We investigate in the context of the scattering equations, how one-loop linear propagator integrands in gauge theories can be linked to integrands with quadratic propagators using a double forward limit. We illustrate our procedure through examples and demonstrate how the different parts of the derived quadratic integrand are consistent with cut-integrands derived from four-dimensional generalized unitarity. We also comment on applications and discuss possible further generalizations.

We study the ground state energy of a gas of 1D bosons with density ρ, interacting through a general, repulsive 2-body potential with scattering length a, in the dilute limit ρ|a| ≪ 1. The first terms in the expansion of the thermodynamic energy density are π 2 ρ 3 /3(1 + 2ρa), where the leading order is the 1D free Fermi gas. This result covers the Tonks-Girardeau limit of the Lieb-Liniger model as a special case, but given the possibility that a > 0, it also applies to potentials that differ significantly from a delta function. We include extensions to spinless fermions and 1D anyonic symmetries, and discuss an application to confined 3D gases.