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

This paper investigates the mechanical properties of pseudoscalar charmonium and bottomonium within the light-front quark model by evaluating gravitational form factors and analyzing their spatial distributions, revealing that while most properties are sensitive to wave function choices near the meson's center, the pressure distribution exhibits a sign-changing node and the force distribution remains positive to support stability.

The Gist

Imagine a proton or a heavy particle like a "charmonium" or "bottomonium" not as a solid marble, but as a tiny, vibrating cloud of energy held together by invisible forces. For a long time, physicists have been able to map out where the electric charge lives inside these particles, kind of like drawing a map of where the "electricity" is concentrated. But they haven't been able to see the "mechanics" of the particle: Where is the pressure? Where is the force pushing things apart? Where is the force pulling them together?

This paper is like taking a high-resolution X-ray of the internal pressure and stress inside two specific types of heavy particles: the charmonium (made of heavy charm quarks) and the bottomonium (made of even heavier bottom quarks).

Here is a breakdown of what the researchers did and found, using simple analogies:

1. The Tool: A "Light-Front" Camera

To see inside these particles, the scientists used a specific mathematical framework called the Light-Front Quark Model.

• The Analogy: Imagine trying to understand a spinning top. If you look at it from the side, it's a blur. But if you could "freeze" time and look at it from a specific angle (the "light-front"), you could see exactly how the parts are moving and where the weight is distributed. This model allows them to calculate the Energy-Momentum Tensor, which is essentially a report card on how energy, pressure, and stress are distributed inside the particle.

2. The Two Maps: Testing Different Shapes

The researchers didn't just draw one map; they drew two. They used two different mathematical "shapes" (called wave functions) to describe how the quarks are arranged inside the particle.

• The Analogy: Think of trying to guess the shape of a cloud. One guess says it's a perfect sphere (Set I), and the other says it's a slightly squashed sphere (Set II). By comparing the results of both, the scientists could see which parts of their map are solid facts and which parts depend on how they guessed the shape.

3. The Findings: What's Happening Inside?

A. The Pressure Map (The "Balloon" Effect)
The most interesting discovery is about pressure.

• The Center: Deep inside the particle, the pressure is positive. Imagine a balloon being squeezed from the outside; the air inside is pushing back hard. This is a repulsive force keeping the quarks from collapsing into each other.
• The Edge: As you move away from the center toward the edge of the particle, the pressure flips. It becomes negative. This is like a magnetic pull or a rubber band stretching, trying to hold the particle together so it doesn't fly apart.
• The "Node": There is a specific ring where the pressure is exactly zero. This is the boundary where the "pushing out" stops and the "pulling in" begins. The researchers found this happens very close to the center (about 0.14 femtometers for charmonium and even closer for bottomonium).

B. The Force Distribution (Stability)
The paper checks if the particle is stable.

• The Analogy: For a building to stand, the forces pushing it up must balance the forces pulling it down. The researchers found that the net force inside these particles always points outward (positive). This confirms that the particles are stable and won't spontaneously fall apart, satisfying a famous physics rule called the "von Laue condition."

C. The "Heavy" Difference
They compared the charmonium (lighter heavy particle) with the bottomonium (heavier heavy particle).

• The Result: The bottomonium is much more compact. Its internal pressure and energy are concentrated in a much smaller area than the charmonium.
• The Analogy: If charmonium is like a fluffy marshmallow, bottomonium is like a dense lead ball. The "fluffy" one has its forces spread out over a wider area, while the "dense" one has all its energy crammed into a tiny core.

D. Sensitivity to the "Shape"
The researchers found that the results near the very center of the particle depend heavily on which "shape" (wave function) they guessed.

• The Analogy: If you are trying to guess the temperature in the very center of a fire, your guess matters a lot. But if you look at the edge of the fire, the temperature is cool regardless of your guess. Similarly, the pressure and energy near the center of the particle change based on the math used, but the behavior at the edges is consistent.

4. Why This Matters (According to the Paper)

The paper doesn't claim this will lead to new engines or medical devices. Instead, it claims to provide a theoretical blueprint.

• It helps physicists understand how nature holds heavy particles together.
• It offers a "stress test" for the laws of physics (Quantum Chromodynamics) in the heavy-quark sector.
• It provides data that future experiments (like the Electron-Ion Collider) and computer simulations (Lattice QCD) can use to check if their own models are correct.

In Summary:
This paper is a detailed stress-test of two heavy, exotic particles. It reveals that inside these tiny worlds, there is a fierce battle between a repulsive force in the center (pushing apart) and an attractive force on the outside (holding together). The heavier the particle, the tighter this battle is packed into a smaller space.

Technical Summary

Technical Summary: Mechanical Distribution of Pseudoscalar Charmonium and Bottomonium on the Light-Front

Problem Statement
While the electromagnetic structure of hadrons is well-established through electromagnetic form factors (EMFFs), their mechanical properties—specifically internal pressure, shear stress, and force distributions—remain less explored. These properties are encoded in the Gravitational Form Factors (GFFs), particularly the $A$-term (related to mass and momentum) and the $D$-term (related to internal stress and stability, often called the "Druck" term). Although significant progress has been made in understanding the mechanical structure of nucleons and pions, studies focusing on the heavy mesonic sector, specifically pseudoscalar quarkonia ($\eta_c$ and $\eta_b$), are limited. Understanding these heavy systems is crucial for probing the interplay between nonperturbative QCD dynamics and heavy-quark effects, offering insights into confinement and the internal mechanical stability of heavy hadrons.

Methodology
The authors investigate the energy-momentum tensor (EMT) of pseudoscalar charmonium ($\eta_c$) and bottomonium ($\eta_b$) within the framework of the Light-Front Constituent Quark Model (LFQM).

• Formalism: The study utilizes a self-consistent LFQM approach, specifically adopting the Bakamjian–Thomas (BT) construction to handle zero-mode contributions and restore covariance. This is necessary because calculating the $D$-term typically involves "bad" current components sensitive to light-front zero modes.
• Wave Functions: To examine the sensitivity of results to the internal quark-antiquark distribution, two distinct Gaussian forms for the spatial part of the light-front wave function (LFWF) are employed:
  1. SET-I ($\phi_{CJ}$): Based on the Chung-Coester-Polyzou form, utilizing a Jacobian factor derived from relating instant-form differentials to light-front variables ($M_0/4x(1-x)$).
  2. SET-II ($\phi_{TK}$): Based on the form introduced in Refs. [76, 77], utilizing a Jacobian factor of $1/2x(1-x)$.
• Calculation: The GFFs ($A$ and $D$) are evaluated as matrix elements of the EMT operator containing non-interacting quark fields. The spatial mechanical distributions (momentum density, pressure, shear stress, force density, and internal energy density) in the transverse plane are obtained via the 2D Fourier transform of these GFFs. This approach respects relativistic invariance and avoids the quantum delocalization issues associated with 3D Breit-frame definitions.
• Parameters: Calculations use constituent quark masses ($m_c = 1.68$ GeV, $m_b = 5.10$ GeV) and harmonic scale parameters ($\beta_{\eta_c} = 0.699$ GeV, $\beta_{\eta_b} = 1.376$ GeV) adopted from previous literature.

Key Contributions and Results

• Gravitational Form Factors:
  • The $A$-term satisfies the momentum sum rule ($A(0)=1$) for both mesons. It decreases with increasing momentum transfer, with a steeper slope for bottomonium, indicating a more concentrated longitudinal momentum distribution compared to charmonium.
  • The $D$-term at zero momentum transfer is negative but significantly smaller in magnitude than the value of $-1$ predicted for Goldstone bosons (pions) in the chiral limit. The calculated values are approximately $-0.29$ for $\eta_c$ and $-0.13$ for $\eta_b$. The magnitude of the $D$-term decreases as the meson mass increases.
• Spatial Mechanical Distributions:
  • Momentum Density: Positive throughout the transverse plane. It is sensitive to the choice of wave function near the center of the meson but converges at larger transverse distances. The density is more concentrated in the central region for the heavier $\eta_b$.
  • Pressure Distribution: Exhibits a characteristic node where the sign changes from positive (repulsive) near the center to negative (attractive) at larger distances, eventually vanishing at the periphery. This behavior satisfies the von Laue stability condition ($\int z_\perp p(z_\perp) dz_\perp = 0$). The peak pressures for both quarkonia are substantially larger than those reported for the proton, pion, and Quark-Gluon Plasma (QGP). The position of the node is largely independent of the wave function choice.
  • Shear Stress: Shows noticeable sensitivity to the wave function choice in the intermediate transverse region, unlike other distributions which are sensitive primarily near the center.
  • Force Density: Defined as $F = p + \frac{1}{2}s$, it remains positive throughout the transverse plane for both mesons and both wave functions, satisfying the local stability condition (outward force).
  • Internal Energy Density: Positive and decreases with distance from the center. It is highly concentrated in the central region for $\eta_b$.
• Radii and Scales:
  • The mechanical radii (associated with the $D$-term) and momentum radii (associated with the $A$-term) are found to be smaller than the charge radii (associated with EMFFs).
  • For charmonium, the hierarchy $\sqrt{\langle r^2_A \rangle} < \sqrt{\langle r^2_F \rangle} < \sqrt{\langle r^2_D \rangle}$ holds, consistent with pion studies. This hierarchy does not hold for bottomonium.
  • The mechanical distributions of bottomonium are confined to a region approximately half the size of those for charmonium.

Significance and Claims
The paper claims to provide a comprehensive description of the internal mechanical distributions of heavy quarkonium systems using light-front dynamics. By employing two distinct wave functions, the authors demonstrate that while most spatial distributions are sensitive to the wave function choice near the meson center, they become nearly insensitive toward the periphery, with the exception of shear stress which shows sensitivity in the intermediate region. The study confirms the stability of these heavy mesons through the satisfaction of both global (von Laue) and local force conditions.

The authors position these findings as useful theoretical inputs for future investigations into gravitational form factors, generalized parton distributions (GPDs), and lattice QCD studies of heavy hadronic systems. They modestly suggest that the work can be extended to excited states, higher Fock sectors, and spin-dependent mechanical properties in future research, but do not propose immediate experimental applications or specific new experimental proposals beyond the context of future Electron-Ion Collider (EIC) capabilities mentioned in the introduction as a general motivation for the field.