Stokes Polytopes : The positive geometry for ${\varphi }^4$ interactions

Abstract

In a remarkable recent work [arXiv : 1711.09102] by Arkani-Hamed et al, the amplituhedron program was extended to the realm of non-supersymmetric scattering amplitudes. In particular it was shown that for tree-level planar diagrams in massless ${\varphi }^3$ theory (and its close cousin, bi-adjoint ${\varphi }^3$ 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 is nothing but residue of 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 ${\varphi }^4$ 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 residues of 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 ${\varphi }^3$ theory, we show how factorization of Stokes polytope implies unitarity and locality of the amplitudes.

arXiv commenttypos corrected, improvement in explanation of Stokes polytopes (combinatorial vs. convex realisation) and the Q-compatibility rule; accepted for publication in JHEP

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• The positive geometry for ${\varphi }^p$ interactions
• Singular Solutions in Soft Limits
• On-Shell Electroweak Sector and the Higgs Mechanism
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• Positive Geometries for all Scalar Theories from Twisted Intersection Theory
• Positive Geometries and Differential Forms with Non-Logarithmic Singularities I
• Cosmological Polytopes and the Wavefuncton of the Universe for Light States
• Amplituhedra, and Beyond
• Sign Flip Triangulations of the Amplituhedron
• Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors
• Stokes Polytopes and Intersection Theory
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra
• Weights and recursion relations for ${\varphi }^p$ tree amplitudes from the positive geometry
• Celebrating Loday's Associahedron
• Towards Positive Geometry of Multi Scalar Field Amplitudes : Accordiohedron and Effective Field Theory
• On Polytopes and Generalizations of the KLT Relations
• Constraining the weights of Stokes Polytopes using BCFW recursions for Phi^4
• Gauge and Gravity Amplitudes on the Celestial Sphere
• Open associahedra and scattering forms
• Weights, Recursion relations and Projective triangulations for Positive Geometry of scalar theories
• One-loop integrand from generalised scattering equations
• On the Positive Geometry of Quartic Interactions III : One Loop Integrands from Polytopes
• Double Soft Theorem for Generalised Biadjoint Scalar Amplitudes
• Geometric Recursion from Polytope Triangulations and Twisted Homology
• Triangulations for ABHY Polytopes and Recursions for Tree and Loop Amplitudes
• Positive Geometry for Stringy Scalar Amplitudes
• Triangular tessellations of one-loop scattering amplitudes in ${\varphi }^3$ theory
• Learning scattering amplitudes by heart