Improved measurement of Born cross sections for $\chi_{bJ}\,\omega$ and $\chi_{bJ}\,(\pi^+\pi^-\pi^0)_{\rm non-\omega}$ ($J$ = 0, 1, 2) at Belle and Belle II

Using data from the Belle and Belle II experiments, this study measures Born cross sections for $\chi_{bJ}\,\omega$ and $\chi_{bJ}\,(\pi^+\pi^-\pi^0)_{\rm non-\omega}$ production, revealing that the $\Upsilon(10753)$ state decays exclusively into $\chi_{bJ}\,\omega$ while the $\Upsilon(10860)$ decays into the non-$\omega$ channel, and provides precise measurements of the $\Upsilon(10753)$ mass, width, and relevant partial width products.

The Gist

Imagine the subatomic world as a giant, chaotic dance floor where particles are constantly colliding, spinning, and transforming into new forms. Physicists at the Belle and Belle II experiments are like high-tech bouncers and choreographers. They smash electrons and positrons (the antimatter twins of electrons) together at incredibly high speeds to see what new "dancers" (particles) pop out of the collision.

This specific paper is about investigating a very specific, rare dance move involving heavy particles called bottomonium (made of a bottom quark and its anti-quark) and some lighter companions.

Here is the breakdown of their discovery, translated into everyday language:

1. The Setting: The High-Energy Dance Floor

The scientists used two massive detectors (Belle and Belle II) at particle accelerators in Japan. They ran a "energy scan," which is like tuning a radio dial. They didn't just listen to one frequency; they slowly turned the dial from 10.73 GeV to 11.02 GeV, looking for specific signals at every step.

2. The Mystery: Two Different "States"

In this energy range, there are three famous "heavy hitters" (resonances) that appear when the energy is just right:

• The Υ(10753): A new, mysterious state discovered recently.
• The Υ(10860): An older, well-known state.
• The Υ(11020): Another heavy state.

The big question was: How do these three states decay? When they break apart, what do they turn into?

3. The Experiment: Sorting the Debris

When these heavy states decay, they often produce a particle called $\chi_b$ (a bottomonium excitation) and a group of pions (light particles). The pions can arrange themselves in two distinct ways:

1. The "Organized" Group ($\omega$): Three pions ($\pi^+\pi^-\pi^0$) huddled together in a tight, specific formation called an omega meson.
2. The "Chaotic" Group ($\text{non-}\omega$): Three pions flying around loosely, not forming an omega.

The scientists wanted to see which heavy state preferred which group.

4. The Big Discovery: A Tale of Two Personalities

The results were strikingly clear, like finding out that two twins have completely different tastes in music:

• The Υ(10753) is a "Clubber":
  When this state decays, it loves the organized $\omega$ group. It produces $\chi_b + \omega$ frequently. However, it almost never produces the chaotic "non-$\omega$" group. It's like a dancer who only performs with a tight, synchronized trio.

• The Υ(10860) is a "Free Spirit":
  This state does the exact opposite. It refuses to produce the organized $\omega$ group. Instead, it loves the chaotic non-$\omega$ group. It's like a dancer who prefers a wild, unstructured jam session.

• The Υ(11020):
  This one also seems to prefer the chaotic "non-$\omega$" style, similar to the 10860.

5. Why This Matters: The "Internal Structure" Clue

In the world of particle physics, how something decays tells you what it's made of.

• If these states were just simple "balls of quarks" (like a standard bottomonium), they should behave similarly.
• But because they have such opposite preferences (one loves order, the other loves chaos), it suggests they have different internal structures.

Think of it like this:

• Υ(10753) might be a standard "quark ball" or a specific type of hybrid.
• Υ(10860) might be something more exotic, perhaps a "tetraquark" (four quarks stuck together) or a molecule of two lighter particles.

The paper suggests that the "chaotic" decay of the 10860 might happen because it first splits into an intermediate "messenger" particle (called a $Z_b$) which then breaks apart into the chaotic pions. The 10753 doesn't seem to use this messenger at all.

6. The Measurements

The scientists didn't just guess; they measured everything precisely:

• They calculated the exact mass and width (how long it lives) of the Υ(10753).
• They calculated the probability (branching fractions) of these decays.
• They confirmed that the Υ(10753) is a real, distinct particle and not just a fluke of data.

Summary

Imagine you have two identical-looking cars (the heavy states). You drive them off a cliff and see what parts fly out.

• Car A (10753) always ejects a neatly packed toolbox ($\omega$).
• Car B (10860) always ejects a pile of loose screws and nuts (non-$\omega$).

Even though they look the same from the outside, the fact that they eject such different parts proves they were built differently inside. This paper provides the strongest evidence yet that the Υ(10753) and Υ(10860) are fundamentally different types of matter, helping physicists solve the mystery of how quarks stick together to form the universe.

Technical Summary

1. Problem and Motivation

The paper addresses the nature of the exotic bottomonium-like states $\Upsilon(10753)$, $\Upsilon(10860)$, and $\Upsilon(11020)$. While the $\Upsilon(10860)$ and $\Upsilon(11020)$ were previously observed, the $\Upsilon(10753)$ was discovered more recently by the Belle experiment as a structure in the $e^+e^- \to \Upsilon(nS)\pi^+\pi^-$ cross-section.

• Theoretical Ambiguity: The nature of these states is debated; they could be conventional $b\bar{b}$ bottomonium, hybrid states, or tetraquarks.
• Decay Patterns: Previous studies suggested that $\Upsilon(10753)$ decays into $\chi_{bJ} \omega$, while $\Upsilon(10860)$ decays into $\chi_{bJ} \pi^+\pi^-\pi^0$. However, the separation between the $\omega$ resonance and the non-resonant $(\pi^+\pi^-\pi^0)_{\text{non-}\omega}$ component was not rigorously quantified in previous measurements.
• Goal: This study aims to precisely measure the Born cross sections for $e^+e^- \to \chi_{bJ} \omega$ and $e^+e^- \to \chi_{bJ} (\pi^+\pi^-\pi^0)_{\text{non-}\omega}$ across a wide energy range to clarify the decay patterns and internal structures of these resonances.

2. Methodology

Data Sets:

• Belle: Analyzed data collected at 18 energy points between $\sqrt{s} = 10.73$ and $11.02$ GeV, plus a large sample at the $\Upsilon(5S)$ peak ($10.8658$ GeV, $122 \text{ fb}^{-1}$). Total integrated luminosity: $142.5 \text{ fb}^{-1}$.
• Belle II: Analyzed data at four energy points ($10.653, 10.701, 10.745, 10.805$ GeV). Total integrated luminosity: $19.8 \text{ fb}^{-1}$.

Event Reconstruction and Selection:

• Decay Chain: $e^+e^- \to \chi_{bJ} \pi^+\pi^-\pi^0$, with $\chi_{bJ} \to \Upsilon(1S)\gamma$ and $\Upsilon(1S) \to \ell^+\ell^-$ ($\ell = e, \mu$).
• Kinematic Fit: A 6C kinematic fit was performed, constraining the four-momentum to the initial $e^+e^-$ center-of-mass energy, the lepton pair mass to the $\Upsilon(1S)$ mass, and the two-photon mass to the $\pi^0$ mass. This significantly improved the mass resolution for $\chi_{bJ}$ candidates (improving resolution by a factor of 1.9 compared to previous Belle II analyses).
• Signal Separation:
  • $\chi_{bJ} \omega$: Selected events where $M(\pi^+\pi^-\pi^0)$ falls within the $\omega$ mass window ($0.75 < M < 0.81$ GeV/$c^2$).
  • $\chi_{bJ} (\pi^+\pi^-\pi^0)_{\text{non-}\omega}$: Selected events where $M(\pi^+\pi^-\pi^0)$ is outside the $\omega$ window.
• Background Suppression: Used likelihood ratios for particle identification (PID) and rejected backgrounds from higher $\Upsilon$ states (e.g., $\Upsilon(2S)\pi^+\pi^-$) by checking recoil masses.

Analysis Techniques:

• High-Population Points: For energy points with sufficient statistics (e.g., $10.745$ GeV and $10.866$ GeV), a 2D unbinned extended maximum-likelihood fit was performed on the distributions of $M(\Upsilon(1S)\gamma)$ and $M(\pi^+\pi^-\pi^0)$.
• Low-Population Points: For sparse data points, event counting was used with background estimation derived from a combined fit of all energies.
• Cross-Feed Correction: A migration matrix was applied to correct for the overlap between $\chi_{b1}$ and $\chi_{b2}$ mass windows.
• Energy Dependence Fit: The cross sections were fitted using a coherent sum of Breit-Wigner amplitudes for $\Upsilon(10753)$, $\Upsilon(10860)$, and $\Upsilon(11020)$, including phase factors and interference terms.

3. Key Contributions

• First Clear Separation: This is the first study to rigorously separate the $\omega$ and non-$\omega$ contributions in the $\chi_{bJ} \pi^+\pi^-\pi^0$ final state across a broad energy scan.
• Improved Precision: Utilized the new precise beam-energy calibration and 6C kinematic fits to significantly improve mass resolution and signal extraction compared to previous Belle II measurements.
• Comprehensive Energy Scan: Combined data from both Belle and Belle II to cover the energy range from $10.73$ to $11.02$ GeV, allowing for a simultaneous fit of multiple resonances.

4. Key Results

Decay Patterns (The "Distinct" Behavior):

• $\Upsilon(10753)$: Decays significantly into $\chi_{bJ} \omega$ but not into $\chi_{bJ} (\pi^+\pi^-\pi^0)_{\text{non-}\omega}$.
  • Measured Mass: $(10756.1 \pm 3.4 \text{ (stat)} \pm 2.7 \text{ (syst)})$ MeV/$c^2$.
  • Measured Width: $(32.2 \pm 11.3 \text{ (stat)} \pm 14.9 \text{ (syst)})$ MeV.
  • Product $\Gamma_{ee}\mathcal{B}$: $(1.57 \pm 0.27 \pm 0.22)$ eV for $\chi_{b1}\omega$ and $(1.39 \pm 0.41 \pm 0.33)$ eV for $\chi_{b2}\omega$.
• $\Upsilon(10860)$: Decays significantly into $\chi_{bJ} (\pi^+\pi^-\pi^0)_{\text{non-}\omega}$ but not into $\chi_{bJ} \omega$.
  • The $\Upsilon(10860) \to \chi_{bJ} \omega$ signal is consistent with zero.
• $\Upsilon(11020)$: Shows a low-significance peak in the $\chi_{bJ} (\pi^+\pi^-\pi^0)_{\text{non-}\omega}$ channel, consistent with the $\Upsilon(10860)$ pattern.

Branching Fraction Ratios:

• The ratio $\mathcal{B}(\Upsilon(10753) \to \chi_{b1}\omega) / \mathcal{B}(\Upsilon(10753) \to \chi_{b2}\omega) = 1.13 \pm 0.38 \pm 0.34$.
• This ratio is inconsistent with the prediction for a pure $D$-wave bottomonium state (expected $\sim 15$) and differs by $1.8\sigma$ from the prediction for an $S-D$ mixed state ($0.2$), suggesting a complex internal structure.

Upper Limits:

• Strict upper limits were set for $\Upsilon(10753) \to \chi_{bJ} (\pi^+\pi^-\pi^0)_{\text{non-}\omega}$ and $\Upsilon(10860) \to \chi_{bJ} \omega$, confirming the mutual exclusivity of these decay modes.

5. Significance

• Structural Evidence: The distinct decay patterns of $\Upsilon(10753)$ and $\Upsilon(10860)$—despite having the same $J^{PC}$ quantum numbers and being separated by only $\sim 100$ MeV—strongly suggest they have different internal structures.
• Support for Exotic Models: The observation that $\Upsilon(10860)$ and $\Upsilon(11020)$ decay to $\chi_{bJ} (\pi^+\pi^-\pi^0)_{\text{non-}\omega}$ supports the hypothesis that these decays proceed via intermediate $Z_b(10610)$ and $Z_b(10650)$ states (tetraquark candidates), i.e., $\Upsilon \to Z_b \pi \to \chi_{bJ} \rho \pi$. The absence of this pattern in $\Upsilon(10753)$ implies it may not couple to these intermediate states in the same way, or possesses a different nature (e.g., a hybrid or a different tetraquark configuration).
• Future Directions: The results provide a clear roadmap for future Belle II studies to investigate the $\chi_{bJ} \rho$ mass spectrum around the $\Upsilon(10860)$ and $\Upsilon(11020)$ resonances to definitively search for $Z_b$ states.

In conclusion, this paper provides definitive experimental evidence that the $\Upsilon(10753)$ and $\Upsilon(10860)$ are distinct entities with mutually exclusive decay modes into $\chi_{bJ}$ mesons, challenging simple conventional bottomonium interpretations and supporting the existence of exotic hadronic structures.