Search for $ψ_0(4360)\rightarrow ηψ(2S)$ through the process $e^+e^- \rightarrow ηηψ(2S)$

Using BESIII data collected at center-of-mass energies of 4.84, 4.92, and 4.95 GeV, no significant signal for the exotic charmonium-like state $\psi_0(4360)$ was observed in the process $e^+e^- \to \eta\eta\psi(2S)$, leading to the establishment of 90% confidence level upper limits on the relevant production cross-sections.

The Gist

Imagine the universe as a giant, high-speed racetrack where tiny particles called electrons and positrons zoom around and crash into each other. When they collide, they sometimes create a brief, fiery explosion of energy that instantly turns into new, heavier particles. Physicists at the BESIII experiment in China act like detectives at this racetrack, trying to figure out what new "vehicles" (particles) are being built in these crashes.

The Mystery: Looking for a "Ghost" Car

For a long time, physicists have known about a family of particles called "charmonium," which are like standard, well-behaved cars made of a charm quark and an anti-charm quark. But recently, they've spotted some strange, "exotic" vehicles that don't fit the standard blueprint. These are the XYZ particles.

One specific mystery they are trying to solve is the existence of a particle called $\psi_0(4360)$.

• The Theory: A few years ago, theorists predicted this particle might exist. They think it's not a standard car, but a "molecular" vehicle—a loose binding of two other particles stuck together, like two cars magnetically linked.
• The Clue: This particle is predicted to have a very specific "shape" (quantum numbers $0^{--}$) that makes it behave differently than the usual suspects. If it exists, it would be a huge clue about how the universe builds matter.

The Experiment: The "Partial Reconstruction" Trick

The team wanted to find this particle by smashing electrons and positrons together at specific high energies (4.84, 4.92, and 4.95 GeV). They were looking for a specific crash pattern:

1. The crash should produce an eta particle ($\eta$) and the mysterious $\psi_0(4360)$.
2. The mysterious $\psi_0(4360)$ should immediately fall apart into another eta particle and a $\psi(2S)$ (a known, heavier cousin of the standard charmonium).

The Challenge:
Detecting every single piece of debris from a crash is like trying to catch every single spark from a firework while wearing blinders. Some particles are hard to see or get lost in the noise.

The Solution (The Analogy):
Instead of trying to catch every single spark, the physicists used a clever trick called "partial reconstruction."

• Imagine you are trying to identify a specific type of car that always drops a red ball and a blue ball when it crashes.
• Instead of trying to find both balls, they only looked for the red ball (one eta particle, which they could see clearly because it turned into two photons).
• They assumed the blue ball (the second eta particle) was there, even if they couldn't see it directly. They calculated where it should be based on the laws of physics (conservation of energy and momentum).
• They also tracked the $\psi(2S)$, which they could see because it left a clear trail of other particles.

The Hunt: What They Found

The team analyzed a massive amount of data (0.9 "inverse femtobarns" of collisions, which is like watching billions of crashes). They looked for the specific "signature" of the $\psi_0(4360)$ in the data.

The Result:

• No Ghost Car Found: They did not see any evidence of the $\psi_0(4360)$ particle. The data looked exactly like what you would expect if only standard particles were being made, with no exotic "ghost" vehicles hiding in the crowd.
• Setting Limits: Since they didn't find it, they didn't just give up. They calculated the maximum possible size (cross-section) that this particle could have and still have gone unnoticed. Think of it like saying, "If this ghost car exists, it must be smaller than a grain of sand, or we would have seen it." They set strict upper limits on how likely it is to be produced.

Why This Matters (According to the Paper)

Even though they didn't find the particle, this is important work.

• Ruling Out Options: By proving the particle isn't there (or is extremely rare) at these specific energies, they are helping theorists refine their blueprints. It tells them that if this "molecular" particle exists, it might be harder to make or have different properties than predicted.
• Future Hope: The paper notes that the energy they used might have been just a little too low to easily create this specific particle (the "threshold" is around 4.9 GeV). They suggest that future upgrades to the machine (BEPCII-U), which will run at higher energies and with more power, will be needed to really solve this mystery.

In short: The physicists ran a high-speed search for a predicted exotic particle using a clever "missing piece" detection method. They didn't find it, but they successfully mapped out exactly where it isn't, narrowing the search for future discoveries.

Technical Summary

Technical Summary: Search for $\psi_0(4360) \to \eta\psi(2S)$ via $e^+e^- \to \eta\eta\psi(2S)$

Problem and Motivation
The paper addresses the search for an exotic charmonium-like state with quantum numbers $J^{PC} = 0^{--}$, denoted as $\psi_0(4360)$. While vector charmonium-like states ($Y$ states, $J^{PC}=1^{--}$) such as $Y(4260)$ and $Y(4360)$ are well-documented, the existence of a $0^{--}$ state is predicted by molecular scenarios where the $\psi(4360)$ is understood as a bound system dominated by $D\bar{D}_1(2420)$ components. Specifically, theoretical models predict a $0^{--}$ bound state of $D^*\bar{D}_1(2420)$ with a mass of approximately $(4366 \pm 18)$ MeV/$c^2$ and a narrow width ($<10$ MeV). Observing this state would provide crucial insights into the internal structure of vector mesons in the 4.2–4.5 GeV/$c^2$ mass range and the nature of other exotic hadrons like $Z_c$ and $P_c$ states. The process $e^+e^- \to \eta\psi_0(4360)$ followed by $\psi_0(4360) \to \eta\psi(2S)$ is proposed as a distinct channel to search for this state, distinguishable from conventional $1^{--}$ vector states by the angular distribution of the outgoing $\eta$ meson.

Methodology
The analysis utilizes data samples collected by the BESIII detector at the BEPCII storage ring at three center-of-mass (c.m.) energies: 4.84, 4.92, and 4.95 GeV, corresponding to a total integrated luminosity of 0.9 fb$^{-1}$.

• Event Selection: A partial reconstruction method is employed. The final state is $e^+e^- \to \eta\eta\psi(2S)$. One $\eta$ meson ($\eta_1$) is reconstructed via its decay to two photons ($\gamma\gamma$). The second $\eta$ ($\eta_2$) is treated inclusively as a missing particle, with its mass determined via recoil against the $\eta_1\psi(2S)$ system. The $\psi(2S)$ candidate is reconstructed through the decay chain $\psi(2S) \to \pi^+\pi^-J/\psi$, with the $J/\psi$ identified via its leptonic decays ($e^+e^-$ or $\mu^+\mu^-$).
• Kinematic Constraints: A one-constraint (1C) kinematic fit is applied to the $\gamma\gamma$ pair to constrain its mass to the nominal $\eta$ mass. The recoil mass of the $\eta_1\psi(2S)$ system is used to identify the inclusive $\eta_2$.
• Signal Definition: For the $\psi_0(4360)$ search, the candidate is reconstructed from the $\eta_h\psi(2S)$ system, where $\eta_h$ is the higher-momentum $\eta$ candidate between $\eta_1$ and $\eta_2$. The signal region for $\psi_0(4360)$ is defined as $[4310.8, 4416.4]$ MeV/$c^2$ ($\pm 3\sigma$ around the predicted mass).
• Background Estimation: The dominant background is estimated to be $e^+e^- \to \pi^0\pi^0\psi(2S)$. The expected background yield is calculated using measured cross-sections, integrated luminosity, detection efficiencies, and branching fractions, corrected for Initial State Radiation (ISR) and vacuum polarization.
• Statistical Analysis: Signal yields are obtained by subtracting estimated backgrounds from observed events. Upper limits on signal yields and cross-sections are calculated at the 90% confidence level (CL) using a frequentist method (implemented via trolke in ROOT) that incorporates unbounded profile-likelihood treatment for systematic uncertainties.

Key Contributions and Results

• Search for $e^+e^- \to \eta\eta\psi(2S)$: No significant signal for the non-resonant process $e^+e^- \to \eta\eta\psi(2S)$ was observed at any of the three energy points. Upper limits on the Born cross section $\sigma(e^+e^- \to \eta\eta\psi(2S))$ were established at the 90% CL.
• Search for $\psi_0(4360)$: No significant signal for the resonant process $e^+e^- \to \eta\psi_0(4360)$ with $\psi_0(4360) \to \eta\psi(2S)$ was observed at c.m. energies of 4.92 and 4.95 GeV.
• Upper Limits: The paper provides the first experimental upper limits on the product of the Born cross section and branching fraction, $\sigma(e^+e^- \to \eta\psi_0(4360)) \cdot \mathcal{B}(\psi_0(4360) \to \eta\psi(2S))$, at the 90% CL for the energies 4.92 and 4.95 GeV.
  • At 4.92 GeV: $\sigma \cdot \mathcal{B} < 6.8$ pb.
  • At 4.95 GeV: $\sigma \cdot \mathcal{B} < 16$ pb.
• Systematic Uncertainties: Comprehensive systematic uncertainties were evaluated, including luminosity (0.6%), tracking (4.0%), photon detection (1.0%), kinematic fits, mass windows, generator models, and branching fractions. The total systematic uncertainty ranges from approximately 4.9% to 7.4% depending on the energy and process.

Significance and Claims
The authors state that no $0^{--}$ molecular candidate $\psi_0(4360)$ was observed at the investigated energies. They note that the production threshold for $\eta\psi_0(4360)$ is approximately 4.9 GeV, suggesting that the process may be highly suppressed at the current BESIII energies due to limited phase space. Consequently, the paper concludes that while no signal was found, the established upper limits constrain theoretical models predicting this state. The authors suggest that future studies with higher energy (up to 5.6 GeV) and larger statistics, anticipated from the BEPCII-U upgrade, will be necessary to further explore this channel and potentially observe the state if it exists.