Improved prediction of the mass splitting for $P$-wave $\Omega$ baryons

Using the QCD sum rule method, we investigate the mass splitting for the spin-orbit partner states of the $\Omega(2012)$ baryon assuming that it is a $P$-wave excitation with $J^P=3/2^-$. This study is an extension of the previous work [1] in which the masses of these states were estimated with uncertainties too large to extract the reliable mass splitting. In the present study, by directly formulating a sum rule for the mass splitting, we obtain an improved prediction, $\delta M = M_{3/2^-} - M_{1/2^-} = -18.0^{+ 33.6}_{-17.1}$ MeV. This result provides a more quantitative insight into the spectrum of $P$-wave $\Omega$ baryons and serves as a useful reference for future experiments.