Symbology for elliptic multiple polylogarithms and the symbol prime

TL;DR

This paper explores the structure of elliptic multiple polylogarithms through their symbols, introduces the symbol prime to reveal identities, and demonstrates these concepts with two-loop Feynman integral examples.

Contribution

It introduces the symbol prime for elliptic symbol letters, providing a new tool to understand identities among elliptic polylogarithms.

Findings

The symbol prime makes elliptic symbol letter identities manifest.

Application to two-loop integrals demonstrates the utility of the symbol prime.

Results include explicit calculations for sunrise and double-box integrals.

Abstract

Elliptic multiple polylogarithms occur in Feynman integrals and in particular in scattering amplitudes. They can be characterized by their symbol, a tensor product in the so-called symbol letters. In contrast to the non-elliptic case, the elliptic letters themselves satisfy highly non-trivial identities, which we discuss in this paper. Moreover, we introduce the symbol prime, an analog of the symbol for elliptic symbol letters, which makes these identities manifest. We demonstrate its use in two concrete examples at two-loop order: the unequal-mass sunrise integral in two dimensions and the ten-point double-box integral in four dimensions. Finally, we also report the result of the polylogarithmic nine-point double-box integral, which arises as the soft limit of the ten-point integral.