Improved measurement of the decays $η' \to π^{+}π^{-}π^{+(0)}π^{-(0)}$ and search for the rare decay $η' \to 4π^{0}$

Using a sample of 10 billion $J/\psi$ events collected by the BESIII detector, this study presents improved measurements of the branching fractions for $\eta' \to \pi^+\pi^-\pi^+\pi^-$ and $\eta' \to \pi^+\pi^-\pi^0\pi^0$, establishes an upper limit for the rare decay $\eta' \to 4\pi^0$, and reports the first extraction of the doubly virtual isovector form factor $\alpha$ from an amplitude analysis of $\eta' \to \pi^+\pi^-\pi^+\pi^-$.

The Gist

Imagine the subatomic world as a bustling, chaotic dance floor where particles are constantly colliding, breaking apart, and reassembling. In this paper, the BESIII Collaboration (a team of scientists from around the world) acts like a team of high-speed detectives, using a massive camera called the BESIII detector to take billions of snapshots of these collisions.

Here is the story of what they found, explained without the heavy math.

The Setup: A Giant Particle Factory

The scientists used a machine called the BEPCII (think of it as a giant racetrack for electrons and positrons). They smashed these particles together to create a very heavy, short-lived particle called the $J/\psi$.

When the $J/\psi$ dies, it often spits out a photon (a particle of light) and a lighter particle called the $\eta'$ (eta-prime). The $\eta'$ is the star of this show. It's a bit of a mystery particle in physics, and the scientists wanted to see exactly how it breaks apart.

The Mission: Three Different Cases

The team looked at three specific ways the $\eta'$ could decay (break apart) into pions (tiny particles that make up protons and neutrons).

Case 1 & 2: The "Common" Breakups (Improved Precision)

The $\eta'$ can break into four pions. Sometimes these are charged (like $+$ and $-$), and sometimes they are neutral (like $0$).

• The Goal: The scientists wanted to count exactly how often this happens.
• The Analogy: Imagine you are trying to count how many people in a stadium wear red hats. In the past, you only had a blurry photo of 1.3 billion people, and you guessed the number. This time, the team had a crystal-clear, high-definition photo of 10 billion people.
• The Result: Because they had so much more data, they could count the "red hats" (the decays) much more accurately.
  • They found that the $\eta'$ breaks into two charged and two neutral pions about 212 times out of every million.
  • They found it breaks into four charged pions about 86 times out of every million.
  • Why it matters: These numbers match what we expected, but now we know them with much higher confidence. This helps physicists test their theories about how the strong nuclear force works.

Case 3: The "Ghost" Hunt (The Rare Decay)

The third case was a search for a "ghost." The scientists looked for the $\eta'$ to break into four neutral pions ($4\pi^0$).

• The Analogy: This is like searching for a specific, incredibly rare type of snowflake in a blizzard. Theoretically, this event should happen, but it's so suppressed (rare) that it's almost impossible to see. It's like looking for a needle in a haystack, where the needle is made of glass and the haystack is made of other needles.
• The Result: They looked at their 10 billion snapshots and found zero clear evidence of this "ghost" event.
• The Takeaway: Since they didn't see it, they set a new "speed limit" for how rare it can be. They concluded that if this decay happens at all, it occurs less than 1 time in 80,000. This is four times stricter than the previous limit. It tells other scientists, "If you think you see this, you need to look harder, because our data says it's even rarer than we thought."

The Deep Dive: The "Amplitude Analysis"

For the first time, the scientists didn't just count the particles; they analyzed the dance steps of the decay.

• The Analogy: Imagine watching a car crash. You can count how many cars were involved (the branching fraction), but you can also analyze how they crashed. Did they hit head-on? Did they spin?
• The Discovery: They analyzed the decay of the $\eta'$ into four charged pions to measure a specific "twist" or "shape" in the physics, called a form factor (represented by the Greek letter $\alpha$).
• The Result: They measured this twist to be 1.22. This number matches the predictions of a famous theory called the Vector Meson Dominance (VMD) model. It's like confirming that the car crash happened exactly the way the physics textbooks predicted it would.

Why Should You Care?

You might ask, "Why do we care about a particle breaking into four pions?"

1. The Muon Mystery: One of the biggest unsolved mysteries in physics is the "muon g-2" experiment (why a particle called a muon wobbles slightly differently than predicted). The $\eta'$ particle plays a role in the calculations for this wobble. By measuring these decays more precisely, the scientists are helping to solve the puzzle of whether our current understanding of the universe is complete or if there is "new physics" hiding in the shadows.
2. Testing the Rules: Every time we measure a particle decay with higher precision, we are stress-testing the Standard Model of physics. If the numbers had been wrong, it would have meant our entire understanding of the universe's building blocks was flawed. Since they matched, the Standard Model gets another gold star.

Summary

In short, the BESIII team used a massive dataset (10 billion collisions) to:

1. Count two types of particle decays with record-breaking precision.
2. Hunt for a super-rare decay and set a new, stricter limit on how often it can happen (finding none).
3. Analyze the internal mechanics of a decay to confirm that our theoretical models of how particles interact are correct.

They didn't find a new particle, but they polished the lens through which we view the subatomic world, making the picture sharper and clearer than ever before.

Technical Summary

1. Problem and Motivation

The $\eta'$ meson is a flavor singlet state arising from the axial $U(1)$ anomaly, playing a crucial role in understanding low-energy Quantum Chromodynamics (QCD). While its dominant decays are well-measured, rare decays remain an open field for investigating symmetry-breaking mechanisms and testing Chiral Perturbation Theory (ChPT).

Specifically, this paper addresses three key areas:

• Precision Measurement of Four-Pion Decays: The decays $\eta' \to \pi^+\pi^-\pi^+\pi^-$ and $\eta' \to \pi^+\pi^-\pi^0\pi^0$ proceed via the internal conversion of two virtual photons ($\gamma^*\gamma^*$). These channels provide critical information on the meson-$\gamma^*\gamma^*$ coupling, which is essential for calculating the hadronic light-by-light contribution to the muon anomalous magnetic moment ($g-2$). Previous measurements by BESIII (2014) were limited by sample size.
• Amplitude Analysis: There was a lack of amplitude analysis for $\eta' \to \pi^+\pi^-\pi^+\pi^-$ to extract the doubly virtual isovector transition form factor, a parameter predicted by Vector Meson Dominance (VMD) models.
• Search for Rare CP-Violating Decay: The decay $\eta' \to 4\pi^0$ is highly suppressed in the Standard Model (S-wave CP-violation is negligible, $10^{-23}$). However, D-wave contributions in ChPT and VMD models predict a branching fraction around $10^{-8}$. Searching for this decay tests the limits of experimental sensitivity and theoretical models.

2. Methodology

Data Source 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.
• Process: The study focuses on the radiative decay chain $J/\psi \to \gamma \eta'$, followed by the subsequent decays of the $\eta'$.
• Simulation: Monte Carlo (MC) samples were generated using GEANT4, incorporating the BESIII detector geometry and response. Signal models were based on ChPT and VMD frameworks.

Event Selection:
Three distinct decay modes were analyzed with specific kinematic constraints:

1. Mode I ($\eta' \to \pi^+\pi^-\pi^+\pi^-$): Selected events with four charged tracks and at least one photon. A 4-constraint (4C) kinematic fit was applied to the $J/\psi \to \gamma \pi^+\pi^-\pi^+\pi^-$ hypothesis. Backgrounds from $\eta' \to \pi^+\pi^-\eta$ and $\eta' \to \pi^+\pi^-e^+e^-$ were suppressed by comparing $\chi^2$ values of different hypotheses.
2. Mode II ($\eta' \to \pi^+\pi^-\pi^0\pi^0$): Selected events with two charged tracks and at least five photons. Two $\pi^0$ candidates were reconstructed via 1C kinematic fits to $\gamma\gamma$ pairs. A 6C kinematic fit was applied to the full $J/\psi \to \gamma \pi^+\pi^-\pi^0\pi^0$ hypothesis. Strict vetoes were applied to reject $\eta$ and $\omega$ resonances in the $\pi^+\pi^-\pi^0$ system.
3. Mode III ($\eta' \to 4\pi^0$): Selected events with at least eight photons. Four $\pi^0$ candidates were reconstructed. An 8C kinematic fit was performed. A veto on the $\eta \to 3\pi^0$ mass combination was applied to reduce background.

Analysis Techniques:

• Branching Fraction Measurement: Signal yields were extracted using unbinned maximum likelihood fits to the invariant mass spectra ($M(\pi^+\pi^-\pi^+\pi^-)$ and $M(\pi^+\pi^-\pi^0\pi^0)$). Signal shapes were modeled by MC simulations convolved with Gaussian functions to account for resolution differences.
• Amplitude Analysis: For $\eta' \to \pi^+\pi^-\pi^+\pi^-$, an amplitude analysis was performed using a PDF constructed from a combination of ChPT and VMD models. The decay amplitude depends on a free parameter $\alpha$ (related to the ratio of coupling constants $c_3/(c_1-c_2)$).
• Rare Decay Search: For $\eta' \to 4\pi^0$, where no signal was observed, a Bayesian approach was used to determine the upper limit on the branching fraction. Systematic uncertainties were incorporated by convolving the likelihood distribution with a Gaussian function representing the total systematic error.

3. Key Contributions

1. Improved Precision: The branching fractions for $\eta' \to \pi^+\pi^-\pi^+\pi^-$ and $\eta' \to \pi^+\pi^-\pi^0\pi^0$ were measured with approximately three times better precision than previous BESIII results.
2. First Amplitude Analysis: This paper presents the first amplitude analysis of $\eta' \to \pi^+\pi^-\pi^+\pi^-$, successfully extracting the doubly virtual isovector form factor parameter $\alpha$.
3. Stringent Upper Limit: The search for $\eta' \to 4\pi^0$ set a new, significantly tighter upper limit on the branching fraction, improving upon the previous BESIII limit by a factor of four.

4. Results

Branching Fractions:
The measured branching fractions are consistent with previous measurements but with reduced uncertainties:

• $\mathcal{B}(\eta' \to \pi^+\pi^-\pi^+\pi^-) = (8.56 \pm 0.25_{\text{stat}} \pm 0.23_{\text{syst}}) \times 10^{-5}$
• $\mathcal{B}(\eta' \to \pi^+\pi^-\pi^0\pi^0) = (2.12 \pm 0.12_{\text{stat}} \pm 0.10_{\text{syst}}) \times 10^{-4}$

Amplitude Analysis:
The amplitude analysis yielded the parameter:

• $\alpha = 1.22 \pm 0.33_{\text{stat}} \pm 0.04_{\text{syst}}$
  This value is in good agreement with the theoretical prediction of unity derived from combined ChPT and VMD models (where $c_3 = 1$ and $c_1 - c_2 = 1$).

Rare Decay Search:

• No significant signal for $\eta' \to 4\pi^0$ was observed in the $M(4\pi^0)$ distribution.
• The upper limit on the branching fraction at the 90% confidence level was determined to be:
  • $\mathcal{B}(\eta' \to 4\pi^0) < 1.24 \times 10^{-5}$
    (This improves the previous limit of $4.94 \times 10^{-5}$).

5. Significance

• Muon $g-2$ Calculation: The improved precision on the branching fractions and the extraction of the form factor $\alpha$ provide more accurate inputs for calculating the hadronic light-by-light scattering contribution to the muon anomalous magnetic moment, a critical test of the Standard Model.
• Validation of Theoretical Models: The agreement of the measured $\alpha$ with VMD/ChPT predictions validates the theoretical description of the $\eta' \to 4\pi$ transition mechanism.
• Constraints on New Physics: The stringent upper limit on the $\eta' \to 4\pi^0$ decay constrains potential sources of S-wave CP-violation and tests the limits of D-wave contributions in effective field theories.
• Experimental Benchmark: The analysis demonstrates the capability of the BESIII detector to perform high-precision measurements of rare hadronic decays using large $J/\psi$ datasets, setting a new standard for future studies of light meson dynamics.