论文标题
使用积极的热带格拉曼尼亚人连接标量振幅
Connecting Scalar Amplitudes using The Positive Tropical Grassmannian
论文作者
论文摘要
Biadjoint标量部分振幅，$ m_n（\ mathbb {i}，\ mathbb {i}）$，可以表示为在积极热带格拉斯曼尼亚语上的单个积分，从而产生了全局的schwinger参数化。这项工作的第一个结果是，使用限制程序在集成汇体中产生指标函数的限制过程。相同的限制过程导致$ ϕ^4 $振幅的积分表示，其中指示器函数变成了Dirac Delta函数。他们的支持分解为$ \ textrm {c} _ {n/2-1} $区域，$ \ textrm {c} _q $ $ q^{\ rm th} $ - 加泰罗尼亚号。每个区域的贡献都用$ m_ {n/2+1}（α，\ mathbb {i}）$振幅确定。我们使用Lagrange反演构造为$ ϕ^4 $振幅提出了一个通用公式，对区域提供了组合描述。我们开始探索$ ϕ^p $理论，发现其区域是用非交叉$（P-2）$和弦图编码的。 $ ϕ^p $振幅的扩展的结构，根据$ ϕ^3 $振幅的幅度与绿色函数相同，而绿色函数在$φ^{p-1} $ matrix模型的平面限制中相同。我们还讨论了基于stokes多面体和Accouniohedra的最新结构的可能联系。
The biadjoint scalar partial amplitude, $m_n(\mathbb{I},\mathbb{I})$, can be expressed as a single integral over the positive tropical Grassmannian thus producing a Global Schwinger Parameterization. The first result in this work is an extension to all partial amplitudes $m_n(α,β)$ using a limiting procedure on kinematic invariants that produces indicator functions in the integrand. The same limiting procedure leads to an integral representation of $ϕ^4$ amplitudes where indicator functions turn into Dirac delta functions. Their support decomposes into $\textrm{C}_{n/2-1}$ regions, with $\textrm{C}_q$ the $q^{\rm th}$-Catalan number. The contribution from each region is identified with a $m_{n/2+1}(α,\mathbb{I})$ amplitude. We provide a combinatorial description of the regions in terms of non-crossing chord diagrams and propose a general formula for $ϕ^4$ amplitudes using the Lagrange inversion construction. We start the exploration of $ϕ^p$ theories, finding that their regions are encoded in non-crossing $(p-2)$-chord diagrams. The structure of the expansion of $ϕ^p$ amplitudes in terms of $ϕ^3$ amplitudes is the same as that of Green functions in terms of connected Green functions in the planar limit of $Φ^{p-1}$ matrix models. We also discuss possible connections to recent constructions based on Stokes polytopes and accordiohedra.
