The elliptic double box and symbology beyond polylogarithms

TL;DR

This work represents the elliptic double-box integral, a two-loop massless contribution to a 10-point N=4 SYM amplitude, in terms of elliptic multiple polylogarithms and computes its symbol. A key finding is that the first-entry structure mirrors that of non-elliptic cases, with elliptic letters confined to the last two entries, and the symbol obeys familiar physical constraints like the Steinmann relations. The authors derive a differential relation at the symbol level that ties the elliptic double-box to a 6D hexagon, implying the possibility of bootstrapping elliptic amplitudes from the hexagon data. They develop a birational torus framework and an eMPL-based formalism, paving the way for extending bootstrap techniques and cluster-inspired ideas to elliptic functions in scattering amplitudes.

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.

Figures (2)

• Figure 1: The elliptic double box and the related 6D hexagon, as well as their dual graphs.
• Figure 2: Four roots of $y^{2}(x)$ in the positive kinematics region and two integration contours. The contour $\gamma_2$ which defines $\omega_2$ runs along the real axis.