  We study bosonic systems in the regime in which the two-body system has a
shallow bound state or, equivalently, a large value of the two-body scattering
length. Using the effective field theory framework as a guide, we construct a
series of potential terms which have decreasing importance in the description
of the binding energy of the systems. The leading order potential terms consist
of a two-body term, usually attractive, plus a three-body term, usually
repulsive; this last term is required to prevent the collapse of systems with
more than two particles. At this order, the parametrization of the two-body
potential is done to obtain a correct description of the scattering length,
which governs the dynamics in this regime, whereas the three-body term fixes a
three-body datum. We investigate the role of the cut-off in the leading order
description and we extend the exploration beyond the leading order by including
the next-to-leading order terms in both, the two- and three-body potentials. We
use the requirement of the stability of the N-body system, whose energy is
variationally estimated, to introduce the three-body forces. The potential
parametrization, as a function of the cut-off, is fixed to describe the energy
of 4 He clusters up to seven particles within the expected accuracy. Finally,
we also explore the possibility to describe at the same time the atom-dimer
scattering length.
