Λ single particle energies

The Λ single-particle energies $B_{\Lambda }$ of hypernuclei (HN) are calculated microscopically using the Fermi hypernetted chain method to obtain for our $\Lambda N$ and $\Lambda \mathrm{NN}$ potentials the Λ binding $D\left(\rho \right)$ to nuclear matter, and the effective mass $m_{\Lambda }^{*}\left(\rho \right)$ at densities $\rho <~{\rho }_0$ $({\rho }_0$ is normal nuclear density), and also the corresponding effective $\Lambda N$ and $\Lambda \mathrm{NN}$ potentials. The Λ core-nucleus potential $U_{\Lambda }\mathrm{}\left(r\right)\mathrm{}$ is obtained by suitably folding these into the core density. The Schrödinger equation for $U_{\Lambda }$ and $m_{\Lambda }^{*}$ is solved for $B_{\Lambda }.$ The fringing field (FF) due to the finite range of the effective potentials is theoretically required. We use a dispersive $\Lambda \mathrm{NN}$ potential but also include a phenomenological ρ dependence allowing for less repulsion for $\rho <{\rho }_0,$ i.e., in the surface. The best fits to the data with a FF give a large ρ dependence, equivalent to an A dependent strength consistent with variational calculations of ${}_{\mathrm{\Lambda }}^5\mathrm{He},$ indicating an effective $\Lambda \mathrm{NN}$ dispersive potential increasingly repulsive with A whose likely interpretation is in terms of dispersive plus two-pion-exchange $\Lambda \mathrm{NN}$ potentials. The well depth is $29\pm 1\mathrm{MeV}.$ The $\Lambda N$ space-exchange fraction corresponds to $m_{\Lambda }^{*}\left(\rho \right)\approx 0.75–0.80$ and a ratio of p- to s-state potentials of $\approx 0.5\pm \mathrm{0.1.}$ Charge symmetry breaking (CSB) is significant for heavy HN with a large neutron excess; with a FF the strength agrees with that obtained from the $A=4\mathrm{}\mathrm{HN}.$ The fits without FF are excellent but inconsistent with the requirement for a FF, with ${}_{\mathrm{\Lambda }}^5\mathrm{He},$ and also with the CSB sign for $A=4.\mathrm{}$