${}_{}{}^3{}_{}{}^{}\mathrm{H}$ bound state for the Reid soft-core potential: Exact calculation by a perturbational approach

The perturbational approach to the Faddeev equation proposed by us is applied to the ${}_{}{}^3{}_{}{}^{}\mathrm{H}$ bound state. The perturbation iteration is found to converge beautifully, leaving no question about its usefulness. Using the Reid soft-core potential, we obtain the ${}_{}{}^3{}_{}{}^{}\mathrm{H}$ binding energy at 6.62 MeV, $P\left(S\right)=90.28\%$, $P\left(S^{\prime }\right)=1.70\%$, and $P\left(D\right)=8.02\%$, for the three-body state consisting of the two-body pair in the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ and the ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}={}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$ state with the third particle in the $s$-state relative to the center-of-mass of the pair. The contribution from the third particle in the $d$-state adds to the binding energy only a small fraction (∼0.05 MeV). The one-body charge form factor of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ has a dip at $q^2\sim 16$ ${\mathrm{fm}}^{-2\mathrm{}}$, and the height of the second peak is about an order of magnitude too low compared to the data. Each Faddeev component of the triton wave function as a function of the relative distance of a pair has a node near the core radius in its ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ and ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ components, but no node is found in the ${}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$ component. The wave function as a function of the spectator coordinate extends over quite large distances without a node in any component. The nodes are attributed to the strong soft-core and the particle exchange effects.

NUCLEAR STRUCTURE Triton bound state. Exact solution of the Faddeev equation by a perturbative approach. Binding energy; 6.62 MeV. Node of the wave function near the core radius.