Observation of $\eta_c(1S)\to \Sigma^0\bar \Sigma^0$ and search for $h_c(1P)\to \Sigma^0\bar \Sigma^0$ via $\psi(3686)$ transitions

Using a large sample of $\psi(3686)$ events collected by the BESIII detector, this study reports the first observation of the decay $\eta_c(1S)\to\Sigma^0\bar{\Sigma}^0$ with branching fractions determined under different interference scenarios, while setting an upper limit on the unobserved decay $h_c(1P)\to\Sigma^0\bar{\Sigma}^0$.

The Gist

Imagine the subatomic world as a bustling, high-energy dance floor where particles are constantly being created, spinning, and then crashing into each other to form new groups. This paper is a report from the BESIII Collaboration, a team of scientists acting like ultra-precise detectives at a massive particle collider in Beijing (the BEPCII). They are trying to solve a mystery about two specific "dancers" in the charmonium family: the $\eta_c$ (eta-c) and the $h_c$ (h-c).

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

The Setup: A Giant Particle Factory

The scientists had a huge stash of data: over 27 million events where a heavy particle called the $\psi(3686)$ (psi-3686) was created. Think of the $\psi(3686)$ as a giant, unstable balloon. When it pops, it doesn't just disappear; it transforms into other particles.

The team was looking for two specific ways this balloon could pop:

1. The $\eta_c$ Case: The balloon pops, emits a flash of light (a photon), and leaves behind an $\eta_c$ particle, which then immediately splits into a pair of "baryon twins": a $\Sigma^0$ and an anti-$\Sigma^0$.
2. The $h_c$ Case: The balloon pops, emits a neutral pion (a different type of particle), and leaves behind an $h_c$ particle, which then tries to split into that same pair of twins.

The Mystery: The "Interference" Dance

The main discovery in this paper is about the first case ($\eta_c \to \Sigma^0 \bar{\Sigma}^0$).

In the quantum world, particles behave a bit like waves. When two waves meet, they can either cancel each other out (destructive interference) or boost each other up (constructive interference). It's like two people trying to push a swing: if they push at the same time, the swing goes high (constructive); if one pushes while the other pulls back, the swing barely moves (destructive).

The scientists found that the $\eta_c$ particle did decay into the $\Sigma^0$ pair, but the number of times it happened depended entirely on this "pushing and pulling" dance:

• Scenario A (Destructive): If the waves canceled out, they saw about 786 events.
• Scenario B (Constructive): If the waves boosted each other, they saw about 358 events.

Because they couldn't be 100% sure which "dance step" nature chose, they reported two different answers for how often this happens. Both answers are significant because this is the first time anyone has ever seen the $\eta_c$ turn into this specific pair of particles.

The Search: The "Ghost" Particle

Next, the team looked for the second case: the $h_c$ turning into the $\Sigma^0$ pair. They scanned their data with the same high-powered microscope.

The Result: They found nothing. No ghosts, no signals, no evidence that the $h_c$ did this specific dance.

Because they didn't see it, they couldn't measure a number. Instead, they set a speed limit (an upper limit). They said, "If the $h_c$ does do this, it happens less than 1 time out of every 10,000 attempts." It's like saying, "We looked for a needle in a haystack and didn't find it, so we know the needle must be smaller than a grain of sand."

Why Does This Matter?

The paper compares their findings to what theoretical physicists predicted using math models (specifically a theory called pQCD).

• The Theory: Predicted that these decays should happen a certain way, based on the rules of how quarks interact.
• The Reality: The numbers the scientists found were inconsistent with the theory. The real world didn't follow the script the theorists wrote.

This is a big deal in physics. It's like a chef following a recipe perfectly, but the cake tastes completely different than the cookbook says it should. This tells the scientists that their current "recipe" (the theory) is missing an ingredient or a step. They need to rewrite the rules of how these particles interact.

Summary in a Nutshell

• The Detective Work: The BESIII team analyzed millions of particle collisions.
• The Success: They found the $\eta_c$ particle turning into a $\Sigma^0$ pair for the first time, but the result depends on a tricky quantum "interference" effect.
• The Miss: They didn't find the $h_c$ particle doing the same thing, setting a strict limit on how often it might happen.
• The Twist: The results don't match the current mathematical predictions, suggesting our understanding of the subatomic "dance" needs an update.

The paper is purely about observing these particles and measuring how often they appear; it does not discuss any medical or technological applications. It is a fundamental study of how the universe works at its smallest scale.

Technical Summary

Technical Summary: Observation of $\eta_c(1S) \to \Sigma^0 \bar{\Sigma}^0$ and Search for $h_c(1P) \to \Sigma^0 \bar{\Sigma}^0$ via $\psi(3686)$ Transitions

Problem and Motivation
The charmonium spectrum below the open-charm threshold contains unresolved questions regarding spin-singlet states, specifically the $P$-wave $h_c$ and the $S$-wave ground state $\eta_c$. While the $\eta_c$ has been studied for decades, many of its decay modes remain unidentified, and measured branching fractions (BFs) for known channels often suffer from large uncertainties. A critical theoretical issue involves the perturbative Quantum Chromodynamics (pQCD) helicity selection rule, which predicts that exclusive charmonium decays into baryon-antibaryon pairs ($B_8 \bar{B}_8$) should be highly suppressed. However, experimental observations have shown contradictions with these predictions (e.g., $\eta_c \to p\bar{p}$ vs. $\eta_c \to \Sigma^+ \Sigma^-$). Furthermore, the isospin partner decay $\eta_c \to \Sigma^0 \bar{\Sigma}^0$ had not been observed, preventing a test of isospin invariance in this sector. Similarly, the decay modes of the $h_c$ remain poorly understood experimentally; aside from its dominant electric dipole transition $h_c \to \gamma \eta_c$, no $B_8 \bar{B}_8$ final states have been observed. This paper addresses these gaps by searching for the first observation of $\eta_c \to \Sigma^0 \bar{\Sigma}^0$ and the first search for $h_c \to \Sigma^0 \bar{\Sigma}^0$.

Methodology
The analysis utilizes a data sample of $(2712.4 \pm 14.3) \times 10^6$ $\psi(3686)$ events collected by the BESIII detector at the BEPCII collider. The study focuses on two cascade processes:

1. $\psi(3686) \to \gamma \eta_c$, followed by $\eta_c \to \Sigma^0 \bar{\Sigma}^0$.
2. $\psi(3686) \to \pi^0 h_c$, followed by $h_c \to \Sigma^0 \bar{\Sigma}^0$.

Both decay chains proceed through $\Sigma^0 \to \gamma \Lambda$ ($\Lambda \to p \pi^-$) and $\bar{\Sigma}^0 \to \gamma \bar{\Lambda}$ ($\bar{\Lambda} \to \bar{p} \pi^+$).

• Event Selection: Charged tracks ($p, \bar{p}, \pi^+, \pi^-$) are identified using the drift chamber (MDC) and time-of-flight (TOF) systems. Photon candidates are reconstructed in the electromagnetic calorimeter (EMC).
• Kinematic Fits:
  • For the $\eta_c$ search, a four-constraint (4C) kinematic fit is performed under the hypothesis $\psi(3686) \to 3\gamma p \bar{p} \pi^+ \pi^-$.
  • For the $h_c$ search, a five-constraint (5C) fit is applied to $\psi(3686) \to 4\gamma p \bar{p} \pi^+ \pi^-$, including a mass constraint for the accompanying $\pi^0$.
• Background Estimation: Backgrounds are estimated using inclusive Monte Carlo (MC) simulations, sideband regions in the $\Sigma^0 \bar{\Sigma}^0$ mass distribution, and data-driven methods for specific backgrounds like $\psi(3686) \to \pi^0 \Sigma^0 \bar{\Sigma}^0$.
• Signal Extraction:
  • For $\eta_c$, an unbinned maximum likelihood fit is performed on the $M(\Sigma^0 \bar{\Sigma}^0)$ distribution. The fit model explicitly includes the interference between the resonant $\eta_c$ amplitude and the non-resonant background (NRB), parameterized by a phase $\phi$.
  • For $h_c$, the signal yield is extracted from a fit to the $\pi^0$ recoil mass distribution, $M_{\text{recoil}}(\pi^0)$.

Key Contributions and Results

1. First Observation of $\eta_c \to \Sigma^0 \bar{\Sigma}^0$:
   The decay is observed for the first time. The analysis reveals that the measured branching fraction is highly sensitive to the interference pattern between the $\eta_c$ resonance and non-resonant processes. Two scenarios are considered:

   • Destructive Interference: Yields a signal of $786 \pm 42$ events, resulting in a branching fraction of $\mathcal{B}(\eta_c \to \Sigma^0 \bar{\Sigma}^0) = (2.59 \pm 0.14_{\text{stat}} \pm 0.44_{\text{syst}}) \times 10^{-3}$.
   • Constructive Interference: Yields a signal of $358 \pm 35$ events, resulting in a branching fraction of $\mathcal{B}(\eta_c \to \Sigma^0 \bar{\Sigma}^0) = (1.18 \pm 0.12_{\text{stat}} \pm 0.21_{\text{syst}}) \times 10^{-3}$.
     The authors note that the significant dependence on the interference pattern highlights the necessity of coherent amplitude analyses in such measurements.

2. Search for $h_c \to \Sigma^0 \bar{\Sigma}^0$:
   No significant signal is observed in the $\pi^0$ recoil mass distribution. Consequently, an upper limit is set on the branching fraction at the 90% confidence level (C.L.):
   $$\mathcal{B}(h_c \to \Sigma^0 \bar{\Sigma}^0) < 1.02 \times 10^{-4}$$

3. Comparison with Theory:
   The measured branching fractions for $\eta_c \to \Sigma^0 \bar{\Sigma}^0$ (even the lower constructive limit) are inconsistent with theoretical calculations based on long-distance contributions via charmed hadron loops, which predict values in the range of $(4.82 \sim 9.56) \times 10^{-4}$. Similarly, the upper limit for $h_c \to \Sigma^0 \bar{\Sigma}^0$ is below the theoretical prediction range of $(5.57 \sim 7.08) \times 10^{-4}$.

Significance
The paper claims that these results provide the first experimental evidence for the $\eta_c \to \Sigma^0 \bar{\Sigma}^0$ decay channel, filling a gap in the charmonium decay landscape. The measurement underscores the complexity of charmonium decays into baryon-antibaryon pairs, demonstrating that interference effects between resonant and non-resonant amplitudes play a decisive role in determining the observed rates. The discrepancy between the experimental results and existing theoretical models suggests that current understanding of the mechanisms governing these exclusive decays, particularly regarding isospin conservation and long-distance contributions, requires further refinement. The null result for $h_c \to \Sigma^0 \bar{\Sigma}^0$ sets a stringent constraint on theoretical models attempting to describe $h_c$ hadronic transitions.