First measurements of the branching fractions for the decay modes $Ξ_c^{0} \to Λη$ and $Ξ_c^0 \to Λη'$ and search for the decay $Ξ_c^{0} \to Λπ^0$ using Belle and Belle II data

Using combined Belle and Belle II data, this study presents the first observation of the decay $\Xi_c^0 \to \Lambda\eta$, provides evidence for $\Xi_c^0 \to \Lambda\eta'$, and sets an upper limit for $\Xi_c^0 \to \Lambda\pi^0$, thereby determining their absolute branching fractions and advancing the understanding of singly Cabibbo-suppressed charm baryon decay mechanisms.

The Gist

Imagine the subatomic world as a bustling, chaotic construction site. At the center of this site are Charmed Baryons (specifically the $\Xi^0_c$), which are like heavy-duty, three-person construction crews made of quarks. These crews are unstable; they don't last long and constantly break apart into smaller, lighter crews (like the $\Lambda$ baryon) and other debris (mesons like $\eta$, $\eta'$, or $\pi^0$).

For a long time, physicists have been trying to predict exactly how these crews break apart and how often they choose specific paths. It's like trying to guess if a falling tree will crack into three large logs or shatter into a million splinters.

This paper is the report from two massive construction sites, Belle and Belle II, located in Japan. These sites are actually giant particle detectors (like high-speed, 360-degree cameras) that watch billions of these particle collisions to see what happens when the "Charmed Baryon" crews break up.

Here is the breakdown of their latest findings, explained simply:

1. The Mission: Catching Rare Breakups

Most of the time, these particles break apart in very common, predictable ways. But physicists are interested in the rare, "forbidden" (or suppressed) breakups.

• Think of it like a magician pulling a rabbit out of a hat. Everyone expects a rabbit. But sometimes, the magician pulls out a duck.
• The "duck" in this experiment is a specific, rare breakup where the $\Xi^0_c$ turns into a $\Lambda$ particle plus a neutral meson ($\eta$, $\eta'$, or $\pi^0$).
• The team wanted to answer three questions:
  1. Does the $\Xi^0_c$ ever turn into $\Lambda + \eta$?
  2. Does it ever turn into $\Lambda + \eta'$?
  3. Does it ever turn into $\Lambda + \pi^0$?

2. The Results: Two Hits, One Miss

The Big Discovery: $\Xi^0_c \to \Lambda\eta$

• The Verdict: Yes! They found it.
• The Analogy: Imagine looking for a specific, rare coin in a jar of a billion other coins. They didn't just find a few; they found enough to say, "We are 99.9% sure this coin exists."
• Significance: This is the first time anyone has ever definitively observed this specific breakup. It's a confirmed discovery.

The "Maybe": $\Xi^0_c \to \Lambda\eta'$

• The Verdict: Strong Evidence, but not quite a full discovery yet.
• The Analogy: They found a lot of footprints that look exactly like the rare coin, but they haven't quite caught the coin itself in the act yet. It's a "strong hint" or "evidence."
• Significance: They are very confident it happens, but they need a bit more data to call it a formal "discovery."

The "Nope": $\Xi^0_c \to \Lambda\pi^0$

• The Verdict: Nothing found.
• The Analogy: They looked very hard for this specific coin, but the jar was empty.
• Significance: While they didn't find it, they set a "limit." They can now say, "If this breakup happens, it's so rare that it happens less than once in every 10,000 tries." This helps rule out some theories that predicted it would happen more often.

3. How They Did It (The "Super-Cameras")

The team used data from two different eras of particle physics:

• Belle (The Veteran): Operated from 1999 to 2010. It's like an old, reliable film camera that took a huge number of photos.
• Belle II (The Modern Pro): Started in 2019. It's a high-definition digital camera with better sensors, taking even more photos, but with higher precision.

By combining the data from both, they had a massive sample size (over 1.4 billion "photos" of collisions). They used sophisticated computer algorithms (like advanced facial recognition software) to sift through the noise and find the specific "faces" (decay patterns) they were looking for.

4. Why Does This Matter?

You might ask, "Who cares about a particle breaking into a $\Lambda$ and an $\eta$?"

• Testing the Rulebook: Physicists have a "Rulebook" called the Standard Model, which predicts how particles should behave. However, for these heavy baryons, the rulebook is a bit fuzzy. There are many different theories (like different architects designing the same building) that predict different probabilities for these breakups.
• The Reality Check: Before this paper, these predictions were all over the place—some said the breakup would happen 100 times, others said 1,000 times.
• The Result: The measurements from Belle and Belle II act as a reality check. They show that most of the "architects" (theoretical models) were actually in the right ballpark. The data fits well with the current understanding of how the "strong force" (the glue holding quarks together) works inside these particles.

Summary

In simple terms, this paper is a milestone report from two of the world's best particle detectors. They successfully caught a rare particle breakup for the first time, found strong evidence for a second rare breakup, and ruled out a third one.

This helps scientists refine their understanding of the fundamental forces that hold the universe together, proving that their current "Rulebook" of particle physics is working correctly, even in these tricky, rare scenarios.

Technical Summary

1. Problem Statement

Charmed baryon spectroscopy is a critical platform for understanding the dynamics of light quarks in the presence of heavy quarks. While the absolute branching fractions of key antitriplet charmed baryon decays (e.g., $\Lambda_c^+ \to pK^-\pi^+$, $\Xi_c^+ \to \Xi^-\pi^+\pi^-$) have been measured, many singly Cabibbo-suppressed (SCS) decay modes of the $\Xi^0_c$ baryon remain unmeasured.

Specifically, the decays $\Xi^0_c \to \Lambda\eta$, $\Xi^0_c \to \Lambda\eta'$, and $\Xi^0_c \to \Lambda\pi^0$ are theoretically significant because their amplitudes involve complex interplays between factorizable (internal $W$-emission) and non-factorizable (inner $W$-emission and $W$-exchange) contributions. Theoretical predictions for these branching fractions vary widely (from $10^{-5}$ to $10^{-3}$) depending on the model used (e.g., SU(3) flavor symmetry, pole models, or irreducible SU(3) approaches). There was a lack of experimental data to constrain these models and clarify the underlying decay mechanisms.

2. Methodology

Data Samples:
The analysis utilized a combined dataset corresponding to an integrated luminosity of 1.42 ab$^{-1}$:

• Belle: 988.4 fb$^{-1}$ collected at the KEKB $e^+e^-$ collider (operating at $\Upsilon(nS)$ resonances).
• Belle II: 427.9 fb$^{-1}$ collected at the SuperKEKB collider (operating at $\Upsilon(4S)$ and near 10.75 GeV).

Reconstruction Strategy:
The study reconstructed the signal decays $\Xi^0_c \to \Lambda h^0$ (where $h^0 = \eta, \eta', \pi^0$) and the normalization mode $\Xi^0_c \to \Xi^-\pi^+$.

• Decay Chains:
  • $\Lambda \to p\pi^-$
  • $\eta \to \gamma\gamma$ and $\eta \to \pi^+\pi^-\pi^0$
  • $\eta' \to \pi^+\pi^-\eta$ (with $\eta \to \gamma\gamma$ or $\pi^+\pi^-\pi^0$) and $\eta' \to \pi^+\pi^-\gamma$
  • $\pi^0 \to \gamma\gamma$
• Unified Framework: All Belle data was converted to the Belle II format using the B2BII software package to allow for a unified analysis within the basf2 framework.

Event Selection:

• Particle Identification (PID): Combined likelihood ratios from detector subsystems were used to identify protons, pions, and kaons.
• Photon Selection: Clusters in the Electromagnetic Calorimeter (ECL) were vetted using energy deposition ratios and a FastBDT classifier to reject fake photons and beam background.
• Kinematic Cuts:
  • $\Lambda$ candidates were required to have a flight distance significance and specific momentum cuts in the center-of-mass (c.m.) frame.
  • $\Xi^0_c$ candidates were required to have a scaled momentum $x_p > 0.60$ to suppress backgrounds from $B$-meson decays and continuum processes.
• Optimization: Selection criteria were optimized using the Punzi figure of merit ($FOM = \epsilon / (3/2 + \sqrt{N_b})$) to maximize sensitivity.

Statistical Analysis:

• Normalization: The branching fractions were measured relative to the normalization channel $\Xi^0_c \to \Xi^-\pi^+$ to cancel common systematic uncertainties (like tracking efficiency and luminosity).
• Fitting: Unbinned extended maximum-likelihood fits were performed on the invariant mass distributions ($M(\Lambda\eta)$, $M(\Lambda\eta')$, $M(\Lambda\pi^0)$).
  • Signal shapes were modeled using double-Gaussian functions (or bifurcated Gaussians for $\Lambda\pi^0$).
  • Backgrounds were modeled using second-order Chebyshev polynomials.
• Simultaneous Fits: Fits were performed simultaneously across different $h^0$ decay channels and datasets (Belle and Belle II), weighted by luminosity, efficiency, and product branching fractions.

3. Key Contributions

1. First Observation of $\Xi^0_c \to \Lambda\eta$: This paper reports the first definitive observation of this decay mode with a statistical significance of 5.3$\sigma$ (5.1$\sigma$ including systematics).
2. First Evidence of $\Xi^0_c \to \Lambda\eta'$: The analysis provides the first evidence for this decay with a significance of 3.3$\sigma$ (3.2$\sigma$ including systematics).
3. First Search for $\Xi^0_c \to \Lambda\pi^0$: No significant signal was found, leading to the first upper limit set on this mode.
4. Comprehensive Systematic Treatment: The study provides a detailed breakdown of systematic uncertainties, including tracking, PID, $\pi^0$ reconstruction, photon detection, and fit procedure variations, specifically comparing Belle and Belle II performance.

4. Results

Branching Fraction Ratios:
The ratios relative to the normalization mode $\Xi^0_c \to \Xi^-\pi^+$ were determined as:

• $\frac{\mathcal{B}(\Xi^0_c \to \Lambda\eta)}{\mathcal{B}(\Xi^0_c \to \Xi^-\pi^+)} = (4.16 \pm 0.91 \pm 0.23)\%$
• $\frac{\mathcal{B}(\Xi^0_c \to \Lambda\eta')}{\mathcal{B}(\Xi^0_c \to \Xi^-\pi^+)} = (2.48 \pm 0.82 \pm 0.12)\%$
• Upper limit (90% C.L.) for $\Xi^0_c \to \Lambda\pi^0$: $\frac{\mathcal{B}(\Xi^0_c \to \Lambda\pi^0)}{\mathcal{B}(\Xi^0_c \to \Xi^-\pi^+)} < 3.5\%$

Absolute Branching Fractions:
Using the world-average value $\mathcal{B}(\Xi^0_c \to \Xi^-\pi^+) = (1.43 \pm 0.27)\%$, the absolute branching fractions were calculated:

• $\mathcal{B}(\Xi^0_c \to \Lambda\eta) = (5.95 \pm 1.30 \text{ (stat)} \pm 0.32 \text{ (syst)} \pm 1.13 \text{ (norm)}) \times 10^{-4}$
• $\mathcal{B}(\Xi^0_c \to \Lambda\eta') = (3.55 \pm 1.17 \text{ (stat)} \pm 0.17 \text{ (syst)} \pm 0.68 \text{ (norm)}) \times 10^{-4}$
• Upper limit (90% C.L.) for $\Xi^0_c \to \Lambda\pi^0$: $\mathcal{B}(\Xi^0_c \to \Lambda\pi^0) < 5.2 \times 10^{-4}$

Comparison with Theory:
The measured values were compared against twelve theoretical predictions.

• The results for $\Lambda\eta$ and $\Lambda\eta'$ are consistent with most models within $1\sigma$ to $3\sigma$.
• Three SU(3)$_F$-based predictions agree with the measurements within $1\sigma$.
• All theoretical predictions lie below the 90% C.L. upper limit for $\Lambda\pi^0$.

5. Significance

• Validation of Theoretical Models: These measurements provide crucial experimental constraints on theoretical models describing non-leptonic weak decays of charmed baryons. The data helps distinguish between different topological diagram contributions (factorizable vs. non-factorizable) and tests the validity of SU(3) flavor symmetry assumptions.
• Advancement in Charmed Baryon Physics: By providing the first measurements for these specific SCS modes, the study fills a gap in the spectroscopy of the antitriplet charmed baryon sector.
• Detector Performance Demonstration: The successful combination of Belle and Belle II datasets demonstrates the capability of the basf2 framework to unify legacy and modern data, leveraging the improved resolution and background rejection of Belle II (particularly for low-energy photons) while maintaining the statistical power of the larger Belle dataset.
• Future Guidance: The results guide future theoretical refinements and suggest that the $\Xi^0_c \to \Lambda\pi^0$ mode, while not observed, remains a target for future high-luminosity runs to further constrain the $W$-exchange mechanisms.