Biadjoint scalar tree amplitudes and intersecting dual associahedra

TL;DR

This paper introduces a novel combinatorial and geometric framework for biadjoint scalar tree amplitudes using dual associahedra in kinematic space, unifying amplitude calculations through intersections and integrals in a toric variety.

Contribution

It presents a new formula for biadjoint scalar amplitudes based on dual associahedra and kinematic space cones, linking geometric intersections to amplitude computations.

Findings

Amplitudes encoded as intersections of dual associahedra in dual kinematic space.

All n-point partial amplitudes derived from integrals over subvarieties in a single toric variety.

Connection established between the geometric construction and the inverse KLT kernel.

Abstract

We present a new formula for the biadjoint scalar tree amplitudes $m(\alpha|\beta)$ based on the combinatorics of dual associahedra. Our construction makes essential use of the cones in 'kinematic space' introduced by Arkani-Hamed, Bai, He, and Yan. We then consider dual associahedra in 'dual kinematic space.' If appropriately embedded, the intersections of these dual associahedra encode the amplitudes $m(\alpha|\beta)$. In fact, we encode all the partial amplitudes at $n$-points using a single object (a fan) in dual kinematic space. Equivalently, as a pleasant corollary of our construction, all $n$-point partial amplitudes can be understood as coming from integrals over subvarieties in a single toric variety. Explicit formulas for the amplitudes then follow by evaluating these integrals using the equivariant localisation (or Duistermaat-Heckman) formula. Finally, by introducing a lattice in kinematic space, we observe that our fan is also related to the inverse KLT kernel, sometimes denoted $m_{\alpha'}(\alpha|\beta)$.