Mechanical distribution of the pseudoscalar charmonium and bottomonium on the light-front

Abstract: We investigate the energy-momentum tensor of pseudoscalar charmonium and bottomonium within the framework of the light-front quark model. The gravitational form factors (GFFs), namely the $A$ and $D$-terms, are evaluated in terms of the light-front wave functions. The corresponding spatial mechanical distributions in the transverse plane are obtained through the Fourier transform of these GFFs. To examine the sensitivity of the results to the internal quark-antiquark distribution inside the meson, two distinct Gaussian forms are employed for the spatial part of the wave function. We analyze several mechanical properties in the transverse plane, including the momentum density, pressure distribution, shear stress, force density, and internal energy density. The pressure distribution exhibits a node where it changes sign from positive (repulsive) to negative (attractive) with increasing transverse distance. The force distribution remains positive throughout the transverse plane, supporting the stability condition proposed in earlier studies. Most of the spatial distributions, except for the shear stress, are found to be sensitive to the choice of the spatial wave function near the center of the meson, while they become nearly insensitive toward the periphery. In contrast, the shear stress distribution exhibits noticeable sensitivity to the choice of wave function in the intermediate transverse region.

• The paper presents a novel application of the LFQM to evaluate EMT form factors in heavy quarkonia, identifying distinct D-term values for charmonium and bottomonium.
• It utilizes two light-front wave function prescriptions to elucidate the sensitivity of momentum, pressure, and shear distributions to the internal meson structure.
• The analysis confirms consistency with lattice QCD results and highlights how concentrated mechanical densities vary with quark mass.

Mechanical Structure of Pseudoscalar Charmonium and Bottomonium on the Light-Front

Introduction

The investigation of hadronic mechanical properties via energy-momentum tensor (EMT) form factors has emerged as a central theme in hadron physics, particularly in the context of the so-called $D$-term, which encodes the distribution of mechanical forces, including pressure and shear, within hadrons. While substantial progress has been made in nucleon and pion studies, the mechanical structure of heavy quarkonia, such as charmonium ($\eta_c$) and bottomonium ($\eta_b$), remains comparatively unexplored due to the theoretical complexities of nonperturbative QCD in the heavy-quark sector. The light-front quark model (LFQM) provides a tractable framework for evaluating mesonic gravitational form factors (GFFs), specifically the $A$ and $D$-terms, by expressing the EMT matrix elements in terms of light-front wave functions (LFWFs). This study extends the formalism to heavy pseudoscalar mesons, examining both the sensitivity of mechanical distributions to the internal quark-antiquark wave function and the correlation between spatial distributions and observables such as pressure, momentum, and internal energy in the transverse plane.

Framework: Light-Front Quark Model and Gravitational Form Factors

The analysis is performed in the LFQM, where the mesonic states are described in the valence quark-antiquark sector using LFWFs obtained via the Brodsky–Huang–Lepage (BHL) prescription. Two classes of spatial wave functions are considered, differing by their so-called "Jacobi factors," to probe model dependence. The GFFs—$A(q^2)$ and $D(q^2)$—are defined through matrix elements of the quark EMT operator, with explicit attention to resolving zero-mode ambiguities via the Bakamjian-Thomas (BT) construction.

Spatial mechanical densities in the transverse plane (pressure, shear, force, energy, and longitudinal momentum) are calculated using covariant two-dimensional Fourier transforms of the corresponding GFFs at fixed light-front time, ensuring relativistic consistency and avoiding quantum delocalization artifacts inherent to three-dimensional Breit-frame densities.

Analysis of Gravitational Form Factors

A-term and D-term Behavior

The $A$-term, associated with the quark and antiquark contributions to internal energy and longitudinal momentum, is shown to satisfy the momentum sum rule $A(0) = 1$ for both charmonium and bottomonium. Critically, the $A$-term decreases with increasing momentum transfer, with greater sensitivity to the wave function at high transfer, particularly for charmonium. The slope of the $\eta_c$0-term at zero momentum transfer indicates that longitudinal momentum is more spatially concentrated in bottomonium, consistent with the heavy quark limit.

Figure 1: Momentum transfer dependence of the $\eta_c$1-term for $\eta_c$2 and $\eta_c$3, comparing two wave function prescriptions.

The $\eta_c$4-term, encoding mechanical stability through pressure and shear, is found to be negative at zero momentum transfer for both systems, with values $\eta_c$5 and $\eta_c$6, far from the $\eta_c$7 chiral limit result for pions, as expected in the non-Goldstone, heavy mass regime. The magnitude of $\eta_c$8 is smaller in bottomonium, and sensitivity to the internal wave function is again stronger for charmonium.

Figure 2: Dependence of the $\eta_c$9-term on squared momentum transfer for $\eta_b$0 and $\eta_b$1.

The electromagnetic form factor (EMFF) is also computed and compared with lattice QCD results for $\eta_b$2, showing satisfactory consistency within uncertainties.

Figure 3: Electromagnetic form factors for $\eta_b$3 and $\eta_b$4 with lattice QCD comparison for $\eta_b$5.

A unified display of all form factors (GFFs and EMFFs) reveals the scale hierarchy between distributions and clarifies the role of each form factor in fixing spatial widths and force profiles.

Figure 4: Momentum transfer dependence of all form factors for $\eta_b$6 and $\eta_b$7; the normalized $\eta_b$8-term is displayed for comparison.

Spatial Mechanical Densities

Light-front Mechanical Distributions

Longitudinal momentum density is maximized at the center of the meson ($\eta_b$9) and shows marked sensitivity to the wave function; the distribution for $A$0 is markedly narrower, reflecting the higher constituent mass.

Figure 5: Light-front longitudinal momentum density for $A$1 and $A$2 as a function of transverse position.

The pressure distribution exhibits a node, transitioning from positive at small $A$3 (repulsive core) to negative at larger $A$4 (attractive surface), as required by the von Laue stability condition. The node occurs at $A$5 fm for $A$6 and $A$7 fm for $A$8, essentially independent of the chosen wave function. Maximum pressures at the center substantially exceed QGP, proton, and pion values, highlighting the intense internal forces in heavy quarkonia.

The pressure is highly sensitive to the wave function near $A$9, with less dependence in the periphery.

Shear, Force, and Internal Energy Densities

The shear stress displays a characteristic peak near the pressure node position; its intermediate-$D$0 behavior is notably sensitive to the wave function. The total mechanical force density, defined as $D$1, remains positive throughout the transverse plane for both mesons, satisfying the local stability criterion.

Figure 6: Shear stress distribution in the transverse plane for $D$2 and $D$3.

Internal energy density peaks at the center, especially for bottomonium, and declines rapidly for increasing $D$4. As for other densities, sensitivity to the wave function is localized near the core.

Numerical Results: Radii and Hierarchies

The analysis yields root-mean-square radii for mechanical (momentum and force) and charge sectors. In the $D$5D transverse plane, momentum and force radii are nearly identical, and all are substantially smaller for bottomonium than for charmonium. The mechanical radii are consistently smaller than the charge radii, indicating that internal mechanical forces are more localized than the overall charge distribution.

The hierarchy $D$6 holds for $D$7, in line with the scaling observed in pions, but breaks down in the heavier bottomonium. This supports the presence of nontrivial QCD dynamics in heavy-flavor systems.

Theoretical and Practical Implications

Mechanically, the quarkonia exhibit concentrated internal energy, pressure, and force densities, with spatial distributions highly sensitive to both the constituent mass and the adopted LFWF shape. The reduction of $D$8 with increasing quark mass underscores the departure from the chiral symmetry-driven structure of light mesons and highlights the influence of heavy-quark symmetry in the EMT sector.

The observed sensitivity of central spatial densities to wave function choice underscores the importance of precision modeling of heavy-meson LFWFs and motivates further investigation of higher Fock sectors, radiative corrections, and model-independent constraints via lattice QCD and GPD physics.

Experimentally, advances in hard exclusive processes (e.g., deeply virtual Compton scattering and generalized parton distribution measurements at future EIC facilities) may eventually provide access to heavy-meson GFFs, making such theoretical investigations essential benchmarks.

Conclusion

This work provides a detailed, two-dimensional light-front characterization of the mechanical structure of heavy pseudoscalar quarkonia, delivering both form-factor data and spatial distributions for energy, momentum, pressure, shear, and force. All results are presented for two distinct LFWF parameterizations, allowing for model sensitivity estimates. The formalism and findings can be directly extended to other heavy mesonic systems, excited states, or to higher Fock sector analyses. These results are expected to inform both the interpretation of future experimental measurements of heavy-hadron structure and theoretical efforts to quantify confinement and QCD mechanical stability in the heavy-flavor sector.