Λ−ΛPotential from Analysis ofBe10ΛΛ

Abstract

The strength of the $\ensuremath{\Lambda}\ensuremath{-}\ensuremath{\Lambda}$ interaction in the $^{1}S_{0}$ state is determined by analyzing the double hypernucleus $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}\mathrm{Be}^{10}$ treated as a four-body system of $2\ensuremath{\alpha}+2\ensuremath{\Lambda}$. The $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ potential chosen fits the $^{1}S_{0}$ phase shifts up to a c.m. energy of about 12 MeV. The $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\Lambda}$ potential is determined from the binding energy of $_{\ensuremath{\Lambda}}\mathrm{He}^{5}$, taking into proper account the size of the alpha particle and the range of the $\ensuremath{\Lambda}$-nucleon interaction. With these potentials, the binding energy of $_{\ensuremath{\Lambda}}\mathrm{Be}^{9}$ is also given correctly. For the $\ensuremath{\Lambda}\ensuremath{-}\ensuremath{\Lambda}$ part, a potential is used which has a hard core of radius 0.4 F and an attractive well of exponential shape. The intrinsic range is chosen as 1.5 F, corresponding to the mechanism of 2-pion exchange. Using a 12-parameter variational function, the $\ensuremath{\Lambda}\ensuremath{-}\ensuremath{\Lambda} ^{1}S_{0}$ potential which yields the observed separation energy of the two $\ensuremath{\Lambda}$ particles in $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}\mathrm{Be}^{10}$ is found to have a well-depth parameter of ${{0.732}_{\ensuremath{-}0.034}}^{+0.027}$, which is about 10% smaller than the value obtained by treating $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}\mathrm{Be}^{10}$ as a three-body system of ${\mathrm{Be}}^{8}+2\ensuremath{\Lambda}$. The scattering length and effective range are equal to --- (${{1.04}_{\ensuremath{-}0.22}}^{+0.20}$) and ${{2.91}_{\ensuremath{-}0.27}}^{+0.46}$ F, respectively. Also, we found that the ${\mathrm{Be}}^{8}$ core in $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}\mathrm{Be}^{10}$ is quite compressed, with the $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ separation about 12% smaller than the corresponding separation in $_{\ensuremath{\Lambda}}\mathrm{Be}^{9}$.