Stokes polytopes: the positive geometry for ϕ4 interactions

Abstract

In a remarkable recent work [1], the amplituhedron program was extended to the realm of nonsupersymmetric scattering amplitudes. In particular it was shown that for tree-level planar diagrams in massless φ 3 theory (and its close cousin, bi-adjoint φ 3 theory) a polytope known as the associahedron sits inside the kinematic space and is the amplituhedron for the theory. Precisely as in the case of amplituhedron, it was shown that scattering amplitude can be obtained from the canonical form associated to the Associahedron. Combinatorial and geometric properties of associahedron naturally encode properties like locality and unitarity of (tree level) scattering amplitudes. In this paper we attempt to extend this program to planar amplitudes in massless φ 4 theory. We show that tree-level planar amplitudes in this theory can be obtained from geometry of objects known as the Stokes polytope which sits naturally inside the kinematic space. As in the case of associahedron we show that the canonical form on these Stokes polytopes can be used to compute scattering amplitudes for quartic interactions. However unlike associahedron, Stokes polytope of a given dimension is not unique and as we show, one must sum over all of them to obtain the complete scattering amplitude.Not all Stokes polytopes contribute equally and we argue that the corresponding weights depend on purely combinatorial properties of the Stokes polytopes. As in the case of φ 3 theory, we show how factorization of Stokes polytope implies unitarity and locality of the amplitudes.

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

“…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.…”

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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.

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“…In kinematic space, attempts were made to extend the amplituhedron program to generic scalar interactions [1,13,14]. In [1] it was shown that planar tree-level amplitudes for massless quartic interactions in any space-time dimensions were intricately tied to another polytope called Stokes polytope. Just as in the case of cubic interactions and the associahedron in kinematic space, it was argued that any n particle amplitude can be obtained from canonical forms on n−4 2 dimensional Stokes polytopes.…”

“…Each Stokes polytope is parametrized by a reference quadrangulation Q of an n-gon such that the corresponding canonical form produces a "partial" scattering amplitude m Q n . It was argued in [1] that a (weighted) sum over m Q n produces the full planar tree-level amplitude. These claims were verified for n = 6, 8, and 10 particle scattering.…”

“…The paper is organised as follows. In Section 2 we give a quick review of the core ideas contained in [1,2] which are needed for the rest of the paper. In Section 3 we use the beautiful ideas of [4] to realise Stokes polytopes as convex polytopes in kinematic space.…”