Hypernuclear Spectroscopy forA=6

The hypernuclei ${}_{\Lambda \Lambda }{}^{}{}_{}{}^{}\mathrm{He}_{}^6{}_{}{}^{}$ and ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{He}_{}^6{}_{}{}^{}$ are studied using the method of Dawson, Talmi, and Walecka. As a check on the calculation, ${\mathrm{He}}^6$ is also studied using a pure central force for the neutron-neutron interaction. By working within a harmonic-oscillator frame work, the Bethe-Goldstone equation is transformed to relative coordinates and solved using the variational procedure developed by Dawson and Walecka. Assuming a pure central potential with a Yukawa shape for the $\Lambda -\Lambda$ interaction, the strengths of the potential yielding the observed binding energy are found for three different ranges. As a check, the binding energy of ${\mathrm{He}}^6$ is estimated assuming central potentials of the Bruckner-Gammel-Thaler type and of the Hamada type. The spin-orbit and tensor parts of the potentials are neglected, because the neutrons interact most strongly in a relative singlet $s$ state. Too much binding is found, presumably because of our use of harmonic-oscillator single-particle potentials. A 15% reduction in the attractive part of the central Hamada-type potential gives reasonable binding for the neutrons. The energy levels of the ground state and excited states of ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{He}_{}^6{}_{}{}^{}$ are estimated using the potentials given by Shemsher Ali, Grypeos, and Kok. A 20 to 25% reduction in the potential yields the observed binding energy of the ground state. In addition to the ground state, which is $1^{-}$, a $2^{-}$ excited state is also found, which may not be bound against breakup into ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{He}_{}^5{}_{}{}^{}+n$.