Separable-potential three-body model of theA=6system. I.He6ground state

Separable-potential three-body ($\alpha \mathrm{nn}$) models of the ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ ground state are investigated. Different separable-potential fits to the $\alpha -n$ phase shifts and the low-energy singlet two-nucleon parameters are discussed, after which integral equations are derived for the spectator functions of the ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ ground-state wave function assuming $S_{\frac{1}{2}}$, $P_{\frac{1}{2}}$, and $P_{\frac{3}{2}}$ components in the $\alpha -n$ interaction. These equations are solved to determine the ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ binding energy and examine its sensitivity to the two-nucleon singlet effective range, its variation upon neglecting different components of the $\alpha -n$ interaction, and its dependence on different analytic forms for the $P$-wave $\alpha -n$ interactions. The spectator functions for the "best" two sets of parameters are generated and tabulated, and for one set the contributions to the ground-state normalization of the various components in the ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ ground-state wave function are computed. With the "best" $S_{\frac{1}{2}}$,$P_{\frac{1}{2}}$, and $P_{\frac{3}{2}} \alpha -n$ interactions, the calculated ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ binding energy is 0.542 MeV for a two-nucleon $s$-wave separable potential fitted to the $n-p {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ effective-range parameters, $r_0=2.73\mathrm{}$ fm, and $a=-23.715\mathrm{}$ fm, and 0.359 MeV for the $n-n {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ effective-range parameters, $r_0=2.84\mathrm{}$ fm, and $a=-17\mathrm{}$ fm, compared to the experimental value of 0.969 MeV.