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Abstract This paper explores a novel revision of the Faddeev equation for three-body (3B) bound states, as initially proposed in Ref. [J. Golak, K. Topolnicki, R. Skibiński, W. Glöckle, H. Kamada, A. Nogga, Few Body Syst. 54, 2427 (2013)]. This innovative approach, referred to as t-matrix-free in this paper, directly incorporates two-body (2B) interactions and completely avoids the 2B transition matrices. We extend this formalism to relativistic 3B bound states using a three-dimensional (3D) approach without using partial wave decomposition. To validate the proposed formulation, we perform a numerical study using spin-independent Malfliet–Tjon and Yamaguchi interactions. Our results demonstrate that the relativistic t-matrix-free Faddeev equation, which directly implements boosted interactions, accurately reproduces the 3B mass eigenvalues obtained from the conventional form of the Faddeev equation, referred to as t-matrix-dependent in this paper, with boosted 2B t-matrices. Moreover, the proposed formulation provides a simpler alternative to the standard approach, avoiding the computational complexity of calculating boosted 2B t-matrices and leading to significant computational time savings. 

This study presents a solution to the Yakubovsky equations for four-body bound states in momentum space, bypassing the common use of two-bodyt− matrices. Typically, such solutions are dependent on the fully-off-shell two-bodyt− matrices, which are obtained from the Lippmann-Schwinger integral equation for two-body subsystem energies controlled by the second and third Jacobi momenta. Instead, we use a version of the Yakubovsky equations that does not requiret− matrices, facilitating the direct use of two-body interactions. This approach streamlines the programming and reduces computational time. Numerically, we found that this direct approach to the Yakubovsky equations, using 2B interactions, produces four-body binding energy results consistent with those obtained from the conventionalt− matrix dependent Yakubovsky equations, for both separable (Yamaguchi and Gaussian) and non-separable (Malfliet-Tjon) interactions. 

We study the ground-state properties of $$\prescript{6}{YY}{\text{He}}$$ double hyperon for $$\lla$$ and $$\ooa$$ nuclei in a three-body model $$(Y+Y+\alpha)$$. We solve two coupled Faddeev equations corresponding to three-body configurations $$(\alpha Y,Y)$$ and $$(YY, \alpha)$$ in configuration space with the hyperspherical harmonics expansion method by employing the most recent hyperon-hyperon interactions obtained from lattice QCD simulations. Our numerical analysis for $$\lla$$, using three $$\Lambda\Lambda$$ lattice interaction models, leads to a ground state binding energy in the domain $(-7.468, -7.804)$ MeV and the separations $$\langle r_{\Lambda-\Lambda} \rangle$$ and $$\langle r_{\alpha-\Lambda} \rangle$$ in the domains $(3.555, 3.629)$ fm and $(2.867 , 2.902 )$ fm, correspondingly. The binding energy of double-$$\Omega$$ hypernucleus $$\ooa$$ leads to $-67.21$ MeV and consequently to smaller separations $$\langle r_{\Omega-\Omega} \rangle = 1.521$$ fm and $$\langle r_{\alpha-\Omega} \rangle = 1.293 $$ fm. Besides the geometrical properties, we study the structure of ground-state wave functions and show that the main contributions are from the $s-$wave channels. Our results are consistent with the existing theoretical and experimental data. 

Abstract The matrix elements of relativistic nucleon–nucleon ( NN ) potentials are calculated directly from the nonrelativistic potentials as a function of relative NN momentum vectors, without a partial wave decomposition. To this aim, the quadratic operator relation between the relativistic and nonrelativistic NN potentials is formulated in momentum-helicity basis states. It leads to a single integral equation for the two-nucleon (2 N ) spin-singlet state, and four coupled integral equations for two-nucleon spin-triplet states, which are solved by an iterative method. Our numerical analysis indicates that the relativistic NN potential obtained using CD-Bonn potential reproduces the deuteron binding energy and neutron-proton elastic scattering differential and total cross-sections with high accuracy.