On Polytopes and Generalizations of the KLT Relations

TL;DR

This paper introduces a novel framework combining polytope theory and intersection theory to generalize KLT relations via new scattering equations on accordiohedra, broadening the understanding of amplitude relations in scalar theories.

Contribution

It develops a new class of double copy relations based on polytopes and twisted intersection theory, extending classical KLT relations to more general scalar theories.

Findings

Generalized scattering equations on accordiohedra.

New intersection number relations generalize KLT.

A natural extension of the BCJ basis for scalar theories.

Abstract

We combine the technology of the theory of polytopes and twisted intersection theory to derive a large class of double copy relations that generalize the classical relations due to Kawai, Lewellen and Tye (KLT) . To do this, we first study a generalization of the scattering equations of Cachazo, He and Yuan. While the scattering equations were defined on $\mathcal{M}_{0,n}$ - the moduli space of marked Riemann spheres - the new scattering equations are defined on polytopes known as accordiohedra, realized as hyperplane arrangements. These polytopes encode as patterns of intersection the scattering amplitudes of generic scalar theories. The twisted period relations of such intersection numbers provide a vast generalization of the KLT relations. Differential forms dual to the bounded chambers of the hyperplane arrangements furnish a natural generalization of the Bern-Carrasco-Johansson (BCJ) basis, the number of which can be determined by counting the number of solutions of the generalized scattering equations. In this work the focus is on a generalization of the BCJ expansion to generic scalar theories, although we use the labels KLT and BCJ interchangeably.