Four-nucleon shell-model calculations in a Faddeev-like approach

We use equations for Faddeev amplitudes to solve the shell-model problem for four nucleons in a model space that includes up to $14ħ\Omega$ harmonic-oscillator excitations above the unperturbed ground state. Two- and three-body effective interactions derived from the Reid93 and Argonne V8’ nucleon-nucleon potentials are used in the calculations. Binding energies, excitation energies, point-nucleon radii, and electromagnetic and strangeness charge form factors for ${}^4\mathrm{He}$ are studied. The structure of the Faddeev-like equations is discussed and a formula for the matrix elements of the permutation operators in a harmonic-oscillator basis is given. The dependence on harmonic-oscillator excitations allowed in the model space and on the harmonic-oscillator frequency is investigated. It is demonstrated that the use of three-body effective interactions improves the convergence of the results.