Heavy mesons with dynamical gluon on the light front

This paper investigates the structure of charmonium, bottomonium, and $\rm B_c$ mesons using the Basis Light-Front Quantization approach with both quark-antiquark and quark-antiquark-gluon Fock sectors to calculate various observables and provide the first BLFQ predictions for gluon parton distribution functions in heavy mesons.

The Gist

Imagine the universe is built out of tiny, invisible Lego bricks. In the world of physics, the most complex structures are called hadrons (like protons, neutrons, and mesons). For a long time, scientists have tried to figure out exactly how these Lego bricks snap together.

This paper is a new, high-tech blueprint for understanding a specific type of heavy Lego structure called heavy mesons. These are made of a heavy "quark" and an equally heavy "anti-quark" (like a charm quark and an anti-charm quark).

Here is the story of what the scientists did, explained simply:

1. The Old Map vs. The New Map

For years, scientists used a map that showed these heavy mesons as just two pieces stuck together: a quark and an anti-quark. It was like drawing a picture of a house and only showing the front door and the back door, ignoring the walls, the roof, and the people inside.

This paper says, "Wait a minute! There's a third piece we've been ignoring: the gluon."

• The Analogy: Imagine the quark and anti-quark are two dancers holding hands. The gluon is the invisible music and energy field that keeps them dancing together. Without the music, they fall apart. The scientists realized that to understand the dance perfectly, you have to include the music (the gluon) in your description, not just the dancers.

2. The "Light-Front" Camera

To study these particles, the team used a special method called Basis Light-Front Quantization (BLFQ).

• The Analogy: Imagine trying to take a photo of a speeding race car. If you take a normal photo, it's blurry. But if you use a "light-front" camera, you are essentially taking a snapshot of the car from the side as it zooms past, freezing time in a way that lets you see exactly how the wheels, engine, and driver are arranged at that exact moment.
• This method allows them to see the "internal structure" of the meson as if they were looking at a 3D hologram of the particles moving at near light speed.

3. The "Heavy" Problem

Heavy mesons are tricky. Because the quarks are so heavy (like bowling balls compared to ping-pong balls), they move slowly. This makes them easier to study than lighter particles, but it also means they are deeply influenced by the "glue" (the strong force) holding them together.

The scientists built a mathematical simulation (a virtual laboratory) where they could:

1. Put the heavy quark and anti-quark together.
2. Add the dynamic gluon (the dancing music).
3. Turn on the "confining potential" (a virtual rubber band that keeps them from flying apart).

4. What They Found

By running their simulation, they calculated several things that act like the "ID card" for these particles:

• The Mass (Weight): They predicted how heavy these particles should be. Their numbers matched real-world experiments very well, proving their "three-piece" model (Quark + Anti-Quark + Gluon) is more accurate than the old "two-piece" model.
• The Size (Radius): They figured out how big these particles are. They found that the excited versions (particles that are vibrating or "jumping" more) are slightly larger and looser than the calm, ground-state versions.
• The "Parton" Distribution (The Crowd): This is the most exciting part. They calculated how the "momentum" (the energy of movement) is shared among the quarks and the gluon.
  • The Analogy: Imagine a pie. In the old model, the pie was just split between the two quarks. In this new model, they found that the gluon actually eats a slice of the pie!
  • They discovered that the gluon tends to hang out in the "small-x" region (a specific way of measuring how much energy it carries). It's like finding out that the DJ at the party (the gluon) is actually carrying a significant amount of the party's energy, even though he isn't the main dancer.

5. Why This Matters

This is a big deal because it's the first time this specific method has been used to include the gluon explicitly for these heavy particles.

• Before: We had a blurry, incomplete picture.
• Now: We have a sharper, more realistic 3D model that includes the "glue" holding everything together.

The scientists say this is just the beginning. Now that they have this new, better blueprint, they can use it to predict things we haven't measured yet, like how these particles behave in high-energy collisions (like those at the Large Hadron Collider). It's like finally getting the full instruction manual for a complex machine, which helps us understand how the whole universe is built.

In short: They took a heavy particle, added the missing "glue" piece to their math, and got a much clearer, more accurate picture of how nature's building blocks fit together.

Technical Summary

Here is a detailed technical summary of the paper "Heavy mesons with dynamical gluon on the light front" by Jiatong Wu et al.

1. Problem Statement

Quantum Chromodynamics (QCD) describes the strong interaction, but calculating hadron structure from first principles remains challenging due to non-perturbative dynamics. While heavy mesons (charmonium, bottomonium, and $B_c$) are often treated as non-relativistic bound states of a quark and antiquark ($|q\bar{q}\rangle$), this approximation neglects the role of dynamical gluons. Although gluon radiation is suppressed in heavy mesons compared to light mesons, gluons act as the mediators of the strong force between heavy quarks. Previous Basis Light-Front Quantization (BLFQ) studies of heavy mesons were limited to the leading $|q\bar{q}\rangle$ Fock sector using effective one-gluon exchange (OGE) interactions. The lack of explicit dynamical gluon degrees of freedom ($|q\bar{q}g\rangle$) limits the accuracy of predictions for parton distribution functions (PDFs) and the understanding of the interplay between perturbative and non-perturbative QCD dynamics.

2. Methodology

The authors employ the Basis Light-Front Quantization (BLFQ) approach, a non-perturbative framework for solving relativistic bound-state problems.

• Fock Space Truncation: The study extends the basis space to include the $|q\bar{q}g\rangle$ sector (quark-antiquark-gluon) alongside the leading $|q\bar{q}\rangle$ sector. This allows for the explicit inclusion of one dynamical gluon.
• Hamiltonian Construction:
  • The input light-front Hamiltonian $P^-$ consists of the QCD kinetic and interaction terms plus a confining potential.
  • Confinement: A holographic-inspired confining potential is used, featuring both transverse ($\kappa_T$) and longitudinal ($\kappa_L$) components to reproduce 3D confinement in the non-relativistic limit.
  • Interactions: The Hamiltonian includes the quark-gluon vertex and the instantaneous gluon exchange interaction derived from light-front QCD in the $A^+=0$ gauge.
  • Renormalization: A sector-dependent renormalization scheme is implemented. A mass counterterm ($\delta m_q$) is introduced in the $|q\bar{q}\rangle$ sector to compensate for self-energy corrections arising from the higher $|q\bar{q}g\rangle$ sector. Effective mass parameters ($m_f$ for the vertex and $m_g$ for the gluon) are used to model non-perturbative effects and regulate infrared divergences.
  • Regularization: To restore rotational symmetry broken by basis truncation, a multiplicative regulating function is applied to the instantaneous gluon exchange interaction.
• Basis Functions: The wave functions are expanded using plane waves in the longitudinal direction and 2D Harmonic Oscillator (2D-HO) functions in the transverse direction. The calculation is performed with truncation parameters $N_{max}=12$ and $K=17$.
• Parameter Determination: Model parameters (quark masses, coupling constants, confinement strengths) are fitted to reproduce the experimental mass spectra of low-lying charmonium and bottomonium states. Parameters for the $B_c$ system are interpolated from the $c\bar{c}$ and $b\bar{b}$ systems.

3. Key Contributions

• First BLFQ Study with Dynamical Gluons for Heavy Mesons: This work represents the first application of the BLFQ framework to heavy quarkonium and $B_c$ mesons including the $|q\bar{q}g\rangle$ Fock sector.
• Realistic Initial Conditions for QCD Evolution: By explicitly including dynamical gluons at the model scale, the results provide a more realistic initial condition for perturbative QCD evolution, leading to more accurate predictions of parton structures at higher scales.
• Systematic Comparison: The study systematically compares results across charmonium, bottomonium, and $B_c$ systems, analyzing the impact of quark mass on gluon distribution and Fock sector probabilities.
• First Predictions for Gluon PDFs in Heavy Mesons: The paper presents the first predictions for gluon Parton Distribution Functions (PDFs) in heavy mesons derived from the $|q\bar{q}g\rangle$ light-front wave function.

4. Key Results

• Mass Spectra: The calculated mass spectra for the eight lowest-lying states of charmonium, bottomonium, and $B_c$ mesons show good agreement with experimental data (PDG) and lattice QCD. The inclusion of the dynamical gluon improves the mass predictions for both S-wave and P-wave states compared to previous OGE-only calculations (reducing RMS deviations to ~17 MeV for charmonium and ~23 MeV for bottomonium).
• Fock Sector Probabilities:
  • The leading $|q\bar{q}\rangle$ sector remains dominant, with probabilities ($N^2_{q\bar{q}}$) ranging from ~0.5 to ~0.8.
  • The probability of finding a dynamical gluon ($1 - N^2_{q\bar{q}}$) is substantial: ~36–50% for charmonium and ~18–31% for bottomonium.
  • The gluon probability decreases as the heavy quark mass increases (bottomonium < charmonium) and increases with excitation energy.
• Light-Front Wave Functions (LFWFs):
  • Excited states exhibit nodal structures. The longitudinal nodal structure is less pronounced than the transverse one, reflecting the dynamical breaking of rotational symmetry on the light front.
  • Bottomonium states are more tightly bound (smaller spatial size) than charmonium states.
• Electromagnetic Form Factors and Radii:
  • Charge radii increase with excitation energy.
  • The inclusion of the $|q\bar{q}g\rangle$ sector leads to systematically larger charge radii for charmonium compared to OGE-only calculations, bringing them closer to lattice QCD results. This is attributed to the spatially extended distribution of the unconfined $|q\bar{q}g\rangle$ component.
• Decay Constants:
  • Calculated decay constants generally agree with PDG data and other theoretical approaches (Lattice, DSE).
  • A discrepancy is noted where the vector decay constant ($J/\psi$) is smaller than the pseudoscalar ($\eta_c$), suggesting that higher Fock sectors might be needed for a complete description of the vector-pseudoscalar splitting.
• Parton Distribution Amplitudes (PDAs):
  • PDAs for bottomonium are narrower and higher than charmonium, consistent with the non-relativistic limit.
  • For excited states (2S), the PDAs show a single-peak profile, differing from the double-peak structure seen in OGE-only models, due to weaker longitudinal nodal structures in the current calculation.
• Parton Distribution Functions (PDFs):
  • Quark PDFs: Symmetric for equal-mass systems (peaked at $x=0.5$) and asymmetric for $B_c$ (peaked at $x>0.5$ for the heavier $b$ quark).
  • Gluon PDFs: The gluon distribution is concentrated in the small-$x$ region.
  • Mass Dependence: Gluon content is suppressed as the heavy quark mass increases.
  • Excitation Dependence: Gluon distributions are enhanced in excited states, particularly in the small-$x$ region, indicating increased gluon emission probability in less tightly bound systems.
  • Momentum Fraction: Gluons carry a small but non-negligible fraction of the total longitudinal momentum (e.g., ~9% for charmonium, ~4% for bottomonium).

5. Significance

This work significantly advances the understanding of heavy meson structure by moving beyond the valence quark approximation.

• Validation of BLFQ: It confirms the feasibility of the light-front Hamiltonian approach for heavy mesons beyond the valence sector, bridging the gap between perturbative and non-perturbative QCD.
• Gluon Dynamics: It quantifies the significant role of dynamical gluons even in heavy systems, challenging the notion that heavy mesons are purely $|q\bar{q}\rangle$ bound states at the non-perturbative scale.
• Phenomenological Impact: The provided wave functions and PDFs serve as crucial inputs for calculating hard exclusive processes and for evolving parton distributions to experimental scales.
• Future Directions: The authors outline a path toward including higher Fock sectors (sea quarks, multiple gluons) and calculating Generalized Parton Distributions (GPDs) and Transverse Momentum Dependent distributions (TMDs) to achieve a full 3D tomography of heavy mesons.