On positive geometries of quartic interactions: Stokes polytopes, lower forms on associahedra and world-sheet forms

Abstract

In [1], two of the present authors along with P. Raman attempted to extend the Amplituhedron program for scalar field theories [2] to quartic scalar interactions. In this paper we develop various aspects of this proposal. Using recent seminal results in Representation theory [3,4], we show that projectivity of scattering forms and existence of kinematic space associahedron completely capture planar amplitudes of quartic interaction. We generalise the results of [1] and show that for any n-particle amplitude, the positive geometry associated to the projective scattering form is a convex realisation of Stokes polytope which can be naturally embedded inside one of the ABHY associahedra defined in [2,5]. For a special class of Stokes polytopes with hyper-cubic topology, we show that they have a canonical convex realisation in kinematic space as boundaries of kinematic space associahedra.We then use these kinematic space geometric constructions to write world-sheet forms for φ 4 theory which are forms of lower rank on the CHY moduli space. We argue that just as in the case of bi-adjoint φ 3 scalar amplitudes, scattering equations can be used as diffeomorphisms between certain n−4

Positive geometry, local triangulations, and the dual of the Amplituhedron

“…where the labels 1 and n can be in either the positive or negative regions 13 (appropriately taking into account the twisted cyclic symmetry as commented on above).…”

“…This region is a quadrilateral with the four vertices (12), (23), (24) and (25) that correspond to the four physical leading singularities accessible from the (AB)=(A2) cut surface. In contrast, the MHV region corresponds to {+, +, +, +} which is the triangle with vertices (12), (23), (13).…”

“…For a review of these geometric ideas, see [8], and for attempts to include other interactions and matter, cf. [9][10][11][12][13]. Recent advances have also uncovered positive geometries in conformal field theory correlation functions [14] and in cosmology [15][16][17].…”

Accordiohedra as positive geometries for generic scalar field theories

“…In this paper, we illustrate our proposal with two examples with mixed vertices, two propagators, and six external lines. More details of the embedding of the accordiohedron for a higher number of propagators and external lines will be given in an upcoming work [17]. For a precise mathematical description of geometric realization of accordiohedron, we refer to Sec.…”

“…we get our constraints (17). Now, we will use similar logic to embed accordiohedron associated with a Feynman diagram in the kinematic space.…”