Studies of hadron spectroscopy at Belle and Belle II

Using a $1.6\,\mathrm{ab}^{-1}$ dataset from Belle and Belle II, this study reports the first evidence for the $h_b(2P)\to \Upsilon(1S)\eta$ transition and the exotic $P_{c\bar c s}(4459)^0 \to J/\psi\Lambda$ decay in $\Upsilon$ decays, while finding no support for other predicted bottomonium transitions.

The Gist

Imagine the universe is a giant, cosmic Lego set. For decades, physicists have been trying to figure out how the smallest, most fundamental pieces (quarks) snap together to build the complex structures we see around us, like protons and neutrons.

This paper is a report card from two massive experiments, Belle and Belle II, located in Japan. Think of these experiments as two high-speed particle colliders that act like giant, ultra-precise "smashers." They crash electrons and positrons (matter and antimatter) together at incredibly high speeds to create a shower of new, short-lived particles. By studying the debris, scientists try to understand the rules of the "Lego set."

Here is a breakdown of what they found, explained simply:

1. The "Heavy Quark" Family Tree

The scientists were specifically looking at a family of particles called bottomonium. Imagine these as heavy, exotic atoms made of a "bottom quark" and its anti-particle, orbiting each other.

In this family, there are two main "personality types" based on how the quarks spin:

• The Singlets: The spins are opposite (like a calm, quiet couple).
• The Triplets: The spins are aligned (like an energetic, dancing couple).

The Big Question: Can a "calm" particle (Singlet) suddenly change into an "energetic" one (Triplet) by spitting out a tiny piece of debris?
According to old rules (the "Quark Model"), this is like asking a quiet librarian to suddenly start breakdancing. It's supposed to be extremely rare or impossible because of a rule called "heavy-quark spin symmetry." However, some scientists thought maybe invisible "ghost loops" (virtual particles popping in and out of existence) could help the librarian breakdance.

The Findings:

• The "Ghost" Theory: They looked for a specific dance move where the quiet particle turns into an energetic one and spits out a neutral pion (a tiny particle). Result: No breakdancing found. The quiet librarian stayed quiet. This suggests the "ghost loops" aren't helping as much as some theories predicted.
• The Surprise Discovery: They did find evidence of a different dance move: the quiet particle turning into an energetic one and spitting out an eta particle. This is the first time this specific transition has been seen!
  • The Catch: They found it, but it happened much less often than the "ghost loop" theory predicted. It's like finding a breakdancing librarian, but they only did it once a year instead of once a minute. This suggests the "ghost loops" aren't the main reason these transitions happen.

2. The "Exotic Zoo" (Pentaquarks)

For a long time, we thought matter was built in two ways:

1. Mesons: Two Lego bricks stuck together (a quark and an anti-quark).
2. Baryons: Three Lego bricks stuck together (three quarks).

But the laws of physics say you could build weird shapes, like a Tetraquark (4 bricks) or a Pentaquark (5 bricks). These are called "exotic" because they are so unusual.

Recently, the LHCb experiment in Europe found a candidate for a Pentaquark made of 4 quarks and 1 anti-quark, specifically one containing a "strange" quark, named $P_{c\bar{c}s}(4459)^0$. It's like finding a 5-piece Lego structure that nobody thought could hold together.

The New Discovery:
The Belle and Belle II teams asked: "Can we find this weird 5-piece structure in the debris of our Japanese smashers?"

• They looked at the wreckage from the $\Upsilon(1S)$ and $\Upsilon(2S)$ resonances (specific heavy particle states).
• Result: They found evidence that this exotic 5-piece structure ($P_{c\bar{c}s}(4459)^0$) is indeed being created in these collisions!
• Why it matters: This is the first time anyone has seen this specific exotic shape being born from these types of collisions. It's like finding a rare, rare bird in a new forest. It proves that nature is even more creative with its Lego building than we thought.

3. The Future: Building a Bigger Net

The paper concludes by looking ahead. The Belle experiment has finished its run, but its successor, Belle II, is just getting started.

• Belle collected about 1,000 "buckets" of data.
• Belle II aims to collect 50 times that amount.

Think of it like fishing. Belle cast a small net and caught a few interesting fish. Belle II is casting a massive industrial trawler. With 50 times more data, they will be able to:

1. Confirm if those rare "breakdancing librarians" (the transitions) really exist or if they were just flukes.
2. Catch even more of those exotic "5-piece Lego monsters" (exotic states) to understand exactly how they are built.

Summary in a Nutshell

• Did they find the "ghost loops"? No, not really. The transitions they found are rarer than the "ghost" theories predicted.
• Did they find new physics? Yes! They found the first evidence of a specific rare transition and, more importantly, the first evidence of a rare exotic 5-quark particle being made in these collisions.
• What's next? With the new, super-powerful Belle II machine, we are about to enter a golden age of discovering new, weird, and wonderful forms of matter.

Technical Summary

1. Problem and Motivation

The paper addresses two primary challenges in high-energy physics:

• Understanding Heavy-Quark Spin Symmetry: In the quark model, bottomonium states ($b\bar{b}$) are characterized by spin-singlet ($S=0$) and spin-triplet ($S=1$) configurations. Transitions between these states (e.g., $h_b \to \Upsilon$) are theoretically suppressed by heavy-quark spin symmetry. However, this suppression may be lifted by "couple-channel effects" arising from hadron loops. Precise measurements of branching fractions for these transitions are required to test effective field theories and constrain these non-perturbative QCD effects.
• Discovery of Exotic Hadrons: While standard mesons ($q\bar{q}$) and baryons ($qqq$) are well understood, QCD permits exotic configurations like tetraquarks and pentaquarks. The paper seeks to identify pentaquark candidates containing a $c\bar{c}$ pair and a strange quark ($P_{c\bar{c}s}$) produced in $\Upsilon(1S, 2S)$ decays, an environment previously unexplored for such states despite the known enhancement of baryon production in these decays.

2. Methodology

The analysis utilizes a dataset of 1.6 ab$^{-1}$ of $e^+e^-$ collision data collected by the Belle and Belle II experiments at center-of-mass energies near the $\Upsilon(nS)$ resonances. The study is divided into two main analytical components:

A. Spin-Singlet to Triplet Transitions

• Data Source: Primarily the $\Upsilon(5S)$ resonance sample (121 fb$^{-1}$) from Belle, where $h_b(1P, 2P)$ states are produced via $e^+e^- \to h_b(2P)\pi^+\pi^-$.
• Reconstruction Strategy:
  • $h_b(2P) \to \Upsilon(1S)\eta$: Reconstructed via $\Upsilon(1S) \to \ell^+\ell^-$ and $\eta \to \gamma\gamma$. A 2D kinematic fit is performed using the invariant mass of the $\gamma\gamma$ system ($M_{\gamma\gamma}$) and the recoil mass against the $\pi^+\pi^-$ system ($M^{rec}_{\pi\pi}$).
  • $h_b(2P) \to \Upsilon(1S)\pi^0$: Similar kinematic selection but targeting the $\pi^0$ mass region.
  • $h_b(2P) \to \chi_{bJ}(1P)\gamma$: Reconstructed via the cascade $h_b(2P) \to \chi_{bJ}\gamma_1 \to \Upsilon(1S)\gamma_2\gamma_1 \to \mu^+\mu^-\gamma_1\gamma_2$. A 2D fit is applied to the difference in invariant masses of the decay products and the recoil mass.
• Statistical Analysis: Unbinned extended maximum likelihood fits are used to extract signal yields, accounting for backgrounds from processes like $\Upsilon(5S) \to \chi_{bJ}\pi^+\pi^-\pi^0$ and combinatorial backgrounds.

B. Search for Exotic $P_{c\bar{c}s}$ States

• Data Source: Inclusive decays of $\Upsilon(1S)$ and $\Upsilon(2S)$.
• Reconstruction Strategy:
  • Target process: $\Upsilon(1S, 2S) \to P_{c\bar{c}s} + X \to J/\psi \Lambda + X$.
  • Decay chains: $J/\psi \to \ell^+\ell^-$ and $\Lambda \to p\pi^-$.
  • Background Estimation: A 2D sideband subtraction method is employed using the invariant masses of the $\Lambda$ and $J/\psi$ decay products to estimate combinatorial and non-resonant backgrounds.
• Signal Extraction: A fit to the $M(J/\psi\Lambda)$ invariant mass distribution is performed. The fit includes a signal component (constrained by LHCb mass/width parameters initially), a non-resonant component, and a background polynomial constrained by sideband data.

3. Key Contributions and Results

A. $h_b$ Transition Results

1. First Evidence for $h_b(2P) \to \Upsilon(1S)\eta$:
   • Significance: 3.5$\sigma$ (including systematics).
   • Branching Fraction: $\mathcal{B} = (7.1^{+3.7}_{-3.2} \pm 0.8) \times 10^{-3}$.
   • Implication: This value is approximately 10 times lower than the theoretical prediction of $\sim 10\%$ derived from $\Upsilon(3S) \to h_b(1P)\pi^0$ data. This discrepancy suggests that hadronic loop effects (couple-channel effects) do not contribute significantly to this decay, challenging previous theoretical models.
2. Null Results for Other Transitions:
   • $h_b(1P, 2P) \to \Upsilon(1S)\pi^0$: No events found in signal regions. Upper limits set at 90% CL: $\mathcal{B} < 1.8 \times 10^{-3}$.
   • $h_b(2P) \to \chi_{bJ}(1P)\gamma$: No events found. Upper limits set at 90% CL:
     • $J=0$: $\mathcal{B} < 2.7 \times 10^{-1}$
     • $J=1$: $\mathcal{B} < 5.4 \times 10^{-3}$
     • $J=2$: $\mathcal{B} < 1.3 \times 10^{-2}$
   • These limits are consistent with relativistic quark model predictions ($10^{-6}$–$10^{-5}$) but are not yet sensitive enough to fully constrain hadronic loop enhancements.

B. Exotic State Results

1. Evidence for $P_{c\bar{c}s}(4459)^0$:
   • Significance: 3.3$\sigma$ (including systematics).
   • Observation: An enhancement in the $M(J/\psi\Lambda)$ spectrum near the nominal mass of the $P_{c\bar{c}s}(4459)^0$.
   • Branching Fractions:
     • $\Upsilon(1S) \to P_{c\bar{c}s}(4459)^0 + X \to J/\psi\Lambda + X$: $(3.5 \pm 2.0 \pm 0.2) \times 10^{-6}$.
     • $\Upsilon(2S) \to P_{c\bar{c}s}(4459)^0 + X \to J/\psi\Lambda + X$: $(2.9 \pm 1.7 \pm 0.4) \times 10^{-6}$.
   • Mass and Width: Measured as $M = 4471.7 \pm 4.8 \pm 0.6$ MeV/$c^2$ and $\Gamma = 22 \pm 13 \pm 3$ MeV/$c^2$, consistent with LHCb measurements.
2. Null Result for $P_{c\bar{c}s}(4338)^0$:
   • No significant signal observed. Upper limits set at 90% CL: $\mathcal{B} < 1.8 \times 10^{-6}$ ($\Upsilon(1S)$) and $< 1.6 \times 10^{-6}$ ($\Upsilon(2S)$).

4. Significance

• Theoretical Constraints: The measurement of the $h_b(2P) \to \Upsilon(1S)\eta$ branching fraction provides a critical constraint on non-perturbative QCD models. The lower-than-expected rate disfavors the hypothesis that hadronic loops significantly lift heavy-quark spin symmetry suppression in these transitions.
• Discovery of Exotics in $\Upsilon$ Decays: This work presents the first evidence of an exotic hadronic state ($P_{c\bar{c}s}(4459)^0$) produced in inclusive $\Upsilon(1S, 2S)$ decays. This confirms that $\Upsilon$ decays are a viable and rich source for discovering strange pentaquarks, expanding the search landscape beyond the $B$-meson and $pp$ collision environments (LHCb).
• Future Prospects: With Belle II aiming for a dataset 50 times larger than Belle's, these results lay the foundation for high-precision spectroscopy, potentially turning the current 3$\sigma$ evidence into definitive discoveries and probing the internal structure of exotic hadrons with unprecedented accuracy.