Elliptic Feynman integrals and pure functions

TL;DR

The paper addresses the challenge of extending the notion of pure, uniformly weighted functions from MPLs to elliptic Feynman integrals. It introduces pure elliptic multiple polylogarithms via a basis of unipotent iterated integrals on elliptic curves, along with explicit integration kernels and a regularisation scheme, and demonstrates that several two-loop elliptic FIs evaluate to pure combinations of these functions with uniform weight. Through detailed analysis of the sunrise, kite, and elliptic three-point functions, it shows that elliptic purity is compatible with known maximal-cut structures and that purity persists across a range of topologies, linking to Eisenstein-series iterated integrals and cusp limits. The results provide a foundation for canonical differential equations in the elliptic setting and suggest broader implications for perturbative QFT, including potential insights for $ ext{N}=4$ SYM amplitudes and future work on higher-genus and more complex geometries.

Abstract

We propose a variant of elliptic multiple polylogarithms that have at most logarithmic singularities in all variables and satisfy a differential equation without homogeneous term. We investigate several non-trivial elliptic two-loop Feynman integrals with up to three external legs and express them in terms of our functions. We observe that in all cases they evaluate to pure combinations of elliptic multiple polylogarithms of uniform weight. This is the first time that a notion of uniform weight is observed in the context of Feynman integrals that evaluate to elliptic polylogarithms.

Figures (1)

• Figure 1: The collection of two-loop Feynman integrals of uniform weight that we have evaluated analytically. Thick lines denote massive propagators.