Simple Woods-Saxon type form for $\Omega \alpha $ and $\Xi \alpha$ interactions using Folding Model

We derive a simple Woods-Saxon type form for the potentials between $Y=\Xi, \Omega$ and $\alpha$ by using a single-folding potential method, based on a separable $Y$-nucleon potential. Accordingly, the potentials $\Xi+\alpha$ and $\Omega+\alpha$ are obtained using the ESC08c Nijmegens $\Xi N$ potential (in $^{3}S_{1}$ channel) and HAL QCD Collaboration $\Omega N$ interactions (in lattice QCD), respectively. In deriving the potential between $Y$ and $\alpha$, the same potential between $Y$ and $N$ is used. Binding energy, scattering length and effective range of $Y$ particle on the alpha particle are approximated by the resultant potentials. The depths of the potentials in $\Omega \alpha$ and $\Xi \alpha$ systems are obtained $-61$ and $-24.4$ MeV, respectively. In the case of $\Xi \alpha$ potential, a fairly good agreement is observed between the single-folding potential method and the phenomenological potential of Dover-Gal model. These potentials can be used in 3-,4- and 5-body cluster structures of $\Omega$ and $\Xi$ hypernuclei.