Elliptic Multiple Polylogarithms with Arbitrary Arguments in \textsc{GiNaC}

TL;DR

This work tackles the numerical evaluation of elliptic multiple polylogarithms (eMPLs) with arbitrary complex arguments at high precision. It introduces a convergent $q$-series representation $\tilde{\Gamma}(n_1\dots z_k; z,\tau)=\sum_m q_{\tau'}^m C_m(\{z_i\}; z,\tau')$ whose coefficients are ordinary MPLs, and develops a multi-step preparation algorithm (shiftsingular loci, regularisation, modular-domain mapping, path-decomposition) to ensure convergence. The authors implement the approach in GiNaC, delivering the first public package capable of evaluating eMPLs for arbitrary arguments and high precision, with practical demonstrations in Feynman-integral computations such as two-loop elliptic Master integrals in Bhabha scattering. The work thus provides a scalable, precise, and broadly applicable numerical tool for elliptic function spaces encountered in modern perturbative QFT. The combination of a robust mathematical framework with a performant, openly available implementation significantly advances phenomenological and mathematical studies of eMPLs.

Abstract

We present an algorithm for the numerical evaluation of elliptic multiple polylogarithms for arbitrary arguments and to arbitrary precision. The cornerstone of our approach is a procedure to obtain a convergent $q$-series representation of elliptic multiple polylogarithms. Its coefficients are expressed in terms of ordinary multiple polylogarithms, which can be evaluated efficiently using existing libraries. In a series of preparation steps the elliptic polylogarithms are mapped into a region where the $q$-series converges rapidly. We also present an implementation of our algorithm into the \texttt{GiNaC} framework. This release constitutes the first public package capable of evaluating elliptic multiple polylogarithms to high precision and for arbitrary values of the arguments.

Figures (3)

• Figure 1: The lattice $\Lambda_\tau$ spanned by the periods $\tilde{\psi}_1 = 1$ and $\tilde{\psi}_2 = \tau$. Each facet of the lattice corresponds to a copy of the torus. The whole complex $\mathds{C}$ is the universal cover of the torus $\mathds{C}/\Lambda_{\tau}$.
• Figure 2: Integration contour in $t$-space (left) and $w$-space (right). The path shown in a solid red line corresponds to the integration in the original eMPL. The canonical path in $w$-space is shown as a dashed line. The image $w_1$ of the singular locus $z_1$ obstructs us from deforming the original contour to the dashed path. The orange segments indicate the integration path that we construct in $w$-space, which passes on the correct sides of all singular loci. The dashed orange line indicates the optimised integration path, which avoids adding a node at $\tilde{w}_2$.
• Figure 3: Runtime of the numerical evaluation of different eMPLs using our implementation, the iterated_integral library and naive numerical integration using NIntegrate in Mathematica, measured on a typical laptop. Upper plot: Four individual eMPLs of different lengths from 1 to 4. For lengths 3 and 4, only a single point could be computed using numerical integration. Lower plot: Elliptic master integral from two-loop Bhabha scattering.