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

TL;DR

This paper explores the positive geometry of scalar theories, showing how weights are determined by factorization, extending recursion relations, and analyzing triangulations of accordiohedra, including one-loop integrands in quartic theories.

Contribution

It introduces a method to determine weights from factorization and extends projective recursion relations to polynomial scalar theories and one-loop integrands.

Findings

Weights in accordiohedra are determined by factorization properties.

Extended recursion relations correspond to projective triangulations.

Analyzed one-loop integrands in quartic scalar theories.

Abstract

The story of positive geometry of massless scalar theories was pioneered in [1] in the context of bi-adjoint $\phi^3$ 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 $\phi^p$ 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.