Weights, Recursion relations and Projective triangulations for Positive Geometry of scalar theories

Abstract

The story of positive geometry of massless scalar theories was pioneered in [1] in the context of bi-adjoint ${\varphi }^3$ theories. Further study proposed that the positive geometry for a generic massless scalar theory with polynomial interaction is a class of polytopes called accordiohedra [2]. Tree-level planar scattering amplitudes of the theory can be obtained from a weighted sum of the canonical forms of the accordiohedra. In this paper, using results of the recent work [3], we show that in theories with polynomial interactions all the weights can be determined from the factorization property of the accordiohedron. We also extend the projective recursion relations introduced in [4,5] to these theories. We then give a detailed analysis of how the recursion relations in ${\varphi }^p$ theories and theories with polynomial interaction correspond to projective triangulations of accordiohedra. Following the very recent development [6] we also extend our analysis to one-loop integrands in the quartic theory.

arXiv commentv2 : 33 pages, 13 figures. A new section (section 7) on one-loop integrands in the quartic theory added, a sentence added to the abstract and references added

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Cited by in corpus (7)

• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Celebrating Loday's Associahedron
• Internal boundaries of the loop amplituhedron
• On Polytopes and Generalizations of the KLT Relations
• Open associahedra and scattering forms
• Geometric Recursion from Polytope Triangulations and Twisted Homology
• BCFW like recursion for Deformed Associahedron