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Numerical Evaluation of Harmonic Polylogarithms

T. Gehrmann, E. Remiddi

TL;DR

The authors address the need for precise, fast numerical evaluation of harmonic polylogarithms (HPLs) up to weight 4, which arise in multi-scale Feynman integrals. They develop an algorithm based on region-specific series expansions and a suite of transformation formulae that map complex arguments to a central, rapidly convergent domain, enabling double-precision accuracy with modest computation. A comprehensive reduction framework reduces most HPLs to a small irreducible basis, while analyticity and cut-structure guide the numerical strategy. The resulting hplog subroutine provides reliable, efficient evaluation for arbitrary real x and weights up to 4, with careful checks on derivatives, continuity across regions, and cross-region consistency, and is ready for extension to higher weights by leveraging known $x=1$ values.

Abstract

Harmonic polylogarithms $\H(\vec{a};x)$, a generalization of Nielsen's polylogarithms ${S}_{n,p}(x)$, appear frequently in analytic calculations of radiative corrections in quantum field theory. We present an algorithm for the numerical evaluation of harmonic polylogarithms of arbitrary real argument. This algorithm is implemented into a {\tt FORTRAN} subroutine {\tt hplog} to compute harmonic polylogarithms up to weight 4.

Numerical Evaluation of Harmonic Polylogarithms

TL;DR

The authors address the need for precise, fast numerical evaluation of harmonic polylogarithms (HPLs) up to weight 4, which arise in multi-scale Feynman integrals. They develop an algorithm based on region-specific series expansions and a suite of transformation formulae that map complex arguments to a central, rapidly convergent domain, enabling double-precision accuracy with modest computation. A comprehensive reduction framework reduces most HPLs to a small irreducible basis, while analyticity and cut-structure guide the numerical strategy. The resulting hplog subroutine provides reliable, efficient evaluation for arbitrary real x and weights up to 4, with careful checks on derivatives, continuity across regions, and cross-region consistency, and is ready for extension to higher weights by leveraging known values.

Abstract

Harmonic polylogarithms , a generalization of Nielsen's polylogarithms , appear frequently in analytic calculations of radiative corrections in quantum field theory. We present an algorithm for the numerical evaluation of harmonic polylogarithms of arbitrary real argument. This algorithm is implemented into a {\tt FORTRAN} subroutine {\tt hplog} to compute harmonic polylogarithms up to weight 4.

Paper Structure

This paper contains 21 sections, 49 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Definitions
  3. The algebra and the reduction equations
  4. The analyticity properties
  5. Identities for HPLs of related arguments
  6. The $x=-y$ transformation
  7. The $x=1/t$ transformation
  8. The $x=(1-r)/(1+r)$ transformation
  9. The numerical evaluation
  10. The central region $-(\sqrt{2}-1)\leq x\leq+(\sqrt{2}-1)$
  11. The region $(\sqrt{2}-1)<x\leq(\sqrt{2}+1)$
  12. The region $(\sqrt{2}+1)<x<+\infty$
  13. The region $-(\sqrt{2}+1)\leq x<-(\sqrt{2}-1)$
  14. The region $-\infty<x<-(\sqrt{2}+1)$
  15. Checks
  16. ...and 6 more sections

Figures (1)

  • Figure 1: Plots of irreducible HPLs in the interval $[-5;5]$. Irreducible HPLs that follow from the above by $x\to -x$ are left out. Solid line: real part, dashed line: imaginary part.