Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet

TL;DR

This work reframes scattering amplitudes as differential forms on kinematic space, uncovering the kinematic associahedron as a central positive geometry whose canonical form reproduces tree-level bi-adjoint phi^3 amplitudes. It reveals deep links between worldsheet and kinematic pictures via the scattering equations, and shows a striking color-kinematics duality where color factors are mirrored by kinematic forms. The analysis extends to Yang-Mills and the nonlinear sigma model through permutation-invariant scattering forms, and develops triangulations, dual geometries, and recursion relations that mirror and illuminate familiar field-theory representations. The framework suggests broad generalizations, including larger big-kine space constructions, massive and loop extensions, and connections to CHY, ambitwistor strings, and the amplituhedron, offering a unifying geometric lens on scattering amplitudes across theories.

Abstract

The search for a theory of the S-Matrix has revealed surprising geometric structures underlying amplitudes ranging from the worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to kinematic space where amplitudes live. In this paper, we propose a novel geometric understanding of amplitudes for a large class of theories. The key is to think of amplitudes as differential forms directly on kinematic space. We explore this picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint cubic scalar, we establish a direct connection between its "scattering form" and a classic polytope--the associahedron--known to mathematicians since the 1960's. We find an associahedron living naturally in kinematic space, and the tree amplitude is simply the "canonical form" associated with this "positive geometry". Basic physical properties such as locality, unitarity and novel "soft" limits are fully determined by the geometry. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between this old "worldsheet associahedron" and the new "kinematic associahedron", providing a geometric interpretation and novel derivation of the bi-adjoint CHY formula. We also find "scattering forms" on kinematic space for Yang-Mills and the Non-linear Sigma Model, which are dual to the color-dressed amplitudes despite having no explicit color factors. This is possible due to a remarkable fact--"Color is Kinematics"--whereby kinematic wedge products in the scattering forms satisfy the same Jacobi relations as color factors. Finally, our scattering forms are well-defined on the projectivized kinematic space, a property that provides a geometric origin for color-kinematics duality.

Figures (25)

• Figure 1: The one-dimensional associahedron (red line segment) as the intersection of the positive region and the subspace $s+t=c$ where $c>0$ is a constant.
• Figure 2: Pictures for $n{=}5$ (left) and $n{=}6$ (right) associahedra, where we have labeled every facet by the corresponding vanishing planar variable.
• Figure 3: The scattering equations provide a diffeomorphism from the worldsheet associahedron to the kinematic associahedron.
• Figure 4: An example of the duality between color factors and differential forms
• Figure 5: Correspondence between a 3-diagonal partial triangulation and a triple cut. Note that the vertices are numbered on the left while the edges/particles are numbered on the right.