Numerical Evaluation of Two-Dimensional Harmonic Polylogarithms

TL;DR

This work develops a numerically stable, double-precision algorithm to evaluate two-dimensional harmonic polylogarithms up to weight 4 in the kinematic region $0\le y\le 1-z$, $0\le z\le 1$, $y+z\le 1$, providing a complete FORTRAN77 implementation (tdhpl). It builds on a rigorous definition and algebra of 2dHPLs, introduces region-specific expansions and cut-separation techniques, and leverages Bernoulli-type and Chebyshev accelerations to achieve high accuracy efficiently. Comprehensive checks, including derivative consistency, cross-reductions to Nielsen’s polylogarithms, and agreement with two-loop doublebox results, validate the method. The resulting tool enables fast, reliable evaluation of 2dHPLs for multi-loop radiative-correction calculations in quantum field theory and is readily extensible to higher weights.

Abstract

The two-dimensional harmonic polylogarithms $\G(\vec{a}(z);y)$, a generalization of the harmonic polylogarithms, themselves a generalization of Nielsen's polylogarithms, appear in analytic calculations of multi-loop radiative corrections in quantum field theory. We present an algorithm for the numerical evaluation of two-dimensional harmonic polylogarithms, with the two arguments $y,z$ varying in the triangle $0\le y \le 1$, $ 0\le z \le 1$, $\ 0\le (y+z) \le 1$. This algorithm is implemented into a {\tt FORTRAN} subroutine {\tt tdhpl} to compute two-dimensional harmonic polylogarithms up to weight 4.

Figures (2)

• Figure 1: Kinematic regions for the evaluation of the 2dHPLs
• Figure 2: Examples for the dependence of 2dHPLs on $y$ and $z$