An Etude on Recursion Relations and Triangulations

Abstract

Following~\cite{Arkani-Hamed:2017thz}, we derive a recursion relation by applying a one-parameter deformation of kinematic variables for tree-level scattering amplitudes in bi-adjoint ${\varphi }^3$ theory. The recursion relies on properties of the amplitude that can be made manifest in the underlying kinematic associahedron, and it provides triangulations for the latter. Furthermore, we solve the recursion relation and present all-multiplicity results for the amplitude: by reformulating the associahedron in terms of its vertices, it is given explicitly as a sum of "volume" of simplicies for any triangulation, which is an analogy of BCFW representation/triangulation of amplituhedron for $\mathcal{N}=4$ SYM.

arXiv comment26 pages, 3 figures

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Cited by in corpus (14)

• Stokes Polytopes : The positive geometry for ${\varphi }^4$ interactions
• UV considerations on scattering amplitudes in a web of theories
• Accordiohedra as positive geometries for generic scalar field theories
• Smoothly Splitting Amplitudes and Semi-Locality
• Amplituhedra, and Beyond
• Sign Flip Triangulations of the Amplituhedron
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Weights and recursion relations for ${\varphi }^p$ tree amplitudes from the positive geometry
• On Positive Geometry and Scattering Forms for Matter Particles
• Open associahedra and scattering forms
• Constraining the weights of Stokes Polytopes using BCFW recursions for Phi^4
• Weights, Recursion relations and Projective triangulations for Positive Geometry of scalar theories
• BCFW like recursion for Deformed Associahedron
• Smooth Splitting and Zeros from On-Shell Recursion