Two- and three-charged-particle nuclear scattering in momentum space: A two-potential theory and a boundary condition model

Abstract

The two- and three-charged-particle nuclear scattering problems are investigated in momentum space. The three-body equations with a short-range nuclear potential, a three-body force potential, and the long-range Coulomb potential are presented in a mathematically rigorous way within a generalized two-potential theory. To remove the serious singularity in the two- and three-body Coulomb problems in momentum space, we have proposed a novel boundary condition to the phase shift that arises from a potential difference, the so-called auxiliary potential (AP), between the Coulomb potential and the screened Coulomb one: ${V}^{\ensuremath{\phi}}={V}^{C}\ensuremath{-}{V}^{R}$. Furthermore, we point out the importance of the off-shell amplitude for the AP, by which one can uniquely obtain the two-body on-shell and off-shell Coulomb amplitude in momentum space. Therefore, this formulation is also useful for atomic systems, which is another benefit. It is recalled that the traditional phase-shift renormalization theory, in which the screened Coulomb amplitude is sandwiched by renormalized phase factors ${e}^{i\ensuremath{\phi}}$, is not consistent with two-potential theory. Some ambiguities or misunderstandings of the traditional methods for handling the Coulomb problem are clarified. Finally, the three-body unitarity relation is proved for the amplitude generated from these three kinds of potential. Moreover, a generalized two-potential theory is presented. It is pointed out that the asymptotic three-body nuclear wave function with the Coulomb potential could not be written by a product of two individual wave functions ${\stackrel{~}{\ensuremath{\psi}}}_{x}(x){\stackrel{~}{\ensuremath{\psi}}}_{y}(y)$ defined in terms of Jacobi coordinates $x$ and $y$, respectively.