多圈Feynman积分的解析计算

目前粒子物理的主要任务之一是精确检验标准模型，寻找超越标准模型的新物理。精确检验标准模型的核心工作是计算电弱观测量的多圈辐射修正，而难点在于准确计算多圈图对应的Feynman积分。利用Feynman积分的Mellin-Barnes表示，我们可以写出Feynman积分满足的GKZ偏微分方程组，并构造它的多重超几何级数表达式。将超几何函数的自变量看作Grassmann流形上的参数，通过相应Grassmann流形上Plucker坐标之间的数学变换，可以得到Feynman积分在整个参数空间的解析表达式。基于这种方法编写的数值程序可以节约内存与机时，提高计算效率。此外，对于圈图散射振幅和形状因子的计算中所出现的主积分，也可以利用这种方法进行解析计算。

A central target for particle physics now is to test the SM and to search for new physics beyond. In order to predict the electroweak observables precisely with dimensional regularization, one should evaluate Feynman integrals exactly in the time-space dimension D=4-2epsilon. Using Mellin-Barnes or Lee-Pomeransky representation of Feynman integral, we derive the GKZ-hypergeometric system of partial differential equations satisfied by Feynman integral, and construct the canonical hypergeometric series solutions in some parameter space. The variables of hypergeometric series are regarded as parameters on Grassmannians. Through the transformation between the Plucker coordinates on Grassmannian, the analytic continuation of Feynman integral can be performed from the convergent parameter space to the whole parameter space. The numerical code based on this method can save memory and CPU time, and improve the calculation efficiency drastically. In addition, this method can also be used to calculate the master integral of scattering amplitude and form factor.