Theoretical study of the $\Omega(2012)$ state in the $\Omega_c^0 \to \pi^+ \Omega(2012)^- \to \pi^+ (\bar{K}\Xi)^-$ and $\pi^+ (\bar{K}\Xi\pi)^-$ decays

We report on a theoretical study of the newly observed $\Omega(2012)$ resonance in the nonleptonic weak decays of $\Omega_c^0 \to \pi^+ \bar{K}\Xi^*(1530) (\eta \Omega) \to \pi^+ (\bar{K}\Xi)^-$ and $\pi^+ (\bar{K}\Xi\pi)^-$ via final-state interactions of the $\bar{K}\Xi^*(1530)$ and $\eta \Omega$ pairs. The weak interaction part is assumed to be dominated by the charm quark decay process: $c(ss) \to (s + u + \bar{d})(ss)$, while the hadronization part takes place between the $sss$ cluster from the weak decay and a quark-antiquark pair with the quantum numbers $J^{PC} = 0^{++}$ of the vacuum, produces a pair of $\bar{K}\Xi^*(1530)$ and $\eta \Omega$. Accordingly, the final $\bar{K}\Xi^*(1530)$ and $\eta \Omega$ states are in pure isospin $I= 0$ combinations, and the $\Omega_c^0 \to \pi^+ \bar{K}\Xi^*(1530)(\eta \Omega) \to \pi^+ (\bar{K}\Xi)^-$ decay is an ideal process to study the $\Omega(2012)$ resonance. With the final-state interaction described in the chiral unitary approach, up to an arbitrary normalization, the invariant mass distributions of the final state are calculated, assuming that the $\Omega(2012)$ resonance with spin-parity $J^P = 3/2^-$ is a dynamically generated state from the coupled channels interactions of the $\bar{K}\Xi^*(1530)$ and $\eta \Omega$ in $s$-wave and $\bar{K}\Xi$ in $d$-wave. We also calculate the ratio, $R^{\bar{K}\Xi\pi}_{\bar{K}\Xi} = {\rm Br}[\Omega_c^0 \to \pi^+ \Omega(2012)^- \to \pi^+ (\bar{K}\Xi \pi)^-] / {\rm Br}[\Omega_c^0 \to \pi^+ \Omega(2012)^- \to \pi^+ (\bar{K}\Xi)^-$]. The proposed mechanism can provide valuable information on the nature of the $\Omega(2012)$ and can in principle be tested by future experiments.