Study of $e^+e^- \to π^+π^-Υ(1D)$ at Belle II

Using 19.6 fb⁻¹ of data collected by the Belle II detector, this study searches for D-wave bottomonium states via the reaction $e^+e^- \to \pi^+\pi^- \Upsilon(1D)$ near the $\Upsilon(10753)$ resonance, finds no significant signals, and sets 90% credibility level upper limits on the relevant cross-section and branching fraction products.

The Gist

The Big Picture: Hunting for a "Ghost" in the Particle Zoo

Imagine the universe is filled with a massive library of particles. Most of the books in this library are well-organized and cataloged. We know exactly what the "S-wave" and "P-wave" particles look like; they are the popular kids in school, well-studied and understood.

But there is a dusty, forgotten section of the library called the "D-wave" section. Scientists have a theory that there should be particles there, but no one has ever actually found them clearly. They are like ghosts in the machine—predicted by math, but elusive in reality.

This paper is about a team of scientists (the Belle II Collaboration) trying to catch one of these ghosts. They are looking for a specific type of heavy particle called bottomonium, which is made of a heavy "bottom" quark and its anti-particle. Specifically, they are hunting for the D-wave version of this particle.

The Setup: The Particle Collision Course

To find these ghosts, the scientists use a giant particle accelerator called SuperKEKB in Japan. Think of this accelerator as a high-speed racetrack where they smash electrons and positrons (anti-electrons) together.

When these particles collide, they create a burst of energy that can momentarily form heavy, unstable particles. The scientists are particularly interested in a specific "resonance" (a sweet spot of energy) called $\Upsilon(10753)$.

Think of the $\Upsilon(10753)$ as a loud, vibrating tuning fork. When you hit it just right, it should ring in a specific way. The scientists are asking: "If we hit this tuning fork, does it ring in a way that produces our missing D-wave ghost particles?"

The Detective Work: How They Looked

The scientists didn't just stare at the collision; they looked for a specific "fingerprint" left behind.

1. The Collision: They smashed particles together at four different energy levels (like tuning the radio to slightly different frequencies) right around the $\Upsilon(10753)$ resonance.
2. The Decay Chain: They hoped the collision would create a D-wave particle, which would then immediately fall apart (decay) into a chain of other particles:
   • First, it splits into two pions (light particles) and a D-wave particle.
   • The D-wave particle then turns into a photon (light) and a "chi" particle ($\chi_b$).
   • The "chi" particle turns into another photon and a very stable bottomonium particle ($\Upsilon(1S)$).
   • Finally, the stable particle splits into a pair of electrons or muons.

It's like a game of musical chairs where the music stops, and you have to identify who is left sitting in the chair by the specific clothes they are wearing. The scientists looked for this specific chain of "clothes" (particles) in their data.

The Result: The Ghost is Still Hiding

After analyzing 19.6 units of data (a massive amount of collision records), the scientists looked at their charts.

• What they expected: If the D-wave particles existed and were being produced, they should have seen a distinct "bump" or peak in their data charts, standing out clearly against the background noise.
• What they found: Nothing. The charts were flat. There were no bumps. The "ghost" was not there.

It's like setting up a camera to catch a rare bird in a forest. You wait for days, take thousands of photos, and look through every single one. You don't see the bird. You only see the trees and the wind.

What Does "No Signal" Mean?

In science, finding "nothing" is actually a very important result. It's not a failure; it's a constraint.

Because they didn't see the particles, they can now say: "If these particles exist, they are much rarer than we thought."

They calculated a "ceiling" (an upper limit). They can say with 90% confidence that the rate of this specific reaction is below a certain number. It's like saying, "We looked for a needle in a haystack, and we didn't find it. So, if the needle is there, there can't be more than 5 needles in the whole haystack."

Why Does This Matter?

This is crucial for understanding the nature of the $\Upsilon(10753)$ itself.

• The Mystery: Physicists are confused about what the $\Upsilon(10753)$ actually is. Is it a standard heavy particle? Is it a "hybrid" particle made of quarks and gluons mixed together? Or is it an "exotic" particle made of four quarks?
• The Test: Different theories predict different behaviors.
  • If $\Upsilon(10753)$ is a standard particle, it should easily produce these D-wave ghosts.
  • If it's an exotic "weird" particle, it might not produce them, or produce them very rarely.
• The Conclusion: Since the scientists found almost no D-wave particles, it suggests that the $\Upsilon(10753)$ might not be a standard particle. It might be something exotic, behaving differently than our old textbooks predicted.

The Takeaway

The Belle II team went on a high-tech treasure hunt for a missing piece of the particle puzzle. They used a massive detector to scan the wreckage of high-speed collisions. They didn't find the treasure (the D-wave particles), but by proving it's not there, they learned something profound: The "tuning fork" they were hitting ($\Upsilon(10753)$) is likely a stranger than we thought.

This result forces physicists to rewrite their theories about how these heavy particles are built, bringing us one step closer to understanding the deep, non-perturbative rules of the universe (Quantum Chromodynamics).

Technical Summary

1. Problem Statement and Motivation

The bottomonium spectrum (bound states of a $b\bar{b}$ quark pair) serves as a critical testing ground for Quantum Chromodynamics (QCD) in the non-perturbative regime. While S-wave ($L=0$) and P-wave ($L=1$) bottomonium states are well-characterized, D-wave states ($L=2$) remain poorly understood.

• The Gap: Theoretical models (e.g., Godfrey-Isgur, Cornell potential) predict the existence of spin-triplet $\Upsilon_J(1D)$ states ($J=1, 2, 3$) and a spin-singlet state. Previous experiments (CLEO, BaBar, Belle) have provided tentative evidence but failed to definitively resolve the individual contributions of these states or their production mechanisms.
• The Target: The study focuses on the $\Upsilon(10753)$ resonance, a vector state ($J^{PC}=1^{--}$) discovered near the $\Upsilon(5S)$ mass. Its nature is controversial; it may be a higher radial excitation, a hybrid state, or a tetraquark.
• The Specific Process: The paper investigates the hadronic transition $\Upsilon(10753) \to \pi^+\pi^-\Upsilon_J(1D)$. If $\Upsilon(10753)$ is a conventional quarkonium state, this decay rate should follow potential model predictions. If it is exotic, the rate or kinematic distributions may differ significantly.

2. Methodology

Data and Detector

• Dataset: The analysis utilizes 19.6 fb$^{-1}$ of data collected by the Belle II detector at the SuperKEKB collider.
• Center-of-Mass Energies ($\sqrt{s}$): Data was taken at four energies surrounding the $\Upsilon(10753)$ resonance: 10.653, 10.701, 10.745, and 10.805 GeV.
• Simulation: Monte Carlo (MC) samples were generated using Geant4 for detector response and EvtGen for decay dynamics. Signal processes were modeled with a phase-space (PHSP) approach, while backgrounds included known processes like $e^+e^- \to \omega\chi_{b1,2}$ and radiative returns.

Event Selection and Reconstruction

The signal chain is:
$$e^+e^- \to \Upsilon(10753) \to \pi^+\pi^-\Upsilon_J(1D) \to \pi^+\pi^- \gamma \chi_{bJ'} \to \pi^+\pi^- \gamma \gamma \Upsilon(1S) \to \pi^+\pi^- \gamma \gamma \ell^+\ell^-$$
(where $\ell = e, \mu$)

Key selection criteria included:

1. Particle Identification:
   • Pions: $p < 1.0$ GeV/c.
   • Leptons: $p > 1.0$ GeV/c, distinguished by ECL energy deposition ($E_{cluster} > 1.0$ GeV for electrons, $< 1.0$ GeV for muons).
   • Photons: Identified in the ECL with energy thresholds (20–22.5 MeV) and quality cuts.
2. Kinematic Fit: A 4-constraint kinematic fit was applied to enforce energy-momentum conservation ($\chi^2/NDOF < 25$).
3. Mass Windows:
   • $\Upsilon(1S)$ reconstructed from $\ell^+\ell^-$ in $[9.430, 9.499]$ GeV/$c^2$.
   • $\chi_{b1}$ and $\chi_{b2}$ reconstructed via a modified invariant mass variable $M[\gamma_H \Upsilon(1S)]$ to improve resolution.
4. Background Suppression:
   • $\gamma$-conversion veto: Rejected events with $\cos(\theta_{\pi^+\pi^-}) > 0.95$ to remove radiative Bhabha/di-muon backgrounds.
   • $\eta$ veto: Rejected events with $M(\gamma\gamma) \in [0.511, 0.576]$ GeV/$c^2$ to suppress $e^+e^- \to \eta\Upsilon(2S)$.

Signal Extraction Strategy

• Observable: The recoil mass against the $\pi^+\pi^-$ pair ($M(\pi^+\pi^-)_{\text{recoil}}$) was chosen over the $\gamma\gamma\Upsilon(1S)$ invariant mass because it offered superior resolution (5.9 MeV/$c^2$ vs. 6.1 MeV/$c^2$) due to precise tracking of low-momentum pions.
• Fitting: An unbinned extended maximum likelihood fit was performed on the recoil mass spectrum.
  • Signal PDFs: Derived from MC simulations for $\Upsilon_J(1D)$ and $\Upsilon(2S)$.
  • Background PDFs: A constant term plus a fixed-shape component for $e^+e^- \to \omega\chi_{b1,2}$ (normalized to previous Belle II measurements).
• Channel Separation:
  • $\Upsilon_2(1D)$: Searched via $\chi_{b1}$ channel (dominant decay).
  • $\Upsilon_3(1D)$: Searched via $\chi_{b2}$ channel (dominant decay).
  • Cross-feed between channels was neglected to ensure fit stability.

3. Key Contributions

• First Dedicated Search: This is the first dedicated search for $\Upsilon(10753) \to \pi^+\pi^-\Upsilon_J(1D)$ at the Belle II experiment, utilizing the high-luminosity environment of SuperKEKB.
• Systematic Rigor: The analysis provides a comprehensive evaluation of systematic uncertainties, including luminosity, tracking, photon efficiency, branching fractions, and decay models (PHSP vs. $f_0(500)$ resonance).
• Hypothesis Testing: The results are presented under two distinct hypotheses for the cross-section line-shape: one assuming production via the $\Upsilon(10753)$ resonance and another via the $\Upsilon(5S)$ resonance.

4. Results

• Observation: No significant signal for $\Upsilon_J(1D)$ ($J=2, 3$) was observed at any of the four center-of-mass energies.
• Upper Limits: 90% credibility level (C.L.) upper limits were set on the product of the cross-section and branching fraction ($\sigma \times \mathcal{B}$) for both channels.
  • $\Upsilon_2(1D)$: Limits range from < 0.18 pb to < 0.52 pb depending on energy.
  • $\Upsilon_3(1D)$: Limits range from < 0.69 pb to < 1.12 pb.
• Comparison with Theory/Previous Data:
  • The authors extrapolated the expected cross-section based on inclusive Belle measurements of $\Upsilon(5S) \to \pi^+\pi^-X$.
  • Under the $\Upsilon(5S)$ hypothesis, the measured upper limits are consistent with (or slightly above) the extrapolated expectations.
  • Under the $\Upsilon(10753)$ hypothesis, the measured upper limits are significantly lower (by a factor of several) than the expected cross-section near the resonance peak ($\sqrt{s} \approx 10.745$ GeV).

5. Significance and Conclusion

• Suppression of Dipion Transitions: The non-observation of $\Upsilon(10753) \to \pi^+\pi^-\Upsilon_J(1D)$ implies a strong suppression of this decay mode for the $\Upsilon(10753)$ resonance compared to standard potential model predictions for a conventional $b\bar{b}$ state.
• Implications for $\Upsilon(10753)$ Nature: This suppression, combined with the known enhancement of $\Upsilon(10753) \to \omega\chi_{b1,2}$, suggests that the $\Upsilon(10753)$ possesses an exotic internal structure (e.g., hybrid, tetraquark, or molecular state) rather than being a simple higher radial excitation of the bottomonium family.
• Future Outlook: The results constrain theoretical models of bottomonium spectroscopy and provide crucial data for understanding the dynamics of heavy quarkonium transitions and the nature of vector states in the 10.7–10.8 GeV region.