On Positive Geometries of Quartic Interactions II : Stokes polytopes, Lower Forms on Associahedra and Worldsheet 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 worldsheet forms for ${\varphi }^4$ theory which are forms of lower rank on the CHY moduli space. We argue that just as in the case of bi-adjoint ${\varphi }^3$ scalar amplitudes, scattering equations can be used as diffeomorphisms between certain $\frac{n-4}{2}$ forms on the worldsheet and $\frac{n-4}{2}$ forms on ABHY associahedron that generate quartic amplitudes.

arXiv comment58 pages, 7 figures, Section 6 revised and a new Appendix (Appendix G) added

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Cited by in corpus (17)

• Accordiohedra as positive geometries for generic scalar field theories
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• Positive Geometries for all Scalar Theories from Twisted Intersection Theory
• Amplituhedra, and Beyond
• Sign Flip Triangulations of the Amplituhedron
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Stokes Polytopes and Intersection Theory
• Celebrating Loday's Associahedron
• Towards Positive Geometry of Multi Scalar Field Amplitudes : Accordiohedron and Effective Field Theory
• Weights, Recursion relations and Projective triangulations for Positive Geometry of scalar theories
• Constraining the weights of Stokes Polytopes using BCFW recursions for Phi^4
• On the Positive Geometry of Quartic Interactions III : One Loop Integrands from Polytopes
• One-loop integrand from generalised scattering equations
• Geometric Recursion from Polytope Triangulations and Twisted Homology
• Double Soft Theorem for Generalised Biadjoint Scalar Amplitudes
• BCFW like recursion for Deformed Associahedron