An Infinite Family of Elliptic Ladder Integrals

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.

Motivic geometry of two-loop Feynman integrals

“…Throughout the same period, physicists have continued to study Feynman integrals in a more direct manner. It has become clear that, in certain families of examples with few edges and vertices, the resulting Feynman integrals tend to be composed of a limited collection of building blocks including multiple polylogarithms [9], elliptic functions, elliptic polylogarithms [14,18], and more generally motivic periods of (singular) Calabi-Yau varieties [6,7,10,11,12,13,15,16,23,24,29,31,39,47]. This suggests that the periods attached to the graphs studied in the works listed above in this paragraph are, up to mixed Tate factors, related to elliptic curves and Calabi-Yau varieties.…”