One-loop integrand from generalised scattering equations

Abstract

Generalised bi-adjoint scalar amplitudes, obtained from integrations over moduli space of punctured ${\mathbb{CP}}^{k-1}$, 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 $\text{Gr}\left(3,6\right)$ cluster algebra, we relate the singularities of $\left(3,6\right)$ 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.

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