Investigation of $\Xi^{-}nn$ ($S=-2$) Hypernucleus in Low-energy Pionless Halo Effective Theory

In the strangeness $S=-2$ sector, we study the $\Xi^{-}nn$ ($I=3/2, J^P={1/2}^+$) three-body system using pionless halo effective field theory (EFT), which provides a systematic model independent framework for assessing the feasibility of light particle-stable three-body bound states, utilizing low-energy universality. Here we take recourse to a simplistic speculation of the three-body system by eliminating the repulsive spin-singlet $\Xi^- n$ sub-system, while retaining the predominantly attractive (possibly bound) spin-triplet $\Xi^{-}n$ and the virtual bound spin-singlet $nn$ sub-system. In particular, a qualitative leading order EFT investigation by introducing a sharp momentum (ultraviolet) cut-off parameter $\Lambda_c$ into the Faddeev-like coupled integral equations, indicates a discrete scaling behavior akin to a renormalization group limit cycle, thereby suggesting the formal existence of Efimov states in the unitary limit, as $\Lambda_c\to \infty$. Our subsequent non-asymptotic analysis indicates that the three-body binding energy $B_3$ is sensitively dependent on the cut-off without the inclusion of three-body contact interactions. Furthermore, our analysis reproduces several values of the binding energy $B_3\sim 3-4$~MeV, predicted in context of existing potential models, with the regulator $\Lambda_c$ in the range, $\sim 350-460$~MeV. Finally, based on these model inputs for $B_3$, a ballpark estimate of the three-body scattering length in the range, $2.6-4.9$~fm, is naively constrained by our EFT analysis, ostensibly demonstrating the universal nature of three-body correlations that is likely to manifest themselves in a halo-bound $\Xi^{-}nn$ system.