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

Abstract

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.

35 pages, 10 figures

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