自从量子场论基本框架建立以来，散射振幅就一直倍受关注。上个世纪人们发展了一系列树图和圈图的方法（Unitarity CUt）来计算散射振幅，从而分析高能物理实验中的multi-jet过程。最近，BCFW方法和散射振幅的on-shell图表示从根本上革新了人们对散射振幅的认识。散射振幅的结构分析已经被用于探索物理系统中的更深层的对称性和一般物理原理。本项目立足于Unitarity Cut方法，BCFW方法和on-shell图表示，拟对N=4 超对称Yang-Mills场论中的非平面散射振幅进行系统深入的研究。首先我们将分析非平面图leading singularity的on-shell图的表示方法，求得其BCFW-Bridge分解方式和对应的Grassmannian矩阵。之后我们将具体构造单圈平面NMHV、双圈平面MHV、单圈非平面MHV、双圈非平面五点散射振幅积分因子的解析表达式。

The scattering amplitude has been attracting much attention from the building up of quantum field theory. In the last century, people have developed a series method for the tree and loop diagram (Unitarity CUt) to calculate the scattering amplitude. These methods are widely used in analyzing the multi-jet processes of high-energy physics experiments. Recently, BCFW method and the on-shell diagram representation revolutionized the understanding of people on the scattering amplitude. The structural analysis of the scattering amplitude has been used to explore the deeper symmetry and general physical principles of a physical system. This project is based on the Unitarity Cut method, BCFW method and on-shell diagram representation. We will study the non-planar scattering amplitude in-depth for N = 4 supersymmetric Yang-Mills field theory. First, we will analyze the representations of the leading singularities of the general non-planar diagrams and seek their BCFW-Bridge decompositions and the corresponding Grassmannian matrices. After that we will construct the analytical expressions of the integrands for the one-loop planar NMHV, double-loop planar MHV, one-loop non-planar MHV, double-loop non-planar five-point scattering amplitudes.