Faddeev Equations and Coulomb Effects inHe3

Abstract

When two particles interact via a potential $V$ which is the sum of a separable potential $V_S$ and a Coulomb potential $V_C$, the two-body $T$ matrix can be obtained exactly and can be split in the usual way into a pure Coulomb $T$ matrix and a "nuclear" $T$ matrix which also contains Coulomb effects. Using the fact that the "nuclear" $T$ matrix is still separable in the off-shell variables, the formalism of Alt, Grassberger, and Sandhas is applied to the calculation of Coulomb effects in the three-nucleon system, with the contribution of the nonseparable Coulomb $T$ matrix fully taken into account. The Coulomb energy ${\Delta }_C$ of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and the probability $P_{\frac{3}{2}}$ of finding the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in an $I=\frac{3}{2}$ state are calculated, using $s$-wave spin-dependent potentials of the Yamaguchi type to describe the two-nucleon interaction. This model yields values for ${\Delta }_C$ which are in reasonable agreement with the binding-energy difference of ${}_{}{}^3{}_{}{}^{}\mathrm{H}$ and ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, but predicts a negligible admixture of the $I=\frac{3}{2}$ state in the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ wave function. The effect of the nonseparable part of the $p-p$ $T$ matrix on the binding energy and wave function of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is discussed. Finally, the possible relevance of hard-core effects is pointed out.