Numerical evaluation of iterated integrals related to elliptic Feynman integrals

TL;DR

The paper presents a GiNaC-based numerical framework to evaluate iterated integrals tied to elliptic Feynman integrals, covering both τ- and z-direction constructions and incorporating modular forms, Kronecker functions, and elliptic MPLs. It formalizes integration kernels for MPLs, modular forms, ELi/Ebar constructs, and user-defined kernels, and provides a robust truncation-based series evaluation within the region of convergence. The implementation includes complete elliptic integrals computed via the AGM and demonstrates the approach through concrete examples including Eisenstein-based integrals, Kronecker-based iterated integrals, elliptic MPLs, and the sunrise Feynman integral, highlighting convergence behavior and q-expansions. The work also discusses practical limitations, such as the need for analytic continuation in some regimes and the use of modular transformations to improve convergence, offering a clear roadmap for extending numerical capabilities in elliptic Feynman integral studies.

Abstract

We report on an implementation within GiNaC to evaluate iterated integrals related to elliptic Feynman integrals numerically to arbitrary precision within the region of convergence of the series expansion of the integrand. The implementation includes iterated integrals of modular forms as well as iterated integrals involving the Kronecker coefficient functions $g^{(k)}(z,τ)$. For the Kronecker coefficient functions iterated integrals in $dτ$ and $dz$ are implemented. This includes elliptic multiple polylogarithms.