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$ΛΛN$-$ΞNN$ $S$ wave resonance

We use an existing model of the $ΛΛN - ΞNN$ three-body system based in two-body separable interactions to study the $(I,J^P)=(1/2,1/2^+)$ three-body channel. For the $ΛΛ$, $ΞN$, and $ΛΛ- ΞN$ amplitudes we have constructed separable potentials based on the most recent results of the HAL QCD Collaboration. They are characterized by the existence of a resonance just below or above the $ΞN$ threshold in the so-called $H$-dibaryon channel, $(i,j^p)=(0,0^+)$. A three-body resonance appears {2.3} MeV above the $Ξd$ threshold. We show that if the $ΛΛ- ΞN$ $H$-dibaryon channel is not considered, the $ΛΛN - ΞNN$ $S$ wave resonance disappears. Thus, the possible existence of a $ΛΛN - ΞNN$ resonance would be sensitive to the $ΛΛ- ΞN$ interaction. The existence or nonexistence of this resonance could be evidenced by measuring, for example, the $Ξd$ cross section.