A Three-Parameter Elliptic Double-Box

TL;DR

This paper models a simplified version of the elliptic double-box integral using elliptic polylogarithms, revealing its symmetry properties and underlying identities in a specific unphysical limit.

Contribution

It introduces a novel elliptic polylogarithm representation of a toy model of the elliptic double-box, highlighting symmetry and functional identities.

Findings

The toy model depends on three cross-ratios.

The function is a pure elliptic polylogarithm.

Permutation symmetry is not manifest in the formalism.

Abstract

We express a toy model of the ten-point elliptic double-box, first characterized in arXiv:1712.02785, in terms of elliptic polylogarithms. This toy model corresponds to a particular unphysical limit of the elliptic double-box in which it depends on only three dual conformal cross-ratios. While the diagram is fully permutation symmetric in the cross-ratios in this limit, this property is not manifest in either of the two elliptic polylogarithm formalisms we use to express it. We observe that the function is a pure elliptic polylogarithm, which is the result of nontrivial identities between elliptic integrals depending on the conformal cross-ratios.