Measurements of branching fractions of $\Lambda_{c}^{+}\to\Sigma^{0}K_{S}^{0}\pi^{+}$ and $\Lambda_{c}^{+}\to\Sigma^{0}K_{S}^{0}K^{+}$

Using a data sample of 6.4 fb$^{-1}$ collected by the BESIII detector, this paper reports the first observation of the singly Cabibbo-suppressed decay $\Lambda_c^+ \to \Sigma^0 K_S^0 \pi^+$ with a statistical significance of 5.9$\sigma$ and provides the first evidence for the decay $\Lambda_c^+ \to \Sigma^0 K_S^0 K^+$ with a significance of 3.7$\sigma$.

The Gist

Imagine the universe as a giant, bustling construction site. In this site, there are tiny, heavy bricks called protons and neutrons that make up everything we see. But deep inside these bricks, there are even smaller, more elusive workers called quarks.

One specific type of worker is the charm quark. It's heavy, short-lived, and when it gets tired, it has to "retire" (decay) into lighter, more stable workers. Usually, it retires by taking the easy path, but sometimes, it takes a difficult, winding path that nature rarely allows.

This paper is like a report from a team of cosmic detectives (the BESIII collaboration) who spent years watching this construction site to catch the charm quark in the act of taking one of these rare, difficult retirement paths.

Here is the story of their discovery, broken down simply:

1. The Detective Work: The BESIII Detector

Think of the BESIII detector as a massive, ultra-high-speed camera and a giant net wrapped around a particle accelerator (a circular racetrack for particles).

• The Racetrack: They smashed electrons and positrons (matter and antimatter) together at incredibly high speeds.
• The Net: When these particles collided, they created a shower of new particles, including the Lambda-c ($\Lambda_c^+$). This particle is like a "heavy-duty truck" made of three quarks (one charm, one up, one down).
• The Job: The truck is unstable. It immediately breaks apart. The detectors recorded billions of these breakups to find the specific, rare ones the scientists were looking for.

2. The Rare Event: The "Impossible" Breakup

The scientists were hunting for two specific ways the Lambda-c truck could break apart.

• The Goal: They wanted to see the truck split into a Sigma-zero ($\Sigma^0$), a neutral Kaon ($K^0$), and either a pion ($\pi^+$) or a kaon ($K^+$).
• The Difficulty: In the world of particle physics, there are "rules" (called the Standard Model) that make some breakups very common and others very rare. These specific breakups are "Cabibbo-suppressed."
  • Analogy: Imagine a door that is usually wide open (easy to walk through). These specific breakups are like trying to squeeze through a keyhole. It can happen, but it's very hard and unlikely.

3. The Big Discovery: Catching the Thief

After analyzing 6.4 "inverse femtobarns" of data (which is a fancy way of saying they looked at a huge pile of collision data—billions of events), they found something amazing:

• The First Success ($\Lambda_c^+ \to \Sigma^0 K_S^0 \pi^+$):
  They found 28 clear examples of the truck breaking apart into a Sigma, a Kaon, and a Pion.

  • The Significance: In the world of statistics, finding this many events when you expect almost none is like winning the lottery five times in a row. The team calculated the odds of this being a fluke as less than 1 in a billion (a 5.9 sigma significance).
  • The Result: They officially announced, "We have seen this for the first time!" They measured exactly how often it happens (the branching fraction), finding it occurs about 0.58 times out of every 1,000 Lambda-c decays.

• The "Almost" Success ($\Lambda_c^+ \to \Sigma^0 K_S^0 K^+$):
  They also looked for the version with two Kaons. They found hints of it (about 8 events), but it wasn't quite enough to be 100% sure.

  • The Significance: This was a 3.7 sigma result. It's like seeing a shadow that might be a person, but you need more light to be certain. They couldn't claim a full discovery, but they set a strict "speed limit" (an upper limit) on how often this could possibly happen.

4. The Mystery of the "Ghost" Particles

One of the coolest parts of the paper is the mention of resonances.

• The Analogy: Imagine the Lambda-c truck doesn't just break apart directly. Maybe it first turns into a different, temporary "ghost" vehicle (like a $K^*$ particle) for a split second before breaking into the final pieces.
• The data suggests that for the first discovery, a significant chunk of the events happened because the truck briefly turned into this "ghost" vehicle ($\Sigma^0 K^{*+}$) first. This helps physicists understand the "internal structure" of these particles—like figuring out if a car fell apart because it hit a wall, or because a specific bolt loosened first.

5. Why Does This Matter?

You might ask, "Why do we care about a truck breaking into a Sigma and a Kaon?"

• Testing the Rules: The Standard Model is our rulebook for how the universe works. But the rulebook is incomplete when it comes to how quarks interact with the "strong force" (the glue holding them together).
• The Puzzle: Theoretical predictions said this breakup should happen about 0.17 times per 1,000 decays. The experiment found 0.58 times.
  • The Takeaway: The real world is doing something the math didn't fully predict! This suggests there are hidden mechanisms (like those "ghost" resonances) at play.
• The Future: By measuring these rare events, scientists are building a better map of the subatomic world. It helps us understand why the universe is made of matter and not just empty space, and it might one day help us find cracks in our current theories that lead to "New Physics."

Summary

In short, the BESIII team acted like cosmic detectives, sifting through billions of particle collisions to find a very rare, difficult-to-achieve breakup of a heavy particle. They successfully caught it in the act for the first time, proving that nature is a bit more complex and interesting than our current math predicted. They found that the "hard path" is taken more often than we thought, likely because the particles are taking a scenic route through a temporary "ghost" state before finally settling down.

Technical Summary

1. Problem Statement and Motivation

The dynamics of charmed baryon decays remain poorly understood compared to charmed meson decays. While meson decays are often dominated by spectator diagrams and well-described by factorization approaches, charmed baryon decays involve additional light quarks, leading to significant $W$-exchange contributions where the factorization approximation often fails.

• Specific Gap: The decay modes $\Lambda_c^+ \to \Sigma^0 K_S^0 \pi^+$ and $\Lambda_c^+ \to \Sigma^0 K_S^0 K^+$ are singly Cabibbo-suppressed. Prior to this work, $\Lambda_c^+ \to \Sigma^0 K_S^0 \pi^+$ had never been observed experimentally, and $\Lambda_c^+ \to \Sigma^0 K_S^0 K^+$ had only an upper limit established by a previous BESIII analysis using a smaller dataset and the double-tag method.
• Theoretical Context: Theoretical predictions based on $SU(3)$ flavor symmetry ($SU(3)_F$) exist for non-resonant contributions, but discrepancies between theory and experiment often arise due to unaccounted resonant intermediate states (e.g., $K^*$, $a_0(980)$). Measuring these branching fractions (BFs) is crucial for validating theoretical frameworks and understanding the internal structure of the $\Lambda_c^+$.

2. Methodology

The analysis utilized data collected by the BESIII detector at the BEPCII collider.

• Dataset: A total integrated luminosity of 6.4 fb$^{-1}$ collected at 13 center-of-mass energy points ranging from 4.600 GeV to 4.950 GeV.

• Event Reconstruction:

  • Signal Topology: The $\Lambda_c^+$ candidates were reconstructed via the decay chain:
    • $\Lambda_c^+ \to \Sigma^0 K_S^0 \pi^+$ (where $\Sigma^0 \to \gamma \Lambda$, $\Lambda \to p \pi^-$, $K_S^0 \to \pi^+ \pi^-$).
    • $\Lambda_c^+ \to \Sigma^0 K_S^0 K^+$ (similar decay chain for $\Sigma^0$ and $K_S^0$).
  • Selection Criteria: Charged tracks were required to be within specific polar angles and impact parameter cuts. Photons were identified in the electromagnetic calorimeter (EMC). Particle Identification (PID) combined $dE/dx$ and Time-of-Flight (TOF) measurements.
  • Kinematic Fits: A four-constraint (4C) kinematic fit was applied to improve signal purity, constraining the recoil $\bar{\Lambda}_c^-$ mass and the intermediate particle masses ($K_S^0$, $\Lambda$, $\Sigma^0$).
  • Background Suppression: The Beam Constrained Mass ($M_{BC}$) was used to identify signal candidates. Peaking backgrounds (e.g., $\Lambda_c^+ \to \Sigma^0 \pi^+ \pi^- \pi^+$) were estimated using Monte Carlo (MC) simulations and PDG branching fractions.

• Analysis Technique:

  • Signal Yield Extraction: A simultaneous maximum likelihood fit was performed on the $M_{BC}$ distributions across all 13 energy points.
  • Shape Modeling: Signal shapes were derived from MC simulations convolved with Gaussian functions to account for mass resolution differences between data and simulation. Backgrounds were modeled using ARGUS functions (for $\pi^+$ mode) or inclusive MC shapes (for $K^+$ mode).
  • Resonance Search: For the $\pi^+$ mode, the $K_S^0 \pi^+$ invariant mass distribution was analyzed to identify resonant contributions (specifically $K^{*+}$).

3. Key Contributions

1. First Observation: This paper reports the first experimental observation of the singly Cabibbo-suppressed decay $\Lambda_c^+ \to \Sigma^0 K_S^0 \pi^+$.
2. Evidence for Rare Decay: It provides the first evidence for the decay $\Lambda_c^+ \to \Sigma^0 K_S^0 K^+$.
3. Resonance Analysis: The study explicitly investigates and quantifies the contribution of the intermediate resonant state $\Lambda_c^+ \to \Sigma^0 K^{*+}$ within the $\Lambda_c^+ \to \Sigma^0 K_S^0 \pi^+$ channel.
4. Improved Precision: By utilizing a larger dataset (6.4 fb$^{-1}$) and the single-tag method, the analysis significantly improves the statistical precision compared to previous upper limits.

4. Results

A. $\Lambda_c^+ \to \Sigma^0 K_S^0 \pi^+$

• Statistical Significance: 5.9$\sigma$ (establishing a first observation).
• Branching Fraction:
  $$\mathcal{B}(\Lambda_c^+ \to \Sigma^0 K_S^0 \pi^+) = (0.58 \pm 0.14_{\text{stat}} \pm 0.04_{\text{syst}}) \times 10^{-3}$$
• Resonant Contribution: A significant contribution from the intermediate state $\Lambda_c^+ \to \Sigma^0 K^{*+}$ (where $K^{*+} \to K_S^0 \pi^+$) was observed.
  • $\mathcal{B}(\Lambda_c^+ \to \Sigma^0 K^{*+}) \times \mathcal{B}(K^{*+} \to K_S^0 \pi^+) = (0.41 \pm 0.19 \pm 0.03) \times 10^{-3}$.
  • This implies that the non-resonant component is smaller than the total measured rate, highlighting the importance of resonant dynamics in this decay.
• Comparison with Theory: The measured total branching fraction is approximately 3.4 times larger than the theoretical prediction for the non-resonant component alone ($(0.17 \pm 0.05) \times 10^{-3}$), confirming the necessity of including resonant contributions in theoretical models.

B. $\Lambda_c^+ \to \Sigma^0 K_S^0 K^+$

• Statistical Significance: 3.7$\sigma$ (evidence level, not yet observation).
• Branching Fraction: The central value is $(0.35 \pm 0.16_{\text{stat}} \pm 0.04_{\text{syst}}) \times 10^{-3}$.
• Upper Limit: Due to the significance being below the 5$\sigma$ discovery threshold, a 90% Confidence Level (C.L.) upper limit was set:
  $$\mathcal{B}(\Lambda_c^+ \to \Sigma^0 K_S^0 K^+) < 1.23 \times 10^{-3}$$
• Comparison: This result is consistent with previous BESIII limits and theoretical predictions based on $SU(3)_F$.

5. Significance

• Validation of Theoretical Models: The results provide critical experimental constraints for $SU(3)_F$ symmetry calculations and other theoretical frameworks describing non-leptonic charmed baryon decays. The discrepancy between the total measured rate and the non-resonant theoretical prediction underscores the dominance of resonant mechanisms (like $K^*$) in these decays.
• Understanding W-Exchange: These measurements help elucidate the role of $W$-exchange diagrams, which are suppressed in mesons but potentially dominant in baryons, offering insights into the strong interaction effects within the Standard Model.
• Future Physics: The observation of these modes opens new avenues for studying light scalar mesons (via potential $a_0(980)$ contributions in the $K^+ K_S^0$ channel) and searching for doubly-charmed baryons, as precise knowledge of singly-charmed decay dynamics is a prerequisite for background suppression in those searches.

In summary, this work represents a major step forward in charmed baryon spectroscopy, transforming a theoretical prediction and an upper limit into a confirmed observation and a significant evidence signal, while highlighting the complex interplay of resonant and non-resonant decay mechanisms.