Study of $\chi_{b1,2}(2P) \to \omega \Upsilon(1S)$ transitions in $\Upsilon(3S) \to \gamma \chi_{b1,2}(2P)$ decays at BaBar

Using a dataset of 121.3 million $\Upsilon(3S)$ decays collected by the BaBar detector, this study presents the first measurements of $\chi_{b1,2}(2P)$ angular distributions and improved branching fractions for $\chi_{b1,2}(2P) \to \omega \Upsilon(1S)$ transitions while finding no evidence for the $\chi_{b0}(2P)$ decay mode.

The Gist

Imagine the universe as a giant, high-energy dance floor. In this paper, the scientists from the BABAR experiment are like detectives trying to figure out exactly how two specific dancers, a heavy "bottom" quark and its anti-quark partner, perform a very specific, tricky move.

Here is the story of their discovery, broken down into simple concepts.

1. The Setting: The Particle Dance Floor

The experiment took place at SLAC, a giant particle accelerator in California. Think of this as a massive, circular racetrack where they smash electrons and positrons (anti-electrons) together at incredible speeds.

When these particles collide, they sometimes create a heavy, short-lived particle called $\Upsilon(3S)$ (pronounced "Upsilon-three-S"). You can think of this particle as a "super-heavy dancer" that is excited and wants to calm down.

2. The Mystery Move: The "Photon Flash"

The excited $\Upsilon(3S)$ wants to lose energy. It does this by flashing a photon (a particle of light) and transforming into a slightly less excited version of itself. This is like a dancer spinning wildly, flashing a strobe light, and then settling into a slower, more graceful spin.

The scientists were looking for a specific sequence:

1. The heavy dancer ($\Upsilon(3S)$) flashes a photon.
2. It turns into a "chi-bottom" particle ($\chi_b$).
3. This new particle then instantly breaks apart into two things:
   • A $\Upsilon(1S)$: The "ground state" dancer (the calmest version).
   • An $\omega$ (omega): A small, three-pion "cloud" (a tiny burst of debris).

The paper focuses on two specific versions of the "chi-bottom" dancer: the $\chi_{b1}$ and the $\chi_{b2}$. They wanted to see how often this specific breakup happens and how the dancers spin during the process.

3. The Detective Work: Finding the Needle in the Haystack

The problem is that the racetrack is messy. For every one of these special moves, there are millions of other random collisions happening. It's like trying to hear a specific whisper in a stadium full of cheering fans.

The BABAR team had to filter through 121 million collisions (their "haystack") to find the 1,651 special events (the "needles") where this specific dance happened.

• The Filter: They looked for specific tracks left by particles (like footprints in the sand) and specific flashes of light.
• The "Veto" (The Bouncer): They knew some fake moves looked very similar to the real thing. They set up a "bouncer" at the door who kicked out any event that looked like a common background trick (specifically, events involving a different particle called $\Upsilon(2S)$). This cleaned up the data significantly.

4. The Results: What They Found

A. The Frequency (Branching Fractions)

The scientists counted how often the dance happened.

• The $\chi_{b1}$ dancer: They found that about 2.56% of the time, this specific dancer broke apart into the $\omega$ and $\Upsilon(1S)$. This is a very precise measurement, much better than previous attempts by other teams (CLEO and Belle).
• The $\chi_{b2}$ dancer: This one was rarer, happening only about 0.69% of the time.

B. The Spin (Angular Distributions)

This was a "first" for science. They didn't just count the dancers; they watched how they spun.

• They measured the angles at which the debris flew out.
• The Result: The way the particles flew matched the theoretical predictions perfectly. It's like if a physicist predicted that a spinning top would wobble in a specific pattern, and when they actually spun it, it wobbled exactly that way. This confirms our understanding of the "rules of the dance" (Quantum Mechanics).

C. The Missing Dancer ($\chi_{b0}$)

The scientists also looked for a third dancer, the $\chi_{b0}$.

• The Prediction: Theory suggested this dancer might exist and do the same move.
• The Reality: They found zero evidence of it.
• The Conclusion: If this dancer exists at all, it is so rare that it happens less than 0.23% of the time. They set a "ceiling" on how often it could possibly happen, effectively saying, "We don't see it, and if it's there, it's hiding very well."

5. Why Does This Matter?

You might ask, "Why do we care about these tiny particles?"

Think of the universe as a giant Lego set. The "rules" of how the Legos snap together are described by a theory called Quantum Chromodynamics (QCD). However, calculating these rules for heavy particles is incredibly hard, like trying to solve a Rubik's cube while riding a rollercoaster.

By measuring exactly how often these particles decay and how they spin, the BABAR team is providing high-precision test data.

• If their measurements match the theory, the theory is correct.
• If they don't match, the theory needs to be rewritten.

In this case, the measurements matched the theory beautifully (except for one ratio that was slightly off, suggesting there might be some subtle physics we haven't fully understood yet). This helps physicists refine their "instruction manual" for how the universe is built.

Summary Analogy

Imagine you are a film critic reviewing a movie.

• The Movie: The collision of particles.
• The Scene: The specific decay of the $\Upsilon(3S)$.
• The Critics (BABAR): They watched 121 million showings of the movie.
• The Review: They confirmed that the "Lead Actor" ($\chi_{b1}$) and "Supporting Actor" ($\chi_{b2}$) performed their scenes exactly as the script predicted. They also confirmed that the "Villain" ($\chi_{b0}$) didn't show up at all.

This paper is a high-precision review that tells us the script of the universe is mostly correct, but we need to keep reading to understand the tiny details.

Technical Summary

1. Problem Statement and Motivation

The paper investigates the radiative transitions of bottomonium states, specifically the decay chain $\Upsilon(3S) \to \gamma \chi_{bJ}(2P) \to \gamma \omega \Upsilon(1S)$.

• Scientific Context: The bottomonium ($b\bar{b}$) system serves as a laboratory for testing Quantum Chromodynamics (QCD), particularly potential models and lattice calculations.
• Previous Observations: The CLEO collaboration previously observed $\chi_{b1,2}(2P) \to \omega\Upsilon(1S)$ transitions, and Belle confirmed these with evidence for a $\chi_{b0}(2P)$ contribution.
• The Gap: While the $\chi_{b1}$ and $\chi_{b2}$ transitions were known, precise measurements of their branching fractions and angular distributions were lacking. Furthermore, the existence of the $\chi_{b0}(2P) \to \omega\Upsilon(1S)$ mode was theoretically predicted but experimentally elusive due to the kinematic threshold near the $\omega\Upsilon(1S)$ mass and potential suppression.
• Objective: To perform high-precision measurements of the branching fractions for $\chi_{b1,2}(2P) \to \omega\Upsilon(1S)$, measure their angular distributions for the first time, and search for the $\chi_{b0}(2P)$ decay mode using a significantly larger dataset than previous experiments.

2. Methodology

Dataset and Detector

• Data Source: The analysis utilizes data collected by the BABAR detector at the PEP-II asymmetric-energy $e^+e^-$ collider at SLAC.
• Integrated Luminosity: $28.0 \text{ fb}^{-1}$, corresponding to approximately $121.3 \times 10^6$ $\Upsilon(3S)$ decays.
• Reconstruction Chain: The study reconstructs the full decay chain:
  $$\Upsilon(3S) \to \gamma_s \chi_{b}(2P) \to \gamma_s \omega \Upsilon(1S)$$
  where $\omega \to \pi^+\pi^-\pi^0$ and $\Upsilon(1S) \to \ell^+\ell^-$ ($\ell = e, \mu$).

Event Selection and Background Suppression

• Track Requirements: Events must contain exactly four well-measured tracks (two leptons, two pions) with total charge zero and transverse momentum $> 0.1 \text{ GeV}/c$.
• Photon and $\pi^0$ Reconstruction: Photons are reconstructed in the electromagnetic calorimeter (EMC). $\pi^0$ candidates are formed from photon pairs with invariant mass in the range $(0.115\text{--}0.155) \text{ GeV}/c^2$.
• Kinematic Constraints:
  • Lepton/Pion Identification: To minimize systematic uncertainties, the two tracks with the highest momentum are assumed to be leptons, and the lower momentum tracks are pions. Particle ID is used only to tag the $\Upsilon(1S)$ decay mode ($\mu^+\mu^-$ vs $e^+e^-$).
  • Recoil Mass Veto: A major background arises from cascade decays $\Upsilon(3S) \to \gamma \chi_{b} \to \gamma \Upsilon(2S) \to \gamma \pi^+\pi^- \Upsilon(1S)$ and $\Upsilon(3S) \to \pi^0\pi^0 \Upsilon(2S)$. These are suppressed by calculating the recoil mass to the $\pi^+\pi^-$ system ($m_{\text{rec}}(\pi^+\pi^-)$) and vetoing events in the region around $9.79 \text{ GeV}/c^2$ (the $\Upsilon(2S)$ mass). This reduces background by a factor of $\approx 2 \times 10^{-3}$.
  • $\Delta^2$ Variable: To handle combinatorial background, the variable $\Delta^2 = m_{\text{rec}}^2(\gamma_s) - m^2(\Upsilon(1S)\omega)$ is used. Events with $|\Delta^2| > 0.4 \text{ GeV}^2/c^4$ are rejected.

Analysis Techniques

• Signal Extraction: An unbinned maximum likelihood fit (Channel Likelihood method) is performed on the recoil mass distribution of the signal photon, $m_{\text{rec}}(\gamma_s)$.
• Lineshape Modeling: The $\chi_{b1,2}(2P)$ signals are modeled as Breit-Wigner functions convolved with detector resolution functions (a sum of a Gaussian and a Crystal-Ball function). The natural width $\Gamma$ was optimized via a scan, finding $\Gamma = 1.4 \pm 0.5 \text{ MeV}$, which accounts for Doppler broadening effects better than the theoretical assumption of $\Gamma \approx 0.1 \text{ MeV}$.
• Angular Analysis: Three angles ($\theta_\omega$, $\theta$, $\theta_\gamma$) are defined to study the decay dynamics. Efficiency corrections are derived from Monte Carlo (MC) simulations and applied to the data to extract angular distributions.
• Branching Fraction Calculation: The branching fractions are calculated using the formula:
  $$\mathcal{B} = \frac{N_{\text{corr}}}{N_{\Upsilon(3S)} \cdot \mathcal{B}(\Upsilon(3S) \to \gamma \chi_{bJ}) \cdot \mathcal{B}(\chi_{bJ} \to \omega\Upsilon(1S))_{\text{meas}}}$$
  where $N_{\text{corr}}$ is the efficiency-corrected yield. Only the $\mu^+\mu^-$ sample is used for the final branching fraction measurement to avoid systematic uncertainties related to electron trigger prescaling.

3. Key Contributions

1. High-Precision Branching Fractions: The paper provides the most precise measurements to date for the branching fractions of $\chi_{b1,2}(2P) \to \omega\Upsilon(1S)$.
2. First Angular Distribution Measurements: This is the first experimental measurement of the angular distributions for these specific transitions, testing theoretical predictions regarding spin-dependent effects.
3. Optimized Resolution Modeling: The study demonstrates that the observed lineshape broadening is better described by an optimized width ($\Gamma \approx 1.4 \text{ MeV}$) rather than the theoretical natural width, attributing the difference to kinematic effects in the decay chain.
4. Systematic Uncertainty Control: A rigorous evaluation of systematic uncertainties (tracking, reconstruction, PID, lineshape modeling) is presented, resulting in total uncertainties of $\sim 7\%$ for $\chi_{b1}$ and $\sim 9\%$ for $\chi_{b2}$.

4. Results

Branching Fractions

The measured branching fractions (in percent) are:

• $\chi_{b1}(2P) \to \omega\Upsilon(1S)$: $(2.56 \pm 0.09_{\text{stat}} \pm 0.18_{\text{sys}} \pm 0.25_{\text{PDG}})\%$
• $\chi_{b2}(2P) \to \omega\Upsilon(1S)$: $(0.69 \pm 0.07_{\text{stat}} \pm 0.06_{\text{sys}} \pm 0.09_{\text{PDG}})\%$

These results are consistent with previous CLEO and Belle measurements but with significantly improved precision.

Ratio of Branching Fractions

The ratio $r_{2/1} = \mathcal{B}(\chi_{b2} \to \omega\Upsilon(1S)) / \mathcal{B}(\chi_{b1} \to \omega\Upsilon(1S))$ is measured as:

• $r_{2/1} = 0.27 \pm 0.03_{\text{stat}} \pm 0.02_{\text{sys}} \pm 0.04_{\text{PDG}}$

Significance: Theoretical models (Voloshin) predict this ratio to be approximately $1.3 \pm 0.3$ based on S-wave phase-space factors. The experimental result of $0.27$ deviates from the prediction by approximately 3.4$\sigma$, suggesting a potential discrepancy in theoretical understanding of the spin-dependent dynamics or phase space in these transitions.

Angular Distributions

• The angular distribution for $\omega \to \pi^+\pi^-\pi^0$ follows the expected $1 - \cos^2\theta_\omega$ behavior.
• The angular distributions for $\chi_{b1,2} \to \omega\Upsilon(1S)$ are consistent with theoretical expectations ($1 + \cos^2\theta$ for $\chi_{b1}$ and $1 - \frac{1}{7}\cos^2\theta$ for $\chi_{b2}$).

Search for $\chi_{b0}(2P)$

• No evidence was found for the $\chi_{b0}(2P) \to \omega\Upsilon(1S)$ decay mode.
• An upper limit is set at 90% confidence level:
  $\mathcal{B}(\chi_{b0}(2P) \to \omega\Upsilon(1S)) < 0.23\%$

5. Significance

This work significantly advances the spectroscopy of the bottomonium system. By providing high-precision branching fractions and the first angular distribution measurements, it offers stringent constraints on QCD potential models. The significant tension (3.4$\sigma$) between the measured ratio $r_{2/1}$ and theoretical predictions highlights a potential gap in the current understanding of spin-dependent forces or relativistic corrections in heavy quarkonium transitions. Additionally, the stringent upper limit on the $\chi_{b0}$ mode helps constrain models predicting its width and decay dynamics near the kinematic threshold.