发表机构
Instituto de Física Corpuscular, Universitat de València – Consejo Superior de Investigaciones Científicas（粒子物理研究所，瓦伦西亚大学-西班牙高等科学研究委员会）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出将散射面元分解为基-纤维几何，利用变形图形zonotopes刻画因果演化，从而无需积分即可获得费曼积分的因果表示与生成树表示。
AI 中文摘要
宇宙学多面体的散射面元（Scattering Facet, SF）是一种由费曼图定义并编码其组合结构的正几何。其正则微分形式重现了标量费曼积分中对圈动量能量分量进行积分的结果，因此与圈-树对偶（Loop-Tree Duality, LTD）密切相关。在本文中，我们提出了SF作为基-纤维几何的新描述。纤维是一族称为变形图形zonotopes的多面体，它们对底层费曼图的因果演化具有自然的解释，并捕获了散射振幅的有趣信息。SF的这种分解引出了正则形式的一种描述，从而得到所谓的因果表示。此外，它提供了新的数学工具，使我们能够刻画其所有正则三角剖分。从这些三角剖分获得的正则形式对应于生成树表示，与原始的LTD振幅（表示为生成树之和）相匹配。这种几何视角为我们提供了获得因果表示和生成树表示的新方法，而无需进行任何积分。
英文摘要
The Scattering Facet (SF) of the Cosmological Polytope is a positive geometry defined from a Feynman diagram and encoding its combinatorics. Its canonical differential form reproduces the result of the integration over the energy components of loop momenta in scalar Feynman integrals, and is therefore closely related to the Loop-Tree Duality (LTD). In this article, we present a novel description of the SF as a base-fiber geometry. The fibers are a family of polytopes called deformed graphical zonotopes, which have a natural interpretation in terms of the causal evolution of the underlying Feynman diagram, and capture interesting information on scattering amplitudes. This decomposition of the SF induces a description of the canonical form leading to the so-called causal representation. Moreover, it provides new mathematical tools that allow us to characterize all its regular triangulations. Canonical forms obtained from these triangulations correspond to spanning tree representations, matching the original LTD amplitudes expressed as a sum over spanning trees. This geometrical viewpoint provides us with new methods to obtain the causal and spanning tree representations, without the need of performing any integration.
Comments59 pages, 7 figures
