Singly Cabibbo-suppressed hadronic weak decays of the $\Omega^-$ hyperon

We study the two-body hadronic weak decays of the $\Omega^-$ hyperon with strangeness $S=-3$, including three singly Cabibbo-suppressed decay modes: $\Xi^0 \pi^-$, $\Xi^-\pi^0$ and $\Lambda K^-$. The decay amplitudes at the quark level, arising from $s\to ud \bar{u}$ transitions (direct pion emission and color-suppressed processes) and $su\to ud$ transitions (pole terms), are calculated in the framework of the non-relativistic constituent quark model.The theoretical results show that the $\Xi^0 \pi^-$ channel is dominated by the color-allowed direct pion emission process, while the $\Lambda K^-$ channel is well described by one type of pole contribution mediated through intermediate $\Xi$ resonances ($1^2S_{1/2^+}$ and $1^2P_{1/2^-}$ states). However, the contribution from tree-level mechanisms alone to the branching ratio of $\Omega^- \to \Xi^- \pi^0$ is small due to its color-suppressed nature. The discrepancy is resolved by including final state interactions through rescattering processes via intermediate states $\Xi^0\pi^-$ and $\Lambda K^-$. This work demonstrates that a unified description of $\Omega^-$ hadronic weak decays necessitates the interplay of quark-level weak vertices, baryon pole structures, and long-distance final state rescattering dynamics. With these mechanisms, the obtained branching ratios are in agreement with the high-precision experimental data from the BESIII. Furthermore, these above decays are found to be dominated by the parity-conserving $P$-wave transitions, thus the asymmetry parameters are almost zero.