Further reduction of the three-body problem with finite-range forces

Abstract

The well-known approach to the two-body problem in nuclear physics, based on the use of a parameter characterizing the finite size of the interaction range, is extended to the three-body problem. It is shown for the general case, without any approximation, that the problem of three particles interacting with arbitrary pair finite-range potentials reduces to the solution of a one-dimensional set of integral equations for functions depending on the spectator variable. The kernels of the obtained equations are determined by the two-particle scattering wave function off the energy shell in the pair interaction range and in turn satisfy the auxiliary two-variable integral equations with the integration over the finite range of three-particle configuration space. The auxiliary set of equations has nonsingular kernels and can be solved by simple methods. In the zero approximation, taking only the pair correlation of particles into account, the kernel of the basic set is known in explicit form, in the zero-range force limit it turns into the Skornyakov-Ter-Martirosyan kernel. On the basis of the performed reduction a new approach to the solution of the three-body problem is proposed. Some examples concerning the simplest model interactions are given. With use of the experimental data on the binding energy of a Σ− hyperdeuteron the binding energy of the three-particle Σ-hypernucleus, consisting of two neutrons and one Σ− hyperon (Σ− hypertriton), is calculated.