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

The J=1/2 iso-doublet double-$\Lambda$-hypernuclei, namely, ${}_{\Lambda\Lambda}^{5}$H and ${}_{\Lambda\Lambda}^{5}$He, are examined as the three-body cluster states, $\Lambda\Lambda t$ ($t\equiv {}^3$H or triton) and $\Lambda\Lambda h$ ($h\equiv {}^3$He or helion), respectively, in a model independent framework utilizing pionless halo effective theory. Both singlet and triplet states of the constituent $\Lambda T$ ($T\equiv t,h$) sub-system are used in the elastic channel for the study of $_\Lambda^4$H$-\Lambda$ and $_\Lambda^4$He$-\Lambda$ 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-$\Lambda$-separation energy ($B_{\Lambda\Lambda}$) 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 $B_{\Lambda\Lambda}$ 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 $2m_\pi$, where low-energy $\Lambda$-$\Lambda$ interactions dominated by $\pi\pi$ or $\sigma$-meson exchange. Finally, to demonstrate the predictability of our effective theory, we present preliminary estimates of the S-wave $\Lambda\Lambda T$ three-body scattering lengths and the $\Lambda$-separation energies.