Observation of $ψ(3686)\to γη(1405)$ via $η(1405)\to f_0(980)π^0$

Using a sample of 2.7 billion $\psi(3686)$ events collected by the BESIII detector, this study reports the first observation of the decay $\eta(1405)\to\pi^+\pi^-\pi^0$ via the $f_0(980)$ intermediate state, measures its associated branching fraction, observes an isospin-violating decay involving the $f_1(1285)$, and sets upper limits on $\eta_c$ production in these channels.

The Gist

Imagine the subatomic world as a massive, high-speed cosmic dance floor. In the center of this floor sits a very heavy, short-lived dancer named $\psi(3686)$ (pronounced "psi-three-six-eight-six"). This dancer is so energetic that it occasionally decides to shed some weight by flashing a burst of light (a photon, or $\gamma$) and spinning off into lighter, faster dancers.

This paper is a report from the BESIII Collaboration, a team of scientists acting like ultra-high-speed cameras at this cosmic dance floor. They watched over 2.7 billion of these heavy dancers ($\psi(3686)$) to see exactly what happens when they flash a light and break apart.

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

1. The Main Discovery: Catching a Ghost in the Machine

The scientists were looking for a specific, tricky dance move:
$$\psi(3686) \rightarrow \gamma + \eta(1405)$$
The $\eta(1405)$ is a very unstable particle (a "ghost") that disappears almost instantly. It doesn't just vanish; it immediately splits into three other particles: a neutral pion ($\pi^0$) and a pair of charged pions ($\pi^+\pi^-$).

The Twist: The scientists found that the $\eta(1405)$ doesn't just split randomly. It seems to take a specific "detour" through a middleman called the $f_0(980)$.

• The Analogy: Imagine the $\eta(1405)$ is a magician. Instead of pulling a rabbit out of a hat directly, it pulls out a box (the $f_0(980)$), and then the box opens to reveal the rabbit and the hat (the pions).
• The Result: This is the first time anyone has seen this specific "box-opening" move happen in the decay of the $\psi(3686)$. They measured how often this happens and found it occurs about 3.77 times out of every 10 million tries.

2. The Mystery of the "Broken Rules"

In the world of particle physics, there is a famous rule of thumb called the "12% Rule." It predicts that if a heavy particle (like the $\psi(3686)$) decays into a certain final state, it should happen roughly 12% as often as a slightly lighter particle (the $J/\psi$) does the same thing.

• The Surprise: When the scientists compared their new data to old data from the lighter $J/\psi$ dancer, they found that the $\psi(3686)$ was breaking the rules. It was doing this specific decay much less often than the 12% rule predicted (only about 2.5% as often).
• Why it matters: This suggests that our current understanding of how these particles interact is incomplete. It's like predicting a car will get 30 miles per gallon, but it actually gets 8. Something hidden is happening under the hood. The paper suggests a complex quantum effect called a "triangle singularity" might be the culprit, acting like a traffic jam that slows down the decay.

3. The "Isospin" Violation (The Identity Crisis)

One of the most fascinating parts of the paper is the observation of Isospin Violation.

• The Concept: In the subatomic world, particles have a property called "isospin," which is like a secret identity badge. Usually, particles respect these badges and don't mix with others that have different badges.
• The Anomaly: The decay observed here involves the $f_0(980)$ and the $\eta(1405)$ mixing in a way that shouldn't happen according to standard rules. It's like seeing a person with a "Red Team" badge suddenly start dancing with the "Blue Team" in a way that breaks the club's strict dress code.
• The Explanation: The scientists suspect this happens because of a "mixing" effect between two similar particles ($a_0$ and $f_0$), but the math shows that this mixing alone isn't strong enough to explain what they saw. There must be other forces at play.

4. The Search for the "Missing" Particle ($\eta_c$)

The team also looked for a different dancer, the $\eta_c$ (eta-charm), which is a heavy particle made of a charm quark and an anti-charm quark.

• The Hunt: They scanned the debris from the $\psi(3686)$ decay to see if the $\eta_c$ was hiding there, decaying into three pions.
• The Result: They didn't find it. No clear signal.
• The Silver Lining: Even though they didn't find it, they set a very strict "speed limit" (an upper limit) on how often it could be happening. They improved the previous limit by a factor of 5, meaning if the $\eta_c$ is doing this dance, it's doing it very, very rarely. This helps rule out some theories about how these particles behave.

5. The Detective Work (How they did it)

To find these tiny signals in a sea of noise, the BESIII team used a massive dataset (2.7 billion events) and acted like forensic detectives:

• The Filter: They used computer algorithms to filter out billions of "fake" events (background noise) that looked similar but weren't the real thing.
• The Fit: They used statistical curves (like fitting a puzzle piece) to see if the remaining data formed a clear peak (a signal) or just random scatter (noise).
• The Confidence: For the main discovery ($\eta(1405)$), they were 100% sure (10.9 sigma significance). For the secondary discovery ($f_1(1285)$), they were about 99% sure (2.9 sigma), which is considered "evidence" but not yet a full "discovery."

Summary

In simple terms, this paper is a triumph of detective work in the quantum realm.

1. They found a new dance move: The $\psi(3686)$ decaying into an $\eta(1405)$ which then splits via an $f_0(980)$.
2. They broke a rule: This decay happens much less often than the "12% rule" predicts, hinting at new physics.
3. They solved a mystery (partially): They confirmed that simple mixing of particles can't explain the weird behavior; something more complex (like a triangle singularity) is likely involved.
4. They cleared the board: They ruled out the presence of the $\eta_c$ in this specific decay channel with much higher precision than before.

It's a story of how scientists use massive amounts of data and clever math to peek behind the curtain of the universe's most fundamental laws.

Technical Summary

1. Problem and Motivation

The paper addresses several open questions in hadron spectroscopy and the dynamics of charmonium decays:

• Nature of $\eta(1405)$ and $\eta(1475)$: These states near 1440 MeV/$c^2$ have long been debated. While $\eta(1475)$ decays predominantly to $K^*\bar{K}$, the nature of $\eta(1405)$ and its strong coupling to $a_0(980)\pi$ and $K\bar{K}\pi$ remains unclear.
• Isospin Violation: The decay $\eta(1405) \to f_0(980)\pi^0$ violates isospin symmetry. Previous measurements in $J/\psi$ radiative decays showed this branching ratio is unexpectedly large compared to $a_0(980)-f_0(980)$ mixing intensity. Theoretical models suggest a triangle singularity mechanism might explain this, but further data is needed.
• The "12% Rule": Perturbative QCD predicts that the ratio of branching fractions for $\psi(3686)$ and $J/\psi$ decaying to the same hadronic final state should be approximately 12% (specifically the ratio of their leptonic widths). Significant violations of this rule have been observed in other channels (e.g., $\rho\pi$), and it is unknown if this rule holds for radiative decays involving $\eta(1405)$ or $f_1(1285)$.
• $f_1(1285)$ Structure: The axial vector meson $f_1(1285)$ is hypothesized to be a $K^*\bar{K}$ molecule or a tetraquark. Its isospin-violating decay to $3\pi$ via $f_0(980)\pi^0$ requires further investigation.
• $\eta_c$ Search: Searching for the isospin-violating decay $\eta_c \to \pi^+\pi^-\pi^0$ provides a test of isospin symmetry in the charmonium sector. Previous limits were set with lower statistics.

2. Methodology

Data Sample:

• The analysis utilizes a sample of $(2712.4 \pm 14.3) \times 10^6$ $\psi(3686)$ events collected by the BESIII detector at the BEPCII collider. This is approximately five times larger than the dataset used in previous BESIII analyses of this channel.

Event Selection:

• Final State: The process $\psi(3686) \to \gamma \pi^+ \pi^- \pi^0$ is reconstructed, where $\pi^0 \to \gamma\gamma$.
• Tracks: Events require two oppositely charged pion tracks and at least three photons.
• Kinematics: A four-constraint (4C) kinematic fit is applied to the $\pi^+\pi^-\gamma\gamma\gamma$ combination.
• Particle Identification (PID): Pions are identified using $dE/dx$ and Time-of-Flight (TOF) information to reject electrons.
• Photon Selection: Isolated electromagnetic calorimeter (EMC) showers with energy $>25$ MeV (barrel) or $>50$ MeV (end cap) are selected.
• Background Rejection:
  • $\pi^0$ candidates are formed from $\gamma\gamma$ pairs with invariant mass close to $m_{\pi^0}$.
  • Events with $\eta$ mass ($0.52-0.57$ GeV/$c^2$) in any $\gamma\gamma$ pair are rejected.
  • Specific veto windows are applied to remove backgrounds from $J/\psi \to \pi^+\pi^-\gamma$ and $\omega \to \pi^+\pi^-\pi^0$.

Analysis Techniques:

• Signal Extraction: Unbinned maximum likelihood fits are performed on the invariant mass distributions:
  • $M(\pi^+\pi^-)$ to identify the $f_0(980)$ resonance.
  • $M(\pi^+\pi^-\pi^0)$ to identify $\eta(1405)$ and $f_1(1285)$ signals.
• Background Modeling:
  • For $\eta(1405)$: The background is modeled using sidebands of the $f_0(980)$ mass region and Chebyshev polynomials.
  • For $\eta_c$: The background includes $J/\psi$ peaks (modeled by Breit-Wigner convolved with Gaussian) and combinatorial backgrounds (polynomials).
• Systematic Uncertainties: Sources include tracking efficiency, photon detection, PID, kinematic fit quality, mass window definitions, signal shape modeling (MC vs. Breit-Wigner), $f_0(980)$ width variations, and fit range choices.

3. Key Contributions

1. First Observation: This is the first observation of the decay $\psi(3686) \to \gamma\eta(1405)$ via the intermediate state $\eta(1405) \to f_0(980)\pi^0$.
2. Evidence for $f_1(1285)$: Provides the first evidence for the isospin-violating decay $f_1(1285) \to f_0(980)\pi^0$ in $\psi(3686)$ radiative decays.
3. Testing the "12% Rule": The study explicitly tests the ratio of branching fractions between $\psi(3686)$ and $J/\psi$ for these specific radiative channels.
4. Improved Limits on $\eta_c$: Sets significantly improved upper limits on the isospin-violating decay $\eta_c \to \pi^+\pi^-\pi^0$.

4. Results

A. $\eta(1405)$ Observation:

• A clear signal is observed in the $M(\pi^+\pi^-\pi^0)$ spectrum with a statistical significance of 10.9$\sigma$.
• Measured Branching Fraction:
  $$\mathcal{B}(\psi(3686) \to \gamma\eta(1405) \to \gamma f_0(980)\pi^0 \to \gamma\pi^+\pi^-\pi^0) = (3.77 \pm 0.43_{\text{stat}} \pm 0.30_{\text{syst}}) \times 10^{-7}$$
• Comparison with Theory: The measured rate is significantly larger than what can be explained solely by $a_0(980)-f_0(980)$ mixing (expected $\sim 4 \times 10^{-8}$). This supports the hypothesis that a triangle singularity mechanism plays a crucial role in this decay.
• "12% Rule" Violation: Comparing with $J/\psi \to \gamma\eta(1405)$, the ratio is:
  $$\frac{\mathcal{B}(\psi(3686) \to \gamma\eta(1405))}{\mathcal{B}(J/\psi \to \gamma\eta(1405))} = (2.51 \pm 0.44)\%$$
  This is significantly lower than the predicted $\sim 12\%$, indicating a violation of the "12% rule" for this channel.

B. $f_1(1285)$ Observation:

• An excess is observed in the $f_0(980)\pi^0$ mass spectrum consistent with $f_1(1285)$.
• Significance: 2.9$\sigma$ (evidence level).
• Measured Branching Fraction:
  $$\mathcal{B}(\psi(3686) \to \gamma f_1(1285) \to \gamma f_0(980)\pi^0 \to \gamma\pi^+\pi^-\pi^0) = (7.36 \pm 2.25_{\text{stat}} \pm 2.36_{\text{syst}}) \times 10^{-8}$$
• "12% Rule" Compliance: Comparing with $J/\psi \to \gamma f_1(1285)$, the ratio is:
  $$\frac{\mathcal{B}(\psi(3686) \to \gamma f_1(1285))}{\mathcal{B}(J/\psi \to \gamma f_1(1285))} = (13.57 \pm 7.41)\%$$
  This result is consistent with the "12% rule" within uncertainties.

C. $\eta_c$ Search:

• No significant signal for $\eta_c \to \pi^+\pi^-\pi^0$ or $\eta_c \to f_0(980)\pi^0$ was found in the $\psi(3686)$ radiative decays.
• Upper Limits (90% CL):
  • $\mathcal{B}(\psi(3686) \to \gamma\eta_c) \times \mathcal{B}(\eta_c \to \pi^+\pi^-\pi^0) < 3.09 \times 10^{-7}$
  • $\mathcal{B}(\psi(3686) \to \gamma\eta_c) \times \mathcal{B}(\eta_c \to f_0(980)\pi^0) \times \mathcal{B}(f_0(980) \to \pi^+\pi^-) < 7.97 \times 10^{-8}$
• These limits represent an improvement by a factor of 5.2 over previous PDG values.

5. Significance

• Dynamics of Isospin Violation: The observation confirms that the isospin-violating decay $\eta(1405) \to f_0(980)\pi^0$ is a dominant mode in $\psi(3686)$ decays, reinforcing the need for non-perturbative mechanisms like triangle singularities to explain the large branching fraction.
• Charmonium Decay Rules: The contrasting results for $\eta(1405)$ (violation) and $f_1(1285)$ (compliance) regarding the "12% rule" suggest that the rule's validity is highly dependent on the specific hadronic final state and the underlying decay dynamics, challenging simple perturbative QCD expectations.
• Spectroscopy: The results provide crucial constraints for theoretical models attempting to distinguish between the $\eta(1405)$ and $\eta(1475)$ and to understand the internal structure of the $f_1(1285)$ meson.
• Experimental Benchmark: The improved upper limits on $\eta_c$ decays set a new standard for future searches in isospin-violating channels.