High-precision numerical evaluation of Lauricella functions

TL;DR

This work tackles the problem of high-precision numerical evaluation of Lauricella functions with indices that depend linearly on $\varepsilon$, via Laurent expansions about $\varepsilon=0$ for applications in Feynman integral calculations. Analytic continuation is performed using Frobenius generalized power-series solutions of the associated Pfaffian differential systems, producing one-dimensional series that offer higher precision than multi-dimensional re-expansions. The $\varepsilon$ dependence is reconstructed from evaluations at lattice values $\varepsilon \in \{\pm mh\}$ and interpolated to obtain the Laurent expansion, enabling straightforward parallelization and controllable accuracy. The approach is implemented in the PrecisionLauricella package and demonstrated on test integrals, with potential extensions to Kampé de Fériet, Horn and GKZ $\mathcal{A}$-hypergeometric functions and direct applications to master integrals.

Abstract

We present a method for high-precision numerical evaluations of Lauricella functions, whose indices are linearly dependent on some parameter $\varepsilon$, in terms of their Laurent series expansions at zero. This method is based on finding analytic continuations of these functions in terms of Frobenius generalized power series. Being one-dimensional, these series are much more suited for high-precision numerical evaluations than multi-dimensional sums arising in approaches to analytic continuations based on re-expansions of hypergeometric series or Mellin--Barnes integral representations. To accelerate the calculation procedure further, the $\varepsilon$ dependence of the result is reconstructed from the evaluations of given Lauricella functions at specific numerical values of $\varepsilon$, which, in addition, allows for efficient parallel implementation. The method has been implemented in the $\texttt{PrecisionLauricella}$ package, written in Wolfram Mathematica language.

Figures (5)

• Figure 1: Massless pentagon integral on the left and two-loop sunset on the right. Dashed lines denote massless propagators, and thick lines represent massive propagators.
• Figure 2: On the left, one of the possible paths with complex singular points is shown. The singular points are shown in red, the end point in green, the starting point is at the origin, and shaded circles represent convergence regions. On the right, the intersection graph of the expansion regions is displayed. The nodes in the intersection graph include expansion regions both at regular and singular points.
• Figure 3: One-loop triangle with six independent scales.
• Figure 4: This figure illustrates the path of the analytic continuation of the system in Eq. \ref{['eq:MTEx']} from the origin $0$ to the evaluation point $t_p = 4$. Red points denote singularities, shaded circles represent convergence regions around the points $\left\{0, \frac{9}{7} - i, \frac{39}{14} - i, \frac{53}{14} - \frac{i}{2}\right\}$, blue points indicate the gluing points of these regions, and the green point is our evaluation point.
• Figure 5: Examples of Feynman integrals on which we tested our method. Dashed lines denote massless propagators, and thick lines represent massive propagators.