Resonant Parameters of Vector Charmonium-like States above 4.4 GeV

This paper presents a simultaneous fit to BESIII cross-section data for various charmonium-like final states above 4.4 GeV, determining the resonant parameters of four structures ($\psi(4230)$, $\psi(4500)$, $\psi(4660)$, and $\psi(4710)$) and finding that the $\psi(4660)$ and $\psi(4710)$ dominantly drive the production of $D_s^{+}D_{s1}^{*-}(2536)$, $D_s^{+}D_{s2}^{*-}(2573)$, and $\phi\chi_{c1,2}$ channels.

The Gist

Imagine the universe as a giant, cosmic orchestra. For decades, physicists have been trying to understand the music this orchestra plays. Most of the notes they expected to hear were made by "standard" instruments: pairs of heavy particles called charm quarks (let's call them "Charm Twins") dancing together. These standard dances are called charmonium.

However, in the last 20 years, the orchestra started playing some strange, new notes that didn't fit the sheet music. These are the exotic states. They are like instruments that shouldn't exist according to the old rules, or perhaps they are complex ensembles of four or more particles playing together.

This paper is a report from a team of physicists (the "conductors") who went to a specific concert hall called BESIII (located in Beijing) to listen closely to these strange notes. Here is what they found, explained simply:

1. The Mystery of the "Missing" Notes

The team was looking at a specific range of energy, above 4.4 GeV. Think of this as a high-pitched section of the orchestra. In this range, they expected to hear four specific "standard" notes (theoretical particles named $\psi(4S)$, $\psi(5S)$, etc.).

Instead, the experimenters saw four distinct "blobs" of activity (resonances) at specific energy levels:

• $\psi(4230)$
• $\psi(4500)$
• $\psi(4660)$
• $\psi(4710)$

The big question was: Are these just the standard notes we expected, or are they something completely new and exotic?

2. The Detective Work: Listening to the Decay

To solve the mystery, the team didn't just listen to the main note; they listened to what happened after the note was played. When these heavy particles decay (break apart), they turn into other, lighter particles.

The team looked at six different "exit doors" (decay channels) where these particles could escape:

• Some turned into a mix of strange and charm particles ($D_s$ mesons).
• Some turned into a phi meson and a charmonium particle ($\phi\chi_c$).
• Some turned into kaons and a J/$\psi$ particle.

They treated the data like a mixture of sounds. Imagine you are in a room with four different speakers playing different songs at the same time. The team used a mathematical "mixing board" (a statistical fit) to try and separate the sound of Speaker A from Speaker B, Speaker C, and Speaker D.

3. The Key Findings

After doing the complex math (the "simultaneous $\chi^2$-minimized fit"), they found some very clear patterns:

• The "Heavy Hitters": The particles $\psi(4660)$ and $\psi(4710)$ are the superstars of this show. They are responsible for almost all the activity in the channels where the particles break apart into specific heavy combinations (like $D_s D^*$).
• The "Ghost" in the Machine: They tried to include the particle $\psi(4230)$ in their model, but it turned out that for the specific channels they studied, this particle wasn't the main driver.
• The "Missing Link": There was one channel they couldn't explain yet: the decay into $D^*_s D^*_s$. It's like trying to solve a puzzle, but one piece is missing. The current four "speakers" (resonances) they have identified just don't produce the right sound for that specific exit door. They need to find a new "speaker" or a new way of mixing the sound to explain it.

4. The Big Picture: Are they Standard or Exotic?

The team compared their findings to the "sheet music" (theoretical models).

• The Problem: The masses (the pitch of the notes) they measured don't perfectly match the predictions for the standard "Charm Twins" dancing alone.
• The Possibility: This suggests these particles might be exotic. They might not just be two quarks dancing; they might be complex "tetraquarks" (four quarks) or molecules made of two mesons stuck together.

The Analogy of the "Coupled-Channel Effect"

The paper mentions a concept called the "coupled-channel effect." Imagine a trampoline.

• Standard Model: You think of the trampoline as a solid surface. When you jump, you bounce up and down in a predictable way.
• Reality (Coupled-Channel): The trampoline is actually made of springs connected to other trampolines below it. When you jump, you don't just bounce up; you pull on the springs, which pull on the trampolines below, which pull back on you. This interaction changes your bounce.
• In Physics: The "standard" charm particles are constantly popping into existence as pairs of lighter particles (virtual loops) and then disappearing again. This "tugging" on the vacuum changes the mass and behavior of the heavy particle, making it look different than the simple theory predicted.

Conclusion

This paper is a major step in mapping the "high-energy" region of the particle zoo.

1. They successfully identified the masses and widths (how heavy and how short-lived) of four specific resonant structures.
2. They proved that $\psi(4660)$ and $\psi(4710)$ are the main actors in several specific decay processes.
3. They highlighted that while we have a good map, there are still discrepancies between what we see and what the old theories predict.

The Takeaway: The universe is playing a more complex song than we thought. These "vector charmonium-like states" are likely not just simple pairs of quarks, but complex, exotic structures that require us to rewrite the rules of how matter interacts at these high energies. The team is now calling for even more precise measurements to solve the final missing piece of the puzzle (the $D^*_s D^*_s$ channel).

Technical Summary

1. Problem Statement

The paper addresses the complex nature of vector charmonium-like states ($J^{PC} = 1^{--}$) with masses above 4.4 GeV. While conventional quark models (potential models) successfully describe charmonium states below the open-charm threshold, they struggle to accurately predict the masses and decay properties of higher excited states (e.g., $\psi(4S)$, $\psi(5S)$, $\psi(3D)$, $\psi(4D)$).

• The Discrepancy: Experimental observations by BESIII, BaBar, and Belle have revealed multiple structures (e.g., $\psi(4230)$, $\psi(4500)$, $\psi(4660)$, $\psi(4710)$) in various decay channels. However, the consistency of these structures across different processes is unclear due to limited measurement precision and potential interference effects.
• The Challenge: It is difficult to determine if these observed structures correspond to the same physical states or to assign them definitively to specific conventional charmonium configurations (like $S$-wave or $D$-wave states) because of the "coupled-channel effects" (virtual charm meson loops) that modify the spectrum.

2. Methodology

The authors performed a simultaneous $\chi^2$-minimized fit to the center-of-mass energy ($\sqrt{s}$) dependent line shapes of cross-sections for seven distinct $e^+e^-$ annihilation processes measured by the BESIII experiment.

• Data Channels Analyzed:

  1. $e^+e^- \to K^+K^-J/\psi$
  2. $e^+e^- \to K^0_S K^0_S J/\psi$
  3. $e^+e^- \to K^+K^-\psi(2S)$
  4. $e^+e^- \to D^+_s D^{*-}_s (2536)$
  5. $e^+e^- \to D^+_s D^{*-}_s (2573)$
  6. $e^+e^- \to \phi\chi_{c1}$
  7. $e^+e^- \to \phi\chi_{c2}$

• Fitting Model:

  • The cross-sections were modeled using a coherent sum of relativistic Breit-Wigner (BW) functions.
  • Four resonant structures were employed to describe the data: $\psi(4230)$, $\psi(4500)$, $\psi(4660)$, and $\psi(4710)$.
  • The BW function included phase space factors ($\Phi$) and orbital angular momentum ($l$). Notably, the $D^+_s D^{*-}_s (2573)$ channel was treated with $l=2$ (D-wave), while others used $l=0$.
  • Constraints: Isospin symmetry was applied to constrain the amplitude ratio between $K^0_S K^0_S J/\psi$ and $K^+K^-J/\psi$ to 0.5.
  • Systematics: Systematic uncertainties were excluded from the fit to avoid correlation issues, focusing instead on statistical errors and line shape descriptions.

• Theoretical Context: The analysis was framed within the context of "unquenched" quark models (incorporating coupled-channel effects) which predict the masses of higher charmonium states (up to ~5.0 GeV) and their decay widths into open-charm channels.

3. Key Contributions

• Simultaneous Global Fit: Unlike previous studies that often analyzed channels individually, this work performed a global fit across seven diverse channels simultaneously. This approach leverages the correlations between channels to better constrain the resonant parameters (mass and width).
• Extraction of Resonant Parameters: The study provides precise mass and width values for the four vector states ($\psi(4230)$, $\psi(4500)$, $\psi(4660)$, $\psi(4710)$) derived from a unified dataset.
• Channel Dominance Analysis: The paper identifies which resonances dominate specific decay modes, clarifying the production mechanisms for processes involving charmed-strange mesons ($D_s$) and light hadrons ($\phi, \chi_c$).
• Identification of Limitations: The authors explicitly noted the inability to describe the $D^{*+}_s D^{*-}_s$ channel with the current set of four resonances, highlighting a gap in the current understanding or the need for additional resonance contributions.

4. Results

The simultaneous fit yielded a good description of the data with a $\chi^2/\text{ndf} = 155.4/118$. The extracted parameters are:

Structure	Mass ($M$) [MeV]	Width ($\Gamma$) [MeV]
$\psi(4230)$	$4225.39 \pm 1.95$	$66.31 \pm 5.04$
$\psi(4500)$	$4496.77 \pm 12.19$	$104.28 \pm 23.80$
$\psi(4660)$	$4604.03 \pm 2.92$	$48.48 \pm 5.02$
$\psi(4710)$	$4736.38 \pm 7.03$	$137.25 \pm 20.11$

Key Observations:

• Dominant Decays: The processes $e^+e^- \to D^+_s D^{*-}_s (2536)$, $e^+e^- \to D^+_s D^{*-}_s (2573)$, and $e^+e^- \to \phi\chi_{c1,2}$ are dominantly produced via the decays of $\psi(4660)$ and $\psi(4710)$.
• Channel Specifics:
  • For $K^+K^-J/\psi$ and $K^0_S K^0_S J/\psi$, three resonances ($\psi(4230)$, $\psi(4500)$, $\psi(4710)$) were sufficient; $\psi(4660)$ was not required.
  • For $K^+K^-\psi(2S)$, the contribution is dominated by $\psi(4710)$, with $\psi(4660)$ being negligible.
• Theoretical Assignment: While the data shows four vector states above 4.4 GeV (candidates for $\psi(4S)$, $\psi(5S)$, $\psi(3D)$, $\psi(4D)$), there remains a discrepancy between the measured masses and theoretical predictions from unquenched models. Consequently, a high-confidence assignment of these states to specific conventional charmonium configurations is not yet possible.

5. Significance

• Refinement of the Charmonium Spectrum: This work provides updated, high-precision parameters for vector charmonium-like states, which are crucial inputs for theoretical models attempting to explain the "exotic" nature of these states.
• Validation of Coupled-Channel Effects: The results support the necessity of coupled-channel effects (unquenched models) to explain the mass shifts and decay patterns of high-mass charmonia, as simple potential models fail to match the observed spectrum.
• Future Directions: The paper highlights that while the current four-resonance model fits most channels well, the failure to describe the $D^{*+}_s D^{*-}_s$ channel suggests the existence of additional resonant structures or more complex interference patterns. This sets a clear agenda for future high-precision measurements and theoretical refinements to fully resolve the nature of states above 4.4 GeV.