  I consider non-relativistic bosons interacting via pairwise potentials with
infinite scattering length and supporting no two-body bound states. To lowest
order in effective field theory, these conditions lead to non-interacting
bosons, since the coupling constant of the Lieb-Liniger model vanishes
identically in this limit. Since any realistic pairwise interaction is not a
mere delta function, the non-interacting picture is an idealisation indicating
that the effect of interactions is weaker than in the case of off-resonant
potentials. I show that the leading order correction to the ground state energy
for more than two bosons is accurately described by the lowest order three-body
force in effective field theory that arises due to the off-shell structure of
the two-body interaction. For natural two-body interactions with a
short-distance repulsive core and an attractive tail, the emergent three-body
interaction is repulsive and, therefore, three bosons do not form any bound
states. This situation is analogous to the two-dimensional repulsive Bose gas,
when treated using the lowest-order contact interaction, where the scattering
amplitude exhibits an unphysical Landau pole. The avoidance of this state in
the three-boson problem proceeds in a way that parallels the two-dimensional
case. These results pave the way for the experimental realisation of
one-dimensional Bose gases with pure three-body interactions using ultracold
atomic gases.
