An Elliptic Generalization of Multiple Polylogarithms

TL;DR

The authors develop an elliptic generalization of multiple polylogarithms, termed E-polylogarithms, defined via elliptic-integral kernels tied to the two-loop sunrise graph. They introduce an E-weight that labels these functions and show the governing system splits into a 2×2 block plus a first-order operator, enabling a bottom-up approach to derive relations by weight. Weight-one relations reduce to products of logarithms with the fundamental elliptic master integrals I_0 and J_0, while weight-two and higher reveal genuine elliptic structure captured by additional master integrals. The framework yields compact representations for the imaginary part of the sunrise amplitude through ε-expansion, illustrating the method's potential for systematic study of multi-loop Feynman integrals with elliptic or higher-order differential equations.

Abstract

We introduce a class of functions which constitutes an obvious elliptic generalization of multiple polylogarithms. A subset of these functions appears naturally in the ε-expansion of the imaginary part of the two-loop massive sunrise graph. Building upon the well known properties of multiple polylogarithms, we associate a concept of weight to these functions and show that this weight can be lowered by the action of a suitable differential operator. We then show how properties and relations among these functions can be studied bottom-up starting from lower weights.