One-loop integrand from generalised scattering equations

TL;DR

This paper explores the connection between generalized scattering equations, cluster algebras, and one-loop integrands in bi-adjoint scalar theories, revealing new geometric and algebraic structures underlying scattering amplitudes.

Contribution

It establishes a link between the singularities of (3,6) amplitudes and four-point one-loop integrands using the Gr(3,6) cluster algebra and D4 cluster polytope.

Findings

Identified the relation between (3,6) amplitude singularities and one-loop integrands.

Analyzed factorization properties of the (3,6) amplitude at various boundaries.

Connected scattering amplitude structures to cluster algebra geometries.

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.