纯量理论正几何的权、递归关系和射影三角剖分
Weights, Recursion relations and Projective triangulations for Positive Geometry of scalar theories
摘要：
无质量标量理论的正几何的故事是在[1]中在双伴随$ϕ^3$理论的背景下开创的。进一步的研究表明，具有多项式相互作用的一般无质量标量理论的正几何是一类称为手风琴的多面体[2]。该理论的三级平面散射振幅可由各手风琴标准形的加权和得到。本文利用最近的工作[3]的结果，证明了在具有多项式相互作用的理论中，所有的权都可以由手风琴面体的因式分解性质来确定。我们还将文[4，5]中引入的射影递归关系推广到这些理论。然后详细分析了$ϕ^p$理论和多项式相互作用理论中的递归关系如何对应于手风琴的射影三角剖分。根据最近的发展[6]，我们也将我们的分析扩展到四次理论中的单圈被积函数。
英文摘要：
The story of positive geometry of massless scalar theories was pioneered in [1] in the context of bi-adjoint $ϕ^3$ 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 $ϕ^p$ 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.
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