We derive an analytic representation of the ten-particle, two-loop double-box integral as an elliptic integral over weight-three polylogarithms. To obtain this form, we first derive a fourfold, rational (Feynman-)parametric representation for the integral, expressed directly in terms of dual-conformally invariant cross ratios; from this, the desired form is easily obtained. The essential features of this integral are illustrated by means of a simplified toy model, and we attach the relevant expressions for both integrals in ancillary files. We propose a normalization for such integrals that renders all of their polylogarithmic degenerations pure, and we discuss the need for a new “symbology” of mixed iterated elliptic and polylogarithmic integrals in order to bring them to a more canonical form.

中文翻译：

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椭圆双箱积分：超越对数的无质量散射振幅

我们得出十粒子二环双箱积分作为重三重对数上的椭圆积分的解析表示。为了获得这种形式，我们首先导出积分的四重有理（Feynman-）参数表示，直接用双保形不变的交叉比表示。由此，容易获得所需的形式。通过简化的玩具模型说明了此积分的基本特征，并在辅助文件中附加了两个积分的相关表达式。我们建议对此类积分进行归一化，以使其所有多对数简并变纯，并讨论对混合迭代椭圆和多对数积分的新“符号学”的需求，以使它们成为更规范的形式。