Prospects for observing the missing $2D$and$1F$ charmonium states around 4 GeV

This paper investigates the mass spectra and strong and radiative decay properties of high-lying $2D$and$1F$ charmonium states around 4 GeV, incorporating unquenched effects to provide theoretical predictions that guide future experimental searches at facilities like BESIII, Belle II, LHCb, and STCF.

The Gist

Imagine the world of subatomic particles as a massive, bustling city called Hadronville. In this city, there are different neighborhoods. One of the most famous neighborhoods is Charmonium Town, where particles made of a "charm" quark and an "anti-charm" quark live together like dance partners.

For decades, physicists have been mapping out the "apartments" in Charmonium Town. They know where the ground-floor units (low-energy states) are, and they've found several high-rise apartments (higher-energy states). But, there are some empty, mysterious floors in the 4 GeV range (a specific energy level) that no one has been able to find yet.

This paper is like a detailed architectural blueprint and a treasure map created by a team of physicists from Lanzhou University. They are trying to predict exactly where these missing "apartments" (specifically the 2D and 1F states) are hiding and how to find them.

Here is the breakdown of their mission, explained simply:

1. The Problem: The Missing Floors

Think of the charm quark and anti-charm quark as two dancers. They can spin around each other in different ways:

• S-wave: They are close together, spinning simply. (We know these well).
• P-wave: They are a bit further apart, dancing a waltz. (We know these too).
• D-wave and F-wave: This is where they get fancy. They are spinning wildly, far apart, doing complex acrobatics.

The scientists know the "1D" dancers (the first level of fancy spinning) have been found. But the 2D (the second level) and 1F (the first level of even more complex spinning) are missing. It's like knowing the 1st, 2nd, and 3rd floors of a building exist, but the 4th and 5th floors are invisible.

2. The Tool: The "Unquenched" Telescope

In the past, physicists used a simple model (like a basic pair of glasses) to predict where these particles should be. But that model was "quenched"—it assumed the dancers were isolated.

In reality, the universe is messy. The dancers can briefly split apart and grab other particles from the "vacuum" (the empty space around them) before coming back together. This is called the "unquenched effect."

The authors used a special, high-tech telescope called the Modified Godfrey-Isgur (MGI) model. Think of this as a camera that accounts for the "noise" of the vacuum. It predicts that these missing particles are hiding around 4.0 to 4.5 GeV (a specific energy height).

3. The Predictions: What Do They Look Like?

The team calculated the "ID cards" for these missing particles. Here is what they found:

• The 2D Family (The "Dancers"):

  • There are three of them, with masses around 4,140 MeV.
  • How they break up (Decay): They are unstable. They love to break apart into pairs of "D-mesons" (like $D\bar{D}^*$). Imagine them as unstable couples that immediately split into two new pairs of dancers.
  • The "X(4160)" Mystery: One of these missing particles might actually be the X(4160), a particle already seen by other experiments but whose identity was a mystery. The authors say, "Yes, that's probably our missing 2D dancer!"

• The 1F Family (The "Acrobats"):

  • There are four of them, huddled very close together at 4,070 MeV.
  • How they break up: They also split into D-mesons, but their "dance moves" (quantum numbers) are different.
  • The "hc3" and "chi" names: They give them names like $hc_3(4074)$ and $\chi_{c2}(4070)$.

4. The Treasure Map: How to Find Them

Knowing where they might be isn't enough; you need to know how to spot them. The paper suggests two main ways to catch these elusive particles:

A. The "Flashlight" Method (Radiative Decay)

Imagine shining a flashlight (a photon) at a high-energy particle to knock it down to a lower energy level.

• The Strategy: Use the BESIII experiment (a giant particle collider in China) to smash electrons and positrons together. This creates a "parent" particle called $\psi(4230)$.
• The Catch: The $\psi(4230)$ can emit a photon and turn into one of our missing 1F particles (like $\chi_{c2}$).
• The Result: The authors predict this is a very promising way to find the $\chi_{c2}(1F)$ particle. It's like hearing a specific ringtone in a noisy room. However, finding the "2D" version this way is very hard because the "signal" is incredibly faint.

B. The "Backdoor" Method (B-Meson Decays)

Sometimes, heavy particles (B-mesons) decay and leave behind these missing charmonium states as "leftovers."

• The Strategy: Look at the debris from LHCb (a detector at CERN) or Belle II (in Japan).
• The Result: This is a great way to find the 2D states, especially if they are hiding in the "D-meson" debris.

5. The Big Picture: Why Does This Matter?

Finding these missing particles is like finding the missing pieces of a giant puzzle.

• Testing the Rules: If we find them exactly where the "Unquenched" model predicts, it proves our understanding of how the strong force (the glue holding quarks together) works, even in messy, high-energy environments.
• The "Y" Problem: There are many strange particles (called Y, Z, and X states) that don't fit the old rules. Understanding these missing 2D and 1F states helps scientists figure out if those strange particles are just normal "high-rise" apartments or something exotic (like a house made of four quarks).

Summary

The authors of this paper are essentially saying:

"We have built a better map using a model that accounts for the messy reality of the quantum world. We predict that the missing '2D' and '1F' charmonium states are hiding around 4 GeV. They are likely to be found by looking for specific decay patterns in the data from BESIII, Belle II, and LHCb. If we find them, we will finally complete the family portrait of the charm quark!"

It's a call to action for experimentalists: "Look here, in these specific channels, and you might just find the missing pieces of the puzzle!"

Technical Summary

Here is a detailed technical summary of the paper "Prospects for observing the missing 2D and 1F charmonium states around 4 GeV" by Liu et al.

1. Problem Statement

The charmonium family ($c\bar{c}$) remains incompletely mapped, particularly regarding high-lying orbital excitations. While the $1D$-wave triplet and the$2^3D_1$state ($\psi(4160)$) have been identified, three members of the$2D$-wave triplet ($2^1D_2, 2^3D_2, 2^3D_3$) and the entire$1F$-wave triplet ($1^1F_3, 1^3F_2, 1^3F_3, 1^3F_4$) remain unobserved.

• The Challenge: Traditional "quenched" potential models (which ignore the coupling to open-charm meson pairs) fail to accurately predict the masses and properties of these high-lying states.
• The Gap: Experimental data from facilities like BESIII, Belle II, and LHCb has revealed complex structures (e.g., $Y$ states) around 4.0–4.5 GeV, suggesting significant "unquenched" effects (coupled-channel dynamics) are at play. There is a critical need for theoretical predictions that incorporate these effects to guide future searches for the missing $2D$and$1F$ states.

2. Methodology

The authors employ a multi-faceted theoretical framework to predict the spectroscopic properties and production mechanisms of the missing states:

• Mass Spectrum Calculation (MGI Model):

  • They utilize the Modified Godfrey-Isgur (MGI) model.
  • Unquenched Effects: To account for the coupling to open-charm channels, they replace the linear confinement potential with a screening potential ($br \to b(1-e^{-\mu r})/\mu$). This mimics the effect of light quark-antiquark pair creation at large distances, a key feature of unquenched QCD.
  • The Hamiltonian includes relativistic corrections, spin-orbit, and tensor interactions.

• Strong Decay Analysis (QPC Model):

  • Quark Pair Creation (QPC) Model: Used to calculate partial decay widths for OZI-allowed two-body strong decays (e.g., $D\bar{D}, D\bar{D}^*, D^*\bar{D}^*$).
  • The transition operator describes the creation of a light $q\bar{q}$ pair from the vacuum.
  • Parameters (specifically the pair-creation strength $\gamma$) are calibrated using experimental data from previous studies.

• Radiative Decay Analysis:

  • Calculated using the constituent quark model with non-relativistic expansion of the electromagnetic interaction Hamiltonian.
  • Focuses on transitions to lower-lying $P$-wave and $D$-wave states.

• Production Mechanism (Hadronic Loop):

  • To evaluate the feasibility of observing these states in $e^+e^-$ annihilation (specifically via $\psi(4230)$), they employ the Hadronic Loop Mechanism.
  • This approach treats the process $e^+e^- \to \psi(4230) \to \gamma X$ as a triangle loop diagram where intermediate charmed mesons ($D, D^*, D_1$) are exchanged.
  • Gauge Invariance: Contact terms are added to the amplitudes to satisfy the Ward-Takahashi identity.
  • Form Factors: A dipole form factor is introduced to regularize the loop integrals and account for off-shell effects.

3. Key Contributions and Results

A. Mass Spectrum Predictions

• $2D$-Wave States: The calculated masses are approximately 4137–4144 MeV.
  • $2^1D_2$($\eta_{c2}$): ~4137 MeV.
  • $2^3D_2$($\psi_2$): ~4137 MeV.
  • $2^3D_3$($\psi_3$): ~4144 MeV.
  • These masses are consistent with recent re-evaluations suggesting the $\psi(4160)$ is lower than previously thought (~4145 MeV) and align with the $X(4160)$ candidate.
• $1F$-Wave States: The calculated masses are approximately 4070–4076 MeV, exhibiting near-degeneracy due to heavy quark spin symmetry.
  • $1^1F_3$($h_{c3}$): ~4074 MeV.
  • $1^3F_2$($\chi_{c2}$): ~4070 MeV.
  • $1^3F_3$($\chi_{c3}$): ~4075 MeV.
  • $1^3F_4$($\chi_{c4}$): ~4076 MeV.
• Mass Dependence: The authors emphasize that decay widths for states near the $D^*\bar{D}^*$ threshold are highly sensitive to small mass variations (within $\pm 30$ MeV).

B. Strong Decay Properties

• Dominant Channels:
  • **$2D$States:** Decay predominantly into **$D\bar{D}^$** and **$D^\bar{D}^*$**. The total widths are predicted to be narrow, ranging from 48 to 58 MeV.
  • $1F$ States: Decay channels vary by spin.
    • $1^1F_3$and$1^3F_3$are dominated by **$D\bar{D}^*$**.
    • $1^3F_2$has significant branching into **$D\bar{D}$** and **$D\bar{D}^*$**.
    • $1^3F_4$is the narrowest ($\sim 38$MeV) due to high orbital angular momentum suppressing phase space, with **$D^\bar{D}^$** being a major channel.
• Implication: These states should be searchable in open-charm final states ($D\bar{D}^*, D^*\bar{D}^*$) at experiments like BESIII and LHCb.

C. Radiative Transitions and Production in $e^+e^-$

• Radiative Decays: The study identifies promising radiative transitions (e.g., $2D \to 2P\gamma$and$1F \to 1D\gamma$) with partial widths in the keV range.
• Production via $\psi(4230)$:
  • The authors calculate the cross-sections for $e^+e^- \to \psi(4230) \to \gamma X$.
  • $\chi_{c2}(1F)$: Produced via an S-wave mechanism, resulting in a relatively large cross-section (~20 pb at $\alpha=2$). This is identified as the most promising candidate for discovery in radiative transitions.
  • $\chi_{c3}(1F)$: Produced via a D-wave mechanism, leading to suppression. Cross-section is ~1 pb, making it challenging but potentially accessible with future high-luminosity upgrades (STCF).
  • $\eta_{c2}(2D)$: Production is extremely suppressed ($\sigma \sim 10^{-5}$ pb) due to heavy-quark spin symmetry breaking requirements (vector to tensor singlet transition). It is unlikely to be seen via radiative decay from $\psi(4230)$.

4. Significance and Outlook

• Completing the Spectrum: This work provides the first comprehensive unquenched prediction for the missing $2D$and$1F$ charmonium multiplets, filling a critical gap in the heavy quarkonium spectrum.
• Experimental Guidance:
  • Search Channels: The paper directs experimentalists to look for $2D$and$1F$states in **$D\bar{D}^$** and **$D^\bar{D}^$* final states.
  • Radiative Search: It highlights $e^+e^- \to \gamma \chi_{c2}(1F)$ as a high-priority channel for the $\chi_{c2}(1F)$ state.
  • Alternative Mechanisms: For the $\eta_{c2}(2D)$, which is suppressed in radiative decays, the authors suggest searching in $B$-meson decays (e.g., $B \to J/\psi \phi K$) or double-charmonium production ($e^+e^- \to J/\psi D\bar{D}^*$), similar to the discovery path of $X(4160)$.
• Future Facilities: The results are tailored for current and upcoming experiments, specifically BESIII (with its high statistics at 4.2 GeV), Belle II, LHCb, and the future Super $\tau$-Charm Facility (STCF). The predictions suggest that with increased luminosity, the discovery of these high-spin, high-mass states is imminent.

In conclusion, the paper establishes a robust theoretical foundation for the existence of missing high-lying charmonia and provides specific, quantitative targets for experimental verification, emphasizing the necessity of unquenched models and hadronic loop mechanisms in describing these states.