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 ℳ0, 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.
Highlights
JHEP12(2020)057 this formalism is the trivialization of the KLT relations as a simple fact of linear algebra
Much work has gone into trying to extend these ideas to theories which have more complicated interactions. The first in this line of work was due to Banerjee, Laddha and Raman [12], in which it was found that a class of polytopes known as Stokes polytopes furnish the positive geometries of quartic scalar theories at tree level in the planar limit
We have studied the implications of the twisted Riemann period relations to the study of accordiohedra
Summary
We will present a generalization of the scattering equation formalism relevant to the study of the amplitudes defined by accordiohedra. Much of the technical material of this section has already appeared in previous works by the author To Stokes polytopes and [28, 29] for elementary applications to accordiohedra). The main goal of this section will be to provide a number of examples which have not yet been considered or only received treatment in passing, providing a coherent synthesis of existing techniques and setting the context for later sections
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