⚛️ phenomenology

Topological diagrams of $\Omega^0_c$ decays in the $SU(3)_F$ limit

This paper investigates the topological amplitudes of $\Omega_c^0$ baryon decays into octet and decuplet baryons within the $SU(3)_F$ limit by presenting complete tree- and penguin-induced diagrams, deriving linear relations between topological and irreducible amplitudes via tensor analysis, and establishing isospin relations to test the Körner-Pati-Woo theorem.

Original authors: Ying-Xin Lai, Di Wang

Published 2026-08-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ying-Xin Lai, Di Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Cosmic Lego Set: Why We Study Tiny, Heavy Particles

Imagine the universe as a giant, cosmic construction site. The basic building blocks of everything you see—stars, planets, and even you—are tiny particles called quarks. Usually, these quarks stick together in groups of three to form "baryons," like protons and neutrons. But sometimes, nature throws a curveball: a quark can be replaced by a "heavy" cousin called a charm quark. When this happens, you get a "charmed baryon." These are unstable, short-lived particles that decay (fall apart) almost instantly, transforming into lighter particles.

Scientists are obsessed with watching these particles fall apart because it's like watching a magic trick in slow motion. By studying exactly how they break, we can test the fundamental rules of physics, specifically a set of rules called the "Standard Model." One of the most important tools for understanding these tricks is something called "flavor symmetry." Think of it like a set of rules for swapping ingredients in a recipe. If you swap a "strange" quark for a "down" quark, the laws of physics say the recipe should still work, just with a slightly different flavor. This paper dives deep into the decay of a very special particle called the $\Omega_c^0$ (Omega-zero-charm), which is unique because it's the only one of its kind that doesn't fall apart via the strong force (the glue that holds nuclei together). Instead, it decays slowly via the weak force, making it a perfect candidate for a detailed investigation.

The Paper's Story: Mapping the Decay Map

In this study, researchers Ying-Xin Lai and Di Wang from Hunan Normal University decided to create a complete "topological map" of how the $\Omega_c^0$ baryon decays. Imagine the decay process as a complex game of billiards where balls (quarks) bounce off each other, swap places, and sometimes loop around in circles before settling into new shapes. The authors used a mathematical framework called $SU(3)_F$ symmetry to categorize every possible way these quarks can interact. They didn't just guess; they systematically drew out every possible "topological diagram"—which are essentially flowcharts showing how quarks move, emit, or loop.

The team focused on two main scenarios: when the $\Omega_c^0$ decays into a "decuplet" baryon (a group of 10 related particles) and a meson, and when it decays into an "octet" baryon (a group of 8) and a meson. They found that there are 10 distinct ways the quarks can arrange themselves in the first scenario and 20 distinct ways in the second. By using a method called "tensor analysis" (which is like a sophisticated accounting system for particle properties), they derived precise mathematical relationships between these different diagrams. This means if you know the strength of one type of decay, you can mathematically predict the strength of another, provided the symmetry rules hold true.

One of the most exciting parts of their work is testing a famous rule in physics called the Körner-Pati-Woo theorem. This theorem suggests that if two quarks produced by a weak interaction end up in the same final baryon, they must be "antisymmetric" (like two people who refuse to stand next to each other). If this theorem were perfectly true, it would force many of the decay diagrams to be zero or equal to each other in very specific ways. The authors derived several specific predictions based on this theorem, such as the idea that the decay rate of $\Omega_c^0 \to \Sigma^{*+} K^-$ should be exactly four times the rate of $\Omega_c^0 \to \Sigma^{*0} K^0$ .

However, the paper suggests that this theorem might not be the whole story. The authors point out that in the real world, particles don't just interact once and stop; they can "rescatter," meaning they bounce off each other again after the initial decay. They argue that these long-distance interactions (like a billiard ball hitting another, then hitting a third, then coming back) might mess up the neat predictions of the Körner-Pati-Woo theorem. In fact, they note that previous studies have found the theorem inconsistent with these rescattering dynamics. The authors propose that future experiments can test this by measuring the decay rates and "CP asymmetries" (a measure of how much matter and antimatter behave differently) of specific channels. If the experimental data matches the strict ratios predicted by the theorem, the theorem wins; if the data shows deviations, it suggests that the messy, long-distance rescattering effects are playing a bigger role than the simple symmetry rules allow.

The paper also highlights some practical numbers. For instance, they estimate that the ratio of the branching fraction (the probability of a specific decay happening) for $\Omega_c^0 \to \Omega^- K^+$ to $\Omega_c^0 \to \Omega^- \pi^+$ should be about $5.33 \times 10^{-2}$ . They note that the LHCb collaboration has already measured this ratio as $(6.08 \pm 0.51 \pm 0.40) \times 10^{-2}$ , which is consistent with their prediction. This consistency gives them confidence in their framework. They also suggest that while the "tree" diagrams (the direct, simple paths) are likely the dominant players, the "penguin" diagrams (complex loops involving virtual particles) are crucial for understanding CP violation, even if they are small in terms of overall probability.

Ultimately, this work doesn't claim to have solved the mystery of charmed baryon decays. Instead, it provides a comprehensive toolkit—a complete set of maps and rules—that experimentalists can use to decode the data coming from massive particle colliders like LHCb and Belle. By laying out all the possible topological diagrams and their relationships, the authors have set the stage for future experiments to determine exactly how strong the "rescattering" effects are and whether the Körner-Pati-Woo theorem needs to be revised or if it holds up under the pressure of real-world data.

Technical Summary: Topological diagrams of $\Omega^0_c$ decays in the $SU(3)_F$ limit

Problem Statement
The $\Omega^0_c$ baryon represents a unique system in heavy-flavor physics as the sole charmed sextet baryon that decays exclusively via the weak interaction, providing a complementary counterpart to the antitriplet charmed baryons ( $\Xi^0_c, \Xi^+_c, \Lambda^+_c$ ). While experimental collaborations (ALICE, Belle, LHCb) have recently measured $\Omega^0_c$ decays, theoretical understanding remains challenging. Standard QCD-inspired approaches like QCD factorization (QCDF), perturbative QCD (PQCD), and soft-collinear effective theory (SCET) struggle in the charm sector due to the expansion parameters $\alpha_s(m_c)$ and $\Lambda_{QCD}/m_c$ not being sufficiently small. Although dynamical models (factorization, pole models, quark models, and QCD sum rules) have been applied, a complete and model-independent analysis of the topological diagram decomposition for $\Omega^0_c$ decays, particularly regarding the linear relations between topological and $SU(3)$ irreducible amplitudes, remains unresolved. Furthermore, the interplay between different spin wave functions in baryon decays complicates the classification of topological diagrams.

Methodology
The authors employ the topological diagram approach within the framework of flavor $SU(3)$ symmetry ( $SU(3)_F$ ) to analyze $\Omega^0_c$ decays into a decuplet baryon and a pseudoscalar meson ( $B_{10}M$ ) and into an octet baryon and a pseudoscalar meson ( $B_8M$ ).

Formalism: The study utilizes tensor analysis to decompose the effective weak Hamiltonian into $SU(3)$ irreducible representations (including 3, 3', 6, and 15). The charmed sextet is treated as a tensor $B_{c6}$ , while light baryons are treated as either a decuplet tensor $B_{10}$ or octet tensors ( $B^S_8$ for symmetric and $B^A_8$ for antisymmetric flavor-spin combinations).
Diagram Classification: The authors systematically construct all possible topological diagrams for the decays. For $B_{c6} \to B_{10}M$ , they identify 10 tree-induced and 13 penguin-induced diagrams. For $B_{c6} \to B_8M$ , due to the mixed symmetry of the octet baryon wave functions, the analysis is split into symmetric ( $B^S_8$ ) and antisymmetric ( $B^A_8$ ) sets, resulting in a larger set of amplitudes (up to $B_{27}$ ).
Derivation of Relations: By inserting the weak operators into these topological structures and utilizing the properties of the Levi-Civita symbol and tensor contractions, the authors derive the linear relations connecting the topological amplitudes to the $SU(3)$ irreducible amplitudes. They also incorporate the K"orner-Pati-Woo theorem to test specific symmetry constraints.

Key Contributions and Results

Complete Decomposition: The paper presents a complete set of topological diagrams for $\Omega^0_c \to B_{10}M$ $Ω_{c}^{0} \to B_{10} M$ and $\Omega^0_c \to B_8M$ $Ω_{c}^{0} \to B_{8} M$ decays.
- For $B_{c6} \to B_{10}M$ , the authors identify 10 independent amplitudes after accounting for the fixed combinations of tree and penguin contributions (specifically, penguin-induced diagrams $A^P_{1-13}$ appear in fixed combinations with tree-induced diagrams $A_{8,9,10}$ , leaving no independent degrees of freedom for the penguin-induced amplitudes alone).
- For $B_{c6} \to B_8M$ , the analysis yields 20 independent amplitudes. The authors explicitly derive the relations between the topological amplitudes constructed from 3-rank tensors and those constructed from (1,1)-rank octet tensors.
Isospin Sum Rules: Three isospin sum rules for $\Omega^0_c$ $Ω_{c}^{0}$ decays are derived:
- $A(\Omega^0_c \to \Delta^+ K^-) - A(\Omega^0_c \to \Delta^0 \bar{K}^0) = 0$
- $A(\Omega^0_c \to \Sigma^{*+} \pi^-) + 2A(\Omega^0_c \to \Sigma^{*0} \pi^0) - A(\Omega^0_c \to \Sigma^{*-} \pi^+) = 0$
- $A(\Omega^0_c \to \Sigma^+ \pi^-) - 2A(\Omega^0_c \to \Sigma^0 \pi^0) + A(\Omega^0_c \to \Sigma^- \pi^+) = 0$
Phenomenological Estimates:
- The authors estimate the ratio of branching fractions for Cabibbo-favored modes, $Br(\Omega^0_c \to \Omega^- K^+)/Br(\Omega^0_c \to \Omega^- \pi^+) \approx |V_{us}|^2/|V_{ud}|^2 \approx 5.33 \times 10^{-2}$ , which is consistent with LHCb measurements.
- They discuss the suppression of certain diagrams (e.g., $A_2, A_3, A_4, A_6$ ) based on Wilson coefficients, suggesting that modes dominated by $A_1$ and $A_5$ should have larger branching fractions.
- The paper analyzes the relationship between topological diagrams and final-state rescattering mechanisms (u-, t-, and s-channels), noting that quark-loop diagrams receive long-distance contributions of the same order as tree diagrams.
Testing the K"orner-Pati-Woo Theorem:
- The authors derive specific relations for branching fractions and CP asymmetries that would hold if the K"orner-Pati-Woo theorem were strictly valid (e.g., $Br(\Omega^0_c \to \Delta^+ K^-) = 0$ ).
- However, they explicitly state that previous findings and their own analysis of rescattering dynamics suggest the theorem is not consistent with the dynamics of singly charmed baryon decays. Specifically, the prediction $A_3 = A_4 = 0$ contradicts non-vanishing contributions from triangle and bubble diagrams. Thus, the derived relations serve as experimental tests to verify or falsify the theorem.

Significance
The paper provides a comprehensive theoretical framework for analyzing $\Omega^0_c$ decays in the $SU(3)_F$ limit, filling a gap in the complete decomposition of topological diagrams and their linear relations. By establishing the connection between topological amplitudes and $SU(3)$ irreducible amplitudes, the work offers a model-independent tool for future experimental analyses. The derived isospin relations and the specific predictions for testing the K"orner-Pati-Woo theorem provide concrete targets for experimental collaborations (such as LHCb and Belle) to extract dynamical information and probe the validity of symmetry constraints in heavy baryon decays. The authors emphasize that while current data is insufficient to determine the magnitudes and strong phases of the amplitudes, the framework is ready for application once sufficient branching fraction and decay parameter measurements become available.

The Cosmic Lego Set: Why We Study Tiny, Heavy Particles

The Paper's Story: Mapping the Decay Map

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