Biadjoint scalar tree amplitudes and intersecting dual associahedra

Abstract

We present a new formula for the biadjoint scalar tree amplitudes m(α|β) 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(α|β). In fact, we encode all the partial amplitudes at n-points using a single object, a ‘fan,’ in dual kinematic space. Equivalently, as a corollary of our construction, all n-point partial amplitudes can be understood as coming from integrals over subvarieties in a toric variety. Explicit formulas for the amplitudes then follow by evaluating these integrals using the equivariant localisation formula. Finally, by introducing a lattice in kinematic space, we observe that our fan is also related to the inverse KLT kernel, sometimes denoted mα′(α|β).