Algorithms and tools for iterated Eisenstein integrals

TL;DR

The paper addresses the need to work with iterated Eisenstein integrals that appear in multi-loop Feynman calculations, developing algorithms for analytic continuation across all kinematic regions and for fast-converging, region-specific series representations. It constructs a parity-real basis for Eisenstein series, derives modular-transformation rules, and then combines regularization with SL(2,$\mathbb{Z}$) actions to map difficult regions to favorable ones, enabling high-precision numerical evaluation. The authors illustrate their framework on elliptic ${}_2F_1$-type functions and the sunrise integral, deriving explicit expressions in terms of iterated Eisenstein integrals and MMVs, and provide practical strategies for asymptotics, region-specific expansions, and numerical convergence. These contributions offer a robust, implementable path to computing elliptic and more general beyond-MPL Feynman integrals across the full phase space, with direct relevance for precision collider phenomenology. The methods have broad potential to generalize to other elliptic and modular-function-dominated Feynman integrals, improving both analytic control and numerical efficiency in high-order perturbative calculations.

Abstract

We present algorithms to work with iterated Eisenstein integrals that have recently appeared in the computation of multi-loop Feynman integrals. These algorithms allow one to analytically continue these integrals to all regions of the parameter space, and to obtain fast converging series representations in each region. We illustrate our approach on the examples of hypergeometric functions that evaluate to iterated Eisenstein integrals as well as the well-known sunrise graph.

Figures (4)

• Figure 1: The real and imaginary parts of the first three orders in the $\epsilon$ expansion of the master integral $T_1$.
• Figure 2: The real and imaginary parts of $\tau$.
• Figure 3: The real and imaginary parts of $\gamma_j^{-1} \cdot \tau^{(a,b)}$.
• Figure 4: The real and imaginary parts of the two master integrals of the sunrise graph.