Elliptic multiscale Feynman integrals

这一更新建议的目标是将我们对单尺度椭圆费曼积分的知识扩展到多尺度椭圆费曼积分。费曼积分是粒子物理中精密计算不可缺少的工具。有一类费曼积分，其求值为多个多对数，是无质量量子场论中积分的重要组成部分。然而，对于涉及希格斯玻色子、顶夸克或W / Z玻色子的过程，我们不想忽略重粒子的质量。对于非零内部质量，我们已经在两个循环中留下了多个多对数的函数空间，并且必须考虑椭圆推广。在第一个资助期，我们成功地研究了单尺度椭圆费曼积分。我们现在计划将我们的知识扩展到多尺度椭圆费曼积分。该更新方案的具体目标是：(1)提供了一种在椭圆多尺度设置中构造合适的主积分基的算法方法。（ii）研究作为模形式推广的主积分的微分方程中的积分核。(3)提供了一种椭圆型多尺度环境下主积分的数值计算方法。

The goal of this renewal proposal is to extend our knowledge of single-scale elliptic Feynman integrals towards multi-scale elliptic Feynman integrals.Feynman integrals are an indispensable tool for precision calculations in particle physics.There is a class of Feynman integrals, which evaluate to multiple polylogarithm and captures a significant part of integrals occurring in massless quantum field theories.However, for processes involving the Higgs boson, the top quark or $W$/$Z$-bosons we do not want to neglect the masses of the heavy particles.With non-zero internal masses we leave already at two loops the function space of multiple polylogarithms and have to consider elliptic generalisations.In the first funding period we studied successfully single-scale elliptic Feynman integrals.We now plan to extend our knowledge towards multi-scale elliptic Feynman integrals.The concrete goals of this renewal proposal are:(i) to provide a method for the algorithmic construction of a suitable basis of master integrals in the elliptic multi-scale setting.(ii) to study the integration kernels appearing in the differential equations for the master integrals as generalisations of modular forms.(iii) to provide a method for the numerical evaluation of the master integrals in the elliptic multi-scale setting.