A spin-isospin dependent Three-Dimensional approach based on momentum vectors for formulation of the three-nucleon bound state is presented in this paper. The three-nucleon Faddeev equations with two-nucleon interactions are formulated as a function of vector Jacobi momenta, specifically the magnitudes of the momenta and the angle between them with the inclusion of the spin-isospin quantum numbers, without employing a partial wave decomposition. As an application the spin-isospin dependent Faddeev integral equations are solved with Bonn-B potential. Our result for the Triton binding energy with the value of −8.152 MeV is in good agreement with the achievements of the other partial wave based methods.

The cross section of neutron-deuteron radiative capture nd → 3 Hγ is calculated at energies relevant to Big-Bang nucleosynthesis ( 20 ≤ E ≤ 200 KeV ) with pionless Effective Field Theory. At these energies, magnetic transition M1 gives the dominant contribution. The M1 amplitude is calculated up to next-to-next-to leading order(N 2 LO) with insertion of three-body force. Results are in good agreement within few percent theoretical uncertainty in comparison with available calculated data below E=200 KeV.

The cross section for the thermal neutron capture by the deuteron is calculated with pionless Effective Field Theory(EFT). No new Three-Nucleon forces are needed up to next-to-next-toleading order in order to achieve cut-off independent results, besides those fixed by the triton binding energy and Nd scattering length in the triton channel. The cross-section is accurately determined to be σ tot = [0.503 ± 0.003]mb. At zero energies, the magnetic M 1-transition gives the dominant contribution and is calculated up to next-to-next-to-leading order (N 2 LO). Close agreement between the available experimental data and the calculated cross section is reached. We demonstrate convergence and cutoff independence order by order in the low-energy expansion.

The four-body bound state with two-body interactions is formulated in Three-Dimensional approach, a recently developed momentum space representation which greatly simplifies the numerical calculations of few-body systems without performing the partial wave decomposition.The obtained three-dimensional Faddeev-Yakubovsky integral equations are solved with two-body spin-independent and spin-averaged potentials. This is the first step toward the calculations of four-nucleon bound state problem in Three-Dimensional approach. Results for four-body binding energies are in good agreement with achievements of the other methods.

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. * )