Λ single particle energies

The {Λ} single-particle energies B_Λ of hypernuclei (HN) are calculated microscopically using the Fermi hypernetted chain method to obtain for our ΛN and ΛNN potentials the {Λ} binding D(ρ) to nuclear matter, and the effective mass m_Λ*(ρ) at densities ρ<~ρ₀ (ρ₀ is normal nuclear density), and also the corresponding effective ΛN and ΛNN potentials. The {Λ} core-nucleus potential U_Λ(r) is obtained by suitably folding these into the core density. The Schr\"odinger equation for U_Λ and m_Λ* is solved for B_Λ. The fringing field (FF) due to the finite range of the effective potentials is theoretically required. We use a dispersive ΛNN potential but also include a phenomenological {ρ} dependence allowing for less repulsion for ρ<ρ₀, 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 _Λ⁵He, indicating an effective ΛNN dispersive potential increasingly repulsive with A whose likely interpretation is in terms of dispersive plus two-pion-exchange ΛNN potentials. The well depth is 29±1MeV. The ΛN space-exchange fraction corresponds to m_Λ*(ρ)≈0.75--0.80 and a ratio of p- to s-state potentials of ≈0.5±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=4HN. The fits without FF are excellent but inconsistent with the requirement for a FF, with _Λ⁵He, and also with the CSB sign for $A=4.$