Integral equations for four strongly interacting particles, two of which are charged

Abstract

The problem of the scattering in the systems of four strongly interacting particles with the Coulomb interaction is considered. By extracting and inverting in the kernels of the Faddeev-Yacubovsky equations the terms with the pure Coulomb two-particle transition operators, the Fredholm equations for the nuclear-Coulomb fourparticle transition operators are formulated. The kernels of thus regularized equations satisfy proper integral equations which are non-Fredholm in general case. The removal from these equations of the main Coulomb singularity arising owing to the interaction between the charges of two pairs of bound particles is performed. As it is known, the integral equations of motion describing the scattering in a fewparticle system become non-Fredholm in the presence of the Coulomb interaction [I] and must be reformulated to be applicable in practice. In the case of three particles a formal modification of the integral equations with the inclusion of the Coulomb potential in the non-perturbed Green function was proposed by Noble [2]. The approximate treatment of the Coulomb interaction in the Noble equations was considered in refs. [2, 3], the corresponding application to the proton-deuteron scattering was given in ref. [4]. The explicit separation from the kernel of the Faddeev equations for three particles of the dominant singularity, associated with the interaction between a charged particle and electric monopole moment of the two-particle subsystem, and the following inversion of it was carried out by Vesselova [5]. The relevant integral equations for three nucleons (p,d scattering) were derived in ref. [6]. The numerical solution of these equations was realized in [7]. A direct regularization of the three-particle integral equations by means of separation and inversion of all singularities being caused by the interaction of a charged particle with all multipoles of the two-particle subsystem was carried out in refs. [8, 9]. It was shown that the equations for the transition operator thus regularized were equivalent to the Noble equations obtained by the formal substitution of the Coulomb Green function for the free one. It was found in this way that the Nobletype equations were the result of the most complete regularization of the integral equations for three particles with Coulomb interaction. The method of direct regularization was found to be apt in the case of the system with the charge-exchange inter