We introduce and study a so-called Wilson-loop d log representation of certain Feynman integrals for scattering amplitudes in \( \mathcal{N} \) = 4 SYM and beyond, which makes their evaluation completely straightforward. Such a representation was motivated by the dual Wilson loop picture, and it can also be derived by partial Feynman parametrization of loop integrals. We first introduce it for the simplest one-loop examples, the chiral pentagon in four dimensions and the three-mass-easy hexagon in six dimensions, which are represented by two- and three-fold d log integrals that are nicely related to each other. For multi-loop examples, we write the L-loop generalized penta-ladders as 2(L − 1)-fold d log integrals of some one-loop integral, so that once the latter is known, the integration can be performed in a systematic way. In particular, we write the eight-point penta-ladder as a 2L-fold d log integral whose symbol can be computed without performing any integration; we also obtain the last entries and the symbol alphabet of these integrals. Similarly we study the symbol of the seven-point double-penta-ladder, which is represented by a 2(L − 1)-fold integral of a hexagon; the latter can be written as a two-fold d log integral plus a boundary term. We comment on the relation of our representation to differential equations and resumming the ladders by solving certain integral equations.

我们引入并研究了某些Feynman积分的所谓Wilson环d对数表示形式，用于散射\（\ mathcal {N} \） = 4 SYM及其以后的振幅，这使得它们的评估完全简单明了。这样的表示是由双重Wilson回路图片所激发的，也可以通过回路积分的部分Feynman参数化来推导。首先，我们以最简单的单环示例为例进行介绍，四个维数为手性五边形，六个维数为三质量易六边形，它们分别由两个和三个对数的对数积分表示，它们相互之间具有很好的相关性。对于多回路示例，我们将L回路广义五梯形写为2（L-1）将某些单环积分的d对数积分对折，因此一旦知道了后者，就可以系统地进行积分。特别是，我们将八点五梯形文字写为2 L d d log对数积分，可以在不执行任何积分的情况下计算其符号。我们还获得这些积分的最后一个条目和符号字母。同样，我们研究七点双五边形阶梯的符号，该符号由六边形的2（L -1）倍积分表示。后者可以写为一个二维d对数积分加一个边界项。我们对表示与微分方程的关系进行评论，并通过求解某些积分方程来恢复梯形图。