Distribution amplitudes and functions of ground-state scalar and pseudoscalar charmonia

Using continuum Schwinger function methods, this paper challenges the simple hydrogen-like atomic model of ground-state scalar and pseudoscalar charmonia by revealing their complex internal structures, including nontrivial orbital angular momentum contributions and non-positive-definite distribution amplitudes, while providing theoretical benchmarks for understanding heavy-quark hadrons.

The Gist

The Big Picture: Are Charmonia Simple "Atomic" Systems?

Imagine you are trying to understand how a car engine works. You might start by looking at a simple toy car. It's easy to see the wheels, the axle, and the motor. It feels like a perfect, simple model.

In the world of particle physics, scientists have long treated charmonia (particles made of a charm quark and an anti-charm quark) like that toy car. Because the charm quark is heavy (heavier than a proton!), physicists assumed these particles were simple, "hydrogen-like" systems. They thought you could describe them using simple, non-relativistic rules, just like a planet orbiting a sun.

This paper says: "Not so fast."

Using a sophisticated mathematical toolkit called Continuum Schwinger Function Methods (think of it as a high-resolution 3D scanner for the quantum world), the authors scanned these particles and found they are actually much more complex, messy, and interesting than the simple toy model suggests.

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1. The Dance of the Dancers (Orbital Angular Momentum)

The Old View:
Physicists used to classify these particles like dancers in a ballroom.

• The $\eta_c$ was thought to be a simple "S-wave" dancer: two partners holding hands and spinning in a circle, perfectly synchronized.
• The $\chi_{c0}$ was thought to be a "P-wave" dancer: two partners doing a more complex move where they are slightly out of sync or moving in a figure-eight.

The New Discovery:
When the authors looked closer, they realized the dancers are doing a chaotic, improvised jazz routine.

• The $\chi_{c0}$ isn't just a simple P-wave. It's a mix of S-waves, P-waves, and interference patterns that cancel each other out in some places and amplify in others.
• The $\eta_c$ isn't just a simple S-wave either. It has hidden P-wave "twists" inside it.

The Takeaway: You can't describe these particles with a simple "spin and orbit" picture. They are a complex superposition of many different motions happening at once.

2. The Shape of the Shadow (Distribution Amplitudes)

Imagine shining a flashlight through a complex, multi-layered stained-glass window. The shadow it casts on the wall (the "Distribution Amplitude") tells you about the window's structure.

• The $\eta_c$ Shadow: This shadow is a nice, smooth hill. It's tall in the middle and tapers off at the edges. It looks like a standard, well-behaved wave.
• The $\chi_{c0}$ Shadow: This is where it gets weird. The shadow isn't just a hill; it has negative and positive regions. Imagine a shadow that is dark in the middle, but then has "anti-dark" (lighter than the background) patches on the sides.
  • Why? Because of the specific rules of Quantum Chromodynamics (QCD), the mathematical description of this particle must have these positive and negative zones that balance each other out. It's like a wave that goes up, then down, then up again, creating a complex pattern that a simple "hill" model would completely miss.

3. The Traffic Report (Parton Distribution Functions)

Now, let's look at the "traffic" inside the particle. A charmonium particle isn't just two quarks; it's a bustling city containing:

• Valence Quarks: The main residents (the charm and anti-charm).
• Glue (Gluons): The roads and infrastructure holding them together.
• Sea Quarks: The tourists popping in and out of existence.

The authors calculated how much "momentum" (energy/motion) each of these groups carries.

The Surprising Result:

• The Heavy Quark Effect: Because the charm quark is heavy, it's like a slow-moving truck on the highway. It doesn't want to emit "glue" (radiate energy) as easily as a light car (like an up or down quark in a pion).
• The Glue Share: In a pion (a light particle), the glue carries about 44% of the momentum. In the charmonia, the glue carries only 40%.
• The "Heavy" Penalty: The heavy quarks keep more of the momentum for themselves (about 49%), leaving less for the glue and the sea.

The Evolution:
The authors also looked at what happens when you "zoom out" or look at the particle with higher energy (scale evolution).

• At low energy, the $\chi_{c0}$ and $\eta_c$ look very different (one has that weird negative shadow, the other doesn't).
• But as you zoom out to higher energies, their differences fade away. They start to look almost identical. The "heavy" nature of the quarks makes them behave similarly to each other, even though they started out looking different.

4. Why Does This Matter?

You might ask, "Who cares? We can't even build a charmonium particle in a lab to test this."

The "Benchmark" Analogy:
Think of this paper as creating a gold standard map for a territory we can't easily visit.

• If you are a cartographer trying to draw a map of a hidden island, you need a reference point.
• This paper provides a rigorous, mathematically consistent map of how heavy quarks behave inside these particles.
• Other scientists can use this map to test their own theories. If their theory predicts a simple "toy car" model, this paper says, "Look, the real map is much more complex."

Summary in One Sentence

This paper proves that heavy quark particles (charmonia) are not simple, hydrogen-like atoms, but rather complex, chaotic systems with intricate internal structures, and it provides a precise mathematical map of their behavior for other scientists to use as a reference.

Technical Summary

1. Problem Statement

The charm quark ($c$) occupies a unique regime in Quantum Chromodynamics (QCD): its effective current mass ($m_c \approx 1.3$ GeV) is roughly 40% larger than the proton mass but not "truly" heavy. Consequently, charmonia ($c\bar{c}$) mesons are often treated as simple, hydrogen-like "atomic" systems where non-relativistic approximations (like Non-Relativistic QCD or potential models) are assumed to be valid. In these models, states are classified strictly by orbital angular momentum ($l$) and spin ($s$), implying that the scalar $\chi_{c0}$ is a pure $P$-wave state ($^3P_0$) and the pseudoscalar $\eta_c$ is a pure $S$-wave state ($^1S_0$).

The authors challenge this hypothesis, questioning whether the internal structure of charmonia is as simple as these models suggest and whether their partonic distributions (how momentum is shared among constituents) differ significantly from lighter mesons like the pion.

2. Methodology

The study employs Continuum Schwinger Function Methods (CSMs), a fully Poincaré-covariant framework that preserves essential QCD symmetries.

• Bethe-Salpeter Equation (BSE): The authors solve the homogeneous BSE to obtain the Bethe-Salpeter Wave Functions (BSWFs) for the ground-state scalar ($\chi_{c0}$) and pseudoscalar ($\eta_c$) charmonia.
• Truncation Scheme: They utilize the Rainbow-Ladder (RL) truncation of the Dyson-Schwinger equations (DSEs). This is justified because the large charm mass suppresses non-planar diagrams and vertex corrections, making RL a reliable approximation for $c\bar{c}$ systems.
• Interaction Kernel: The interaction is modeled using an effective charge and gluon propagator product (Eq. 7) that successfully describes heavy quark baryons and $B_c$ transitions. Parameters are fixed to reproduce experimental masses and decay constants.
• Scale Evolution:
  • Hadron Scale ($\zeta_H$): Calculations are performed at a low scale where hadron properties are carried entirely by valence degrees of freedom.
  • Evolved Scale ($\zeta_3 = 3.2$ GeV): Distribution Functions (DFs) are evolved to a higher scale using the All-Orders (AO) scheme with mass-dependent splitting functions. This approach accounts for the suppression of glue emission by heavy quarks compared to light quarks.

3. Key Contributions

This paper provides the first comprehensive analysis within a single framework of:

1. Rest-frame Orbital Angular Momentum (OAM) decomposition of $\chi_{c0}$ and $\eta_c$ BSWFs.
2. Leading-twist Distribution Amplitudes (DAs) for both states.
3. Valence, Glue, and Sea Distribution Functions (DFs) at both the hadron scale and evolved scales.
4. A direct comparison of these heavy-quark systems with light-quark mesons (specifically the pion) to test the limits of non-relativistic interpretations.

4. Key Results

A. Orbital Angular Momentum (OAM) Structure

Contrary to the simple quark model picture:

• $\chi_{c0}$ (Scalar): The wave function is not a simple $P$-wave system. While $P$-wave components ($G_0, H_0$) are present, the $S$-wave components ($E_0, F_0$) and, crucially, $S \otimes P$ interference terms contribute significantly to the normalization. The $P \otimes P$ contributions are large but largely cancel each other out.
• $\eta_c$ (Pseudoscalar): The wave function is not a pure $S$-wave system. It contains significant $P$-wave components and strong $S \otimes P$ interference.
• Conclusion: A simple quark model wave function is a poor approximation for both states; the internal structure is highly nontrivial and relativistic.

B. Distribution Amplitudes (DAs)

• $\chi_{c0}$ DA ($\phi_0$): Due to charge-conjugation symmetry ($0^{++}$), the DA is not positive definite. It possesses domains of balanced negative and positive support (it crosses zero). It is also strongly suppressed at the endpoints ($x \to 0, 1$) compared to the QCD asymptotic form $\phi_{as} = 6x(1-x)$.
• $\eta_c$ DA ($\phi_5$): This DA is positive definite but is contracted (narrower) compared to the asymptotic form and exhibits strong endpoint suppression.
• Reconstruction: The authors successfully reconstructed these DAs using Gegenbauer expansions, confirming the endpoint suppression expected for heavy quarks.

C. Distribution Functions (DFs)

• Hadron Scale ($\zeta_H$):
  • The valence DFs are found to be approximately proportional to the square of the DAs ($c(x) \propto \phi^2(x)$), supporting the hypothesis of a separable Light-Front Wave Function (LFWF).
  • The $\chi_{c0}$ valence DF exhibits a "two-humped" structure (due to the zero in the DA), while the $\eta_c$ DF is single-peaked but narrower than the pion.
• Evolved Scale ($\zeta_3 = 3.2$ GeV):
  • Glue and Sea: Evolution generates non-zero glue and sea quark distributions.
  • Convergence: The differences between the scalar ($\chi_{c0}$) and pseudoscalar ($\eta_c$) valence DFs diminish significantly under scale evolution.
  • Glue Content: The glue DFs for $\chi_{c0}$ and $\eta_c$ become practically indistinguishable. However, both differ markedly from the pion.
  • Momentum Fractions: At $\zeta_3$, the momentum fraction carried by valence $c\bar{c}$ in charmonia is 49%, while glue carries 40%. In contrast, the pion has 44% valence and 44% glue.
  • Key Finding: The glue momentum fraction in charmonia is 10% lower than in the pion. This is attributed to the heavier charm mass suppressing the emission of glue by valence quarks.

5. Significance and Implications

• Refutation of Simplicity: The study demonstrates that charmonia are not simple non-relativistic atomic systems. Their wave functions contain complex mixtures of orbital angular momenta and interference terms that cannot be captured by simple $S$- or $P$-wave assignments.
• Benchmark for Theory: While direct experimental measurement of charmonia DFs is currently unfeasible (due to the difficulty of producing charmonia targets), these results provide rigorous theoretical benchmarks. They test the consistency of QCD-based approaches (like DSEs) across different mass regimes.
• Mass Dependence of QCD Evolution: The results highlight the importance of mass-dependent splitting functions in evolution equations. The suppression of glue emission by heavy quarks leads to a distinct momentum partition between valence and glue sectors compared to light-quark hadrons.
• Unified Framework: The work successfully unifies the description of nucleon and meson structure functions, showing that the same underlying QCD mechanisms (dynamical chiral symmetry breaking and confinement) govern both light and heavy hadrons, albeit with different manifestations due to quark mass.

In summary, the paper argues that while charmonia are valuable probes of heavy-quark QCD, they require a sophisticated, fully relativistic treatment to understand their structural features, challenging the common reliance on non-relativistic potential models.