Unified Description of Pseudoscalar Meson Structure from Light to Heavy Quarks

TL;DR

This paper presents a unified, symmetry-preserving algebraic light-front model for pseudoscalar mesons that connects leading-twist PDAs, LFWFs, GPDs, PDFs, EFFs, charge radii, and IPS-GPDs through a shared Bethe–Salpeter amplitude formulated via the Nakanishi Integral Representation. By anchoring the framework in DSE/BSE insights and employing a rainbow–ladder truncation, the authors capture dynamical chiral symmetry breaking and confinement while retaining analytic tractability. The study systematically analyzes light, heavy-light, and heavy-heavy sectors, revealing a clear mass-dependent transition from broad, asymmetric momentum distributions to symmetric, compact spatial configurations; results show good qualitative and quantitative alignment with lattice QCD and continuum approaches. The model thus offers a transparent, three-dimensional picture of hadron structure across quark-mass regimes and provides a versatile tool for exploring correlations among diverse observables within a single coherent framework.

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 ($π$, $K$), heavy-light ($D$, $D_s$, $B$, $B_s$, $B_c$), and heavy-heavy ($η_c$, $η_b$) 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.

Figures (17)

• Figure S1: Pion and kaon parton distribution amplitudes (PDAs) evaluated at the hadronic scale $\zeta_H$. All distributions were obtained within the Schwinger–Dyson equations (SDE) framework Cui:2020tdfDing:2015rkn and parametrized following Eqs. \ref{['eq:PDAsSDE']}. For reference, the asymptotic form $\phi_{\text{asy}}(x) = 6x(1-x)$ is also displayed.
• Figure S2: Light-front wave functions of the pion and kaon calculated from Eq. \ref{['LFWF']}. The wave functions are shown as $\psi_{M}(x,k_\perp^2) \to \psi_{M}(x,k_\perp^2)/(16\pi^2 f_{M})$. All masses are expressed in GeV.
• Figure S3: Valence quark GPDs for the $D$, $D_s$, $B$, $B_s$, and $B_c$ mesons obtained from Eq. \ref{['GPD_pseudo']} for $\xi=0$. Mass units are given in GeV.
• Figure S4: Valence-quark PDFs at the hadronic scale $\zeta_H$. 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 $q_{sf}(x) = 30 x^2 (1-x)^2$ for comparison.
• Figure S5: 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.