Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, 35131 Padova, Italy.
INFN, Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy.
Phys Rev Lett. 2019 Nov 15;123(20):201602. doi: 10.1103/PhysRevLett.123.201602.
Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for the construction of multivariate intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter. We apply it to the decomposition of a few Feynman integrals at one and two loops, as first steps toward potential applications to generic multiloop integrals. The proposed method can be more generally employed for the derivation of contiguity relations for special functions admitting multifold integral representations.
费曼积分服从由交点数控制的线性关系，交点数作为向量空间之间的标量积。我们提出了一种用于构造与费曼积分相关的多元交点数的通用算法，并首次展示了如何利用它们通过投影将积分简化为主要积分的基，并直接推导出后者所满足的泛函方程。我们将其应用于一阶和二阶的一些费曼积分的分解，作为潜在应用于一般多圈积分的初步步骤。所提出的方法可以更一般地用于推导具有多重积分表示的特殊函数的邻接关系。
