An improved measurement of $η^\prime\rightarrow e^{+}e^{-}ω$

Using a large sample of $J/\psi$ events collected by the BESIII detector, this study presents an improved measurement of the branching fraction for the decay $\eta^{\prime}\rightarrow e^{+}e^{-}\omega$ and reports the first determination of its transition form factor cutoff parameter, providing valuable insights into the internal structure of the $\eta^{\prime}$ meson.

The Gist

Imagine the subatomic world as a bustling, chaotic city where tiny particles are constantly born, dance together, and then explode into new forms. In this paper, a team of scientists from the BESIII Collaboration (a massive international group of physicists) acts like high-tech detectives, zooming in on one very specific, rare "crime scene" to understand the secrets of a particle called the eta-prime ($\eta'$).

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

1. The Crime Scene: A Rare Decay

The $\eta'$ is a heavy, unstable particle. Most of the time, when it dies (decays), it breaks apart in predictable, common ways. But occasionally, it does something very strange: it transforms into an omega particle ($\omega$) and a pair of electrons and positrons ($e^+e^-$).

Think of the $\eta'$ as a magician. Usually, it pulls a rabbit out of a hat. But in this rare trick, it pulls out a whole orchestra (the omega) and a pair of twins (the electron and positron) from thin air. This specific trick is called $\eta' \to e^+e^-\omega$.

Why do they care? Because the way the twins ($e^+e^-$) are born tells us about the internal structure of the magician. It's like looking at the footprints left behind to figure out what kind of shoes the magician was wearing.

2. The Magnifying Glass: The BESIII Detector

To catch this rare event, the scientists used the BESIII detector, a giant, high-tech camera located in Beijing, China.

• The Data: They didn't just look at a few events; they watched 10 billion collisions of particles ($J/\psi$ events). Imagine trying to find a specific, rare snowflake in a blizzard that has lasted for a decade. They had a massive library of snowflakes to search through.
• The Simulation: Before looking at the real data, they built a "virtual reality" version of the detector on a computer. They simulated millions of these rare events to know exactly what the "signal" should look like, so they wouldn't get fooled by fake clues.

3. The Investigation: Filtering the Noise

The real challenge was that for every real "magician trick," there were thousands of "fake tricks" (background noise).

• The Fake Clues: Sometimes, a photon (a particle of light) would accidentally hit the wall of the detector and split into an electron-positron pair. This looked exactly like the real event but wasn't.
• The Detective Work: The scientists used clever rules to filter these out.
  • The "Distance" Test: They checked where the electron pair was born. If they were born far away from the center of the collision (like a fake ID card), they were thrown out.
  • The "Mass" Test: They checked the weight (mass) of the particles. If the math didn't add up perfectly to the weight of an $\eta'$, it was a fake.

After filtering out the noise, they were left with 609 confirmed real events.

4. The Findings: Two Big Discoveries

Discovery A: How Often Does the Trick Happen? (Branching Fraction)

The scientists calculated exactly how often this rare trick happens compared to all the other ways the $\eta'$ can die.

• The Result: They found it happens about 1.79 times out of every 10,000 decays.
• Why it matters: Previous measurements were a bit fuzzy (like a blurry photo). This new measurement is 2.7 times sharper. It's like going from a grainy black-and-white photo to a crisp 4K image. This confirms that our current theories about how these particles work are correct.

Discovery B: The "Shape" of the Magic (Transition Form Factor)

This is the brand-new discovery. The scientists wanted to know how the $\eta'$ creates the electron pair. Is it a sudden pop? Or a slow fade?

• The Analogy: Imagine the $\eta'$ is a speaker playing a song. The "Transition Form Factor" is the equalizer on the speaker. It tells us how the sound (the energy) changes as the volume (the momentum) changes.
• The Result: They measured a specific number (called $\Lambda^{-1}$) that describes this shape. This is the first time anyone has measured this specific "equalizer setting" for this particle. It gives theorists a new piece of the puzzle to understand the "internal wiring" of the $\eta'$.

5. Why Should You Care?

You might ask, "Who cares about a particle decaying into electrons?"

• The Big Picture: The $\eta'$ is a unique particle because it helps scientists understand Quantum Chromodynamics (QCD), the force that holds the nucleus of an atom together. It's like studying the glue that holds the universe together.
• The Future: By measuring these tiny, rare events with extreme precision, scientists can test if our current "laws of physics" are perfect or if there are cracks in the theory that point to new, undiscovered physics.

Summary

In short, the BESIII team used a massive dataset to find a needle in a haystack of 10 billion particles. They successfully measured how often a rare particle transformation happens and, for the first time, mapped out how it happens. This provides a clearer, sharper picture of the subatomic world, helping us understand the fundamental rules that govern everything from atoms to stars.

Technical Summary

Here is a detailed technical summary of the paper "An improved measurement of $\eta' \to e^+e^-\omega$":

1. Problem Statement and Motivation

The paper addresses the need for a more precise measurement of the rare decay process $\eta' \to e^+e^-\omega$ and the first-ever determination of its Transition Form Factor (TFF) cutoff parameter.

• Physics Context: The $\eta'$ meson is a crucial probe for understanding meson internal structure and low-energy Quantum Chromodynamics (QCD). While hadronic decays are well-studied, radiative decays involving virtual photons (like $\eta' \to V e^+e^-$) provide direct access to the meson's electromagnetic structure via the TFF.
• Previous Limitations: A previous measurement by the BESIII collaboration (2015) using $1.31 \times 10^9$$J/\psi$events yielded a branching fraction (BF) of$(1.97 \pm 0.34 \pm 0.17) \times 10^{-4}$. The statistical uncertainty was significant, and the TFF parameter had not been measured.
• Goal: To improve the precision of the branching fraction measurement by a factor of $\sim 2.7$ and to extract the TFF cutoff parameter ($\Lambda^{-1}$) using a significantly larger dataset.

2. Methodology

Data Sample and Detector

• Dataset: The analysis utilizes a sample of $(10087 \pm 44) \times 10^6$ $J/\psi$ events collected by the BESIII detector at the BEPCII collider. This represents a roughly 7.7-fold increase in statistics compared to the previous BESIII study.
• Detector: The BESIII detector features a Helium-based Multi-Drift Chamber (MDC), a Time-of-Flight (TOF) system, and a CsI(Tl) Electromagnetic Calorimeter (EMC) within a 1.0 T (or 0.9 T for part of the data) solenoidal magnetic field.
• Simulation: Monte Carlo (MC) simulations based on GEANT4, KKMC, EVTGEN, and PHOTOS were used to model detector response, initial/final state radiation, and signal/background processes.

Event Selection and Reconstruction

The signal process is reconstructed via the chain:
$$J/\psi \to \gamma \eta', \quad \eta' \to e^+e^-\omega, \quad \omega \to \pi^+\pi^-\pi^0, \quad \pi^0 \to \gamma\gamma$$

• Final State: Four charged tracks ($e^+, e^-, \pi^+, \pi^-$) and at least three photons (one radiative $\gamma$ from $J/\psi$, two from $\pi^0$).
• Kinematic Constraints:
  • Charged tracks must pass within $\pm 20$ cm along the beam and $\pm 2$ cm transversely.
  • Photon energy thresholds: $>25$ MeV (barrel) or $>50$ MeV (end cap).
  • The most energetic photon is identified as the radiative photon ($E_\gamma \approx 1.4$ GeV).
  • A 4-constraint (4C) kinematic fit is applied to the $J/\psi \to \gamma \eta'$ hypothesis.
• Particle Identification (PID): Combined $dE/dx$ (MDC) and TOF information is used to distinguish electrons from pions. The combination with the smallest $\chi^2_{4C+PID}$ is retained.
• Background Suppression:
  • Photon Conversion: A major background arises from $\eta' \to \gamma\omega$ where a photon converts to $e^+e^-$. This is suppressed by requiring the transverse distance of the $e^+e^-$ vertex from the beam axis ($R_{xy}$) to be $< 2$ cm, or applying specific angular cuts for $R_{xy} \ge 2$ cm.
  • Other Decays: Vetoed regions in invariant mass distributions ($M(\gamma e^+e^-)$) to suppress backgrounds from $\eta \to \gamma e^+e^-$ and $\eta' \to \eta \pi^+\pi^-$.
  • Continuum: Off-resonance data showed no significant continuum background.

Analysis Techniques

• Branching Fraction (BF) Extraction: An unbinned maximum-likelihood fit is performed on the distribution of $M(\pi^+\pi^-\pi^0 e^+e^-) - M(\pi^+\pi^-\pi^0)$. This variable effectively separates the signal from peaking backgrounds (like $\eta' \to \gamma\omega$) by smoothing their distributions.
• TFF Extraction: A cleaner signal sample is selected with tighter mass windows. The TFF is extracted by fitting the $e^+e^-$ invariant mass spectrum using a likelihood function that incorporates the theoretical squared amplitude (Eq. 1) and detection efficiency. Background events are assigned negative weights in the likelihood fit to account for contamination.

3. Key Contributions

1. High-Precision Branching Fraction: The study provides the most precise measurement of the $\eta' \to e^+e^-\omega$ branching fraction to date, reducing the statistical uncertainty significantly.
2. First TFF Measurement: This is the first experimental determination of the transition form factor cutoff parameter ($\Lambda^{-1}$) for the $\eta' \to e^+e^-\omega$ decay.
3. Advanced Background Handling: The paper details sophisticated techniques to handle photon conversion backgrounds, which are the dominant systematic challenge in this channel, utilizing vertex reconstruction and specific kinematic vetoes.
4. Systematic Uncertainty Budget: A comprehensive breakdown of systematic uncertainties (tracking, PID, photon detection, kinematic fits, background modeling) is provided, demonstrating robust control over experimental effects.

4. Results

• Branching Fraction:
  $$\mathcal{B}(\eta' \to e^+e^-\omega) = (1.79 \pm 0.09 \text{ (stat)} \pm 0.12 \text{ (syst)}) \times 10^{-4}$$

  • This result is consistent with the previous BESIII measurement but improves the precision by a factor of 2.7.
  • It is also consistent with theoretical predictions based on Vector Meson Dominance (VMD) and chiral perturbation theory.

• Transition Form Factor Parameter:
  $$\Lambda^{-1} = (2.92 \pm 0.83 \text{ (stat)} \pm 0.15 \text{ (syst)}) \text{ GeV}^{-1}$$

  • This parameter characterizes the slope of the TFF at $q^2=0$. The measurement provides a new constraint for phenomenological models of the $\eta'$ meson's internal structure.

• Signal Yield: The fit yielded $609.1 \pm 27.7$ signal events.

5. Significance

• Validation of QCD Models: The precise branching fraction and the new TFF parameter serve as critical benchmarks for testing low-energy QCD models, specifically Vector Meson Dominance (VMD) and chiral perturbation theory.
• Internal Structure: The TFF measurement offers unique insights into the spatial distribution of charge and the internal structure of the $\eta'$ meson, which is a complex state involving gluonic components and $SU(3)$ flavor mixing.
• Future Prospects: The paper notes that the current TFF measurement is statistically limited. The methodology and results pave the way for future high-precision studies with larger datasets, potentially at the BESIII or future colliders, to further refine our understanding of pseudoscalar meson dynamics.

In summary, this work represents a significant advancement in the experimental study of rare $\eta'$ decays, offering a definitive measurement of the branching fraction and a pioneering look at the electromagnetic transition form factor of the $\eta'$ meson.