论文标题
迈向大规模标量场理论的积极几何形状
Towards Positive Geometries of Massive Scalar field theories
论文作者
论文摘要
在[1]中的先前工作的基础上，我们在运动学空间中定位了一个积极的几何形状家族，这是Associahedron的一类特定凸实现。这些实现是通过缩放和翻译由Arkani-Hamed，Bai，He和Yan（Abhy）发现的运动学空间协会获得的。我们称之为所得的多面体，是联合体的变形实现。变形的实现为CHY公式提供了新的灯光。 [2]中的一个引人注目的发现之一是，CHY散射方程在（压缩的）CHY模量空间和Abhy AssociaHedron之间产生差异。正如我们所说，在散射方程下，相关铁体的变形实现也可以解释为CHY模量空间的差异图像，我们称之为变形的散射方程。因此，运动学空间中的规范形式再次是帕克 - 泰勒形式的推动力。我们分析的自然分离是Parke-Taylor形式的普遍性，作为一类（树水平和平面）多尺度场振幅的CHY积分。这些想法有助于我们证明某些特定多量表相互作用的积极几何形状的存在。我们证明，在具有无质量和巨大的双聚体标量字段的田间理论中，通过立方相互作用相互作用，树级的S-matrix与无质量的外部状态，最多有一个巨大的传播器是由协会的某些变形实现所定义的规范形式的加权总和。最后，我们表明这些想法承认了一环的扩展。特别是，一个循环s-matrix Integrand具有最多一个巨大的繁殖器，这是对[3,4]中发现的类型D型群集多层型的变形型实现的规范形式的加权总和。
Building on the prior work in [1] we locate a family of positive geometries in the kinematic space which are a specific class of convex realisations of the associahedron. These realisations are obtained by scaling and translating the kinematic space associahedron discovered by by Arkani-Hamed, Bai, He and Yan (ABHY). We call the resulting polytopes, deformed realisations of the associahedron. The deformed realisations shed new light on the CHY formula. One of the striking discoveries in [2] was the fact that the CHY scattering equations generate diffeomorphism between the (compactified) CHY moduli space and the ABHY associahedron. As we argue, the deformed realisation of the associahedron can also be interpreted as an diffeomorphic image of the CHY moduli space under scattering equations that we call deformed scattering equations. The canonical form in the kinematic space is thus once again the push-forward of the Parke-Taylor form . A natural off-shoot of our analysis is the universality of the Parke-Taylor form as a CHY Integrand for a class of (tree-level and planar) multi-scalar field amplitudes. These ideas help us in proving the existence of positive geometries for certain specific multi-scalar interactions. We prove that in a field theory with a massless and a massive bi-adjoint scalar fields which interact via cubic interaction, the tree-level S-matrix with massless external states and at most one massive propagator is a weighted sum over the canonical forms defined by certain deformed realisations of the associahedron. Finally, we show that these ideas admit an extension to one-loop. In particular, the one loop S-matrix integrand with at most one massive propagator is a weighted sum over canonical forms of a family of deformed realisations of the type-D cluster polytope, discovered in [3,4].
