We describe in detail how a d log representation of Feynman integrals leads to simple differential equations. We derive these differential equations directly in loop momentum or embedding space making use of a localization trick and generalized unitarity. For the examples we study, the alphabet of the differential equation is related to special points in kinematic space, described by certain cut equations which encode the geometry of the Feynman integral. At one loop, we reproduce the motivic formulae described by Goncharov [1] that reappeared in the context of Feynman integrals in [2–4]. The d log representation allows us to generalize the differential equations to higher loops and motivates the study of certain mixed-dimension integrals.

中文翻译：

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费曼积分的对数形式和微分方程

我们详细描述了费曼积分的 ad log 表示如何导致简单的微分方程。我们利用定位技巧和广义幺正性直接在循环动量或嵌入空间中推导出这些微分方程。对于我们研究的例子，微分方程的字母表与运动空间中的特殊点有关，由某些切割方程描述，这些切割方程编码费曼积分的几何形状。在一个循环中，我们重现了 Goncharov [1] 描述的动机公式，该公式在 [2-4] 中的费曼积分上下文中再次出现。d log 表示允许我们将微分方程推广到更高的循环，并激发对某些混合维积分的研究。