Measurement of the singly Cabibbo-suppressed decay $Λ_c^+\to pη'$ with Deep Learning

Using 4.5 fb$^{-1}$ of $e^+e^-$ collision data from the BESIII detector and a Transformer-based deep learning classifier, this study reports the first observation of the singly Cabibbo-suppressed decay $\Lambda_c^+\to p\eta'$ with a statistical significance of $3.4\sigma$ and measures its branching fraction ratio relative to $\Lambda_c^+\to p\omega$ to be $0.55\pm 0.22_{\rm{stat.}} \pm 0.05_{\rm{syst.}}$.

The Gist

The Big Picture: Finding a Needle in a Haystack

Imagine you are at a massive, chaotic music festival (the BESIII detector). Thousands of people are dancing, shouting, and bumping into each other. Among this crowd, you are looking for two specific people who are doing a very rare, secret handshake (the decay of a particle called $\Lambda_c^+$ into a proton and an $\eta'$ meson).

The problem? The crowd is huge, and most people look exactly the same. In the past, scientists tried to find these two people by checking every single person's ID card one by one (the Double-Tag method). This was very accurate but incredibly slow, and they often missed the secret handshake because they couldn't check enough people.

In this new paper, the scientists tried a different approach. They let the crowd flow freely (the Single-Tag method), which means they saw way more people, but the crowd became even messier and noisier. To solve this, they built a super-smart AI detective (a Deep Learning model) to scan the crowd and instantly spot the secret handshake, ignoring the noise.

The Cast of Characters

1. The Particle ($\Lambda_c^+$): Think of this as a "Charm Baryon." It's a heavy, unstable particle that lives for a split second before breaking apart.
2. The Decay ($\Lambda_c^+ \to p\eta'$): This is the "secret handshake." The heavy particle breaks into a proton ($p$) and an $\eta'$ meson. This is a "singly Cabibbo-suppressed" decay.
   • Analogy: Imagine a rare coin flip where you usually get Heads, but sometimes, very rarely, you get Tails. This experiment is trying to count exactly how many times that rare "Tails" happens.
3. The Background Noise: In particle physics, "background" is everything that looks like your signal but isn't. It's like the thousands of festival-goers who just happen to be standing next to each other by accident, making it look like they are doing the secret handshake when they aren't.

The New Tool: The AI Detective

The scientists used a type of Artificial Intelligence called a Transformer-based Neural Network (the same technology behind advanced chatbots).

• How it works: Instead of looking at one piece of data at a time, the AI looks at the "shape" of the entire event. It considers the angles, speeds, and energy of every particle flying out of the collision, just like a detective looking at the body language of a whole group of people to see who is acting suspicious.
• The Training: They fed the AI millions of computer simulations. They showed it examples of the "secret handshake" (signal) and millions of examples of "accidental crowd gatherings" (background).
• The Result: The AI learned to distinguish the real signal from the noise with incredible precision. It was able to filter out the background noise by a factor of 100 (two orders of magnitude), while still keeping about 40% of the real signals.

The Experiment: What Did They Do?

1. The Data: They used data from the BESIII detector in China, which smashes electrons and positrons together at high speeds. They looked at 4.5 billion collisions (4.5 inverse femtobarns).
2. The Strategy: They didn't try to reconstruct the whole event. They just looked for the specific particles coming from the $\Lambda_c^+$ decay and let the AI sort the rest.
3. The Comparison: To be sure their numbers were right, they compared this rare decay to a more common, well-known decay ($\Lambda_c^+ \to p\omega$). It's like comparing the number of rare "Tails" flips to the number of "Heads" flips to get a ratio. This cancels out many errors.

The Findings

• Did they find it? Yes! They found evidence of the rare decay with a statistical significance of 3.4 sigma.
  • Analogy: If you flipped a coin 1,000 times and got 600 heads, you might suspect the coin is rigged. In science, "3.4 sigma" means there is a very high probability (about 99.9%) that this isn't just a random fluke, though it's not quite the "5 sigma" (99.9999%) gold standard required to officially claim a "discovery." It's a very strong "evidence."
• The Ratio: They measured how often this rare decay happens compared to the common one. The ratio is 0.55. This means the rare decay happens about half as often as the common one.

Why Does This Matter?

For a long time, scientists have been trying to understand the "rules of the game" for how these heavy particles break apart. There are different theories (like the Constituent Quark Model or SU(3) Symmetry) that try to predict these rates.

• The Conflict: Some theories predicted this rare decay would be very rare. Others predicted it would be more common.
• The Verdict: This new measurement helps settle the debate. The result suggests that the theories based on symmetry and topology (how the particles are arranged) are likely correct, while the older "constituent quark" models might need some tweaking.

The Takeaway

This paper is a triumph of modern technology meeting old-school physics. By using a cutting-edge AI (Deep Learning) to cut through the noise of a chaotic particle collision, the BESIII team was able to see a rare event that was previously too hard to spot. It's like using a high-tech metal detector to find a specific coin in a pile of sand, proving that with the right tools, we can see the invisible.

Technical Summary

1. Problem Statement

The paper addresses the challenge of measuring the branching fraction (BF) of the singly Cabibbo-suppressed (SCS) decay $\Lambda_c^+ \to p\eta'$.

• Theoretical Context: Weak decays of charmed baryons are crucial for understanding strong and weak dynamics. Unlike charmed mesons, charmed baryon decays involve significant non-factorizable contributions (specifically $W$-exchange topologies), making theoretical predictions difficult. Various models (constituent-quark, topological, SU(3) flavor symmetry) predict the BF of $\Lambda_c^+ \to p\eta'$ in the range of $O(10^{-4})$ to $O(10^{-3})$, but with large discrepancies.
• Experimental Challenge: The primary obstacle is the huge combinatorial background in $e^+e^-$ collisions. Previous attempts using the Double-Tag (DT) method (reconstructing the recoil $\bar{\Lambda}_c^-$) were effective at background suppression but suffered from low efficiency and limited statistics. Conversely, the Single-Tag (ST) method (reconstructing only the signal side) offers higher efficiency but introduces overwhelming hadronic backgrounds that traditional cut-based selection methods cannot effectively separate.

2. Methodology

The authors employed a novel approach combining the ST strategy with Deep Learning (DL) to overcome the background limitations.

• Data Sample: 4.5 fb$^{-1}$ of $e^+e^-$ collision data collected by the BESIII detector at center-of-mass energies between 4.600 and 4.699 GeV (above the $\Lambda_c^+\bar{\Lambda}_c^-$ threshold).
• Analysis Strategy:
  • Single-Tag (ST) Approach: The signal $\Lambda_c^+ \to p\eta'$ is reconstructed without constraints on the recoil side. The $\eta'$ is reconstructed via the dominant mode $\eta' \to \pi^+\pi^-\gamma$.
  • Normalization Channel: The decay $\Lambda_c^+ \to p\omega$ (with $\omega \to \pi^+\pi^-\pi^0$) is used as the reference channel to cancel systematic uncertainties.
  • Deep Learning Classifier: To distinguish signal from the massive hadronic background, the authors utilized a Transformer-based neural network (specifically inspired by the Particle Transformer (ParT) architecture).
    • Input: The model takes a "point cloud" representation of all charged tracks (from the MDC) and isolated photon showers (from the EMC) in an event. Features include kinematic variables, particle identification (PID) likelihoods, and shower properties.
    • Training: A three-class classification task was trained on Monte Carlo (MC) simulations:
      1. Signal ($\Lambda_c^+ \to p\eta'$ or $p\omega$).
      2. $\Lambda_c^+\bar{\Lambda}_c^-$ non-signal background.
      3. Hadronic background ($e^+e^- \to q\bar{q}$).
    • Ensemble Technique: Ten independent DNNs were trained with randomized initialization and dropout mechanisms. Their outputs were averaged to form an ensemble model, enhancing robustness and generalization.
    • Discriminator: A composite variable $S = \text{score}_{\text{sig}} \times (1 - \text{score}_{\text{hadronic}})$ was defined. Events with $S > 0.98$ were retained.

3. Key Contributions

• First Application of Transformer-based DL in BESIII for this Channel: The paper demonstrates the successful application of advanced deep learning architectures (ParT) to charmed baryon physics, moving beyond traditional cut-based or simpler neural network approaches.
• Background Suppression: The DL classifier achieved a background reduction of more than two orders of magnitude while retaining approximately 40% of the signal. This level of suppression made it possible to extract the signal from the ST sample, which was previously impossible due to the overwhelming background.
• Independent Analysis: Although using the same dataset as a previous BESIII paper (Ref. [16]), this analysis is methodologically independent, utilizing ST + DL rather than DT, providing a cross-check and higher statistical yield.

4. Results

• Signal Observation: The $\Lambda_c^+ \to p\eta'$ signal was observed with a statistical significance of 3.4$\sigma$.
• Branching Fraction Ratio: The ratio of branching fractions was measured as:
  $$\frac{\mathcal{B}(\Lambda_c^+ \to p\eta')}{\mathcal{B}(\Lambda_c^+ \to p\omega)} = 0.55 \pm 0.22_{\text{stat}} \pm 0.05_{\text{syst}}$$
• Absolute Branching Fraction: Using the world-average $\mathcal{B}(\Lambda_c^+ \to p\omega) = (1.11 \pm 0.21) \times 10^{-3}$, the absolute branching fraction was determined:
  $$\mathcal{B}(\Lambda_c^+ \to p\eta') = (6.15 \pm 2.45_{\text{stat}} \pm 0.53_{\text{syst}} \pm 1.16_{\text{ref}}) \times 10^{-4}$$
• Systematic Uncertainties: The total systematic uncertainty was 8.7%, dominated by the fit model (6.2%) and the neural network performance (4.6%).

5. Significance

• Theoretical Impact: The measured branching fraction favors predictions based on SU(3)-flavor symmetry and topological diagram models over the constituent-quark model. This provides crucial experimental input to refine theoretical descriptions of non-factorizable contributions in charmed baryon decays.
• Methodological Advancement: The success of the ST method combined with Transformer-based deep learning validates a powerful new strategy for high-energy physics analyses. It allows for the extraction of rare signals from high-background environments where traditional methods fail, potentially applicable to other rare decay searches in the charm and beauty sectors.
• Consistency: The result is consistent with previous measurements by Belle and BESIII (within 1$\sigma$), but achieved with a significantly higher signal yield due to the efficiency of the ST method.