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Comparative study of three-nucleon force models in $A=3,4$ systems

Using modern nucleon-nucleon interactions in the description of the $A=3,4$ nuclei, it is not possible to reproduce both the three- and four-nucleon binding energies simultaneously. This is one manifestation of the necessity of including a three-nucleon force in the nuclear Hamiltonian. In this paper we will perform a comparative study of some, widely used, three-nucleon force models. We will analyze their capability to describe the aforementioned binding energies as well as the $n-d$ doublet scattering length. A correct description of these quantities can be considered a stringent requirement for a nuclear Hamiltonian containing two- and three-nucleon interaction terms. As we will show, this requirement is not fulfilled by several of the models available in the literature. To satisfy it, we propose modifications in the parametrization of the three-nucleon forces and we study their effects on few selected $N-d$ low energy scattering observables.