An Infinite Family of Elliptic Ladder Integrals

TL;DR

This paper introduces two new families of ten-point Feynman diagrams expressed through elliptic multiple polylogarithms, providing integral representations and differential equations that facilitate high-loop order calculations.

Contribution

It generalizes the elliptic double box to two families of diagrams with all-loop elliptic polylogarithm expressions and new integral forms.

Findings

Two families of ten-point diagrams expressed in elliptic polylogarithms.

Linearly reducible integral representations for all but one variable.

Diagrams satisfy second-order differential equations.

Abstract

We identify two families of ten-point Feynman diagrams that generalize the elliptic double box, and show that they can be expressed in terms of the same class of elliptic multiple polylogarithms to all loop orders. Interestingly, one of these families can also be written as a dlog form. For both families of diagrams, we provide new 2l-fold integral representations that are linearly reducible in all but one variable and that make the above properties manifest. We illustrate the simplicity of this integral representation by directly integrating the three-loop representative of both families of diagrams. These families also satisfy a pair of second-order differential equations, making them ideal examples on which to develop bootstrap techniques involving elliptic symbol letters at high loop orders.