Integral equations and binding energies for Λ − NN and Λ − NNN systems

Abstract

A systematic treatment of the procedure used to construct and solve the Faddeev-Yakubovsky integral equations for three and four different particles with nonzero spin is introduced. The four-body T-matrices are written in terms of solutions of the three-body Faddeev integral equations in total angular momenta representation. General spin-angular structure of the four-body Faddeev-Yakubovsky equations is written down and reduced to a spin-angular structure of the three-body system by elimination of one body. Partial-wave expansion performed for the system of four-body integral equations gives an infinite system of one-dimensional coupled equations with integral kernels containing four types of partial components of the three-body Faddeev integral equation solutions. The given method is tested on Λ−NN and Λ−NNN systems with two-body potential written down in only separable form. Both separable NN potential and ΛN potential rewritten in separable form for s-wave are used to determine the Λ-hyperon binding energies in Λ−NN and Λ−NNN systems. Calculation also includes the hyperon conversion in ΛN→ΣN process. Λ-hyperon separation energies were calculated from binding energies and turned out to be 0.147 MeV in Λ−NN system and 2.04 MeV in Λ−NNN system, respectively, that is in close coincidence with experimental ones.