ΛΛHypernucleusBe10ΛΛand theΛ−ΛInteraction

The $\Lambda \Lambda$ hypernucleus ${}_{\Lambda \Lambda }{}^{}{}_{}{}^{}\mathrm{Be}_{}^{10}{}_{}{}^{}$ has been analyzed by use of a four-body $\alpha -\alpha -\Lambda -\Lambda$ model which allows for distortion of the core by the $\Lambda$ particles. In particular, the dependence of the internal energy of the core on the rms separation of the $\alpha$ particles is required. This was obtained from three-body $\alpha -\alpha -\Lambda$ calculations for ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{Be}_{}^9{}_{}{}^{}$. Several types of $\alpha -\alpha$ potentials, whose $s$-wave phase shifts had been previously obtained, were considered. Calculations for ${}_{\Lambda \Lambda }{}^{}{}_{}{}^{}\mathrm{Be}_{}^{10}{}_{}{}^{}$ were made for a singlet $\Lambda -\Lambda$ Yukawa potential (I) of intrinsic range $b=1.48\mathrm{}$ F, appropriate to the exchange of two pions, and for a hard-core Yukawa potential (II) with a hard-core radius $r_c=0.42\mathrm{}$ F and $b=2.66\mathrm{}$ F, appropriate to a range corresponding to two pion masses for the attractive Yukawa part. Results are also given for a hard-core meson-theory potential (III) which has $r_c=0.42\mathrm{}$ F and $b=1.48\mathrm{}$ F. Calculations for III were made for ${}_{\Lambda \Lambda }{}^{}{}_{}{}^{}\mathrm{He}_{}^6{}_{}{}^{}$, and the results were adapted to ${}_{\Lambda \Lambda }{}^{}{}_{}{}^{}\mathrm{Be}_{}^{10}{}_{}{}^{}$. For $\alpha -\alpha$ potentials which give $s$-wave phase shifts consistent with experiment, it is found that (almost independently of the details of the $\Lambda -\Lambda$ potential) the effects of core distortion account for rather more than a third of the experimental additional binding energy of 4.5±0.5 MeV which is obtained after the $\Lambda$ separation energy of ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{Be}_{}^9{}_{}{}^{}$ has been allowed for. Slightly more than half the contribution due to core distortion comes from the core energy of ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{Be}_{}^9{}_{}{}^{}$. The remainder is due to the further distortion of the core by the second $\Lambda$, which causes approximately a 10% decrease in the rms $\alpha -\alpha$ separation relative to the value for ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{Be}_{}^9{}_{}{}^{}$. The effects of core distortion weaken the resulting $\Lambda -\Lambda$ potential quite appreciably. For $b=1.48\mathrm{}$ F, one obtains the scattering length $a_{\Lambda \Lambda }\approx -1\mathrm{}\pm 0.3$ F and the effective range $r_{0\Lambda \Lambda }\approx 3.3\pm 0.6$ F, approximately independent of the shape of the $\Lambda -\Lambda$ potential. For II, one gets $a_{\Lambda \Lambda }={{-2.3\mathrm{}}_{-0.5\mathrm{}}}^{+0.8\mathrm{}}$ F and $r_{\Lambda \Lambda }={{4.9}_{-0.7\mathrm{}}}^{+1.1\mathrm{}}$ F. The well-depth parameters are 0.45±0.08, 0.675±0.065, and 0.77±0.04 for I, II, and III, respectively. These values are about 35%, 20%, and 12%, respectively, less than the values obtained for a rigid core with a three-body ${\mathrm{Be}}^8-\Lambda -\Lambda$ model. The $\Sigma -\Lambda -\pi$ coupling constant, obtained with III, is close to the value obtained from the singlet $\Lambda -N$ interaction for the same hard-core radius.