Biadjoint scalar tree amplitudes and intersecting dual associahedra

Understand

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.