One-loop scattering equations and amplitudes from forward limit

We show that the forward limit of tree-level scattering equations with two massive particles yields the SL(2,C)-covariant form of the one-loop scattering equations recently proposed by Geyer et al. We clarify several properties about these equations and the formulas at one loop. We then argue that in the bi-adjoint scalar theory, such a forward limit yields the correct one-loop massless amplitudes, which leads to a new formula for the latter.