Biadjoint scalars and associahedra from residues of generalized amplitudes

Abstract

In the Grassmannian formulation of the S-matrix for planar $$ \mathcal{N} $$ N = 4 Super Yang-Mills, Nk−2MHV 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 $$ \mathcal{N} $$ N = 4 SYM with generalized biadjoint scalar amplitudes $$ {m}_n^{(3)} $$ m n 3 as introduced by Cachazo-Early-Guevara-Mizera (CEGM) [1]. 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 $$ {m}_n^{(3)} $$ m n 3 an entire biadjoint scalar partial amplitude $$ {m}_n^{(2)} $$ m n 2 , that is a sum over all tree-level Feynman diagrams for a fixed planar order. We make a proposal which would do the same for k ≥ 4.

Holonomic representation of biadjoint scalar amplitudes

“…It would be interesting to study canonical holonomic representations of generalizations of biadjoint amplitudes, which have been proposed in refs. [44,45]. Similar Berends-Giele recursions to those used here also exist for Yang-Mills [46] and were also important for the derivation of second order differential operators in ref.…”

“…[29] so it would be interesting to construct canonical holonomic representations for these amplitudes and determine whether the resulting differential equations manifest a dependence on sub-amplitudes. (12,3,4,5,6), m 6 (1,23,4,5,6), m 6 (1,2,34,5,6), m 6 (1,2,3,45,6), m 6 (1, 2, 3, 4, 56), m 4 (1, 2, 3)m 5 (123,4,5,6), m 4 (2, 3, 4)m 5 (1,234,5,6), m 4 (3, 4, 5)m 5 (1,2,345,6), respectively, where we have used m 3 (w 1 , w 2 ) = 1.…”

Planar matrices and arrays of Feynman diagrams

“…In this paper, we have studied an array of Feynman diagrams consistent with a global notion of planarity, which is closely related to the positive part of the tropical Grassmannian, Trop G + (k, n). The notion of generalized Feynman diagrams was first introduced in [6] and refined in [50,51] to formalize the notion of local planarity or generalized color ordering. Generalized Feynman diagrams are expected to relate to the whole the tropical Grassmannian, Trop G(k, n), and it would be interesting to see how many properties of planar arrays of Feynman diagrams still hold there.…”

“…As explored in [20], by forcing a given planar array of Feynman diagrams to explore its degenerations of highest codimension one finds planar arrays of degenerate Feynman diagrams which encode the information of the poles of the contributions of this planar array to the amplitudes. Initial investigations into the residues of amplitudes at these poles are presented in [31][32][33], revealing unexpected characteristics of the standard k = 2 biadjoint amplitudes. However, a comprehensive analysis of the factorization behavior of generalized biadjoint amplitudes remains a subject for future research.…”

“…While the third version of this paper was being prepared, some new results on local planarity appeared [50,51]. In this paper, we have studied an array of Feynman diagrams consistent with a global notion of planarity, which is closely related to the positive part of the tropical Grassmannian, Trop G + (k, n).…”