A Realistic Formalism for 4N Bound State in a Three-Dimensional Yakubovsky Scheme

Abstract

A spin-isospin dependent Three-Dimensional formalism based on the momentum vectors for the four-nucleon bound state is presented. The four-nucleon Yakubovsky equations with two-and three-nucleon interactions are formulated as a function of the vector Jacobi momenta. Our formalism, according to the number of spin-isospin states that one takes into account, leads to only a strictly finite number of the coupled three dimensional integral equations to be solved. The evaluation of the transition and permutation operators as well as the coordinate transformations due to considering the continuous angle variables instead of the discrete angular momentum quantum numbers are less complicated in comparison with partial wave representation. With respect to partial wave the present formalism with the smaller number of equations leads to higher dimensionality of the integral equations. We have concluded that three dimensional formalism is less cumbersome for considering the three-nucleon forces. * )

Solutions of the bound-state Faddeev-Yakubovsky equations in three dimensions by usingNNand3Npotential models

“…The details of the numerical algorithm for solving the coupled three-dimensional integral equations can be found in Refs. [10,13,14].…”

“…These have been evaluated in detail in Ref. [10]. It is useful to mention that one needs the free 4N basis states |A; γ , where the spin-isospin parts γ are given as |γ ≡ |γ S γ T T γβ (see Ref.…”

“…It is useful to mention that one needs the free 4N basis states |A; γ , where the spin-isospin parts γ are given as |γ ≡ |γ S γ T T γβ (see Ref. [10]). After the abovementioned operators and coordinate transformations are carried out, the coupled Yakubovsky equations can be obtained explicitly: …”

“…For three-nucleon (3N) and four-nucleon (4N) bound states, the FY equations with two-and three-nucleon interactions have been recently formulated in a realistic 3D approach [10]. The formalism, according to the number of spin-isospin states that are taken into account, leads to a finite number of coupled three-dimensional integral equations to be solved.…”

“…[10]. We report results obtained for three-and four-nucleon binding energies by employing spin-isospin-dependent NN potential models along with a scalar two-meson exchange 3NF.…”