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The hypernucleus $\nuclide[6][\Lambda\Lambda]{He}$ is studied as a three-body($\Lambda\Lambda\alpha$) cluster system in cluster effective field theory atleading order. We find that the three-body contact interaction exhibits thelimit cycle when the cutoff in the integral equations is sent to the asymptoticlimit, and thus it should be promoted to leading order. We also derive adetermination equation of the limit cycle which reproduces the numericallyobtained 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 roleof the scale in this approach is also discussed.

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