Binding Energy of aΛ-Particle in Nuclear Matter and theΛ-Nucleon Interaction

The binding energy $D$ of a $\Lambda$-particle in nuclear matter is calculated with the independent-pair approximation for seven central two-body $\Lambda$-nucleon potentials. These potentials are consistent with the binding energy of ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{He}_{}^5{}_{}{}^{}$ and therefore represent the spin-averaged $\Lambda$-nucleon interaction in $S$ states; they have hard cores of radius 0.4 F or 0.6 F and two-parameter attractive wells with ranges suggested by consideration of the two-pion-exchange mechanism. A simple approximation to the Bethe-Goldstone function is suggested; its use permits $D$ and the partial-wave contributions to $D$ to be evaluated easily. When the $S$-wave $\Lambda$-nucleon potentials are assumed to be appropriate to all angular momentum states, the calculated values of $D$, corresponding to a nucleon density equal to the central density in heavy nuclei, are consistent with empirical estimates in the range 30-40 MeV for most of the potentials considered. If the correct value of $D$ is close to 30 MeV, some reduction in the strength of the longer ranged potentials may be required in odd-parity states (at least in $P$ states) to bring about agreement; for the shorter ranged potentials considered, no such reduction would be required. If the correct value of $D$ is close to 40 MeV, odd-parity suppression would not be indicated even for the longer ranged potentials. The first three partial-wave contributions to $D$, as well as $D$ itself, are given for each potential, and the dependence of these on the hard-core radius and on the shape and range of the attractive well is discussed.