Energy-dependent pole expansions for the effective potentials in the four-body integral equations with tensor forces

Abstract

We investigate the accuracy of the energy-dependent pole expansion for the (3 + 1) and (2 + 2) subamplitudes in the calculation of the binding energy of the $\alpha$ particle, $E_{\alpha }$, for separable $\mathrm{NN}$ potentials with tensor components. We employ the truncated $t$-matrix ($t_{00}$) approximation and compare our results for $E_{\alpha }$ to those obtained, independent of any separable expansion, by Gibson and Lehman and to the results for $E_{\alpha }$ obtained with the Hilbert-Schmidt expansion of the subamplitudes. It is shown that the energy-dependent pole expansion is both more economical and converges faster than the Hilbert-Schmidt expansion, even one term of the energy-dependent pole approximation already being accurate to better than 1.5%.