超對稱自旋多項式與雅可比函數

Abstract

自旋蓋根鮑爾多項式展開提供一個正交基底讓我們可以分析任何不 帶質量四點散射幅度內含的重粒子以及自旋。 本論文將以兩個三點構 成四點在殼的超對稱散射幅度為被分析物，對此作自旋蓋根鮑爾多項 式展開。 這將幫助我們檢驗散射幅度內的超對稱多重態。一般來說作 此多項式展開需要內積被分析物後處以歸一化常數，而本篇卻繞過此 方式。 我們利用自旋雅可比多項式的遞迴關係可將超對稱散射幅分裂 成數個超對稱多重態的自旋雅可比多項式，並提取多重態的係數。

Spinning Gegenbauer polynomials provide us orthogonal basis of on- shell tree amplitude with massive spin propagating between 4 massless par- ticle in 4D. In this thesis, we describe the residue of N = 1 supersymmetric 4 massless vertex with mass, scattering angle, and Grassmann variable, re- ferred to supersymmetric N = 1 spinning polynomials. And these can be expanded to spinning Gegenbauer polynomials. This helps us split different spin of components of massive supermultiplet out of supersymmetric spin- ning polynomials. In general, expansion coefficients are derived from inner product of this residue with each orthogonal basis over normalized constant. While supersymmetric spinning polynomials are not pure Jacobi polynomials (orthonormal special functions). We can’t compute the expansion coefficients before constructing orthonormal basis of them. This paper provide another idea of expansion computation. Following the recurrence relation, property of Jacobi polynomials, each algebraic Jacobi polynomial is split to two or more. Therefore,component of residue of N = 1 supersymmetric 4 mass- less vertex can be split to different spinning Gegenbauer polynomials with expansion coefficients.