Biadjoint scalar tree amplitudes and intersecting dual associahedra

TL;DR

This work introduces a geometric framework for biadjoint scalar amplitudes by embedding dual associahedra into dual kinematic space and intersecting them to recover $m(\alpha|\beta)$. A single combinatorial object, a fan in $\mathcal{K}_n^*$, encodes all $n$-point partial amplitudes, which are computed as valuations (via equivariant localisation) of intersections of dual associahedra, with a toric-variety interpretation. The construction naturally yields a lattice-based link to the inverse KLT kernel $m_{\alpha'}(\alpha|\beta)$, and it recovers known diagonal terms through lattice sums while offering a new viewpoint on the double copy and string-theoretic connections. The authors also discuss open questions and speculative directions, including connections to open string moduli space tilings and generalized permutohedra, suggesting a rich toric-geometric structure behind scattering amplitudes. Overall, the paper provides a compact, symmetric geometric description of $m(\alpha|\beta)$ that unifies polytopal intersections, toric geometry, and string-inspired kernels, with potential implications for Yang-Mills and gravity via the double copy.

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 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 $m_{α'}(α|β)$.

Figures (18)

• Figure 1: The three associahedra $\mathcal{ A }(\alpha)$ defined by AHBHY's construction at four points. The associahedra lie in a plane, kinematic space, inside $\mathbb{ R }^3$.
• Figure 2: The cones $\mathcal{ C }(1234)$ and $\mathcal{ C }(2314)$ do not intersect in kinematic space, $\mathcal{ K }_4$. However, the dual cones $\mathcal{ C }(1234)^*$ and $\mathcal{ C }(2314)^*$ do intersect, and their intersection can be regarded as encoding the amplitudes $m(1234|2314)$.
• Figure 3: The three cones $\mathcal{ C }(\alpha)$ in the kinematic space $\mathcal{ K }_4$ for four points.
• Figure 4: The three realisations $\mathcal{ A }(\alpha)$ of the associahedron in the kinematic space $\mathcal{ K }_4$ for four points.
• Figure 5: The three dual cones $\mathcal{ C }(\alpha)$ in $\mathcal{ K }_4^*$.

Theorems & Definitions (30)

• Theorem 4.1
• proof
• Corollary 4.2
• Example 4.3
• Example 4.4
• Example 4.5
• Example 4.6
• Example 4.7
• Lemma 5.1
• proof