Electromagnetic structure of Bc and heavy quarkonia in the light-front quark model

This paper investigates the electromagnetic structure of heavy quarkonia and the $B_c$ meson using a light-front quark model with variational wave functions, finding that the electromagnetic radii of $2S$ and $3S$ states are approximately 1.5 and 1.9 times larger than their $1S$ counterparts, respectively, in agreement with lattice QCD data and other model predictions.

The Gist

Imagine the universe is built out of tiny, invisible Lego bricks called quarks. Usually, these bricks snap together in pairs or triplets to form larger structures called mesons (like tiny, heavy balls). Most of the time, these bricks are made of the same material, but sometimes, you get a "mixed" pair: a heavy brick and a lighter brick stuck together.

This paper is like a team of physicists acting as cosmic architects. They are trying to figure out exactly how big these heavy "Lego balls" are and how their electric charge is spread out inside them. They are looking at three specific types of these balls:

1. Charmonia: Two heavy "charm" bricks stuck together.
2. Bottomonia: Two very heavy "bottom" bricks stuck together.
3. The $B_c$ Meson: A heavy "bottom" brick and a lighter "charm" brick stuck together.

The Problem: We Can't Take a Photo

The problem is that these particles are incredibly short-lived. They pop into existence and vanish almost instantly. It's like trying to measure the size of a soap bubble that pops before you can even touch it. Because we can't take a direct photo or measure them with a ruler in a lab, the scientists have to build a virtual simulation.

The Tool: The "Light-Front" Camera

To do this, the authors used a special mathematical tool called the Light-Front Quark Model (LFQM).

Think of a normal movie camera. It takes a picture of an object from the side. But because these particles move at nearly the speed of light, a side view is blurry and confusing.
The "Light-Front" model is like a special camera that takes a snapshot from the front, looking straight down the path of the particle's motion. This gives a crystal-clear, frozen-in-time view of how the quarks are arranged inside the meson, even while they are zooming at light speed.

The Experiment: Shaking the Bubble

The scientists wanted to know: If we poke these particles with a beam of energy, how do they react?

In the real world, you can't poke a particle. So, in their computer simulation, they "poke" the particles with a mathematical beam of energy (called momentum transfer, or $Q^2$).

• The Reaction: They watched how the "electric charge" inside the particle shifted.
• The Result: They calculated something called the Charge Radius. Think of this as the "average distance" between the two quarks. It tells us how "fluffy" or "compact" the particle is.

The Findings: The "Onion" Effect

Here is what they discovered, using some fun analogies:

1. The Size Matters (Ground State vs. Excited States)
Imagine an onion.

• The 1S state (the ground state) is the tight, small core of the onion. It's very compact.
• The 2S state is like adding a new, larger layer of skin around it. It's bigger and fluffier.
• The 3S state is like adding two more layers. It's the biggest and most spread out.

The paper found that as you go from the 1S to the 2S and 3S states, the particles get significantly larger.

• The 2S particles are about 1.5 times bigger than the 1S ones.
• The 3S particles are about 1.9 times bigger.

This makes sense! Just like a balloon gets bigger when you blow more air into it, these quark pairs spread out more when they are in a higher energy state.

2. The "Heavy" vs. The "Mixed"

• Bottomonia (Two heavy bricks): These are the tightest, smallest balls. Because both bricks are so heavy, they hold onto each other very tightly, like two magnets glued together. They are the most compact.
• Charmonia (Two medium bricks): These are a bit more "fluffy" and spread out.
• The $B_c$ Meson (Heavy + Light): This is the interesting middle child. Because one brick is heavy and one is light, they don't sit perfectly in the middle. The heavy brick stays near the center, while the light brick dances around it more. The size of this particle sits right in between the other two types.

3. The "Wiggly" Wave
The scientists also noticed something cool about the bigger particles (2S and 3S). Because they have "nodes" (points where the wave function flips from positive to negative, like a wave crashing and going underwater), their electric charge distribution gets a bit wiggly.

• Imagine shaking a rope. If you shake it gently (1S), it's a smooth wave.
• If you shake it harder (2S), a knot forms in the middle.
• If you shake it really hard (3S), you get two knots.
  These "knots" cause the electric charge to cancel itself out in some spots, making the particle react differently when hit with high energy.

Why Does This Matter?

You might ask, "Who cares about the size of a particle that vanishes in a nanosecond?"

This is crucial for understanding the rules of the universe.

• Testing the Theory: The scientists compared their "virtual photos" with data from Lattice QCD (which is like a super-computer simulation of the strong force that holds quarks together). Their results matched up very well! This proves their mathematical model is a good way to describe reality.
• Future Predictions: Since we can't measure the bigger particles (2S and 3S) easily in a lab yet, this paper acts as a forecast. It tells other scientists, "If you build a machine to catch these particles, expect them to be this big."

The Bottom Line

This paper is a success story of using math to see the invisible. By using a special "front-view camera" (Light-Front model), the team successfully mapped out the internal "furniture" of heavy mesons. They confirmed that as these particles get more excited, they grow larger and fluffier, and they found that the "mixed" $B_c$ meson is a unique hybrid, sitting right between the heavy and light extremes of the quark world.

Technical Summary

1. Problem Statement

Understanding the internal spatial charge distributions and Quantum Chromodynamics (QCD) dynamics of heavy mesons remains a significant challenge due to the non-perturbative nature of QCD. While heavy quarkonia (charmonia $c\bar{c}$ and bottomonia $b\bar{b}$) and the $B_c$ meson ($b\bar{c}$) serve as ideal testing grounds for QCD models, direct experimental measurements of their electromagnetic form factors (EMFFs) are currently unavailable due to their short lifetimes and production difficulties. Consequently, there is a need for robust phenomenological models to predict these properties and benchmark them against Lattice QCD (LQCD) calculations to validate theoretical frameworks. Specifically, the behavior of excited radial states (2S, 3S) and the unique dynamics of the flavor-asymmetric $B_c$ meson require further investigation.

2. Methodology

The authors employ the Light-Front Quark Model (LFQM) to investigate the electromagnetic structure of heavy mesons. The key methodological components include:

• Hamiltonian and Wave Functions: The meson system is treated as a bound state of dressed valence quark-antiquark pairs satisfying an eigenvalue equation with a QCD-motivated effective Hamiltonian ($H = H_0 + V_{q\bar{q}}$). The potential includes confinement, Coulomb, and spin-spin interactions.
• Variational Approach: The radial wave functions are constructed using a variational approach with a Harmonic Oscillator (HO) basis. The study extends up to the third radial excitation (3S states). The basis functions are:
  • $\phi_{1S}$: Gaussian (no nodes).
  • $\phi_{2S}$: Linear combination with one node.
  • $\phi_{3S}$: Quadratic combination with two nodes.
• Light-Front Dynamics (LFD): The model utilizes Lorentz-invariant internal variables ($x, k_\perp$) and incorporates relativistic effects naturally through the Melosh transformation, which connects the ordinary spin-orbit wave function to the light-front helicity wave function.
• Electromagnetic Form Factors (EMFFs):
  • Calculated using the Drell-Yan-West frame ($q^+ = 0$) to ensure frame independence and avoid zero-mode contributions.
  • The EMFFs are derived from the convolution of initial and final state Light-Front Wave Functions (LFWFs) and the quark current matrix elements.
  • Both helicity-flip and non-flip contributions are considered, though only non-flip contributes in the $+$ component.
• Charge Radii: The root-mean-square (rms) charge radii are extracted from the slope of the EMFFs at zero momentum transfer ($Q^2 \to 0$) using the relation $\langle r^2 \rangle = -6 \frac{dF(Q^2)}{dQ^2}|_{Q^2=0}$.
• Parameters: Model parameters (quark masses $m_c, m_b$ and oscillator parameters $\beta$) were adopted from previous work where they were fitted to reproduce meson spectra and decay constants using a screened Cornell potential.

3. Key Contributions

• Systematic Study of Radial Excitations: The paper provides a comprehensive calculation of EMFFs and charge radii for the ground state (1S) and the first two radial excitations (2S, 3S) for three distinct systems: charmonia ($\eta_c$), bottomonia ($\eta_b$), and the $B_c$ meson.
• Analysis of Nodal Structures: The study explicitly analyzes how the nodal structure of the wave functions (zero crossings in 2S and 3S states) affects the form factors, leading to oscillations at high momentum transfer and rapid falloffs compared to the ground state.
• Flavor Asymmetry in $B_c$: The authors investigate the unique dynamics of the $B_c$ meson, where the unequal masses of the bottom and charm quarks lead to asymmetric radial node distributions and distinct form factor behaviors compared to the symmetric $c\bar{c}$ and $b\bar{b}$ systems.
• Benchmarking: The results are rigorously compared with available Lattice QCD data (specifically B1 and C1 ensembles) and other phenomenological models (BLFQ, BSE, and previous LFQM studies).

4. Results

• Form Factor Behavior:
  • Ground States (1S): Show a steady, monotonic decline in $F(Q^2)$ as $Q^2$ increases.
  • Excited States (2S, 3S): Exhibit more rapid initial declines and oscillatory behavior at high $Q^2$ due to the interference caused by radial nodes.
  • Convergence: The form factors for excited states converge to a common value at high $Q^2$ (approx. 10 GeV$^2$ for charmonia, 36 GeV$^2$ for bottomonia).
• Charge Radii Trends:
  • Growth with Excitation: There is a consistent increase in rms charge radius with higher radial quantum numbers.
    • Charmonia ($\eta_c$): $0.249 \to 0.369 \to 0.466$ fm. The 3S state is $\approx 1.87$ times larger than the 1S state.
    • $B_c$ Meson: $0.235 \to 0.355 \to 0.447$ fm. The 3S state is $\approx 1.9$ times larger than the 1S state.
    • Bottomonia ($\eta_b$): $0.118 \to 0.179 \to 0.226$ fm. The 3S state is $\approx 1.91$ times larger than the 1S state.
  • System Comparison: Bottomonia are the most compact (strongest binding), charmonia are the most diffuse, and the $B_c$ meson occupies an intermediate position.
• Comparison with Other Models:
  • The results for ground states are generally consistent with Lattice QCD and other models, though slight deviations exist (e.g., LQCD predicts a larger radius for $\eta_c(2S)$, attributed to the sensitivity of the nodal wave function to long-range tails).
  • The $B_c(1S)$ radius is found to be smaller than some previous predictions but consistent with the expectation of a tightly bound heavy-heavy system.
  • Predictions for 3S states are provided as they lack comparative data in current literature.

5. Significance

This work significantly advances the understanding of heavy meson structure by:

1. Validating the LFQM: Demonstrating that the Light-Front Quark Model, when combined with a variational HO basis, can reliably reproduce the spatial structure of heavy quarkonia and the $B_c$ meson, aligning well with Lattice QCD benchmarks.
2. Predicting Unmeasured Quantities: Providing the first theoretical predictions for the charge radii and form factors of the 3S excited states for these systems, which are crucial for future experimental searches and lattice simulations.
3. Clarifying Radial Dynamics: Quantifying the "size growth" factor (approx. 1.5x for 1S$\to$2S and 1.9x for 1S$\to$3S) across different heavy flavor systems, confirming the expected expansion of the wave function due to radial nodes.
4. Highlighting Flavor Effects: Illustrating how mass asymmetry in the $B_c$ meson modifies the electromagnetic structure compared to symmetric quarkonia, offering insights into the interplay of heavy quark dynamics in mixed-flavor systems.

The study concludes that the LFQM offers a clear partonic picture and a tractable framework for studying the three-dimensional structure of hadrons, serving as a vital tool for interpreting future high-energy physics data.