Proposed mixing between $2P$ and $1F$ wave charmonia

This paper investigates the significant mixing between $\chi_{c2}(2P)$ and $\chi_{c2}(1F)$ charmonium states driven by coupled-channel effects, predicting specific decay widths and production mechanisms to guide future experimental verification.

The Gist

Imagine the subatomic world as a giant, cosmic apartment building called Charmonium. Inside this building, tiny particles called "charm quarks" live in specific rooms. These rooms are organized by floors (energy levels) and shapes (how the quarks dance around each other).

Usually, scientists have a good map of the lower floors (the "S-waves" and "P-waves"). But as they look higher up, the map gets fuzzy. This paper is about two specific rooms on the 4th floor that are so close together, they might actually be sharing a wall, or even merging into a hybrid apartment.

Here is the story of Sun, Gao, Man, and Liu and their investigation into these "roommates."

1. The Mystery of the Twin Rooms

In this building, there are two types of rooms:

• The 2P Room: A room where the quarks are in their second "excited" dance (like a dancer doing a second spin).
• The 1F Room: A room where the quarks are in a very complex, high-energy dance (an "F-wave" dance).

According to the old blueprints (standard physics models), these two rooms should be distinct. However, the math suggests they are located almost exactly next to each other in the building. When two things are that close, they tend to influence each other. In physics, we call this mixing.

Think of it like two singers standing on a stage right next to each other. If they start singing the same note, their voices blend. You can't tell where one voice ends and the other begins. That's what's happening to these particles.

2. The Old Theory vs. The New Discovery

The Old Theory (The Weak Handshake):
Scientists previously thought these two rooms were connected by a "tensor force." Imagine this as a very weak rubber band connecting the two rooms. The authors calculated this rubber band and found it was so weak (like a piece of dental floss) that it couldn't really make the rooms mix. The "mixing angle" (how much they blend) was tiny, almost zero.

The New Theory (The Open Door):
The authors realized the rubber band wasn't the whole story. They decided to look at the hallway.
In the real world, particles don't just sit in their rooms; they constantly pop out, turn into other particles (like pairs of D-mesons), wander down the hallway, and pop back in. This is called the coupled-channel effect.

Imagine the two rooms have open doors leading to a busy hallway. Because the doors are open, the "ghosts" of the particles can wander out, meet in the hallway, and come back in. This hallway traffic creates a much stronger connection than the weak rubber band.

3. The Big Reveal: A Significant Mix

When the team opened the doors and let the particles wander the hallway (using a super-complex computer model called an "unquenched calculation"), the result was surprising.

The two rooms weren't just touching; they were blending significantly.

• The lower-energy mixed particle (let's call it Room A) is about 7.5% of the "F-dance" and 92.5% of the "P-dance."
• The higher-energy mixed particle (let's call it Room B) is about 15.4% of the "F-dance."

This is a huge deal! It means the "F-dance" is playing a much bigger role in the lower-energy particle than anyone thought. It's like realizing that the person you thought was just a dancer is actually 15% a trapeze artist.

4. How to Spot Them (The Detective Work)

So, how do we know this is true? We can't just walk into the apartment building and look. We have to listen to the music they make when they leave.

The authors predicted two specific "songs" these particles would sing:

1. The Two-Photon Song: When the particle decays, it can emit two beams of light (photons).
2. The Two-Gluon Song: It can also emit two "glue" particles (gluons).

Because the two mixed rooms have different amounts of "F-dance" in them, they sing these songs at different volumes.

• Room A sings the light song loudly (0.14 keV).
• Room B sings it very quietly (0.05 keV).

This difference is the "fingerprint" scientists need to look for. If they measure the volume of these songs and find it matches the prediction, they prove the mixing is real.

5. The Production Plan: Catching Them in the Act

Finally, the paper suggests how to actually find these particles in experiments like Belle II (a giant particle collider in Japan).

They propose smashing two beams of light (photons) together to create these particles.

• Room A is easy to catch. It's like a loud drum; it will show up clearly on the detector.
• Room B is a whisper. It's very hard to hear because it's so quiet and gets lost in the background noise. Finding it will require a massive amount of data (like listening to a whisper in a hurricane for a very long time).

The Bottom Line

This paper is a detective story.

• The Crime: We have two charmonium particles that look like they are mixing, but the old rules say they shouldn't.
• The Clue: The old rules ignored the "hallway traffic" (coupled-channel effects).
• The Solution: When you account for the hallway traffic, the mixing becomes real and significant.
• The Verdict: We now have a clear plan to prove this by listening to the specific "songs" (decay rates) these particles sing.

It's a reminder that in the quantum world, particles aren't just isolated actors on a stage; they are constantly interacting with the world around them, and sometimes, that interaction changes who they are.

Technical Summary

1. Problem Statement

The paper addresses a specific gap in charmonium spectroscopy: the potential mixing between the second radial excitation of the P-wave state, $\chi_{c2}(2P)$, and the ground state of the F-wave, $\chi_{c2}(1F)$ (specifically the $^3F_2$ state).

• Context: While $S-D$ wave mixing (e.g., in $\psi(3770)$) is well-established, $P-F$ mixing has received less attention despite the predicted proximity in mass between $\chi_{c2}(2P)$ and $\chi_{c2}(1F)$ in the $J^{PC}=2^{++}$ sector.
• The Puzzle: Conventional potential models suggest that the direct tensor force (a component of the quark-antiquark potential) is too weak to generate a significant mixing angle between these states. However, neglecting such mixing could lead to discrepancies in predicting the properties of observed states like $\chi_{c2}(3930)$ and undiscovered higher-mass states.
• Goal: To determine if coupled-channel effects (unquenched dynamics) can induce a sizable $2P-1F$ mixing angle, thereby resolving the mass spectrum and decay properties of these states.

2. Methodology

The authors employ a comprehensive theoretical framework combining a potential model with unquenched coupled-channel analysis.

• Bare Mass Calculation (Godfrey-Isgur Model):

  • They utilize the semi-relativistic Godfrey-Isgur (GI) potential model to calculate the "bare" masses of the charmonium states before mixing.
  • The Hamiltonian includes a confining potential, one-gluon exchange, spin-orbit, and hyperfine interactions (including the tensor term $H_T$).
  • Result: The bare mass of $\chi_{c2}(2P)$ is calculated at $\sim 3975$ MeV and $\chi_{c2}(1F)$ at $\sim 4079$ MeV. The tensor term alone yields a negligible mixing angle of only $0.3^\circ$.

• Coupled-Channel Analysis (Unquenched Approach):

  • To account for the "low mass puzzle" (where experimental masses are lower than quenched predictions), the authors introduce a coupled-channel formalism.
  • Hamiltonian: $H = H_0 + H_{BC} + H_I$, where $H_0$ is the bare state, $H_{BC}$ represents the continuum of open-charm meson pairs ($D\bar{D}$, $D\bar{D}^*$, $D_s\bar{D}_s$), and $H_I$ describes the coupling between bare states and the continuum.
  • Interaction Mechanism: The coupling is modeled using the Quark Pair Creation (QPC) model (also known as the $^3P_0$ model).
  • Mass Shifts: The physical masses and mixing are determined by solving the eigenvalue equation of a reduced matrix containing self-energy corrections ($\Delta(M)$) derived from hadronic loops. The authors use a once-subtracted dispersion relation to handle the loop integrals, summing over allowed channels with thresholds below the bare masses.

• Decay and Production Calculations:

  • Strong Decays: OZI-allowed two-body strong decay widths are calculated using the wave functions derived from the coupled-channel analysis.
  • Electromagnetic/Gluonic Decays: Two-photon ($\Gamma_{\gamma\gamma}$) and two-gluon ($\Gamma_{gg}$) widths are calculated using the wave functions at the origin (and their derivatives), incorporating the mixing angles.
  • Production: The production cross-sections via $\gamma\gamma$ fusion (specifically $\gamma\gamma \to J/\psi \omega$) are calculated using a hadronic loop mechanism with effective Lagrangians and dipole form factors.

3. Key Contributions

• Demonstration of Dominant Coupled-Channel Mixing: The paper establishes that while the static tensor force is negligible, the coupled-channel effects are the dominant mechanism driving $2P-1F$ mixing in charmonium.
• Quantitative Prediction of Mixing Angles: The study provides specific, non-negligible mixing angles derived from first principles within the unquenched framework.
• Identification of Physical States: The authors map the theoretical mixed states to physical observables, identifying the lower-mass mixed state as the experimentally observed $\chi_{c2}(3930)$ and predicting the properties of a higher-mass partner yet to be discovered.
• Experimental Signatures: The paper provides distinct predictions for decay widths and production cross-sections that can differentiate between the two mixed states, offering a roadmap for future experiments.

4. Key Results

A. Mixing Angles and Masses

The inclusion of coupled-channel effects shifts the bare masses downward and induces significant mixing:

• Lower Mixed State ($\chi'_{c2}$): Identified as $\chi_{c2}(3930)$.
  • Physical Mass: 3919.3 MeV (consistent with experiment).
  • Mixing Angle: $\theta_1 = 7.5^\circ$.
  • Composition: Dominantly $2P$ with a small $1F$ admixture.
• Higher Mixed State ($\chi''_{c2}$): A predicted state yet to be discovered.
  • Physical Mass: 4030.0 MeV.
  • Mixing Angle: $\theta_2 = 15.4^\circ$.
  • Composition: Dominantly $1F$ with a significant $2P$ admixture.

B. Decay Widths

• Strong Decays:
  • $\chi'_{c2}$: Total width $\approx 22.7$ MeV (dominated by $D\bar{D}$ at 62.9%). This matches the experimental width of $\chi_{c2}(3930)$.
  • $\chi''_{c2}$: Total width $\approx 113.2$ MeV (dominated by $D\bar{D}$ at 79.5%).
• Two-Photon and Two-Gluon Widths:
  • The mixing significantly alters these widths due to the different radial wave function derivatives at the origin for P-wave vs. F-wave states.
  • $\chi'_{c2}$: $\Gamma_{\gamma\gamma} = 0.14$ keV, $\Gamma_{gg} = 0.33$ MeV.
  • $\chi''_{c2}$: $\Gamma_{\gamma\gamma} = 0.05$ keV, $\Gamma_{gg} = 0.13$ MeV.
  • Note: The suppression in $\chi''_{c2}$ is due to its larger F-wave component, where the wave function at the origin is smaller.

C. Production in $\gamma\gamma$ Fusion

The authors analyze the production of these states via $\gamma\gamma \to J/\psi \omega$:

• Cross-Section Disparity: There is a massive difference in production rates due to the interplay of $\Gamma_{\gamma\gamma}$ and the hadronic loop decay width $\Gamma_{J/\psi\omega}$.
  • $\chi'_{c2}$: Peak cross-section $> 100$ pb (readily observable at Belle II).
  • $\chi''_{c2}$: Peak cross-section $< 0.5$ pb (heavily suppressed, requiring high-statistics datasets like Super Tau-Charm Facility).
• The cross-sections depend on a cutoff parameter $\alpha$ (ranging 3–5), but the order-of-magnitude difference between the two states remains robust.

5. Significance

• Resolution of Spectroscopy: This work clarifies the fine structure of the charmonium spectrum around 4.0 GeV, confirming that the $\chi_{c2}(3930)$ is a mixed state and predicting a partner state at $\sim 4030$ MeV.
• Validation of Unquenched Models: It reinforces the necessity of moving beyond the quenched approximation in hadron spectroscopy, demonstrating that coupled-channel dynamics are essential for explaining mass shifts and mixing in heavy quarkonia.
• Experimental Guidance: The paper provides concrete, testable predictions (masses, widths, and specific cross-sections) for upcoming experiments at Belle II and the proposed Super Tau-Charm Facility (STCF). The distinct difference in $\gamma\gamma$ production rates offers a clear strategy to search for the elusive higher-mass mixed state.
• Theoretical Framework: It establishes a robust methodology for calculating $P-F$ mixing, which can be extended to other heavy quarkonium systems (e.g., bottomonium) where similar orbital angular momentum mixing might occur.