Abreu Samuel, Britto Ruth, Duhr Claude, Gardi Einan
Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104 Freiburg, Germany.
School of Mathematics, Trinity College, Dublin 2, Ireland.
Phys Rev Lett. 2017 Aug 4;119(5):051601. doi: 10.1103/PhysRevLett.119.051601. Epub 2017 Jul 31.
We study the algebraic and analytic structure of Feynman integrals by proposing an operation that maps an integral into pairs of integrals obtained from a master integrand and a corresponding master contour. This operation is a coaction. It reduces to the known coaction on multiple polylogarithms, but applies more generally, e.g., to hypergeometric functions. The coaction also applies to generic one-loop Feynman integrals with any configuration of internal and external masses, and in dimensional regularization. In this case, we demonstrate that it can be given a diagrammatic representation purely in terms of operations on graphs, namely, contractions and cuts of edges. The coaction gives direct access to (iterated) discontinuities of Feynman integrals and facilitates a straightforward derivation of the differential equations they admit. In particular, the differential equations for any one-loop integral are determined by the diagrammatic coaction using limited information about their maximal, next-to-maximal, and next-to-next-to-maximal cuts.
我们通过提出一种运算来研究费曼积分的代数和解析结构，该运算将一个积分映射为由一个主被积函数和相应的主轮廓得到的积分对。此运算是一种余作用。它可简化为对多重多项对数的已知余作用，但应用更广泛，例如适用于超几何函数。该余作用也适用于具有任意内部和外部质量配置的一般单圈费曼积分，且适用于维数正规化。在这种情况下，我们证明它可以纯粹用图上的运算，即边的收缩和切割，给出一种图示表示。该余作用可直接得到费曼积分的（迭代）间断，并有助于直接推导它们所满足的微分方程。特别地，任何单圈积分的微分方程由图示余作用利用关于其最大、次最大和次次最大切割的有限信息来确定。
