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${}_{ΛΛ}^{5}$H and ${}_{ΛΛ}^{5}$He Hypernuclei reexamined in Halo/Cluster Effective Theory

The J=1/2 iso-doublet double-$Λ$-hypernuclei, namely, ${}_{ΛΛ}^{5}$H and ${}_{ΛΛ}^{5}$He, are examined as the three-body cluster states, $ΛΛt$ ($t\equiv {}^3$H or triton) and $ΛΛh$ ($h\equiv {}^3$He or helion), respectively, in a model independent framework utilizing pionless halo effective theory. Both singlet and triplet states of the constituent $ΛT$ ($T\equiv t,h$) sub-system are used in the elastic channel for the study of $_Λ^4$H$-Λ$ and $_Λ^4$He$-Λ$ scattering processes. A prototypical leading order investigation using a sharp momentum cut-off regulator, irrespective of the type of the elastic channel chosen, yields almost identical cut-off dependence of the three-body binding energy or the double-$Λ$-separation energy ($B_{ΛΛ}$) is obtained for the mirror partners, evidently suggesting good isospin symmetry in these three-body systems. Subsequently, upon normalization of our solutions to the integral equation with respect to a single pair of input data from an {\it ab initio} potential model analysis for each mirror hypernuclei, yields $B_{ΛΛ}$ which agrees fairly well with various erstwhile regulator independent potential models for our choice of the cut-off regulator, $\sim 200$ MeV. This is either consistent with pionless effective theory or with its slightly augmented version with a hard scale around $2m_π$, where low-energy $Λ$-$Λ$ interactions dominated by $ππ$ or $σ$-meson exchange. Finally, to demonstrate the predictability of our effective theory, we present preliminary estimates of the S-wave $ΛΛT$ three-body scattering lengths and the $Λ$-separation energies.