The elliptic double box and symbology beyond polylogarithms

TL;DR

This paper analyzes the elliptic double-box integral in massless QFTs, expressing it with elliptic polylogarithms, studying its symbol, and revealing a structure that relates it to a 6D hexagon integral, advancing understanding of elliptic integrals in scattering amplitudes.

Contribution

It provides a novel representation of the elliptic double-box integral using elliptic polylogarithms and uncovers its symbol structure, linking it to a 6D hexagon integral for potential bootstrap approaches.

Findings

The integral's symbol has a rich structure with elliptic letters in the last two entries.

The first symbol entry is expressed in terms of logarithms of dual-conformal cross-ratios.

A differential equation relates the double-box integral to a 6D hexagon integral.

Abstract

We study the elliptic double-box integral, which contributes to generic massless QFTs and is the only contribution to a particular 10-point scattering amplitude in N=4 SYM theory. Based on a Feynman parametrization, we express this integral in terms of elliptic polylogarithms. We then study its symbol, finding a rich structure and remarkable similarity with the non-elliptic case. In particular, the first entry of the symbol is expressible in terms of logarithms of dual-conformal cross-ratios, and elliptic letters only occur in the last two entries. Moreover, the symbol makes manifest a differential equation relating the double-box integral to a 6D hexagon integral, suggesting that it can be bootstrapped based on the latter integral alone.