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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 assessingthe feasibility of light particle-stable three-body bound states, utilizinglow-energy universality. Here we take recourse to a simplistic speculation ofthe 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. Inparticular, a qualitative leading order EFT investigation by introducing asharp momentum (ultraviolet) cut-off parameter $\Lambda_c$ into theFaddeev-like coupled integral equations, indicates a discrete scaling behaviorakin to a renormalization group limit cycle, thereby suggesting the formalexistence of Efimov states in the unitary limit, as $\Lambda_c\to \infty$. Oursubsequent non-asymptotic analysis indicates that the three-body binding energy$B_3$ is sensitively dependent on the cut-off without the inclusion ofthree-body contact interactions. Furthermore, our analysis reproduces severalvalues of the binding energy $B_3\sim 3-4$ MeV, predicted in context ofexisting potential models, with the regulator $\Lambda_c$ in the range, $\sim350-460$ MeV. Finally, based on these model inputs for $B_3$, a ballparkestimate of the three-body scattering length in the range, $2.6-4.9$ fm, isnaively constrained by our EFT analysis, ostensibly demonstrating the universalnature of three-body correlations that is likely to manifest themselves in ahalo-bound $\Xi^{-}nn$ system.

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