Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes

Abstract: In the Grassmannian formulation of the S-matrix for planar $\mathcal{N}=4$ Super Yang-Mills, $N^{k-2}MHV$ scattering amplitudes for $k$ negative and $n-k$ positive helicity gluons can be expressed, by an application of the global residue theorem, as a signed sum over a collection of $(k-2)(n-k-2)$-dimensional residues. These residues are supported on certain positroid subvarieties of the Grassmannian $G(k,n)$. In this paper, we replace the Grassmannian $G(3,n)$ with its torus quotient, the moduli space of $n$ points in the projective plane in general position, and planar $\mathcal{N}=4$ SYM with generalized biadjoint scalar amplitudes $m^{(3)}_n$ as introduced by Cachazo-Early-Guevara-Mizera (CEGM). Whereas in the Grassmannian formulation residues of the Parke-Taylor form correspond to individual BCFW, or on-shell diagrams, we show that each such $(n-5)$-dimensional residue of $m^{(3)}_n$ is an entire biadjoint scalar partial amplitude $m^{(2)}_n$, that is a sum over all tree-level Feynman diagrams for a fixed planar order. We propose a generalization which would give rise to identifications of $m^{(2)}_n$ inside $m^{(k)}_n$ for $k\ge 4$, via $(k-2)(n-k-2)$-dimensional residues. Our proof for $k=3$ uses the CEGM formula for $m^{(3)}_n$; it predicts a new Minkowski sum realization of the associahedron in terms of certain positroid polytopes in the second hypersimplex $\Delta_{2,n}$.