Identifying triple-strangeness $\Omega$ hyperons in light of recent experimental results

In this work, we systematically calculate the mass spectra and decay widths of $\Omega$ baryons with $N\leq3$, where $N=2n_\rho+2n_\lambda+l_\rho+l_\lambda$ is the shell of the three-body harmonic oscillator. In our scheme, the newly observed $\Omega(2109)$ is identified as a $N=2$ candidate, with its quantum numbers tentatively assigned as $\Omega(2S,3/2^+)$. However, the experimentally measured mass exhibits a deviation of approximately 50 MeV below the theoretical predictions. The possible reason for this deviation, along with the suggestion for experiments, are discussed in this paper. Additionally, we give possible assignments of $\Omega(2012)^-$, $\Omega(2250)^-$, $\Omega(2380)^-$, and $\Omega(2470)^-$. Furthermore, we calculate the masses and widths of other missing $\Omega$ states, which could provide valuable clues for experimental searches aimed at discovering these states.