It has recently been shown that two-loop kite-type diagrams can be computed analytically in terms of iterated integrals with algebraic kernels. This result was obtained using a new integral representation for two-loop sunset subgraphs. In this paper, we have developed a similar representation for a three-loop banana integral in $d=2-2ϵ$ dimensions. This answer can be generalized up to any given order in the $ϵ$-expansion and can be calculated numerically both below and above the threshold. We also demonstrate how this result can be used to compute more complex three-loop integrals containing the three-loop banana as a subgraph.

中文翻译：

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三环香蕉图的积分表示

最近已经表明，可以根据代数核的迭代积分来解析计算双环风筝型图。该结果是使用两环日落子图的新积分表示法获得的。在本文中，我们已经开发了一个类似的表示，用于三环香蕉积分$d=2-2\epsilon$方面。这个答案可以推广到任何给定的顺序$\epsilon$-expansion 并且可以在阈值以下和以上进行数值计算。我们还演示了如何使用此结果来计算包含三环香蕉作为子图的更复杂的三环积分。