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The $\Lambda\Lambda$-$\Xi N$-$\Sigma\Sigma$ coupling in$^{~6}_{\Lambda\Lambda}$He is studied with the [$\alpha$ + $\Lambda$ +$\Lambda$] + [$\alpha$ + $\Xi$ + $N$] + [$\alpha$ + $\Sigma$ + $\Sigma$] model,where the $\alpha$ particle is assumed as a frozen core. We use the Nijmegensoft-core potentials, NSC97e and NSC97f, for the valence baryon-baryon part,and the phenomenological potentials for the $\alpha-B$ parts ($B$=$N$,$\Lambda$, $\Xi$ and $\Sigma$). We find that the calculated $\DeltaB_{\Lambda\Lambda}$ of $^{~6}_{\Lambda\Lambda}$He for NSC97e and NSC97f are,respectively, 0.6 and 0.4 MeV in the full coupled-channel calculation, theresults of which are about half in comparison with the experimental data,$\Delta B^{exp}_{\Lambda\Lambda}=1.01\pm0.20^{+0.18}_{-0.11}$ MeV.Characteristics of the $S=-2$ sector in the NSC97 potentials are discussed indetail.Comment: 18 pages, 4 figure

physical review. c. nuclear physics/physical review. c, nuclear physics

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>Λ</mml:mi><mml:mi>Λ</mml:mi><mml:mtext>−</mml:mtext><mml:mi>Ξ</mml:mi><mml:mi>N</mml:mi><mml:mtext>−</mml:mtext><mml:mi>Σ</mml:mi><mml:mi>Σ</mml:mi></mml:mrow></mml:math>coupling in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mi mathvariant="normal">He</mml:mi><mml:mprescripts /><mml:mrow><mml:mi>Λ</mml:mi><mml:mi>Λ</mml:mi></mml:mrow><mml:…

ΛΛ−ΞN−ΣΣcoupling inHeΛΛ