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Scattering Amplitudes and the Associahedron

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We establish a direct connection between scattering amplitudes for bi-adjoint scalar theories and a classic polytope--the "associahedron"--known to mathematicians since the 1960s. We find an associahedron naturally living in kinematic space. The tree level scattering amplitude is simply a geometric invariant of the associahedron called its "canonical form" [2], which is a differential form on kinematic space with logarithmic singularities on the boundaries of the associahedron. We show that basic physical principles like locality and unitarity are "rediscovered" as properties of the geometry, and certain "soft" limits can be converted to geometric statements as well.


The associahedron in kinematic space plays an important role in the context of scattering equations. We discover that the scattering equations act as a diffeomorphism between the interior of the open string moduli space (yet another associahedron) and the associahedron in kinematic space. This observation provides the key to a novel derivation of the bi-adjoint CHY formula. Finally, we emphasize the importance of the scattering amplitude as a differential form, as suggested by the "canonical form" construction. We argue on general grounds that every color-dressed amplitude is dual to a form on kinematic space. This is motivated by the surprising observation that "projective" forms on kinematic space satisfy Jacobi identities and therefore have novel implications for color-kinematics duality.



[1] Nima Arkani-Hamed, Yuntao Bai, Song He & Gongwang Yan. In preparation.

[2] Nima Arkani-Hamed, Yuntao Bai & Thomas Lam. Positive Geometries and Canonical Forms. arXiv 1703.04541