The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case

Abstract

We present the result for the finite part of the two-loop sunrise integral with unequal masses in four space-time dimensions in terms of the $\mathcal{O}({\epsilon }^0)$-part and the $\mathcal{O}({\epsilon }^1)$-part of the sunrise integral around two space-time dimensions. The latter two integrals are given in terms of elliptic generalisations of Clausen and Glaisher functions. Interesting aspects of the result for the $\mathcal{O}({\epsilon }^1)$-part of the sunrise integral around two space-time dimensions are the occurrence of depth two elliptic objects and the weights of the individual terms.

Comments: 31 pages