The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms

TL;DR

This paper presents an analytic elliptic integral representation for a complex two-loop amplitude involving ten particles, extending beyond traditional polylogarithmic functions, and introduces a new normalization and symbolic framework for such integrals.

Contribution

It derives a novel elliptic integral form for the ten-particle two-loop double-box integral using a rational parametric approach and proposes a new 'symbology' for elliptic/polylogarithmic integrals.

Findings

Derived a four-fold rational parametric representation of the integral.

Expressed the integral as an elliptic integral over weight-three polylogarithms.

Proposed a normalization that simplifies polylogarithmic degenerations.

Abstract

We derive an analytic representation of the ten-particle, two-loop double-box integral as an elliptic integral over weight-three polylogarithms. To obtain this form, we first derive a four-fold, rational (Feynman-)parametric representation for the integral, expressed directly in terms of dual-conformally invariant cross-ratios; from this, the desired form is easily obtained. The essential features of this integral are illustrated by means of a simplified toy model, and we attach the relevant expressions for both integrals in ancillary files. We propose a normalization for such integrals that renders all of their polylogarithmic degenerations pure, and we discuss the need for a new 'symbology' of iterated elliptic/polylogarithmic integrals in order to bring them to a more canonical form.