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

The hypernucleus ${}_{\mathrm{\Lambda \Lambda }}{}^6\mathrm{He}$ is studied as a three-body ($\mathrm{\Lambda \Lambda }\alpha$) 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 $B_{\mathrm{\Lambda \Lambda }}^{}$ of ${}_{\mathrm{\Lambda \Lambda }}{}^6\mathrm{He}$ and the scattering length $a_{\mathrm{\Lambda \Lambda }}^{}$ of the $S$-wave $\mathrm{\Lambda \Lambda }$ scattering. The role of the scale in this approach is also discussed.

Figure 1

The coupling $g\left({\Lambda }_c\right)$ as a function of ${\Lambda }_c$ for $a_{\mathrm{\Lambda \Lambda }}^{}=-0.6$, $-1.2$, $-1.8$ fm where the values of $g\left({\Lambda }_c\right)$ are fitted by $B_{\mathrm{\Lambda \Lambda }}=6.93\mathrm{MeV}$ of ${}_{\mathrm{\Lambda \Lambda }}{}^6\mathrm{He}$.

Figure 2

The two-$\Lambda$ separation energy $B_{\mathrm{\Lambda \Lambda }}$ as a function of the cutoff ${\Lambda }_c$ for $a_{\mathrm{\Lambda \Lambda }}^{}=-0.6,-1.2,-1.8$ fm without the three-body contact interaction. The experimental data $B_{\mathrm{\Lambda \Lambda }}=6.93\mathrm{MeV}$ is included as a reference line.

Figure 3

The two-$\Lambda$ separation energy $B_{\mathrm{\Lambda \Lambda }}$ as a function of $1/a_{\mathrm{\Lambda \Lambda }}^{}$ for ${\Lambda }_c=170$, 300, $430\mathrm{MeV}$, where $g\left({\Lambda }_c\right)$ is renormalized at the point of $B_{\mathrm{\Lambda \Lambda }}=6.93\mathrm{MeV}$ and $1/a_{\mathrm{\Lambda \Lambda }}^{}=-2.0$ ${\mathrm{fm}}^{-1}$ that is marked by a filled square. Open squares are the results from the potential models in Table 5 of Ref. [12].