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

Abstract: The $Ω<sup>0_c$ baryon is a unique charmed sextet baryon as it decays through weak interaction. In this work, we investigate the topological amplitudes of $Ω_c<sup>0$ decays in the $SU(3)_F$ limit. The tree- and penguin-induced diagrams contributing to $Ω_c<sup>0$ decays into decuplet and octet baryons are presented completely. The linear relations between the topological amplitudes and the $SU(3)$ irreducible amplitudes are derived via tensor analysis. Several isospin relations are obtained, and some relations are derived to test the Körner-Pati-Woo theorem.

• The paper provides a complete tensor decomposition of Ωc⁰ decays into decuplet- or octet-baryon plus pseudoscalar final states, identifying 10 and 20 independent amplitudes, respectively.
• Its SU(3)F analysis yields testable results, including the ratio Br(Ω⁻K⁺)/Br(Ω⁻π⁺) ≈ 5.33 × 10⁻², consistent with the LHCb value, as well as isospin sum rules and additional branching-fraction targets.
• The rescattering framework predicts direct CP asymmetries of roughly 10⁻⁴–10⁻³ in selected singly Cabibbo-suppressed channels and exposes tension between rescattering effects and the Körner–Pati–Woo theorem.

This paper by Lai and Wang presents a complete tensor-analytic decomposition of the topological amplitudes governing the nonleptonic weak decays of the $\Omega_c^0$ baryon into decuplet-plus-pseudoscalar and octet-plus-pseudoscalar final states, in the exact $SU(3)_F$ limit (2608.12742). The $\Omega_c^0$ is the only member of the charmed sextet that cannot decay strongly, so its weak decays provide a clean probe of nonperturbative baryonic transitions at the charm scale, complementary to the well-studied antitriplet baryons. The work extends a framework previously developed for doubly charmed, charmed, and bottom baryon decays, and addresses two gaps left open by earlier topological analyses: the exhaustive enumeration of diagrams and the derivation of linear relations connecting topological amplitudes to $SU(3)$ irreducible amplitudes.

Framework and completeness of the decomposition

The analysis starts from the effective weak Hamiltonian for charm decay, with the magnetic penguin contribution folded into the Wilson coefficients $C_{3\text{–}6}$ via standard substitutions. In the $SU(3)$ tensor language, the weak operator $\mathcal{O}^k_{ij}$ decomposes into irreducible representations $15$, $\overline{6}$, $3$, and $SU(3)_F$0. For $SU(3)_F$1 transitions, the amplitude is written as a sum of thirteen tensor contractions $SU(3)_F$2; the first seven are tree-level topologies and the last six involve quark loops (penguin topologies). The authors prove completeness by a counting argument: with five free indices, there are $SU(3)_F$3 possible contractions, and the symmetry properties of the sextet (two symmetric indices) and decuplet (three symmetric indices) partition these exactly into the thirteen terms, $SU(3)_F$4.

For decays into octet baryons, the Pauli principle forces the flavor and spin wave functions to be coupled: the mixed-symmetry flavor octets $SU(3)_F$5 and $SU(3)_F$6 must appear in the combination $SU(3)_F$7. Consequently the octet topology space splits into two independent sets, $SU(3)_F$8 (33 diagrams) and $SU(3)_F$9 (27 diagrams), each with tree and penguin variants. The paper also introduces a $\Omega_c^0$0-rank octet tensor representation and derives the explicit linear map relating the $\Omega_c^0$1 amplitudes to the $\Omega_c^0$2 amplitudes constructed from that representation — a nontrivial bookkeeping result that underlies the subsequent $\Omega_c^0$3 decomposition.

Irreducible amplitudes and the penguin redundancy

The linear relations between topological and $\Omega_c^0$4 irreducible amplitudes are derived for both the decuplet and octet channels. A key structural result follows from CKM unitarity: since $\Omega_c^0$5, the tree and penguin singlet-type irreducible amplitudes appear only in the fixed combinations $\Omega_c^0$6. Translating this back to diagram space, the penguin diagrams are never independent degrees of freedom; they always accompany specific tree diagrams in fixed combinations (e.g., $\Omega_c^0$7 for the decuplet case). The final count is 10 independent amplitudes for $\Omega_c^0$8 and 20 independent amplitudes for $\Omega_c^0$9. The authors note that while penguin and quark-loop tree diagrams are negligible in branching fractions because $SU(3)$0, they remain relevant to direct CP asymmetries through the weak phase of $SU(3)$1.

Isospin relations and phenomenology

Because $SU(3)$2 is an isospin singlet, three isospin sum rules follow directly, including the familiar $SU(3)$3 relation $SU(3)$4 and its decuplet analogue. $SU(3)$5-spin sum rules cannot be established, since the other members of the $SU(3)$6 $SU(3)$7-spin triplet decay strongly — a limitation inherent to the system rather than the method.

Using $SU(3)$8 and $SU(3)$9 in the NDR scheme, the authors show that only the diagrams $C_{3\text{–}6}$0 and $C_{3\text{–}6}$1 carry unsuppressed Wilson-coefficient combinations ($C_{3\text{–}6}$2), while $C_{3\text{–}6}$3, $C_{3\text{–}6}$4, $C_{3\text{–}6}$5, and $C_{3\text{–}6}$6 are coefficient-suppressed ($C_{3\text{–}6}$7 to $C_{3\text{–}6}$8). This yields the prediction

$C_{3\text{–}6}$9

which is consistent with the LHCb measurement $SU(3)$0. A further prediction, $SU(3)$1, and two suggested measurable ratios involving $SU(3)$2 and $SU(3)$3 are proposed as near-term experimental targets.

Rescattering interpretation and CP asymmetries

The paper connects the quark-loop topologies to long-distance final-state rescattering using leading-order chiral Lagrangians ($SU(3)$4, $SU(3)$5, $SU(3)$6, $SU(3)$7) and tensor completeness relations. The resulting dictionary shows that each quark-loop diagram receives triangle ($SU(3)$8) and bubble ($SU(3)$9) rescattering contributions of the same order as the tree diagrams, while the color-favored emitted diagram is roughly an order of magnitude larger. The immediate phenomenological implication is that singly Cabibbo-suppressed channels without a color-favored emitted contribution should exhibit direct CP asymmetries of order $\mathcal{O}^k_{ij}$0–$\mathcal{O}^k_{ij}$1, whereas channels containing such a contribution have suppressed CP asymmetry. The authors acknowledge that with current data the magnitudes and strong phases of the topological amplitudes cannot yet be extracted; the rescattering correspondence is what will allow quark-loop amplitudes, and hence CP asymmetries, to be estimated once branching fractions and decay parameters are measured.

Testing the Körner–Pati–Woo theorem

A central and somewhat provocative result concerns the Körner–Pati–Woo (KPW) theorem, which requires the two quarks emitted from the weak vertex to be flavor-antisymmetric when both enter the same baryon. In the tensor framework this implies $\mathcal{O}^k_{ij}$2 and $\mathcal{O}^k_{ij}$3 for decuplet modes, plus analogous constraints for the octet sets. These lead to sharp branching-fraction relations beyond isospin, e.g. $\mathcal{O}^k_{ij}$4, $\mathcal{O}^k_{ij}$5, and the vanishing predictions $\mathcal{O}^k_{ij}$6 and $\mathcal{O}^k_{ij}$7.

However, the rescattering analysis contradicts the theorem: the triangle and bubble diagrams contribute nonvanishingly to $\mathcal{O}^k_{ij}$8 and $\mathcal{O}^k_{ij}$9, in direct conflict with the KPW prediction $15$0. The paper is explicit that this tension, already noted for singly charmed baryons in prior work, is reinforced here. The proposed resolution is empirical: the KPW relations, along with associated CP-asymmetry equalities (including partial-wave relations in $15$1, $15$2, $15$3), are all testable in the isospin limit, and future measurements will adjudicate whether the theorem survives as an approximation or fails dynamically.

Limitations and open questions

Several limitations are acknowledged or implicit. The entire analysis is exact only in the $15$4 limit; $15$5-breaking effects, which are known to be sizable in charmed-hadron decays, are not quantified. The Wilson-coefficient hierarchy argument relies on a specific renormalization scheme and scale, and the rescattering estimates depend on chiral Lagrangian couplings and on the inclusion of both ground-state and excited intermediate hadrons, whose treatment is qualitative at this stage. Most significantly, the conflict between the KPW theorem and the rescattering dynamics is identified but not resolved — the paper leaves open which description the data will favor.

Conclusion

The paper supplies the complete, symmetry-verified set of topological amplitudes for $15$6 two-body nonleptonic decays, the explicit maps to $15$7 irreducible amplitudes, and the resulting counting of independent amplitudes (10 for decuplet, 20 for octet final states). It delivers experimentally accessible predictions, including a branching-fraction ratio consistent with the LHCb measurement, three isospin sum rules, and a battery of Körner–Pati–Woo test relations — while flagging a concrete inconsistency between that theorem and rescattering dynamics that only data can settle.