LAUSR

The $J=1/2$ iso-doublet double-$\mathrm{\ensuremath{\Lambda}}$-hypernuclei, namely, $_{\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}}^{5}\mathrm{H}$ and $_{\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}}^{5}\mathrm{He}$, are examined as the three-body cluster states, $\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}t$ ($t\ensuremath{\equiv}{}^{3}$H or triton) and $\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}h$ ($h\ensuremath{\equiv}{}^{3}$He or helion), respectively, in a model-independent framework utilizing pionless halo effective theory. Both singlet and triplet states of the constituent $\mathrm{\ensuremath{\Lambda}}T$ ($T\ensuremath{\equiv}t,h$) subsystem are used in the elastic channel for the study of ${}_{\mathrm{\ensuremath{\Lambda}}}^{4}$H-$\mathrm{\ensuremath{\Lambda}}$ and ${}_{\mathrm{\ensuremath{\Lambda}}}^{4}$He-$\mathrm{\ensuremath{\Lambda}}$ scattering processes. A prototypical leading-order investigation using a sharp momentum cutoff regulator ${\mathrm{\ensuremath{\Lambda}}}_{c}$ in the coupled integral equations for each type of the $\mathrm{\ensuremath{\Lambda}}T$ subsystem spin states yields identical renormalization group limit cycle behavior when the respective three-body contact interactions are taken close to the unitary limit. Furthermore, irrespective of the type of the elastic channel chosen, almost identical cutoff dependence of the three-body binding energy or the double-$\mathrm{\ensuremath{\Lambda}}$-separation energy (${B}_{\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}}$) 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 ab initio potential model analysis for each mirror hypernuclei, yields ${B}_{\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}}$ which agrees fairly well with various erstwhile regulator independent potential models for our choice of the cutoff, ${\mathrm{\ensuremath{\Lambda}}}_{c}\ensuremath{\sim}200$ MeV. This is either consistent with pionless effective theory or with its slightly augmented version with a hard scale of ${\mathrm{\ensuremath{\Lambda}}}_{H}\ensuremath{\gtrsim}2{m}_{\ensuremath{\pi}}$, where low-energy $\mathrm{\ensuremath{\Lambda}}\ensuremath{-}\mathrm{\ensuremath{\Lambda}}$ interactions dominated by $\ensuremath{\pi}\ensuremath{\pi}$ or $\ensuremath{\sigma}$-meson exchange. Finally, to demonstrate the predictability of our effective theory, we present preliminary estimates of the $S$-wave $\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}T$ three-body scattering lengths and the $\mathrm{\ensuremath{\Lambda}}$-separation energies by using a range of currently accepted values of the double-$\mathrm{\ensuremath{\Lambda}}$ scattering length from a variety of existing phenomenological predictions that is constrained by the recent experimental data from relativistic heavy-ion collisions.