${}^{\ \ \ 6}_{\Lambda\Lambda} He$ in cluster effective field theory

TL;DR

This paper models the hypernucleus $^{6}_{\Lambda\Lambda}He$ as a three-body cluster system using effective field theory, revealing a limit cycle in the three-body interaction and exploring correlations between separation energy and scattering length.

Contribution

It introduces the promotion of the three-body contact interaction to leading order due to the limit cycle behavior in the effective field theory framework.

Findings

Identification of the limit cycle in the three-body contact interaction.

Derivation of a determination equation for the limit cycle.

Analysis of correlations between separation energy and scattering length.

Abstract

The hypernucleus $\nuclide[6][\Lambda\Lambda]{He}$ is studied as a three-body ($\Lambda\Lambda\alpha$) cluster system in cluster effective field theory at leading order. We find that the three-body contact interaction exhibits the limit cycle when the cutoff in the integral equations is sent to the asymptotic limit, and thus it should be promoted to leading order. We also derive a determination equation of the limit cycle which reproduces the numerically obtained limit cycle. We then study the correlations between the double $\Lambda$ separation energy $B_{\Lambda\Lambda}^{}$ of $\nuclide[6][\Lambda\Lambda]{He}$ and the scattering length $a_{\Lambda\Lambda}^{}$ of the $S$-wave $\Lambda\Lambda$ scattering. The role of the scale in this approach is also discussed.