One-loop integrand from generalised scattering equations

Generalised bi-adjoint scalar amplitudes, obtained from integrations over moduli space of punctured ${\mathbb{C}\mathbb{P}}^{k-1}$, are novel extensions of the CHY formalism. These amplitudes have realisations in terms of Grassmannian cluster algebras. Recently connections between one-loop integrands for bi-adjoint cubic scalar theory and ${\mathcal{D}}_n$ cluster polytope have been established. In this paper using the $\text{Gr}(3,6)$ cluster algebra, we relate the singularities of $(3,6)$ amplitude to four-point one-loop integrand in the bi-adjoint cubic scalar theory through the ${\mathcal{D}}_4$ cluster polytope. We also study factorisation properties of the $(3,6)$ amplitude at various boundaries in the worldsheet.