Study of $\chi_{cJ}\to \eta \eta \eta^\prime$ via intermediate charmed meson loop mechanisms and its implications for non-observation of $\eta_1(1855)$ in $\chi_{cJ}$ decays

This study employs an effective Lagrangian approach with charmed meson loop mechanisms to successfully reproduce the BESIII experimental branching fractions and invariant mass spectra for $\chi_{cJ} \to \eta \eta \eta^\prime$ decays, thereby offering a theoretical explanation for the absence of the $\eta_1(1855)$ signal in these channels.

The Gist

Imagine the subatomic world as a bustling, chaotic kitchen where tiny particles are the chefs, and the recipes they follow are the laws of physics. Recently, a team of scientists at the BESIII laboratory (a giant particle collider in China) was trying to find a very specific, rare ingredient in their pantry: a mysterious particle called $\eta_1(1855)$.

This particle is special because it has "exotic" quantum numbers—it's like a chef who can juggle three eggs with one hand while cooking with the other. It's a type of particle that shouldn't exist according to the old, simple recipes, suggesting it's made of something more complex (like a mix of quarks and gluons).

The Mystery: The Missing Ingredient

The scientists knew how to make this exotic particle in one specific dish: $J/\psi \to \gamma \eta \eta'$. They found it there with high confidence. So, they decided to try making it in a different dish: $\chi_{cJ} \to \eta \eta \eta'$.

They cooked up a massive batch of these particles and looked for the exotic ingredient in the leftovers. Result? Nothing. The exotic particle was nowhere to be found.

This left the scientists scratching their heads: Is the exotic particle a ghost that only appears in one specific recipe? Or is there a different reason it's hiding in this new dish?

The Investigation: The "Loop" Theory

This is where the authors of the paper you provided step in. They decided to act like culinary detectives. Instead of assuming the exotic particle should be there, they asked: "What else could be happening in the kitchen to create these leftovers?"

They proposed a theory involving intermediate loops.

The Analogy: The Detour

Imagine you want to get from Point A (the starting particle, $\chi_{cJ}$) to Point B (the final three particles: $\eta, \eta, \eta'$).

• The Direct Route: You might expect the particle to just split directly into the three final pieces.
• The Loop Route (The Paper's Idea): The paper suggests the particle takes a "scenic detour." It briefly transforms into heavy, temporary "delivery trucks" (called charmed mesons) that drive around in a circle (a loop) before dropping off the final ingredients.

There are two types of detours they looked at:

1. The Triangle Loop: Three trucks driving in a triangle shape.
2. The Box Loop: Four trucks driving in a square shape.

They also included a "traffic controller" in the middle of the loop called the $f_0(1500)$, which helps organize the traffic.

The Experiment: Simulating the Kitchen

The researchers used a computer model (a "virtual kitchen") to simulate these loops. They plugged in the known rules of physics (mathematical formulas called Lagrangians) and asked: "If the particles take these detours, what does the final dish look like?"

The Results were surprising and satisfying:

1. Perfect Match: When they calculated the results of these "loop detours," the numbers matched the actual experimental data from BESIII almost perfectly. They predicted exactly how many particles were made and how their energies were distributed.
2. The "Full Kitchen" Effect: Their calculations showed that these loop detours were so efficient and dominant that they essentially filled up the entire "production line."
3. The Conclusion: Because the "loop detours" (the charmed mesons and the $f_0(1500)$) are doing 100% of the work to create the final particles, there is no room left for the exotic $\eta_1(1855)$ to show up.

Why the Exotic Particle is "Missing"

Think of it like a crowded concert hall.

• The $\eta_1(1855)$ is a famous celebrity trying to get on stage.
• The Loop Mechanism is a massive, enthusiastic crowd of fans that has already filled every single seat and blocked every exit.

The paper argues that the reason the scientists didn't see the $\eta_1(1855)$ in this specific decay isn't because the particle doesn't exist or can't be made. It's simply because the "loop mechanism" is so loud and busy that it drowns out the signal. The exotic particle might be there, but it's hidden behind a wall of other particles created by the charmed meson loops.

The Takeaway

This study is a triumph of "process of elimination." By proving that a known, standard mechanism (the charmed meson loops) can fully explain the experimental data, the authors provide a strong reason why the exotic particle wasn't seen.

It tells us:

• The physics of the "loops" is working exactly as we thought.
• The search for the exotic particle isn't over; it just means we need to look in a different "dish" or find a way to cut through the "crowd" of the loop mechanism to see it clearly.

In short, the paper says: "We didn't find the ghost because the house was already full of very real, very busy delivery trucks."

Technical Summary

1. Problem Statement

The BESIII Collaboration recently reported the first observation of the exotic $1^{-+}$ isoscalar state $\eta_1(1855)$ in the radiative decay $J/\psi \to \gamma \eta\eta'$. However, a subsequent search for this state in the decay processes $\chi_{cJ}(1P) \to \eta\eta\eta'$ (where $J=0, 1, 2$) yielded no significant signal for the $\eta_1(1855)$ in the $\eta\eta'$ invariant mass spectrum.

• The Core Question: Why is the $\eta_1(1855)$ not observed in $\chi_{cJ} \to \eta\eta\eta'$ decays despite its observation in $J/\psi$ decays? Is the non-observation due to limited statistics, or does the underlying decay mechanism of $\chi_{cJ}$ suppress the production of this exotic state?
• Goal: To investigate the decay mechanisms of $\chi_{cJ} \to \eta\eta\eta'$ using effective Lagrangian approaches to determine if standard hadronic loop contributions can saturate the observed decay rates, thereby explaining the absence of the $\eta_1(1855)$ signal.

2. Methodology

The authors employ an effective Lagrangian approach to model the decay processes $\chi_{cJ} \to \eta\eta\eta'$ via intermediate charmed meson loops.

• Theoretical Framework:

  • Processes: The study focuses on $\chi_{c1} \to \eta\eta\eta'$ and $\chi_{c2} \to \eta\eta\eta'$. The $\chi_{c0}$ channel is excluded as it yields no loop contribution under the adopted leading-order Lagrangians.
  • Loop Mechanisms:
    • Box Loops: Involving intermediate charmed mesons ($D, D^*$) and scalar mesons.
    • Triangle Loops: Specifically for $\chi_{c1}$, involving an intermediate scalar resonance $f_0(1500)$ coupled to the charmed meson loop. This is motivated by BESIII partial wave analysis (PWA) indicating the $\eta\eta'$ structure is dominated by $f_0(1500)$.
  • Lagrangians:
    • Interactions between $\chi_{cJ}$ and charmed mesons ($D, D^*$) are described using heavy quark symmetry.
    • Interactions between charmed mesons and light pseudoscalars ($\eta, \eta'$) use chiral Lagrangians.
    • Couplings for the scalar $f_0(1500)$ to charmed mesons and light mesons are defined phenomenologically.
  • Amplitude Calculation: The total amplitude is the coherent sum of box and triangle loop amplitudes: $\mathcal{M}_{tot} = \mathcal{M}_{box} + e^{i\theta}\mathcal{M}_{tri}$, where $\theta$ is a relative phase angle.
  • Regularization: A form factor $F(q, M) = \frac{M^2 - \Lambda^2}{q^2 - \Lambda^2}$ is introduced to handle ultraviolet divergences, with the cutoff parameter $\Lambda = M + \alpha \Lambda_{QCD}$. The parameter $\alpha$ is varied to estimate uncertainties.

• Parameter Determination:

  • Coupling constants for $\chi_{cJ}D\bar{D}$ and $D^{(*)}D^{(*)}P$ are derived from experimental data and QCD sum rules.
  • The coupling $g_{f_0}$ (scalar to charmed mesons) is not well-known. The authors determine it by fitting the experimental branching fraction of $\chi_{c1} \to \eta\eta\eta'$, considering only box loops first, and then adjusting the triangle loop contribution to match the data. This yields $g_{f_0} \approx 6.87$.

3. Key Contributions

1. Mechanism Identification: The paper establishes that the decay $\chi_{cJ} \to \eta\eta\eta'$ is dominated by hadronic loop mechanisms (specifically charmed meson loops) rather than direct production of exotic resonances.
2. Quantitative Reproduction: The model successfully reproduces the experimental partial decay widths and branching fractions measured by BESIII for both $\chi_{c1}$ and $\chi_{c2}$ channels without invoking the $\eta_1(1855)$.
3. Explanation of Non-Observation: The study provides a theoretical explanation for the absence of the $\eta_1(1855)$ signal: the decay rate is already saturated by the standard $f_0(1500)$ and charmed meson loop contributions. Any additional contribution from $\eta_1(1855)$ would be sub-dominant or comparable to the background, making it difficult to distinguish with current statistics.
4. Predictions: The authors provide detailed predictions for the invariant mass spectra of $\eta\eta'$ and $\eta\eta$ pairs in $\chi_{c1}$ and $\chi_{c2}$ decays, which serve as benchmarks for future experimental verification.

4. Results

• $\chi_{c1} \to \eta\eta\eta'$:
  • The calculated partial decay width matches the BESIII measurement of $(0.117 \pm 0.014)$ keV for cutoff parameters $\alpha$ in the range $0.76 \leq \alpha \leq 0.94$.
  • The triangle loop contribution (via $f_0(1500)$) is slightly larger than the box loop contribution.
  • The relative phase angle $\theta$ between the two amplitudes is constrained to $150^\circ \sim 210^\circ$ to match the shape of the invariant mass distributions observed by BESIII.
  • The predicted invariant mass distributions for $\eta\eta'$ and $\eta\eta$ pairs are consistent with BESIII data, including the peak structure around 1.5 GeV attributed to $f_0(1500)$.
• $\chi_{c2} \to \eta\eta\eta'$:
  • Only box loops contribute in this channel.
  • The model reproduces the experimental width of $(0.087 \pm 0.019)$ keV at a cutoff parameter $\alpha \approx 2.0$.
  • Predictions for invariant mass distributions are provided for future experimental comparison.
• Implications for $\eta_1(1855)$:
  • Since the loop mechanisms saturate the decay width, the branching fraction for the exotic channel is constrained.
  • The authors estimate an upper limit: $B[\chi_{c1} \to \eta_1(1855)\eta^{(\prime)} \to \eta\eta\eta'] \leq 1.05 \times 10^{-4}$.
  • This is consistent with the BESIII upper limit of $9.79 \times 10^{-5}$, suggesting that if $\eta_1(1855)$ is produced, its contribution is negligible compared to the $f_0(1855)$ and loop backgrounds in this specific channel.

5. Significance

• Understanding Exotic States: The work clarifies the production dynamics of the $\eta_1(1855)$, suggesting that while it is observable in $J/\psi$ radiative decays, it is suppressed or masked in $\chi_{cJ}$ decays by dominant conventional hadronic loop mechanisms.
• Validation of Loop Mechanisms: It reinforces the importance of intermediate charmed meson loops in charmonium decays into light mesons, a mechanism often overlooked in favor of direct resonance production.
• Guidance for Future Experiments: The detailed predictions for invariant mass spectra in $\chi_{c2}$ decays offer a clear target for future BESIII analyses to validate the box-loop dominance hypothesis.
• Theoretical Consistency: The study bridges the gap between the observation of the $\eta_1(1855)$ in one channel and its non-observation in another, providing a coherent physical picture based on effective field theory.

In conclusion, the paper argues that the non-observation of $\eta_1(1855)$ in $\chi_{cJ} \to \eta\eta\eta'$ is not necessarily evidence against the existence of the state, but rather a consequence of the decay being dominated by well-understood charmed meson loop processes involving the $f_0(1500)$.