Unitary Pole Approximation and Binding Energy of the Trinucleon

Abstract

The unitary pole approximation (UPA) uses the two-body binding energy and wave function to determine the form factor for the UPA separable $t$ matrix. We develop the UPA for Tabakin's 1965 spin-independent model potential for the trinucleon, and obtain a trinucleon energy within 0.2 MeV of his result. We then develop the UPA for Tabakin's 1964 spin-singlet potential, and for the Schrenk-Mitra singlet. We combine these with Yamaguchi shapes and also a modified Hulth\'en shape for the spin-triplet central and tensor potentials. These choices give trinucleon energies within 0.3 MeV of the experimental value, provided that we fit the deuteron with 4% $D$ state. We further study the dependence of trinucleon energy on the percent $D$ state in the range $0.78%\ensuremath{\le}{P}_{D}\ensuremath{\le}7%$. We also use Tabakin's recent rank-1 separable fit to singlet phase shifts. This separable potential gives a trinucleon energy 1.5 MeV higher than the singlet choices above because of its relatively weak attraction in off-shell $t$-matrix elements.