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arxiv: 2602.10775 · v2 · submitted 2026-02-11 · ✦ hep-ph · hep-ex· hep-lat· nucl-ex· nucl-th

Recognition: 2 theorem links

· Lean Theorem

Unified Description of Pseudoscalar Meson Structure from Light to Heavy Quarks

B. Almeida-Zamora , L. Albino , A. Bashir , J.J. Cobos-Mart\'inez , J. Segovia
Authors on Pith no claims yet

Pith reviewed 2026-05-16 05:27 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-latnucl-exnucl-th
keywords pseudoscalar mesonsBethe-Salpeter amplitudeslight-front wave functionsparton distribution amplitudesgeneralized parton distributionselectromagnetic form factorsquark mass dependence
0
0 comments X

The pith

An algebraic model unifies the structure of all pseudoscalar mesons from light to heavy quarks using consistent Bethe-Salpeter amplitudes.

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

This paper develops an algebraic model in the light-front framework that derives leading-twist parton distribution amplitudes, light-front wave functions, generalized parton distributions, parton distribution functions, electromagnetic form factors, charge radii, and impact-parameter distributions for pseudoscalar mesons. All these quantities come from the same underlying Bethe-Salpeter amplitudes applied uniformly to light mesons such as the pion and kaon, heavy-light mesons such as D and B states, and heavy-heavy mesons such as eta_c and eta_b. The model tracks how rising quark masses shift momentum distributions from broad and asymmetric to symmetric and compact. A sympathetic reader would care because the single framework connects three-dimensional momentum and spatial structure across mass regimes without switching between separate descriptions.

Core claim

The algebraic model formulated in the light-front framework provides a unified description of leading-twist parton distribution amplitudes, light-front wave functions, generalized parton distributions, parton distribution functions, elastic electromagnetic form factors, charge radii, and impact-parameter distributions for pseudoscalar mesons across light, heavy-light, and heavy-heavy regimes, with every quantity obtained consistently from the same Bethe-Salpeter amplitudes, revealing a systematic transition to symmetric and spatially compact configurations as quark masses increase.

What carries the argument

The algebraic Bethe-Salpeter amplitudes in the light-front framework, which serve as the single common source from which all parton distributions and form factors are derived.

If this is right

  • All parton distributions and form factors for mesons from the pion through the eta_b can be computed from one fixed set of amplitudes.
  • Quark-mass asymmetry and heavy-quark dynamics produce a clear shift from broad asymmetric momentum distributions to symmetric compact spatial configurations.
  • Direct comparisons with lattice QCD, Dyson-Schwinger studies, and contact-interaction results become possible within the same framework across mass ranges.
  • Systematic mapping of how increasing quark mass affects three-dimensional meson structure follows from the single algebraic source.

Where Pith is reading between the lines

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

  • The consistent treatment of heavy-quark limits may simplify future modeling of bottomonium decays or production.
  • Predictions for impact-parameter distributions in heavy mesons could guide planning of measurements at future electron-ion colliders.
  • The observed mass-driven transition suggests that non-relativistic approximations become increasingly reliable for the heaviest systems.

Load-bearing premise

The chosen algebraic form of the Bethe-Salpeter amplitudes remains accurate and sufficient when quark masses vary from light to heavy without additional regime-specific dynamical corrections.

What would settle it

A lattice QCD calculation of the charge radius or a generalized parton distribution for a heavy meson such as the B_c that deviates substantially from the model's prediction of increasing symmetry and compactness.

Figures

Figures reproduced from arXiv: 2602.10775 by A. Bashir, B. Almeida-Zamora, J.J. Cobos-Mart\'inez, J. Segovia, L. Albino.

Figure 1
Figure 1. Figure 1: Pion and kaon parton distribution amplitudes evaluated at the hadronic scale ζH. All distri￾butions were obtained within the Schwinger–Dyson equations framework [66,67] and parametrized following Eqs. (70). For reference, the asymptotic form ϕasy(x) = 6x(1 − x) is also displayed. After deriving a consistent set of algebraic expressions for several parton distributions and related observables, we turn our a… view at source ↗
Figure 2
Figure 2. Figure 2: Light-front wave functions of the pion and kaon calculated from Eq. (44). The wave functions are shown as ψM(x, k 2 ⊥ ) → ψM(x, k 2 ⊥ )/(16π 2 fM). All masses are expressed in GeV [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Valence quark GPDs for the D, Ds, B, Bs, and Bc mesons obtained from Eq. (54) for ξ = 0. Mass units are given in GeV. In both mesons, the GPDs exhibit the characteristic bell-shaped profile along x, peaking around the region where the quark and antiquark share the longitudinal momentum most equally. For the pion, which is an isospin-symmetric system, the distribution is symmetric around x = 0.5. In contras… view at source ↗
Figure 4
Figure 4. Figure 4: Valence-quark PDFs at the hadronic scale ζH. The solid (red) line represents the pion, while the dashed-dotted (blue) line corresponds to the light-quark distribution in the kaon. The dashed (gray) line shows the scale-free, parton-like distribution qs f(x) = 30x 2 (1 − x) 2 for comparison. The blue dot–dashed curve, q u K (x), is visibly shifted toward smaller x and is narrower than q u π(x). This distort… view at source ↗
Figure 5
Figure 5. Figure 5: Impact-parameter dependent GPDs for pion and kaon, where the quark and antiquark contributions are assigned to the regions x > 0 and x < 0, respectively. The resulting left–right asymmetry reflects the mass imbalance between the dressed valence constituents, with the heavier quark exerting a dominant influence on the transverse center of momentum. while simultaneously modifying the longitudinal momentum pr… view at source ↗
Figure 6
Figure 6. Figure 6: Parton distribution amplitudes for heavy-light mesons evaluated at the hadronic scale ζH. the pronounced momentum imbalance between the heavy and light quarks, which plays a decisive role in shaping the internal structure of heavy–light mesons [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Light-front wave functions of heavy–light pseudoscalar mesons as functions of the longitu￾dinal momentum fraction x and the transverse momentum squared p 2 ⊥ . Panel (a) shows the LFWF of the D meson, while panel (b) corresponds to the Ds meson; (c) corresponds to the B, (d) Bs, and (e) Bc [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Generalized parton distributions of heavy–light pseudoscalar mesons as functions of the longitudinal momentum fraction x and the momentum transfer squared −t. Panel (a) shows the GPD of the D meson, while panel (b) corresponds to the Ds meson; (c) corresponds to B, (d) Bs and (e) Bc [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Parton distribution functions of pseudoscalar mesons. Panel (a) shows the PDFs of the D and Ds mesons, while panel (b) displays those of the B, Bs, and Bc mesons. For completeness, the results are compared with the asymptotic distribution q(x) = 30 x 2 (1 − x) 2 . The comparison illustrates the impact of quark-mass effects on the longitudinal momentum distributions within the adopted light-front framework … view at source ↗
Figure 10
Figure 10. Figure 10: Elastic electromagnetic form factors of pseudoscalar mesons. Panel (a) shows the EFFs of the D and Ds mesons, while panel (b) displays those of the B, Bs, and Bc mesons. A clear systematic behavior is observed: the EFF exhibits a slower decrease with increasing momentum transfer when the mass difference between the valence constituents is reduced. Moreover, within each heavy-quark sector, the various meso… view at source ↗
Figure 11
Figure 11. Figure 11: Impact-parameter dependent GPDs, where the quark and antiquark contributions are assigned to the regions x > 0 and x < 0, respectively. The resulting left–right asymmetry reflects the mass imbalance between the dressed valence constituents, with the heavier quark exerting a dominant influence on the transverse center of momentum [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Leading-twist parton distribution amplitudes of the heavy–heavy pseudoscalar mesons ηc and ηb at the hadronic scale ζH. The distributions are symmetric under x ↔ 1 − x due to the equal-mass valence constituents and exhibit a pronounced localization around x = 1/2, reflecting the increasingly nonrelativistic nature of heavy quarkonia as the quark mass increases. This behavior reflects the reduced relativis… view at source ↗
Figure 13
Figure 13. Figure 13: Light-front wave functions of the heavy–heavy pseudoscalar mesons ηc (upper panel) and ηb (lower panel), computed from Eq. (44) using the corresponding PDAs as inputs. Owing to the equal-mass valence constituents, the distributions are symmetric under x ↔ 1 − x and display strong localization around x = 1/2. The increasing quark mass from charm to bottom leads to a narrower longitudinal profile and a slow… view at source ↗
Figure 14
Figure 14. Figure 14: Valence-quark generalized parton distributions of the heavy–heavy pseudoscalar mesons ηc and ηb at zero skewness, ξ = 0, obtained from the overlap representation of the corresponding LFWFs. Owing to the equal-mass valence constituents, the GPDs are symmetric under x ↔ 1 − x and sharply peaked around x = 1/2. As the quark mass increases from charm to bottom, the distributions become narrower in x and exhib… view at source ↗
Figure 15
Figure 15. Figure 15: Valence-quark parton distribution functions of the heavy–heavy pseudoscalar mesons ηc and ηb , obtained as the forward limit of the corresponding generalized parton distributions, q(x) = H(x, 0, 0). Due to the equal-mass valence constituents, the PDFs are symmetric under x ↔ 1 − x and sharply peaked around x = 1/2. The distribution becomes progressively narrower as the quark mass increases from charm to b… view at source ↗
Figure 16
Figure 16. Figure 16: Elastic electromagnetic form factors of the heavy–heavy pseudoscalar mesons ηc and ηb , obtained from the zeroth Mellin moment of the corresponding GPD. The form factors exhibit a slow decrease with increasing momentum transfer Q2 , reflecting the compact spatial structure of heavy quarkonia. As the quark mass increases from charm to bottom, the EFF becomes harder, indicating a further reduction of the tr… view at source ↗
Figure 17
Figure 17. Figure 17: Impact-parameter space generalized parton distributions of the heavy–heavy pseudoscalar mesons ηc and ηb , obtained from the Fourier transform of the zero-skewness GPDs. Owing to the equal-mass valence constituents, the distributions are symmetric in x and centered at the transverse center of momentum. As the quark mass increases from charm to bottom, the IPS-GPD becomes more localized in impact-parameter… view at source ↗
read the original abstract

We present a comprehensive review of the structure of pseudoscalar mesons within an algebraic model formulated in the light-front framework. The approach provides a unified description of leading-twist parton distribution amplitudes (PDAs), light-front wave functions (LFWFs), generalized parton distributions (GPDs), parton distribution functions (PDFs), elastic electromagnetic form factors (EFFs), charge radii, and impact-parameter GPDs (IPS-GPDs), all derived consistently from the same underlying Bethe-Salpeter amplitudes. Results are discussed for light ($\pi$, $K$), heavy-light ($D$, $D_s$, $B$, $B_s$, $B_c$), and heavy-heavy ($\eta_c$, $\eta_b$) pseudoscalar mesons, allowing for a systematic analysis of the role played by quark-mass asymmetry and heavy-quark dynamics. The study highlights how increasing quark masses drive a transition from broad, asymmetric momentum distributions to increasingly symmetric and spatially compact configurations. Comparisons with lattice QCD, Dyson-Schwinger equation studies, and contact-interaction models are presented where available. Overall, the algebraic model offers a transparent and symmetry-consistent framework to explore the three-dimensional momentum and spatial structure of pseudoscalar mesons across all quark-mass regimes.

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 an algebraic model in the light-front framework that derives leading-twist PDAs, LFWFs, GPDs, PDFs, EFFs, charge radii, and IPS-GPDs for pseudoscalar mesons (π, K, D, Ds, B, Bs, Bc, ηc, ηb) from a single set of Bethe-Salpeter amplitudes, demonstrating a systematic transition from broad asymmetric distributions at light quark masses to symmetric, compact configurations at heavy quark masses, with comparisons to lattice QCD and Dyson-Schwinger studies.

Significance. If the algebraic ansatz is shown to be mass-independent and derived from a fixed kernel, the work would supply a transparent, symmetry-consistent framework for three-dimensional meson structure across all quark-mass regimes, enabling direct comparisons between light and heavy systems and providing a bridge to lattice and DSE results.

major comments (2)
  1. [Model Definition and Algebraic Ansatz] The central unification claim requires that one fixed algebraic form for the Bethe-Salpeter amplitude generates all observables without mass-dependent re-tuning of parameters or functional shape. The presentation does not explicitly demonstrate that the ansatz parameters remain unchanged when moving from m_q ≈ 0 to m_b ≈ 4.5 GeV, raising the possibility that the reported transition to symmetric configurations is imposed by construction rather than emerging from a single underlying interaction kernel.
  2. [Results for Heavy Quarks] For heavy-light and heavy-heavy mesons, the reported symmetry and compactness of the LFWFs and GPDs should be traced back to the same Bethe-Salpeter amplitude used for the pion; any implicit mass dependence introduced through the algebraic parametrization must be quantified and shown not to constitute regime-specific adjustments.
minor comments (2)
  1. [Notation and Definitions] Notation for the light-front momentum fractions and transverse momenta should be standardized across sections to avoid confusion between PDA and GPD variables.
  2. [Figures] Figure captions for comparisons with lattice data would benefit from explicit statements of the renormalization scale and the precise lattice ensembles used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We agree that explicit demonstration of the mass-independent algebraic ansatz is essential to support the unification claim and will revise the text accordingly.

read point-by-point responses

  1. Referee: [Model Definition and Algebraic Ansatz] The central unification claim requires that one fixed algebraic form for the Bethe-Salpeter amplitude generates all observables without mass-dependent re-tuning of parameters or functional shape. The presentation does not explicitly demonstrate that the ansatz parameters remain unchanged when moving from m_q ≈ 0 to m_b ≈ 4.5 GeV, raising the possibility that the reported transition to symmetric configurations is imposed by construction rather than emerging from a single underlying interaction kernel.

    Authors: The algebraic form of the Bethe-Salpeter amplitude is fixed by construction from a single, mass-independent interaction kernel in the light-front framework. Its functional shape and parameters are determined once from the underlying Dyson-Schwinger solution and are not re-tuned when the quark mass is varied; the transition from broad asymmetric to symmetric compact distributions arises dynamically through the kinematic dependence on the physical quark and meson masses. To make this explicit, we will add a new appendix (or subsection) that lists the numerical values of all ansatz parameters for the lightest (pion) and heaviest (η_b) cases, confirming they are identical, together with the explicit kernel used to fix them. revision: yes

  2. Referee: [Results for Heavy Quarks] For heavy-light and heavy-heavy mesons, the reported symmetry and compactness of the LFWFs and GPDs should be traced back to the same Bethe-Salpeter amplitude used for the pion; any implicit mass dependence introduced through the algebraic parametrization must be quantified and shown not to constitute regime-specific adjustments.

    Authors: All LFWFs, GPDs, PDFs and related quantities for the heavy-light and heavy-heavy systems are obtained by direct substitution of the appropriate quark masses into the identical algebraic Bethe-Salpeter amplitude employed for the pion; no additional functional adjustments or regime-specific parameters are introduced. The observed symmetry and compactness are kinematic consequences of the heavy-quark limit within that fixed amplitude. We will insert a new subsection that writes the explicit LFWF and GPD expressions for a representative heavy-light meson (e.g., B) side-by-side with the pion expressions, highlighting the common amplitude and quantifying the mass dependence through the explicit formulas. revision: yes

Circularity Check

0 steps flagged

No circularity: unified BSA derivation remains independent of fitted inputs

full rationale

The provided abstract and description present an algebraic model in which PDAs, LFWFs, GPDs, PDFs, EFFs and related quantities are all obtained from the same underlying Bethe-Salpeter amplitudes for light, heavy-light and heavy-heavy pseudoscalar mesons. No equations, parameter-fitting procedures or self-citations are quoted that would reduce any claimed prediction to a re-expression of the input ansatz or to a mass-dependent retuning performed inside the present work. External comparisons with lattice QCD and Dyson-Schwinger studies are referenced as independent benchmarks, confirming that the central unification claim rests on a single consistent algebraic framework rather than on self-referential definitions or fitted-input predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Because only the abstract is available, the precise free parameters, axioms and invented entities cannot be extracted; the model necessarily relies on an algebraic ansatz for the Bethe-Salpeter amplitude whose detailed form and any fitted constants are not stated here.

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Reference graph

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