Niels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark.
Department of Physics, Brown University, 182 Hope Street, Providence, Rhode Island 02912, USA.
Phys Rev Lett. 2018 Mar 23;120(12):121603. doi: 10.1103/PhysRevLett.120.121603.
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.
我们推导出了一个十粒子、两圈双盒积分的解析表示，它是一个关于权为三的多对数的椭圆积分。为了得到这个形式，我们首先推导出了这个积分的四重有理（费曼）参数表示式，它直接用对偶共形不变的交叉比来表示；从这里，很容易得到所需要的形式。通过一个简化的玩具模型来说明这个积分的基本特征，并在辅助文件中附上两个积分的相关表达式。我们提出了这样的积分的归一化方案，使得它们所有的多对数退化都是纯的，并讨论了需要一种新的“符号学”来表示混合的迭代椭圆和多对数积分，以便将它们转化为更规范的形式。
