$\mathrm{\Omega }\mathit{NN}$ and $\mathrm{\Omega }\mathrm{\Omega }N$ states

The lattice QCD analysis of the HAL QCD Collaboration has recently derived $\mathrm{\Omega }N$ and $\mathrm{\Omega }\mathrm{\Omega }$ interacting potentials with nearly physical quark masses ($m_{\pi }\simeq 146$ MeV and $m_K\simeq 525$ MeV). They found an attractive interaction in the $\mathrm{\Omega }N{}^{5}S_2$ channel which supports a bound state with a central binding energy of 1.54 MeV. The $\mathrm{\Omega }\mathrm{\Omega }{}^{1}S_0$ channel shows an overall attraction with a bound state with a central binding energy of 1.6 MeV. In this paper we looked closely at the $\mathrm{\Omega }\mathit{NN}$ and $\mathrm{\Omega }\mathrm{\Omega }N$ three-body systems making use of the latest HAL QCD Collaboration $\mathrm{\Omega }N$ and $\mathrm{\Omega }\mathrm{\Omega }$ interactions. Our results show that the $\mathrm{\Omega }d$ system in the state with maximal spin $\left(I\right)J^P=\left(0\right)5/2^{+}$ is bound with a binding energy of about 20 MeV. The $\left(I\right)J^P=\left(1\right)3/2^{+}\mathrm{\Omega }nn$ state presents a resonance decaying to $\mathrm{\Lambda }\mathrm{\Xi }n$ and $\mathrm{\Sigma }\mathrm{\Xi }n$, with a separation energy of $\sim 1$ MeV. The $\left(I\right)J^P=(1/2)1/2^{+}\mathrm{\Omega }\mathrm{\Omega }N$ state also exhibits a resonance decaying to $\mathrm{\Lambda }\mathrm{\Xi }\mathrm{\Omega }$ and $\mathrm{\Sigma }\mathrm{\Xi }\mathrm{\Omega }$ with a separation energy of $\sim 4.6$ MeV. We have calculated the contribution of the Coulomb potential to differentiate among the different charged states.