AI 中文总结
本文针对费曼积分分部积分约化中几何序基的主积分被积函数相交矩阵展开研究，明确了不同基下相交矩阵的元素形式，提出可系统消除转换时引入的辅助超越函数的算法，最小化了计算量。
AI 中文摘要
在费曼积分的分部积分约化中，拉波尔塔（Laporta）算法中的序关系决定了一组主积分。本文研究由几何序关系得到的主积分被积函数的相交矩阵。通过对被积函数及其对偶进行恰当定义，我们发现这类相交矩阵比预期更简单：对于与滤过相容的基，相交矩阵的元素是维数正规化参数ε的洛朗多项式；对于ε分解基，若恰当选取旋转辅助函数的边界值，则元素为整数，仅差一个ε的整体幂次。这具有实际意义：我们可系统消除从滤过相容基转换到ε分解基时引入的某些辅助超越函数。本文提供了一种算法，可在执行该消除操作的同时，最小化所需计算的数量。
英文摘要
In integration-by-parts reduction of Feynman integrals, the order relation in the Laporta algorithm determines a set of master integrals. In this paper we investigate the intersection matrices of the integrands of the master integrals that are obtained from a geometric order relation. With an appropriate definition of integrands and their duals, we find that the intersection matrices are simpler than expected: For a filtration-compatible basis, the entries of the intersection matrix are Laurent polynomials in the dimensional regularisation parameter $\varepsilon$. For an $\varepsilon$-factorised basis, the entries are instead integers, up to an overall power of $\varepsilon$, if the boundary values for the auxiliary functions of the rotation are chosen appropriately. This has practical consequences: We can systematically eliminate certain auxiliary transcendental functions, introduced in going from a filtration-compatible basis to an $\varepsilon$-factorised basis. We provide an algorithm that performs this elimination while minimising the number of required calculations.
Comments36 pages
