论文标题
$ ϕ^p $树的重量和递归关系来自正面几何形状
Weights and recursion relations for $ϕ^p$ tree amplitudes from the positive geometry
论文作者
论文摘要
最近，提出了运动空间中的ACORDIOHEDRON，作为平面树级散射幅度的正数几何形状$ ϕ^p $理论\ cite \ cite {raman：2019utu}。散射幅度以适当权重的某些Accorpiohedra的规范形式给予加权总和。通过要求加权和对应于散射幅度来确定这些权重。这意味着我们需要来自量子场理论的其他数据来计算几何形状的幅度。即使在这种$ ϕ^p $理论中，散射幅度也仅从几何形状中完全获得，这是一个重要的问题。在本文中，我们表明这些权重完全取决于AccorioHEDRON的分解特性。这意味着AccordioHedron的几何形状足以确定这些权重。除此之外，我们还研究了$ ϕ^p $振幅的单参数递归关系。 $ ϕ^3 $振幅的单参数“ BCFW”类似递归关系是从abhy-associahedron \ cite {arkani-hamed：2017tmz}的三角剖分中获得的。此后，通过\ cite {arkani-hamed：2019Vag，Yang：2019ESM}的广义Abhy-Associahedron的预测三角剖分提出了新的递归关系。我们将这些单参数递归关系推广到$ ϕ^p $振幅，并将其解释为Accordiohedra的三角形。
Recently, the accordiohedron in kinematic space was proposed as the positive geometry for planar tree-level scattering amplitudes in the $ϕ^p$ theory \cite{Raman:2019utu}. The scattering amplitudes are given as a weighted sum over canonical forms of some accordiohedra with appropriate weights. These weights were determined by demanding that the weighted sum corresponds to the scattering amplitudes. It means that we need additional data from the quantum field theory to compute amplitudes from the geometry. It has been an important problem whether scattering amplitudes are completely obtained from only the geometry even in this $ϕ^p$ theory. In this paper, we show that these weights are completely determined by the factorization property of the accordiohedron. It means that the geometry of the accordiohedron is enough to determine these weights. In addition to this, we study one-parameter recursion relations for the $ϕ^p$ amplitudes. The one-parameter "BCFW"-like recursion relation for the $ϕ^3$ amplitudes was obtained from the triangulation of the ABHY-associahedron \cite{Arkani-Hamed:2017tmz}. After this, a new recursion relation was proposed from the projecting triangulation of the generalized ABHY-associahedron in \cite{Arkani-Hamed:2019vag, Yang:2019esm}. We generalize these one-parameter recursion relations to the $ϕ^p$ amplitudes and interpret as triangulations of the accordiohedra.
