Momentum-space solution of a bound-state nuclear three-body problem with two charged particles

Abstract

Momentum-space wave function equations are derived for the three-body system of one neutral and two charged particles where the separable interaction is spin and charge independent. The three-body wave function is decomposed so that the equations for the pure nuclear components contain the two-nucleon $t$ matrix, as usual, but the equation for the additional Coulomb component is formulated in terms of the Coulomb potential rather than introducing the Coulomb $t$ matrix. The relationship of these equations to the scattering-amplitude equation of Veselova, in which the Coulomb $t$ matrix appears explicitly, as applied to the same problem by Kok et al., is given. After partial-wave decomposition, the wave function equations are solved numerically with partial waves through $l=4$ retained. The logarithmic singularity that appears when the Coulomb potential is expressed in momentum space is handled for each $l$ by a subtraction technique originated by Lande. Because of the symmetry of the problem, only even values of $l$ arise throughout and only the $l=0$ and $l=2$ partial-wave contributions are required to predict the binding energy (and wave-function components) to four significant figures. An order $\ensuremath{\alpha}$ (fine-structure constant) approximation (i.e., decoupling the Coulomb wave function component from itself) is made to check its validity. It is shown that this $O(\ensuremath{\alpha})$ approximation is equivalent to replacing the Coulomb $t$ matrix by the Coulomb potential in the Veselova equations. The wave functions without Coulomb effects and with Coulomb effects [exact and $O(\ensuremath{\alpha})$] are used to calculate the expectation value of the Coulomb operator to examine perturbation theory, cross check the numerical results, and to clarify the physics. We conclude that the Coulomb interaction can be incorporated easily into momentum-space three-nucleon, bound-state calculations.