#### Abstract

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.

DOI: http://dx.doi.org/10.1103/PhysRev.138.B644

- Received 28 December 1964
- Published in the issue dated May 1965

© 1965 The American Physical Society