Search for the low-lying excited baryon $\Sigma^*(1/2^-)$ through process $\Lambda^+_c \to \Lambda K^0 \pi^+$

Motivated by recent BESIII data, this study investigates the decay $\Lambda^+_c \to \Lambda K^0 \pi^+$ by incorporating contributions from the dynamically generated $\Sigma^*(1/2^-)$ and other resonances, successfully reproducing existing mass distributions and predicting a distinct cusp structure around 1.43 GeV that could confirm the existence of this low-lying excited baryon.

The Gist

Imagine the universe is a giant, bustling construction site where tiny building blocks called quarks are constantly being assembled into larger structures called baryons (which include protons and neutrons). Most of the time, these blocks snap together in predictable ways. But sometimes, they form weird, temporary, or "exotic" structures that are hard to spot and even harder to understand.

This paper is a detective story about finding one specific, elusive "ghost" structure in the subatomic world: a particle called $\Sigma^*(1/2^-)$.

Here is the breakdown of the paper's story, using simple analogies:

1. The Mystery: A Missing Piece in the Puzzle

Scientists have a "family photo album" of all known particles (called the Particle Data Group). In this album, there is a spot for a specific type of excited baryon called $\Sigma^*(1/2^-)$. However, this entry is marked with a big question mark and a "one-star" rating, meaning the evidence is very weak. It's like knowing a cousin exists because of an old family rumor, but no one has ever actually seen them at a party.

The paper suggests this "cousin" might be hiding in plain sight, created by the way particles bounce off each other, rather than being a solid, permanent object.

2. The Crime Scene: A Specific Particle Decay

To find this ghost, the authors looked at a specific event: a heavy particle called the $\Lambda_c^+$ (a charmed baryon) breaking apart.

• The Setup: Think of the $\Lambda_c^+$ as a fragile glass vase. When it shatters, it usually breaks into three pieces: a $\Lambda$ (Lambda), a $K^0$ (a neutral kaon), and a $\pi^+$ (a positive pion).
• The Clue: Recently, the BESIII experiment (a giant particle detector in China) took photos of this shattering. They saw the pieces, but they only looked at how the $K^0$ and $\pi^+$ were moving together. They saw a bump there, but they didn't look closely at how the $\Lambda$ and the $\pi^+$ were moving together.

3. The Theory: The "Echo" Effect

The authors propose a new way to look at the data. They suggest that when the $\Lambda_c^+$ breaks, the pieces don't just fly away; they sometimes crash into each other immediately after.

• The Analogy: Imagine two people (the $\Lambda$ and the $\pi^+$) running away from a crash. As they run, they might bump into each other, creating a temporary "echo" or a ripple in the air before they separate.
• The Prediction: The authors used a complex mathematical model (the "chiral unitary approach") to simulate this. They predicted that if the elusive $\Sigma^*(1/2^-)$ exists, it would act like a cusp (a sharp, sudden spike or a "V" shape) in the data.
• The Location: They predict this sharp spike will appear when the combined mass of the $\Lambda$ and $\pi^+$ is around 1.43 GeV (a specific energy level).

4. The Investigation: Running the Simulation

The authors built a computer model to simulate the shattering of the $\Lambda_c^+$, taking into account three main "actors":

1. The Tree Diagram: The basic, direct break-up.
2. The $K^*(892)$: A known, well-behaved particle that acts as a middleman.
3. The $N(1535)$: Another known particle that interacts with the pieces.
4. The $\Sigma^*(1/2^-)$: The ghost they are hunting.

The Results:

• Matching the Known: When they ran the simulation, the model perfectly matched the existing photos from the BESIII experiment regarding the $K^0$ and $\pi^+$ pieces. This proved their model was working correctly.
• Finding the Ghost: When they looked at the $\Lambda$ and $\pi^+$ combination in their simulation, they saw the predicted sharp "cusp" at 1.43 GeV. It was a distinct, jagged feature that wouldn't be there if the ghost particle didn't exist.

5. The Conclusion: A Call for a Better Camera

The paper concludes that the $\Sigma^*(1/2^-)$ is likely real and is responsible for this sharp "cusp" structure. However, the current photos from the BESIII experiment aren't sharp enough to see this cusp clearly yet.

The Final Message:
The authors are telling the scientific community: "We have a very strong map showing where the treasure is buried. We need the BESIII, Belle II, and the future Super Tau-Charm Facility to take higher-resolution photos of this specific decay. If they look closely at the $\Lambda$ and $\pi^+$ pieces, they should see this sharp spike, which would finally confirm the existence of this long-missing particle."

In short: We think we know where the missing particle is hiding. We just need better eyes to see the sharp spike it leaves behind.

Technical Summary

Technical Summary of "Search for the low-lying excited baryon $\Sigma^*(1/2^-)$ through process $\Lambda_c^+ \to \Lambda K^0 \pi^+$"

Problem Statement
The paper addresses the "mass reverse problem" in the light baryon sector, where the experimentally observed mass of the $N(1535)$ ($J^P = 1/2^-$) is higher than the radially excited $N(1440)$ ($J^P = 1/2^+$), contrary to some theoretical expectations. A central puzzle remains the experimental and theoretical status of the low-lying excited baryon $\Sigma^*(1/2^-)$. While the chiral unitary approach predicts a $\Sigma^*(1/2^-)$ dynamically generated from $S$-wave pseudoscalar meson-octet baryon interactions near the $\bar{K}N$ threshold, its existence lacks definitive confirmation. The Particle Data Group (RPP) lists $\Sigma(1620)$ with $J^P=1/2^-$ but assigns it only one star (poor evidence), omitting it from summary tables. Recent BESIII measurements of singly Cabibbo-suppressed decays $\Lambda_c^+ \to \Lambda K^+ \pi^0$ and $\Lambda_c^+ \to \Lambda K_S^0 \pi^+$, combined with Belle observations of cusp structures in $\Lambda \pi$ invariant mass distributions, suggest a potential signal for $\Sigma^*(1/2^-)$, yet a dedicated investigation in the $\Lambda_c^+ \to \Lambda K^0 \pi^+$ channel is required to confirm its role.

Methodology
The authors investigate the singly Cabibbo-suppressed decay $\Lambda_c^+ \to \Lambda K^0 \pi^+$ using a theoretical framework that combines quark-level weak decay mechanisms with final-state interactions (FSI) described by the chiral unitary approach.

1. Weak Decay Mechanism: The process is modeled via dominant color-favored $W^+$ external emission and color-suppressed $W^+$ internal emission diagrams. The hadronization of quark pairs into meson-baryon states is calculated using $SU(3)$ flavor symmetry, considering the mixing of $\eta-\eta'$. The authors explicitly exclude channels where terms cancel due to the $[P, \partial_\mu P]W^\mu$ structure of the $WPP$ vertex.
2. Final State Interactions (FSI): The model incorporates three primary mechanisms contributing to the decay amplitude:
   • Tree Level: Direct production of the final state.
   • $\Sigma^*(1/2^-)$ Generation: The $\pi^+ \Lambda$ final state interaction dynamically generates the $\Sigma^*(1/2^-)$ resonance. This is treated by solving the Bethe-Salpeter equation $T = [1 - VG]^{-1}V$ within the chiral unitary approach, coupling the $\bar{K}N$, $\pi\Sigma$, and $\pi\Lambda$ channels.
   • $N(1535)$ Generation: The $K^0 \Lambda$ final state interaction dynamically generates the $N(1535)$ resonance, involving coupled channels such as $\pi^- p$, $\pi^0 n$, $\eta n$, and $\Lambda K^0$.
   • Intermediate Resonance: The contribution of the intermediate vector meson $K^*(892)$ via the process $\Lambda_c^+ \to \Lambda K^*(892) \to \Lambda K^0 \pi^+$ is included.
3. Amplitude Construction: The total decay amplitude is constructed as a coherent sum of the tree, $N(1535)$, $\Sigma^*(1/2^-)$, and $K^*(892)$ contributions, including relative phase angles ($\phi, \phi'$) and normalization constants. The loop functions are calculated using dimensional regularization for the $\Sigma^*(1/2^-)$ sector and a cutoff method for the $N(1535)$ sector, with parameters tuned to reproduce known resonance properties.
4. Fitting Procedure: The model parameters (specifically the phase angles and a normalization constant) are fitted to the BESIII experimental data regarding the $K_S^0 \pi^+$ invariant mass distribution.

Key Contributions and Results

• Reproduction of Experimental Data: The model successfully reproduces the BESIII measurements of the $K_S^0 \pi^+$ invariant mass distribution, which shows a peak structure around 0.9 GeV associated with the $K^*(892)$. The fit yields a $\chi^2/\text{d.o.f} = 0.98$.
• Prediction of $\Sigma^*(1/2^-)$ Signature: The analysis predicts a distinct cusp structure in the $\pi^+ \Lambda$ invariant mass distribution around 1.43 GeV. This feature is explicitly associated with the dynamically generated $\Sigma^*(1/2^-)$ resonance.
• $N(1535)$ Contribution: A threshold enhancement structure is observed in the $K^0 \Lambda$ invariant mass distribution, attributed to the $N(1535)$ resonance.
• Dalitz Plot Analysis: The authors present Dalitz plots for the process, which clearly display the signals of the $\Sigma^*(1/2^-)$ and $K^*(892)$ resonances. No clear structure for $N(1535)$ is visible in the Dalitz plot as it lies below the relevant threshold in that projection.
• Parameter Determination: The study determines the relative phase angles $\phi = (1.73 \pm 0.11)\pi$ and $\phi' = (1.57 \pm 0.06)\pi$ and provides specific values for the relative strengths of the tree and resonance mechanisms.

Significance and Claims
The paper claims that its analysis provides a robust theoretical basis for searching for the elusive $\Sigma^*(1/2^-)$ baryon. By demonstrating that the $\Lambda_c^+ \to \Lambda K^0 \pi^+$ decay channel is sensitive to this state through a distinct cusp structure in the $\pi^+ \Lambda$ mass spectrum, the authors argue that this process serves as a crucial laboratory for testing the existence of $\Sigma^*(1/2^-)$.

The authors emphasize that future high-precision measurements of this specific decay channel at BESIII, Belle II, and the proposed Super Tau-Charm Facility (STCF) are essential. Confirming the predicted cusp structure would not only establish a new low-lying baryon resonance but also advance the understanding of non-perturbative Quantum Chromodynamics (QCD) and the light baryon spectrum, specifically resolving the nature of states dynamically generated from meson-baryon interactions. The paper remains modest, noting that while the model fits existing data well, the definitive confirmation of the $\Sigma^*(1/2^-)$ relies on future experimental data with higher precision.