–575

    1994

  51. [58]

    H. J. Munczek, Dynamical chiral symmetry break- ing, Goldstone’s theorem and the consistency of the Schwinger-Dyson and Bethe-Salpeter Equations, Phys. Rev. D 52 (1995) 4736–4740

    1995

  52. [59]

    Bender, C

    A. Bender, C. D. Roberts, L. von Smekal, Goldstone The- orem and Diquark Confinement Beyond Rainbow- Lad- der Approximation, Phys. Lett. B 380 (1996) 7–12

    1996

  53. [60]

    S.-X. Qin, C. D. Roberts, Resolving the Bethe-Salpeter kernel, Chin. Phys. Lett. Express 38 (7) (2021) 071201

    2021

  54. [61]

    Navas, et al., Review of particle physics, Phys

    S. Navas, et al., Review of particle physics, Phys. Rev. D 110 (3) (2024) 030001

    2024

  55. [62]

    Gell-Mann, F

    M. Gell-Mann, F. E. Low, Quantum electrodynamics at small distances, Phys. Rev. 95 (1954) 1300–1312

    1954

  56. [63]

    S. J. Brodsky, A. Deur, C. D. Roberts, The Secret to the Strongest Force in the Universe, Sci. Am. 5 (May) (2024) 32–39

    2024

  57. [64]

    Chang, Y.-X

    L. Chang, Y.-X. Liu, C. D. Roberts, Y.-M. Shi, W.- M. Sun, H.-S. Zong, Chiral susceptibility and the scalar Ward identity, Phys. Rev. C 79 (2009) 035209

    2009

  58. [65]

    S.-X. Qin, L. Chang, Y.-X. Liu, C. D. Roberts, D. J. Wilson, Interaction model for the gap equation, Phys. Rev. C 84 (2011) 042202(R)

    2011

  59. [66]

    C. S. Fischer, R. Williams, Probing the gluon self- interaction in light mesons, Phys. Rev. Lett. 103 (2009) 122001

    2009

  60. [67]

    Chang, C

    L. Chang, C. D. Roberts, Sketching the Bethe-Salpeter kernel, Phys. Rev. Lett. 103 (2009) 081601

    2009

  61. [68]

    S.-X. Qin, L. Chang, Y.-X. Liu, C. D. Roberts, D. J. Wil- son, Investigation of rainbow-ladder truncation for ex- cited and exotic mesons, Phys. Rev. C 85 (2012) 035202

    2012

  62. [69]

    Chang, C

    L. Chang, C. D. Roberts, Tracing masses of ground-state light-quark mesons, Phys. Rev. C 85 (2012) 052201(R)

    2012

  63. [70]

    Chang, Y.-X

    L. Chang, Y.-X. Liu, C. D. Roberts, Dressed-quark anomalous magnetic moments, Phys. Rev. Lett. 106 (2011) 072001

    2011

  64. [71]

    Maris, C

    P. Maris, C. D. Roberts, π and K meson Bethe-Salpeter amplitudes, Phys. Rev. C 56 (1997) 3369–3383

    1997

  65. [72]

    Maris, P

    P. Maris, P. C. Tandy, QCD modeling of hadron physics, Nucl. Phys. Proc. Suppl. 161 (2006) 136–152

    2006

  66. [73]

    Krassnigg, Excited mesons in a Bethe-Salpeter ap- proach, PoS CONFINEMENT 8 (2008) 075

    A. Krassnigg, Excited mesons in a Bethe-Salpeter ap- proach, PoS CONFINEMENT 8 (2008) 075

    2008

  67. [74]

    Schlessinger, C

    L. Schlessinger, C. Schwartz, Analyticity as a Useful Computation Tool, Phys. Rev. Lett. 16 (1966) 1173– 1174

    1966

  68. [75]

    Schlessinger, Use of Analyticity in the Calculation of Nonrelativistic Scattering Amplitudes, Phys

    L. Schlessinger, Use of Analyticity in the Calculation of Nonrelativistic Scattering Amplitudes, Phys. Rev. 167 (1968) 1411–1423

    1968

  69. [76]

    R. A. Tripolt, I. Haritan, J. Wambach, N. Moiseyev, Threshold energies and poles for hadron physical prob- lems by a model-independent universal algorithm, Phys. Lett. B 774 (2017) 411–416

    2017

  70. [77]

    Z.-F. Cui, D. Binosi, C. D. Roberts, S. M. Schmidt, Hadron and light nucleus radii from electron scattering, Chin. Phys. C 46 (12) (2022) 122001

    2022

  71. [78]

    Z. F. Cui, M. Ding, J. M. Morgado, K. Raya, D. Binosi, L. Chang, J. Papavassiliou, C. D. Roberts, J. Rodr ´ ıguez- Quintero, S. M. Schmidt, Concerning pion parton distri- butions, Eur. Phys. J. A 58 (1) (2022) 10

    2022

  72. [79]

    Z. F. Cui, M. Ding, J. M. Morgado, K. Raya, D. Bi- nosi, L. Chang, F. De Soto, C. D. Roberts, J. Rodr ´ ıguez- Quintero, S. M. Schmidt, Emergence of pion parton dis- tributions, Phys. Rev. D 105 (9) (2022) L091502

    2022

  73. [80]

    Raya, Z.-F

    K. Raya, Z.-F. Cui, L. Chang, J.-M. Morgado, C. D. Roberts, J. Rodr ´ ıguez-Quintero, Revealing pion and kaon structure via generalised parton distributions, Chin. Phys. C 46 (26) (2022) 013105

    2022

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