Study of $e^{+}e^{-}\to h^{+}h^{-}J/ψ~(h=π,~K,~p)$ via initial-state radiation at Belle~II

Using 427.9 fb$^{-1}$ of data collected by the Belle~II detector, this study measures the cross sections for $e^{+}e^{-}\to h^{+}h^{-}J/\psi$ ($h=\pi, K, p$) via initial-state radiation, confirming previous results for pion and kaon channels while presenting the first investigation of the proton channel and searching for vector charmonium-like structures.

The Gist

Imagine the universe as a giant, high-speed racetrack. On this track, tiny particles called electrons and positrons (the antimatter twins of electrons) zoom toward each other at nearly the speed of light. When they crash, they don't just bounce off; they annihilate, turning their energy into pure "stuff"—new particles that didn't exist a moment before.

This paper is a report from the Belle II experiment, a massive particle detector in Japan (like a high-tech, 360-degree security camera) that watches these crashes. The scientists are trying to solve a mystery: What are these strange, exotic particles that keep popping up?

Here is a breakdown of what they did and found, using everyday analogies:

1. The Mystery: The "Exotic" Family

For a long time, physicists had a simple family tree for particles. They thought everything was made of either two or three "bricks" (quarks).

• The Standard Model: Think of a normal car made of an engine and four wheels.
• The Exotic Anomalies: Since 2003, scientists have found "cars" that seem to have five wheels, or engines made of weird materials. These are called exotic states (like tetraquarks or pentaquarks). They don't fit the old family tree. The paper is trying to figure out if these are just weird combinations of normal bricks, or if they are entirely new types of "vehicles."

2. The Method: The "Flashlight" Trick

To study these particles, the scientists use a clever trick called Initial-State Radiation (ISR).

• The Analogy: Imagine you are trying to take a photo of a tiny, fragile insect, but your camera flash is too bright and will blow it away. So, you take a photo where the insect accidentally drops a bright flashlight (a photon) before you snap the picture. Because it dropped the flashlight, it slows down.
• In Physics: When the electron and positron crash, sometimes one of them shoots off a photon (the flashlight) before they collide. This means the actual collision happens at a lower energy. By measuring the energy of that "dropped" photon, the scientists can reconstruct exactly what happened at that lower energy, allowing them to scan a wide range of energies without needing to change the speed of the racetrack.

3. The Experiment: Building the "H" Family

The scientists looked for three specific types of crashes, where the result is a J/ψ particle (a heavy, stable particle) plus two other particles (h+h−).

• The Recipe: They looked for:
  1. J/ψ + two Pions (π)
  2. J/ψ + two Kaons (K)
  3. J/ψ + two Protons (p) — This one was brand new!

They analyzed 427.9 fb⁻¹ of data. To put that in perspective, that's like watching 427.9 trillion potential crashes, but only a few hundred actually produced the specific "recipe" they were looking for.

4. The Findings: What Did They See?

A. The Familiar Faces (Pions and Kaons)

When they looked at the Pion and Kaon crashes, the results were like seeing a familiar face in a crowd.

• The Result: They saw a big "bump" (a spike in data) around 4.26 GeV. This confirms the existence of a known exotic particle called Y(4260) (or Y(4230)).
• The "Ghost" Sighting: They also saw a tiny, faint hint of another bump near 4.1 GeV. It wasn't strong enough to be a confirmed discovery (only a 2-sigma signal, which is like a "maybe" in science), but it matches a prediction that there might be a particle related to the ψ(4040). It's like hearing a whisper in a noisy room; you think you heard something, but you need to listen again to be sure.

B. The New Frontier (Protons)

This was the most exciting part. No one had ever successfully measured the crash producing two protons and a J/ψ before.

• The Result: They found very few events (almost a "ghost town"). There was no clear "bump" or structure.
• The Takeaway: They couldn't find a new exotic particle here, but they set a strict limit. They said, "If this particle exists, it must be rarer than we thought." This is valuable because it tells other scientists, "Don't waste time looking for it here; look elsewhere."

C. The "Middleman" Search (Zc particles)

Sometimes, the crash doesn't happen all at once. It might go through a "middleman" particle.

• The Analogy: Imagine a relay race. Instead of A passing to C directly, A passes to B, and B passes to C.
• The Result: In the Pion crashes, they found a very strong signal (5.6 sigma) for a middleman called Zc(3900). This is a confirmed exotic particle! However, in the Kaon crashes, the relay race seemed to go straight from A to C with no middleman.

5. Why Does This Matter?

Think of the universe as a giant Lego set. For decades, we thought we knew all the shapes of the bricks. Then, we started building structures that didn't make sense with the old shapes.

• This paper is like a detailed inventory check. It confirms some of the weird structures we suspected were there (the Zc and Y particles).
• It also clears the air by saying, "We looked for this other weird thing (protons), and it's not there (or it's incredibly rare)."
• The Bottom Line: The more we map these "exotic" particles, the better we understand the fundamental rules of how matter is built. The Belle II team is essentially the cartographers drawing the map of this strange, new territory.

In short: The scientists used a "flashlight trick" to scan a massive amount of data. They confirmed some known exotic particles, found a strong candidate for a new "middleman" particle, and successfully mapped the unexplored territory of proton collisions, even if it turned out to be quieter than expected.

Technical Summary

1. Problem Statement

The traditional quark model successfully describes conventional charmonium states ($c\bar{c}$), but since 2003, numerous "exotic" resonant states (such as the $Y$ family and $Z_c$ states) have been discovered. These states exhibit properties (masses, quantum numbers, decay modes) that cannot be explained by simple $c\bar{c}$ configurations, suggesting structures like tetraquarks, pentaquarks, or hadronic molecules.

Key challenges in understanding these states include:

• Limited Data: Comprehensive understanding is hindered by a lack of high-statistics experimental data across various energy ranges.
• Unexplored Channels: While the $e^+e^- \to \pi^+\pi^-J/\psi$ and $K^+K^-J/\psi$ channels have been studied, the baryonic channel $e^+e^- \to p\bar{p}J/\psi$ remains largely unexplored, despite theoretical predictions suggesting it could offer insights into pentaquark production mechanisms.
• Intermediate States: The nature of intermediate resonances (e.g., $Z_c(3900)^\pm$) in these three-body final states requires further confirmation and precision measurement.

2. Methodology

The study utilizes data collected by the Belle II detector at the SuperKEKB collider.

• Dataset: A total integrated luminosity of 427.9 fb$^{-1}$ collected at or near the $\Upsilon(4S)$ and $\Upsilon(10753)$ resonances.
• Process: The analysis focuses on Initial-State Radiation (ISR) events where the $e^+e^-$ pair emits a photon, reducing the center-of-mass (c.m.) energy ($\sqrt{s}$) available for the production of the final state $h^+h^-J/\psi$ (where $h = \pi, K, p$).
• Energy Range:
  • $\pi^+\pi^-J/\psi$: 3.8 GeV to 5.5 GeV.
  • $K^+K^-J/\psi$: Threshold to 6.0 GeV.
  • $p\bar{p}J/\psi$: Threshold to 7.0 GeV.
• Event Selection & Reconstruction:
  • Tracks: Events are required to have exactly four charged tracks with net zero charge.
  • Particle ID (PID): Likelihood-based PID is used to identify pions, kaons, and protons with high efficiency (~90-95%).
  • $J/\psi$ Reconstruction: Reconstructed via $J/\psi \to e^+e^-$ and $J/\psi \to \mu^+\mu^-$. Mass windows are defined around the nominal $J/\psi$ mass.
  • ISR Tagging: Events are selected by requiring the recoil mass squared $|M^2_{recoil}| < 1 (\text{GeV}/c^2)^2$.
  • Background Suppression: Conversion backgrounds (e.g., $\gamma \to e^+e^-$) are suppressed using invariant mass cuts and PID criteria.
• Efficiency & Systematics:
  • Detection efficiencies are determined using Monte Carlo (MC) simulations generated with phokhara (ISR) and EvtGen (decays), corrected using data-driven reweighting based on Dalitz plot distributions and control channels (e.g., $\psi(2S) \to \pi^+\pi^-J/\psi$).
  • Systematic uncertainties include tracking, trigger, luminosity, PID, and modeling of the $h^+h^-$ dynamics.
• Analysis Techniques:
  • Cross-section Measurement: Calculated using the formula $\sigma = N_{sig} / (L_{eff} \cdot \epsilon \cdot \mathcal{B})$.
  • Intermediate State Search: Dalitz plot analysis and unbinned extended maximum likelihood fits to search for structures in the $h^\pm J/\psi$ invariant mass distributions.

3. Key Contributions

• First Measurement of $p\bar{p}J/\psi$: This paper presents the first experimental investigation of the $e^+e^- \to p\bar{p}J/\psi$ process via ISR at Belle II, setting upper limits on its cross-section.
• Precision Cross-Sections: Provides independent, high-precision measurements of $\sigma(e^+e^- \to \pi^+\pi^-J/\psi)$ and $\sigma(e^+e^- \to K^+K^-J/\psi)$, comparable to previous Belle results but with a new dataset.
• Combined Results: Combines the new Belle II data with previous Belle results to produce world-leading combined cross-section measurements for the $\pi$ and $K$ channels.
• Intermediate State Confirmation: Re-confirms the presence of the $Z_c(3900)^\pm$ structure in the $\pi^\pm J/\psi$ system with high statistical significance.

4. Results

A. Cross-Section Measurements

• $\pi^+\pi^-J/\psi$:
  • Measured cross-sections show an enhancement around 4.26 GeV, consistent with the $Y(4230/4320)$ structures.
  • A small excess is observed near 4.1 GeV (local significance 2.0$\sigma$), potentially consistent with a $\psi(4040)$-like enhancement predicted by coupled-channel analyses.
  • Results are consistent with previous Belle and BaBar measurements within uncertainties.
• $K^+K^-J/\psi$:
  • Measured cross-sections are consistent with previous Belle results.
  • No significant structures (like the $Y(4500)$ or $Y(4710)$ observed by BESIII) are observed, though the statistical power is limited compared to energy-scan methods.
• $p\bar{p}J/\psi$:
  • No significant signal is observed.
  • Upper Limits: 90% Confidence Level (C.L.) upper limits on the Born cross-section are established from threshold up to 7.0 GeV.
  • Near 6 GeV, the upper limit is < 0.16 pb, which is consistent with the theoretical prediction of $\mathcal{O}(4 \text{ fb})$ but significantly constrains models predicting larger cross-sections.

B. Intermediate State Searches

• $\pi^\pm J/\psi$ System:
  • A clear signal for $Z_c(3900)^\pm$ is observed in the $M_{max}(\pi J/\psi)$ distribution within the $Y(4230/4320)$ region.
  • The statistical significance is calculated to be 5.6$\sigma$ (nominal fit) and remains above 5.3$\sigma$ after accounting for systematic uncertainties.
• $K^\pm J/\psi$ System:
  • No obvious structure is found in the $M_{max}(K J/\psi)$ distribution.
• Dalitz Plots:
  • Dalitz plot projections confirm the $Z_c(3900)^\pm$ band in the $\pi J/\psi$ system but show no corresponding bands in the $K J/\psi$ system.

5. Significance

• Validation of Exotic Models: The confirmation of the $Z_c(3900)^\pm$ with high significance reinforces the existence of charged charmonium-like states, supporting tetraquark or molecular interpretations.
• Constraints on Theory: The null result and strict upper limits for $p\bar{p}J/\psi$ provide crucial constraints for theoretical models attempting to describe pentaquark production and baryonic charmonium-like states.
• Benchmark for Future Studies: The combined cross-sections from Belle and Belle II serve as a robust benchmark for future energy-scan experiments (like those at BESIII) and theoretical calculations.
• Methodological Advancement: The successful application of data-driven reweighting techniques to correct MC efficiencies demonstrates a mature analysis framework for ISR studies at Belle II, paving the way for more complex searches with larger datasets.

In conclusion, this study significantly advances the understanding of exotic hadrons by providing precise cross-section measurements, confirming known structures ($Z_c$), and establishing the first experimental limits on the baryonic channel $p\bar{p}J/\psi$.